Mechanical Springs Wahl
Mechanical Springs Wahl
Mechanical Springs Wahl
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Mechanical
Springs
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macHiNE Design
SERIES
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Mechanical
Springs
by
A. M. Wahl
First Edition
PI BUSHED BY
CLEVELAND, OHIO
1944
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Copyright, 1944
CLEVELAND, OHIO
Printed in U.S.A.
FOREWORD
vices, in the past such springs too often have been designed on
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drawn upon. Much of the material in the book has also been based
several years.
siderable detail, but much emphasis also has been laid upon
torsion, spiral, volute and ring springs have been treated. Be-
is not possible to cover all the factors which enter into the
author feels that for best results in any particular design, close
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entation.
A. M. Wahl
May 4, 1944
VI
CONTENTS
Chapter I
Chapter II
of Pitch Angle
Chapter III
Chapter IV
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Chapter V
Chapter VI
Chapter VII
vn
CONTENTS
Chapter VIII
Chapter IX
Test Data
Chapter X
Chapter XI
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Chapter XII
Application of Formulas
Chatter XIII
Chapter XIV
Loading
vi it
CONTENTS
CHarTER XV
DeflectionsSimplified Calculation
Chapter XVI
Chapter XVII
Chapter XVIII
Coils in Contact
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Chapter XIX
Calculation
Chapter XX
Chapter XXI
Chapter XXII
Chapter XXIII
rx
LIST OF SYMBOLS
a = constant
b constant
b = width inch
c, C = constants
c = distance inch
c spring index
J, D diameter inch
F = force lb
h = thickness inch
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k, K = constants'
l = length inch
nt = integer, constant
n = constant, integer
n = factor of safety
N = normal force lb
P = load lb
LIST OF SYMBOLS
q = sensitivity index
q ratio
r, R = radius , inch
R = radial load lb
t = thickness inch
t = time sec
I = temperature degrees
V = volume hi.
\V = weight lb
I, y, z = rectangular co-ordinates
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z = constant
P constant, ratio
7 = constant, ratio
S deflection inch
f = small quantity
* = curvature >n-
* = small quantity -
M = Poisson's ratio
M = coefficient of friction
p = radius inch
1> constant
f, t = co-ordinates
xi
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MECHANICAL SPRINGS
CHAPTER I
GENERAL CONSIDERATIONS
IN SPRING DESIGN
distort under load, they are not all considered as springs. Thus
remain rigid.
spring.
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the deflection will be doubled. The relation will hold true even
.-1
MECHANICAL SPRIXGS
FUNCTIONS OF SPRINGS
of impact when the car goes over a bump. These also function as
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spring supplies the force which holds the valve follower against
DEFLECTION
used under nuts and bolt heads also function essentially as springs
deflection curves
DEFLECTION f
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is present.
MECHANICAL SPRINGS
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pivot. Because of low internal friction such pivots often have real
SPRING MATERIALS
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Since both the deflection and load for most springs are propor-
MECHANICAL SPRINGS
by using one per cent carbon steel and heat treating after form-
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springs are usually heat treated after forming, the latter being
done hot. Smaller sizes of helical springs, on the other hand, are
'Typical specifications for different spring materials, including data on heat treat-
pounds per square inch for 1/32-inch wire. Phosphor bronze also
being obtained.
TYPES OF LOADING
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MECHANICAL SPRINGS
times during the life of the spring. Such springs are known as
where the valve is expected to pop off but a few times during
its life; springs for producing gasket pressure, typified by the con-
anisms where the breaker operates but a few times in its life.
amount this means that as time goes on, the load should not drop
off by more than a small amount (usually a few per cent). This
laxation of the material after a period of time, the valve will pop
off at lower pressure than that for which it was designed. Similar-
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sure usually may be tolerated, too much renders the spring in-
stant deflection and if the stress is too high, there will be a slow
kept well below the elastic limit or yield point of the material,
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in
MECHANICAL SPRINGS
gested that for the usual spring material which has some duc-
stress at the edge of the hole is around three times the stress
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does not remain constant, but varies with time. For example, an
loading. Fig. 12 shows such stress cycles for such a spring sub-
tions A.S.M.E., May, 1941. Page 363. Aln "Relaxation Resistance of Nickel Allov
page 312. gives a further discussion of stress concentration; also Theory of Elasticity.
11
constant stress tr1. equal to half the sum of maximum and minimum
able stress is equal to half the difference between a,x and ami*,
to torsion stresses.
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12
MECHANICAL SPRINGS
true of freight car and locomotive springs. In such cases the de-
since fatigue test data giving results for cases where the variable
the minimum stress is known as the stress range; this is also twice
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of the spring.
nal of Applied Mechanics, December, 1937, gives a further discussion of this problem.
Also "Damage and Ovcrstress in the Fatigue of Ferrous Metals", by Russell and
13
when the variable stress is low, tests show that the curve tends
I-
STATIC
STRESS
TIME
loss in load.
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equidistant from the dot and dash line c which represents the
variable stress
that at any point P the ordinate of the mean stress line c repre-
14
MECHAXICAL SPRINGS
distance between the mean stress line and either the upper or
lower curve gives the variable stress component o>. The upper
a and b as indicated.
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plicity.
15
vided the values av and a, are taken as the yield point and
the yield stress. On the basis of this elliptical law, the factor
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of safety n becomes
Fig. 16Simplification of
STATIC STRESS
16
MECHANICAL SPRINGS
Equation 1.
rather high values, they are among the few available for such
55
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however, that the straight line law, Fig. 16, is safer to use in
17
60,
40 60
StEADY StRESS
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stress. The results of one such test1 on .7 per cent carbon steel
bars are given in Fig. 18, the full lines being results for speci-
l<lFederstaehleHoudremont and Bennek, Stahl und Eisen, Vol. 52, page 660.
18
MECHANICAL SPRINGS
must be admitted that available fatigue test data made for the
that, until further test data are at hand, stress increases due to
not exceed the yield point since the material will merely yield
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INFREQUENT OPERATION
nTo take into account the fact that the material may not be fully sensitive to
stress concentration (i.e., that the actual stress range as found by test may be greater
than the calculated figure usinu theoretical stress concentration factors), a reduction
in the stress range may be made, provided test data are available. Further discussion
and Comparison with Fatigue Tests" Peterson and Wahl, Journal of Applied Mechanics,
March, 1936.
19
Fig. 19. It appears that for this type of stress application, the
eoooor
when the loadings are relatively few. In most cases the increased
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20
MECHANICAL SPRINGS
base of the crack. Actual tests have shown that a very thin layer
inch. When the surface was left untouched, the endurance range
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the stress data do show how important are the surface conditions.
had been heat treated after grinding to size. In this case there
steel and given the same heat treatment but having a thin layer
ther tests were made on specimens which had been heat treated
""Fatigue Strength of Carbon and Alloy Steel Plates as Used for Laminated
oratory Tests"Hankins and Ford- Journal Iron and Steel Institute, 1929, No. 1,
Page 317.
These investigators found that, where the wires had been ma-
in
in
160000
- ft
80000
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I20O00
St.
RBVCE MA
CHINE
,__SURFAC
E UNI
cue
HED
40000
L.
0s io5 o'
22
MECHANICAL SPRINGS
untouched and those having the surface layer ground off. Prob-
ably this may be explained by the fact that this material has a
was small.
pelling small steel shot at high velocity against the spring sur-
ently the peening action of steel shot propelled against the sur-
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face of the spring tends to cold work and thus increase the
CORROSION EFFECTS
tigue tests must be carried out for many more than the usual
Nov. 1940, Page 62. Also Lessells and Murray"The Effect of Shot Blasting and Its
23
the value was obtained for a spring steel with a plating than
place.
VARIATIONS IN DIMENSIONS
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and wire diameter of only 1 per cent will result in a 7 per cent
slightly reducing the coil diameter. In most cases, this slight re-
24
MECHANICAL SPRINGS
FACTOR OF SAFETY
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CHAPTER II
Among the reasons for wide acceptance and general use are
the following:
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The present chapter discusses the theory for stress and deflec-
25
26
MECHANICAL SPRINGS
the deflections per coil are not too large (not more than half
HELIX ANGLE
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large and where the helix angle is small. Since the elementary
theory does not take into account the difference in fiber length
between the inside and outside of the coil which arises because
or moderate indexes.
'Effects of large initial pitch angles combined with large deflections are considered
in Chapter III.
HELICAL SPRINGS 27
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2S
MECHANICAL SPRINGS
.(4)
lePr
ird3
indexes for two reasons: (1) The effect of direct shear stress
axially loaded
the axis (Fig. 24), this element, after deformation, will rotate
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t...
16Pr
.(5;
HELICAL SPRINGS
2')
the bar with respect to the other will be, using Equation 5,
J^Jrnr y pit
(6)
64PH/i
&=pr=
Gd'
.(7)
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(Stress is greater at h
than at a)
IP \
^////////yy//////'
,^y^v////////.
ii
H|
for fairly small spring indexes and for large helix angles. Tests
30
MECHANICAL SPRINGS
in Chapter IV.
that the failure starts from a fatigue crack near the inside of the
pected that the maximum stress occurs at the inside of the coil
near point a', Fig. 26a. The reasons for the existence of the maxi-
mum stress at this point are: First, the fiber length along the
inside of the coil is much less which means that a higher shear-
cross sections. Thus in Fig. 26b, if the radial sections bb' and
aa' rotate through a small angle with respect to each other and
about the bar axis, the inside (and shorter) fiber a'b' will be
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stress due to the direct axial load P is added to that due to the
is that the stresses on the inside of the coil reach values around
shown both by test (Chapter IV) and theory; for larger indexes
(elementary theory)
This type of fracture with the fractured surface making an angle of 45 degrees
with the axis is typical of fatigue fracture of a straight cylindrical bar under alter-
nating torsion.
HELICAL SPRINGS
31
accurate for practical use (within about 2 per cent for practical
27a, the forces acting on this element are resolved into a twisting
nuerschnitt." V.D.I. 1913. Page 1907, but the final numerical results are only slightly
different. See also author's paper "Stresses in Heavv Closely Coiled Helical Springs",
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32
MECHANICAL SPRINGS
sections aa' and bb' will rotate with respect to each other through
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some point O', Fig. 29, which is displaced toward the axis of the
on the axis aa' Fig. 27b. Point O' may be found as follows:
When the sections aa' and bb' have rotated through a small angle
axially loaded
HELICAL SPRINGS
33
(b)
r= (8)
Tx xGdp
(r-y x)d8
, 19)
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34
MECHANICAL SPRINGS
*J 1 y
16r d1 1
V + TeW
16r
.(10)
practical springs d/2r seldom greater than 1/3 and hence t/2/16r-
xGdu
d-
(11)
x)de
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de
32Mr
-*dtG~
(12)
HELICAL SPRINGS
35
wd'fr
32xMr
~ 16r
(13)
a.
in
u. i
<
_ 16M / 4c-1 \
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(14)
16M / 4c+l
/ 4c+l \
7Td2 V 4c+4 /
(15)
36
MECHANICAL SPRINGS
direct axial shear force P, assuming that the pitch angle is small.
shearing stress at a and a! (Fig. 27b) due to the direct axial load
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(16)
(17)
16Pr
(18)
HELICAL SPRINGS
37
4c-1
4c-4
.615
.(19)
that for a spring of index 3 the factor K=1.58, which means that
IjOL
- 2r
8 10 12
= SPRING INDEX
14
16
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the case, the maximum stress may be less than that calculated.
However, even for such cases where yielding occurs, the formula
38
MECHANICAL SPR1NGS
EXACT THEORY
being p and z and if the pitch angle is small so that the elements
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+ ., + 0 (20)
dp dz p
at , + JK_-z>.0 (21)
(22)
sektor", Ing.-Arch. Vol. 2, 1931, Page 381; and "Die Berechnung Zylindrische Schraub-
HELICAL SPRINGS 39
troduced. Taking
dp \ dp' dz- p dp /
( + -)= 0 (25,
dz \ dp- dz- p dp /
, + +2c' = 0 (26)
dp- dz- p dp
as follows:
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+ + + 2c' = 0 (27)
a? d(- Jt \ ai
1 f-
1 + - f -+ - "" (28)
! 1 r r-
40
MECHANICAL SPRINGS
*-*rr*i+*H (29)
where
d? df2
*i satisfies = 0
d? dp r d*
dp d{2 r d$ r2 d{
0 - f) st 0 - 7) at
The total twisting moment acting over the spring cross sec-
tion will be
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1 2 3P
rr1 + - + -r + <32)
(,.!), '"
springs.
pitch angle is
HELICAL SPRINGS
41
16Pr
/ c 1 .J_\
(33)
where c2r/d.
\6Prcos a
TCP
1(d\1(dV
2p'
1+
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2p,
~3~
16
+.
tan'a
-{-hi
(34i
ing the pitch angle into account. The first term of this equation
MECHANICAL SPRINGS
ing formulas may be used with an accuracy within one per cent:
.(36)
16Pr cos a / 5 7 1 \
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ing moment Pr cos <x and the direct shear P cos a. However, there
spring this bending stress o-,.r must be combined with the tor-
HELICAL SPRINGS
43
pitch angles a
1.04,
I 03
g .9B|
Or
--
10*
'
as*
5*
III
II
II
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^ c d WFE DIAMETER
used for the calculation of torsion stress t,,,j. Final results are:
25m3+41m2+28m+8
4Hm2(m
r-28m+8 \
+ 1) )
-() +
1) \2p'/
V 2p' / 1
}...(3S)
32Pr sin a
wd3
.64
0 + -87+
V 2P' / Id
(}--&)
(39)
2p'J J
where p'r/cos2cx. The last term in the brackets d/8r yields the
'Ing. Archie, 1931, Pace 381. Also Theory of ElasticityTimoshenko, Page 361.
44 MECHANICAL SPRINGS
4 P sin a/ircP; the remainder yields the stress due to the bending
cessive approximations.
pitch angles and small indexes. For the usual case the following
I' - ,
25m3+41m"-+28m+8
48m2(m + l)c5 J
(40)
f mox ~
/ 1.12 .64\ , , (
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stress t,ai and the bending stress o-,uj. which act at a given
bending stress am<1i and a shearing stress tm<u act at a given point,
theory is1
4(rm0x)2
.(42)
Hencky theory). This theory states that failure will occur when
12Timoshenko, loc. cit. Part 2, Page 479 gives a further discussion of this theory.
HELICAL SPRINGS
45
foregoing is:
In this case a, may refer either to the yield point or the endur-
ance limit.
compression.
r.-rm.,Jl + (44)
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*--^+^j^-+(r,)' (45)
, = ^-y(--+{t^Y (46)
t,=0 (47)
tion 44 is obtained.
"The derivation of those formulas for principal stress for combined tension and
46
MECHANICAL SPRINGS
apply rigidly only as long as the deflections per turn are small
(relative to the coil diameter) so that the coil radius and pitch
stresses are set up when the deflections per turn approach the
37 may be used. Then cos < x = .978 and tan a = .2126. Using
Equation 37
32Pr
'r23
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Hence
1.444
= 2 tan a-
1.551
16Pr
This differs but slightly from the value obtained by using the
HELICAL SPRINGS
47
tion per turn is not large11 relative to the coil radius and that the
tion isir'
64Pr"7t
G* I _3
16
(^))
(48)
where c is the spring index. It is seen that this is simply the or-
spring index, the nearer will this term approach unity. How-
ever, even for the exceptionally small index of 3, the term in the
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the deflection will be about 2.3 per cent smaller than that cal-
where a and a are the final and initial pitch angles, respectively.
This assumes that the deflection is small so that the coil radius
ture, which is
COS2 a COS2 a
A = (49)
"The case of large deflections (which may occur without excessive stress only for
Page 421.
48 MECHANICAL SPRINGS
COS a
3- Q (51)
where
COS a ZLr .
3 cos* a E
1+
16 c2-l
the pitch angle sc. In Fig. 32 values of the constant <p have been
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negligible effect on the value of i^. The dashed curve for zero
It may be seen that for practical springs where the pitch angle
is usually less than 10 degrees and for the smaller indexes the
value of ip is less than unity, which means that the actual deflec-
This seems surprising at first since one would expect that the
direct shear would act to increase the deflection over that given
in actual springs the effect is usually very small; thus for indexes
HELICAL SPRINGS
49
will be under 2% per cent. For most springs where a<10 de-
grees and c>4 the difference is under one per cent, a figure which
sions: Outside diameter % in., mean coil radius r=.286 in., wire
initial pitch angle 7% degrees, working load 140 lb, the deflec-
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CHAPTER III
THEORY
die initial pitch angle is under 10 degrees and the deflection per
turn less than, say, half the coil radius. However, for cases
where the initial pitch angle is large or where the deflection per
turn is large, some error in the use of the usual formula, Equa-
grees and deflections per turn equal to the initial coil radius. The
reason for this error in the usual formula is partly that the
tion shown in Fig. 33a to that shown in Fig. 33b, the mean coil
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which takes the pitch angle into account. If the spring deflec-
tions per turn are large, however, this formula will also be some-
.50
51
rotate with respect to the other about the spring axis. If this
hooked ends where the hooks are not rigidly held but have some
freedom to rotate about the spring axis. If, however, the ends
about the axis of the coil are set up which tend to prevent this
2. The sanie except that ends are fixed so that rotation about the
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the ends being assumed free to rotate about the coil axis. If
a is the helix or pitch angle and r the actual coil radius, the
16Pr cos ,
r z (53)
32Pr sin a , (
'i#"(54)
52
MECHANICAL SPRINGS
direct shear load P cos a and the tension P sin a will be neglected
tc is
r.=- (55)
stress becomes
16Pr . 16Pr
xa1 to1
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Table I /
0 1.000
5 1.001
10 l.OOfi
IS 1.012
20 1.020
30 1.040
viding the coil radius r is taken as that actually existing when the
Trrf3" 3
53
and under 4 per cent for angles below 30 degrees, ( Table I).
Id) UNLOADED
(b) LOADED
flection per turn and initial coil radius, and of the initial pitch
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51
MECHANICAL SPRINGS
64PrM
'= GdT(58)
16Pr 16PrK,
Fig. 34Open-coiled
axial load
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per turn 8/n is not more than the initial coil radius r, the
errors in the stress formula due to pitch angle changes are under
6 per cent. This error may reach 15 per cent if the angle a
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56
MECHANICAL SPRINGS
15 degrees and 8, nr< .5, the error is under 3 per cent and,
and 20, this factor will vary from 1.07 to 1.14 and is thus of im-
to use expression for t,ni and amax given by Equations 34 and 38,
Chapter II.
ly loaded with the ends free to rotate as the spring deflects, Fig.
COS'a COS2a
r r
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An- (60)
where r and r are the coil radii corresponding to the initial and
Pr sin a=/(Ax)
EI
;"--) (6D
From elastic theory1 it may also be shown that the twist Af?
in the wire per unit length, as the spring deflects from a pitch
57
A6 =
(62)
round wire) will yield the twisting moment mt. This latter is,
Prcos a = GIp(A6)
or using Equation 62
r cos av
.(63)
, /COs'a COS-a\
^EL M 1 .)
r- r \ sin a I
.(64)
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EI
1 G/.
-COS'a
COS'al
EI
"g7
(65)
+tan'a
DEFLECTED
POSITION
-Zwn r
Fig. 36. From the geometry of this figure as the angle changes
58 MECHANICAL SPRINGS
S = l(sin asin a)
cos <*
130
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59
64Pr3n ,.
-lM.=*i7,^ (67)
The factor >pJ depends on the initial pitch angle x. , the ratio E/G
between bending and shear moduli, and on the ratio 8/nr be-
tween nominal deflection per turn and initial coil radius. Values
2.8
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below 10 degrees and deflections per turn less than half the coil
60
MECHANICAL SPRINGS
radius, 8/nr < .5, the error in the usual spring deflection formula
is not over about 3te per cent, i.e., does not differ from unity
by more than about 3% per cent. For pitch angles around 20 de-
grees and deflections per turn equal to the coil radius, however,
the given load P using the ordinary spring formula, Equation 58,
24
20
a.
'12
<n
Ui
UJ
id
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a.
. TO"l
AL T
WIST
fOE
GREt
<*.=
NITI
AL P
TCH
ANG
-E
<
iy
'i
0.
S. .
10
15
ZD
2.5
3.0
ratio 8o/nr may be found. Then knowing this and the initial pitch
angle x, the factor </<, may be read from Fig. 37. The maximum
straight line for various initial pitch angles a and for various
using Equation 67, for tension springs where the ends are fastened
61
rate (in pounds per inch deflection) with load. This would be
expected for tension springs since the coil radius r decreases with
(68)
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/ COS a
V r r I
<k= 1 I (701
(71)
62
MECHANICAL SPRINGS
where ^2 is the twist per turn in degrees and may be read from
Fig. 39 if a and 8/nr are known. It will be noted from this fig-
ure that i/-2 becomes slightly negative for small values of 8/nr,
and for initial pitch angles greater than zero. This means that
for small deflections the spring has a slight tendency to wind up.
This is due to the fact that the distortions of the elements of the
coil under axial load are in such a direction as to cause this wind-
i.e., prevented from rotating about the axis of the spring during
represented by vectors
in
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where the friction between the ends and the supporting plate
for the case where rotation occurs without restraint. In this case,
63
A* =y (74)
EI r r
A9=-Fr7- = (75)
GIP r ro
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r ro /
r V r r /
Mo= 1 I +
V r r /
/ co&a co fa. \
EI COS a
64
MECHANICAL SPRINGS
120
Vl.10
^.00
.95
.60
= 20<
5!
COM
'RES
SIOK
TENSION
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-3-2-1 0 12 3
COS a COS n
or
r=r-
COS a
COS ao
(78)
sin asin a
r. . EI; T(?9)
L GIp J
65
and 67, the factor ip,' may be expressed in terms of initial pitch
angle a and S<,/nr as before, and the results are given on the
the difference between the two cases, i.e., ends fixed or free, is not
some deviations.
springs with fixed ends by using Equation 79 are given in Fig. 42.
2.8
24
20
ORDINARY
FORMULA
/'
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24
66 MECHANICAL SPRINGS
restrained from rotation about the spring axis are given in Fig. 43.
hooks which fit into a hole in a plate, so that, when the spring is
2.8
2.4
2.0
18
i '2
the spring is fitted with spring ends which are in turn fastened in
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stress from Equation 55, for a circular wire cross section becomes
87
16 .
7' rf-^
.(80)
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68
MECHANICAL SPRINGS
proach 15 per cent for initial pitch angles near 20 degrees and de-
flections per turn equal to the coil radius. For usual applications
where the initial pitch angle is under 10 degrees and the deflec-
tion per turn less than half the coil radius, the results indicate
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neglected.
CHAPTER IV
inches and a bar diameter of IV2 inches. This low index was
STRAIN MEASUREMENTS
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two steel arms were welded to the semicoil as shown in Fig. 45.
is shown in Fig. 46. To measure the torsion stress in the coil, the
to the axis, Fig. 45, of the bar2. From these strain measure-
equal compression stress at right angles thereto; both of these stresses being at 45 degrees
to the shear stress. Thus strain measurements taken at 45 degrees to the shear stress
69
70
MECHANICAL SPRINGS
length (one centimeter), the peak stress can be found with suffi-
spring is simulated.
tained from the strain measurements on the outside and the in-
side of a semicoil are shown by the full lines in Fig. 47. Similar
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Page 52.
stress at the inside of the coil and from Equation 17 for the
stress at the outside are also shown. It will be noted that the
within a few per cent with the experimental results. The dashed
inside of coil
less pronounced.
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The open circles represent stress on one side, the full circles
72
MECHANICAL SPRINGS
seen that the stress on one side is about 10 per cent higher than
that on the opposite side at the higher loads. The reason for this
coils, the load will be slightly eccentric to the spring axis (further
STRESS COMPUTED BY
ORDINARY HELICAL
SPRING FORMULA
Equation 17.
In Fig. 50 the average test curve (which gives the stress duo
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These tests thus indicate that for small indexes the simple
73
DEFLECTION TESTS
years ago under the author's direction1. These tests also serve
which takes into account effects due to spring index and pitch
angle.
springs with indexes of 9.5, 4.7 and 2.7 from a single bar of car-
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punch marks a-a' and b-b' in the body of the spring to eliminate
74
MECHANICAL SPRINGS
test curve for a spring of large index is shown in Fig. 52, the
sent test points on one side of the spring, the crosses represent
Bar Nn.
3.
Table II
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Modulus of Rigidity, G
(Ib./sq. in.)
11.45 X 10"
11.46 X 10"
11.50 X 10"
rigidity for the three bars tested are shown in Table II.
% per cent between the different bars and, hence, that the mate-
. 10000
in
to
a.
a.
8000]
<
<
r-
r-
STRESS
DUE TO-
AD P
AXIAL LO
ONLY
4r
75
of Fig. 32, the actual deflection was in most cases slightly less
less than unity. A typical test curve for a spring of small index
is shown by the full line of Fig. 53, this curve representing the
is shown dashed.
indexes of 2.7 or 4.7 is given in Table III, which shows the per-
centage deviation between the test curves and the curves cal-
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springs made from three bars 1, 2, and 3) was 1.7 per cent
for springs of index 2.7 and 1.0 per cent for springs of index
for the known pitch angle and spring index were 2.4 per cent
and 0.7 per cent. It is thus seen that the average test values
70
MECHANICAL SPRINGS
Table III
"All deviations negative, i.e., deections were slightly less than calculated from
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for any given material may vary from an average figure by sev-
after the material has stood for some time. Adams5 found that
77
cover its initial value after the spring has stood for a consider-
PUNCH MARK
PUNCH MARK
EYEBOLT TO APPLY
LOAD TO SPRING.
PUNCH MARK
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PUNCH MARK
EYEBOLT
terial will act like low-carbon steel and will yield at a relatively
""The Effect of Overstrain on Closely Coiled Helical Springs and the Variation of
the Number of Active Coils with Load" by Pletta, Smith and Harrison, Eng. Exp.
78
MECHANICAL SPRINGS
low load. At higher loads the spring will, therefore, act to some
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less for hot-rolled carbon spring steel than for hard-drawn ma-
79
to account for this lower modulus value. Such a layer may easily
200
600
2200 2600
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that within this range for most materials the modulus of rigidity
G=Go(l-mt) (81)
'Bureau of Stds. Jl. of Res.. Vol. 10. 1933, Page 305. See also Brombacher k
80
MECHANICAL SPRINGS
Table IV
(20C to 50C)
Coefficient rn
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t Hard drawn.
degrees Fahr. may be obtained from the curves of Fig. 54. These
81
64Prsn
(82)
1200QOOO1-
5.000,000]
-200
TEMPERATURE-F
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800
82
MECHANICAL SPRINGS
Usually this means that the spring must be cut up after the test
terial as coiled into a spring and heat treated and straight bars
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diameter.
clamping jaws near the ends of the specimen and this may intro-
"This equation is easily derived from the known formula for angular twist of a
'L\K, <e5>
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the source of the data together with the method used is indi-
84
MECHANICAL SPRINGS
pounds per square inch by this treatment gives an idea of the re-
duction which may result from stressing helical springs far beyond
Table V
Wire
or Bar
Modulus
Diam-
of
Heat
eter
Rigidity
No.
Material
(in.)
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Treatment
(Ib./sq.in.XlO<)
Investigator
11.6
Edgerton1
1% C steel
Q & T"
i .,
11.82
Adams2
1% C steel
v"
11.14
Adams3
1% C steel
Q & T->
9, 16
11.2
Wahl,
1% C steel
0 & T*
11.47
Wahl5
.67% C steel
Oil tempered
.028-.08 11.12
Sayre"
Music wire
.035
11.4
Brombachcr
& Melton'
Hard-drawn wire
11.4
Sayre"
85
values of G found for the other cases are probably due largely
Table VI
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Wilson2
'Carnegie Scholarship Memoirs, Iron & Steel Institute, 1937, Pages 1-55. Direct method
'Proceedings A.S.T.M., 1930, Part II, Page 357. Direct method used.
'Proceedings Inst. Mech. Engrs., 1938, Page 460. Direct method used.
"N.A.C.A. Technical Report No. 358, 1930, Page 568. Direct method used.
square inch for chrome vanadium steels, 10.6 X 10" for stainless
steel (18% Cr, 8% Ni), 9 X 10" for Monel metal, 6.4 X 10" for
FATIGUE TESTS
86
MECHANICAL SPRINGS
springs shown in Fig. 55. The results of such tests are of direct
tion engines.
reason for this lies mainly in the fact that the endurance limit
reason for variation in the results obtained lies in the fact that
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and the line of minimum stress'-. The circles with the arrows
cause failure within ten million cycles, while the plain circles
represent ranges which did. On the basis of these tests the es-
87
pounds per square inch. This means that the spring could be ex-
coils, and spring index are given together with values of the
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4-
t *.
99
8cc
111
o90
ec
coo
1 ii
oc
III
--T
oo
-co
e fi 1
co e
.e
I-coo
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MECHANICAL SPRINGS
--
'
-s
1:
!=
RS
oo 9 lo
e0
-1
CO
NO
cn m
m r- co
'n
1-
r-
t-
S3-
333
53
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"a
4-
+-
.N
CO
11
ei
90
MECHANICAL SPRIXGS
and those made under the direction of the special research com-
I40000i
inch. This table covers the smaller wire sizes. Thus the limiting
stress range of 60.000 for cold-wound carbon steel wire means that
the spring will withstand a range of 60,000 pounds per square inch
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Page 12.
91
same or similar materials. The reason for this is that the quality
Table VIII
Limiting
Approx.
Index
Stress
(D/d)
Rangej
Investigate!
(inches)
(Ib./sq. in.)
M3-.16
10-11
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41000 1
31000 ]
Cold-drawn wire )
.16
60000
Zimmerli
Tatnall
.135
11
46000
.25
56000
Hengstenberg
.063
76000
''
.148
6-7
70000
Zimmerli
8-7
.148
1150001
.162
6.5
75000
1150001
.65% C
.135
14
68000
Tatnall
.148
7.4
60000
Zimmerli
.148
7. t
52000
.135
14
53000
Tatnall
(Cold wound)
92
MECHANICAL SPRINGS
per square inch for springs without the shot blasting treatment.
(Table VIII).
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Table IX
Limiting Stress
Range
Material
Bar Diameter
Index
Investigator
(in.)
(D/d)
(lb./sq.in.)
56
4.8
68000
Jobnson
.75
5.0
72700
Edgerton
Cr.-vanadium steel
56
4.8
77000
Johnson
Beryllium bronze
56
4.8
33000
Johnson
factory: For 'i-inch wire diameter springs, shot size .040 and
1940. Page 62. "New Trails in Surface Finishing", Steel, July 5, 1943, Page 102.
Feb., May, Sept. 1943. "Improving; Fatigue Strength of Machine Parts", Mechanical
'"Chapter XXIII gives more data on endurance limits of spring materials (as dis-
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in Table IX.
MECHANICAL SPRINGS
somewhat lower values may be expected and vice versa for the
.148-inch diameter valve spring wire (SAE 6150, index 7.4) a life
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The fatigue test data given in this chapter apply only for
that the full values of stress ranges found in fatigue tests should
11 Private communication.
CHAPTER V
between stress and strain do not apply rigidly after the elastic
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95
9R
MECHANICAL SPRINGS
few times dining the life of the spring. A spring loaded less
than about 1000 times during its life would usually be con-
cited, such as the springs in lightning arresters Fig. 58. Here the
change.
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are subject to static loads; in such cases the designer must guard
i.e., the ratio between coil diameter and wire diameter. Where
STRESS CALCULATIONS
From Fig. 59a it is seen that, for a spring of small index, the
stress ab on the inside of the coil is much larger than the stress
a'c on the outside of the coil, i.e., most of the high stress is con-
area near a.
REGION OF PEAK
of helical spring, small index at a and large index at b (for the spring of
small index most of the stress is concentrated near the inside of the coil at a)
side of the coil is only a little larger than the stress a'c on the
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98
MECHANICAL SPRINGS
a relatively small region near the inside of the coil as is the case
where the index is small (Fig. 60b). In other words, in the case
200 X
ii
spring steel
01 02 03 .04
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for this is that, after the elastic limit is passed and yielding be-
99
case of the small index spring, where only a small part of the
greater slope after the yield point has been passed) it appears
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effects due to the end turns and pitch angle, the torsion moment
at any point along the bar will be equal to Pr while the direct
in Fig. 62a while the peak torsion stress t, due to this moment
alone will be that given by the usual formula (Equation 4). Thus
16iV (86)
id?
similar diagrams.
100
MECHANICAL SPRINGS
the direct shear load P, which for our purposes may be consid-
4P
n - (87)
Tfl'
tm=r, + tJ + (88)
ia' 7rCt-
rm l^-K. (89)
where
K.= l+ (90)
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stress but since this will have a similar effect to that of stress concentration due
() (b) (O
the spring, the working load being then taken as a certain per-
Fig. 61, it may be expected that after exceeding the yield point
something like that shown in Fig. 64a for a spring of small index
and in Fig. 64b for one of large index. Actually it may be ex-
<r
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102
MECHANICAL SPRINGS
curve after passing the yield point will take place due to the
hand, if the index is large, these two areas will be about the
same, Fig. 64/;. Due to this effect, the point O' where the stress
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the resultant shear stress at all points of the cross section under
where O' represents the point of zero stress (the line BB' rep-
O' from the center O. From this the spring index may be found.
tween O'B and O'B', the radius p' of this circle will be
O'D
p=
1-
(91)
Using this formula for any point D along O'B, a series of circles
dA-pdpdB (92)
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(a) (b)
section for low index helical spring under yielding. Point of zero stress O'
104
MECHANICAL SPRINGS
moments over the whole cross section. Thus the moment My for
ing the circles as shown in Fig. 65a, the value of sin ip can be
m Fig. 65a and plotting the value of tp2sin ,p along each radius
the angular spacing of the radii and adding the total. In this
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Equation 92 in this
The total vertical shear force P acting over the section for
P= - / / r, sinii-BipdpdO (96)
plotting the function tp sin (i/< 0) along each radius, and find-
ing the area under each curve. By adding these with proper
coil radius r may be found for a given e and d. From this the
moment M,' (or load P') for complete yielding of a large index
mate of the effect of the direct shear load is possible. The ratio
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(97)
Table X
Wire
Dism. inches
1/8
5 32
3 16
14
5 16
38
7 16
12
5/8
34
7/8
1-1 8
.014
1'
.720
.0378
.590
.0570
.411
106
.352
.172
.293
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1-1 4
. 2 VI
.0228
.016
1.372
.0180
1.09
.0319
.900
0178
.664
.0910
520
.1465
. 118
376
.302
.218
.018
1.98
.0162
1 .55
.0273
1.275
0111
.952
.0794
753
.1285
.625
. 191
.538
.266
.470
.352
Table X (continued)
Wirt
Dim. inch
1-3 8
1-1 H
1-5/8
1-3/4
1-7 8
2-14
2-1 It
2-3/4
3-1 2
4-1 It
(MS
.99!
.051
3.89
.937
3.55
1.12
.055
4.85
4.45
4.11
Mi1
1.03
1.22
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3.25
-yj
6.01
5.50
.955
5.08
1.13
values
r turn)
. De-
.782
.063
7.33
.737
6.71
6.20
1.05
5.75
1.23
.887
7.45
6.90
1.15
108
MECHANICAL SPRINGS
high and since strain hardening effects come into the picture.
tions 89 and 7 for various standard outside coil diameters and wire
sizes. The music wire gage is used for sizes up to .090 and the
this stress is used in the table for convenience only and is not
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pounds per square inch, the loads and deflections given in the
wire, from Table X the load and deflection per turn are 103
pounds per square inch in this size and a yield point in tension
pounds per square inch. This means that the permissible loads
per cent of those given in the table. In the case cited, the
Based on a stress of 100,000 lb./sq. in. To find load at any other stress r, loads given
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I 10
MECHANICAL SPRINGS
coil diameters. Thus for a wire size of .106-inch and a coil out-
results from the charts of Figs. 66 and 67 and for best accuracy
used provided that the stress on which the table or chart is based
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to the direct shear of the axial load. This follows since K KCK.
inch. This latter would then be compared with the yield point.
For any other stress t, values should bo multiplied by T/100,000. Also, if the modulus
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112
MECHANICAL SPRINGS
coil and wire diameters, free and solid heights, are determined.
index 15, the factor K, = 1.06 from Fig. 68. Again assuming
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and left for a period of time varying from three days for the
carbon and low-alloy steel springs to ten days for the stainless
steel springs. After this heating, the springs were removed from
I-
in
ii
o 10 I 1 1 1 1 1 1 1 1 1 1 * 1 1
* 3 4 5 6 7 8 9 10 II 12 13 14 15
SPRING INDEX C=
the test fixture and the loss in free height determined. From this
steel springs where the tests were run ten days. The stresses were
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cent lower. The actual tests were made with springs which had
the lower bluing temperatures not all the coiling stresses are
removed; when these latter are combined with the load stresses
rection, about ten per cent load loss may be expected within three
days for music wire or the .6 per cent carbon steel wire, when
IN
MECHANICAL SPRINGS
less steels of the 18-8 type showed a load loss of only about
four per cent at 350 degrees Fahr. which increased to 11.5 per
cent at 550 degrees Fahr. at the same stress (100,000 pounds per
square inch). These latter tests were run for ten days. For very
spring steels are reliable when stressed to not more than 80,000
pounds per square inch (or to about 72,000 pounds per square
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Table XI
Loss in
Loss in
Load at
Load at
80,000
100,000
lb./sq.in.
Tem-
Bluing
Ib./sq.in.
Rock-
Oiameter perature
Temp.t
Stress"
Stress*
well
Material
(in.)
i Fl
(F)
(%)
(%)
Hardness
Music wire . . .
.148
250
T00
2.5
4.7
48
.91% C .31% Mn
!062
350
700
10
IS
Music wire
250
7(10
2.5
3.5
51
.91% C .31% Mn
.'l48
350
115
per unit length along the wire, the twisting moment will be
7 = />9 (98)
y = y'+y"=pe (99)
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Equation 101,
(100)
1 dr
G ~dt
+g(t> = pd
(102)
s"The Creep of Metals Under Various Stress Conditions'*-A. Nadai, Th. ton
1 16
MECHANICAL SPRINGS
M 27t pa
P= / rp'dp (103)
r r Jo
This has been found to agree with tests over limited ranges of
grating
(3+*)r (15)
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r- (106)
assumed constant:
= 0 or I
at "' at
fftWdp -.(107)
1/
u- at O.l
2Prp
TO4
INITIAL STRESS
DISTRBUTION
STEADY
DISTRBUTION
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not change with time, the angle of twist per unit length 6
(109)
118
MECHANICAL SPRINGS
(110)
8t
+GCt" = 0
(111)
r="Rra-i )CGtyi~^(113'
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1 (n-l)CGt
tn-, ,..o-]
(112)
(114)
CHAPTER VI
SPRINGS
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load; in other words that the stresses computed this way are too
119
120
MECHANICAL SPRINGS
occur (even for fatigue loading) lies in the fact that some ma-
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layer which occurs during heat treatment and the effects of shot-
METHODS OF CALCULATION
mental Stress Analysis, Vol. 1, No. 1, Page 118, discuss this. Also article by Peterson
and Wahl, Journal of Applied Mechanics, March, 1938, Page A-15 and discussion De-
lected
These factors will be discussed more fully later. For the present,
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'r
STATIC STRESS T,
For this case (0 to maximum stress) both the static and vari-
122
MECHANICAL SPRINGS
stress range from tt(), to tmax where these stresses are figured by
tmnz teiin
.(115)
becomes
(116)
t(BBi4"tm in
since this factor has already been used in figuring rmax and t,,;
1.6
<o
1.4
to
u'
cc
to
ij
\Z
i N
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in
/to
Tw N
/f
rmin/tmax.
ty/r'e = 2 and q 1
cIf the yield point is not sharply defined, as an approximation it may be taken
as that point where the plastic strain is .2 per cent. See "Concerning the Yield Point
some cases this would give results in closer agreement with tests, but the results
obtained by using the yield point will, in general, be on the safe side.
. (117)
+-
t t,
~2
'-( 5 )
2.5 and q = 1
.(118)
2 .4 6
T7mn./T7Bax.
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tnat=C^T,' (119;
"It is assumed that the variable component of stress is not Hrea*T than the static
component, i.e., that only stress conditions corresponding to the line PA in Fig. 71 an?
124
MECHANICAL SPRINGS
2 ty
Cm
M-0('~)
(120)
becomes
Fig 71, parallel to the line PA and intersecting the axis of ab-
given values of tv/t,-' are given in Figs. 72, 73 and 74. These
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in Chapter V.
been determined for the given material and wire size by actual
(121)
K/=l and from Equation 122, q=0. For materials fully sensi-
tive, K/=KC and hence q=l. Thus the more sensitive mate-
tion 122,
It is, however, assumed that t,ux and tmi have already been cal-
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K,-l+q(K.-l)
(123)
(124)
(t,.,-t,-) [l+q(Ke-l)]
2 Kc
(125)
2r
(126)
l+gdC-l)
126
MECHANICAL SPRINGS
C = 2tu/t/ -(127)
sensitivity index q = Vi and for t/t,' equal to 1.5, 2.0, and 2.5 are
2-l 1 1 1 1 1 1 1 1 1 1
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60,000 pounds per square inch, while torsion tests show a yield
rmin./Tmax.
K).
From this chart for an index c=3, C, = 1.53 when tmin/tmax -^>-
61,000 pounds per square inch (the stress being figured by using
the factor K). If the spring index were 10 instead of 3, the fac-
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128
MECHANICAL SPRINGS
this limitation.
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the value of t/ by tests on the actual wire size used, since the
larger sizes.
springs of small and large index assuming a given wire size. Al-
129
after coiling, since the effect of the heat treatment may be dif-
2.4r
posite sign so that when the spring is under the working load,
stresses. For this reason the tendency will be for the fatigue
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130
MECHANICAL SPRINGS
of large index.
due to the end coils have been neglected. These effects may in-
the shape and form of the end turns, and on the total number of
of tests yet made to check the effect .of spring index on endur-
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mum points (rmin and tmaJ-) of the limiting stress ranges as found
by Zimmerli for the various indexes are given in the second and
for a spring of large index, the test results for c=11.9 are used
square inch. Since the yield point in torsion for the material will
with sufficient accuracy allowing the use of Fig. 72. Using the
stress tmas were computed using the chart of Fig. 72 and Equa-
tion 119. The computed values of tmax thus obtained are given
cates that the test and calculated values of limiting stress range
differ by only a few per cent. This offers some indication that
Table XII
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two groups of springs coiled from %-inch diameter bar stock, one
were practically the same for the two groups of springs, while
the bar curvature nor the direct shear stress have any effect on
132
MECHANICAL SPRINGS
minimum loads Pmax and P,,, then the range in load will be
from the range in load using the full curvature correction factor
be used in design.)
Peak stress tmax is then calculated from the load Pmax using
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will limit the peak stress to this value. Then the actual limiting
stress range is taken as the test value with the peak stress equal
below the torsional yield point, then the actual endurance range
exceed the yield point in torsion, there will not be much varia-
tion in the value of the endurance range for various peak stresses.
range equal to the range with the peak stress equal to the yield
Design Stresses for a Standard Code" Transactions A.S.M.E., July 1942, Page 476.
133
practical use.
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Equation 89, is 61,000 pounds per square inch. Since the ex-
110,000 pounds per square inch, Table VII Page 88, the factor
CHAPTER VII
COMPRESSION SPRINGS
available. In such cases the use of the spring tables given here
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where these methods are used, however, the choice of the proper
this chapter.
will be advisable.
134
Table XIII
For springs of good-quality spring steel. All stresses based on the use of a curva-
ture correction factor. The table does not hold where corrosion effects or high tem-
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perature are present For phosphor bronze springs 50 per cent and for rust-resisting
spring requirements.
From Table XIII it may be seen that lower stresses are used
for the larger wire sizes and for severe service in accordance
service.
13d
MECHANICAL SPRINGS
000 for %-inch diameter to 80,000 for 1-inch diameter bar. These
Table XIV
(lb./sq.in.) (lb./sq.in.)
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Such high stress values should not be used where long life or
Table XV
Wire diameter
Music Wire
Tempered Steel
Hard-Drawn Steel
(in.)
(lb./sq in.)
(lb./sq.in.)
(lb./sq.in.)
.020 to .030
100,000
100,000
90,000
.031 to .092 .
90.000
100,000
80,000
.093 to .176
90,000
90.000
80,000
.177 to .282 . .
90,000
70,000
.283 to .436 . .
85,000
.437 to .624
80,000
.437 to .624
90,000
.625 to .874 . .
90,000 I
Brass 40,000
SPRING TABLES
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cent of those for carbon steel. The shear modulus used in com-
Table
Severe
- Outside Diameter
Wire
Diam.
'A
H_
014
493
0123
0209
334
0319
.252
0606
203
0990
169
146
.016
.726
0100
. 588
0174
494
0266
.375
.0514
. 302
0838
253
126
.217
.174
018
1 03
00840
836
0147
700
0227
534
0445
.428
.0730
360
.310
153
271
203
no
. 020
1 39
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398
XVI
Service*
of Spring (in.)
H1
1H
IK
1H
1H
IK
1H
Wire
[Ham.
.014
.016
.018
.020
.022
.024
.026
028
030
869
348
032
1 04
906
442.
.034
.036
.038
.325
1 24
1 08
305
.406
1 46
287
1.27
.382
1.14
.490
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140
MECHANICAL SPRINGS
Table XVI
Severr
Outside Diameter
Dlam.
"4
VH
IK
1H
\H
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086
3".. 7
.00532
31 6
00981
-iv, \
21 1
0229
20 '6
0413
17 4
0652
15.0
0945
13.2
.130
118
10.7
216
9 72
.267
8.95
325
8 26
387
.0157
.170
.090
/-
38 9
mi 17 1
34 6
00879
30.8
0143
27 6
0210
22 u
0384
19.2
.0610
16 6
0889
14 6
1 22
13.1
160
118
(G)iilinucd)
Service *
of Spring (in.)
1H'
IK
2H
2'<
3H
5H
Win-
Dii. 1t>.
7 70
156
528
6 76
.608
mi
.086
.782
8 52
432
7 96
501
7.47
.575
6 66
742
h (III
090
.930
9 26
123
8 66
491
8.13
565
7.25
.728
,. 53
.911
091.-,
12 8
11.9
432
U.J
498
10 0
9 02
11 21)
i'
in;
372
.642
.808
.987
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7 19
1-42
MECHANICAL SPRINGS
Table
Severe
Outside Diameter
Wire
Dtam.
yt
"4
.014
.369
.0100
.299
.0170
251
.0260
.189
0494
.152
.0805
.127
.119
.015
.452
.00901
.370
.0155
.311
0238
.233
.0456
.187
.0742
.156
.110
.134
.152
.0162
.565
00801
.459
0139
.388
0213
.292
.0412
. 237
.0674
.196
.100
.169
.140
.0173
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XVII
Service*
of Spring (in.)
IK
1H
1H
i';
1H
Wire
Diam.
.014
.0162
.0173
.0181
.0204
.0230
.0258
.0286
652
283
.032
851
.748
341
.035
256
1 10
234
.952
.311
.853
.399
.038
1 28
221
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.015
141
MECHANICAL SPRINGS
Tabic XVII
Svere
- Outside Diameter
ir.
Dlam.
'A
1H
IK
1H
IH
II91
/'
31 6
00371
28 1
00697
25.0
0112
22.4
0167
in i
15.6
.0484
13 6
0708
119
0969
10 6
.128
9 68
. 163
8.78
.202
8 10
.245
7 47
293
0304
.106
II 1
37 5
00822
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1H
31O
28 1
0237
23 9
.0384
20 II
0570
IK I
16 4
105
118
13 (.
.168
12 5
[Continued)
Service*
if Spring (in.)
i<. VH
2H
2H
3H
'5
5H
Wire
Diuni.
6 94
.344
6.50
460
5 44
.593
4 90
.742
0915
too
10 7
287
10 0
9 45
386
8.40
498
7 58
628
6 90
.769
6 35
.928
.106
.335
IS 9
.244
14 9
286
14 0
12 4
427
11 2
10.3
662
9 45
.800
.121
329
539
.212
ii
20 7
249
19 1
17 2
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6 10
146
MECHANICAL SPRINGS
Table
t Sever*
Outside Diameter
Wire
Dlam.
HA
* 1 * 1 f*
.0142
.0159
0179
. 255
.0114
.0196
.174
.0297
132
0568
.106
.0921
0887
. 137
.357
.00968
.290
.0167
.243
.0257
.184
0492
.148
0810
.124
.120
107
.167
.504
.00805
.410
.0141
.346
.0220
.262
0426
211
. 0702
. 177
.105
152
.146
133
. 194
.0201
.0226
702
.00662
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.208
XVIII
Service*
f Spring (in.)
1 V.
1H
l'y.
IX
!2
^ ire
DUm.
.0142
.0159
.0179
.0201
p-
.0226
.0254
.0285
.032
134
Ml
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S18~
293
.541
386
036
~I4I~
257
.741
.342
.660
.440
.040
1.21
224
1 05
298
.938
.384
,84'i
.48!
.773
148
MECHANICAL SPRINGS
Tabic XVIII
S<'> rr
Outside l>iItmrtn
Wire I
Dinm.
1*8
1H
\H
1H
.091
20 7
00439
II! 1
00819
16 4
14 8
1)196
12 2
0361
10 2
0568
8 90
0833
7 86
7 112
.151
6 32
192
5.76
.237
5 28
.287
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1H
4 <H
>
0133
.115
. 344
.102
24.7
00620
22 6
0104
II. 8
113
0480
12.4
0706
11 0
9 80
129
II IIi
165
8 10
7.45
(Continued)
Service*
of Spring (in.)
IN
IK
2K
2'i
3M
4X
5H
\\ ir.-
I>ium.
I 4 56
4 26
468
3 99
539
3 56
697
3 20
870
.091
6 40
5 95
.405
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404
.-> 60
5 00
.610
4 51
768
4.10
938
3 78
1.13
/'
.102
353
.474
8 90
8 31
7 82
413
7 00
6 28
.672
5.74
.828
5.26
.998
III
307
.358
.538
12 9
264
12.0
307
11 .3
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152
MECHANICAL SPRINGS
pitting Table XVII for stainless steel was 10.5 X 10" while that
severe service, from Table XVI for .263-inch wire diameter and
2 inch outside diameter the allowable load is 161 pounds and the
tion would require .8/. 124 = 6.45, say &k, active turns or about
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.8 = 3.26 inches free length, allowing 10 per cent extra length for
inches, and the solid length about 10 per cent less or about 2.2
inches. The stress at the solid length would be about 10 per cent
DESIGN CHARTS
These charts were published by Wallace Barnes Co. in The Mainspring for June
and August, 1940, and are reproduced through the courtesy of this company.
100 pounds, that of Fig. 79, between 100 and 10,000 pounds.
pounds per square inch torsion stress, the abscissas, inches de-
flection per pound of load per active coil. Thus the abscissa,
rocal of the spring constant in pounds per inch. The set of lines
any line of one set with that of the other set fixes the load at
100,000 pounds per square inch stress and the deflection per
pound of load per active turn. Thus, for example, if the wire
size is .04-inch and the outside coil diameter '/i-inch, the load
at 100,000 pounds per square inch stress will be 9.3 pounds and
there are 10 active turns the spring constant will be 1/.025 =40
pounds per inch. The load at any other stress t different from
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= 5.58 pounds.
If the load and working stress are known the required spring
size may easily be read from the charts of Figs. 78 and 79. Thus,
seen that a wide variety of sizes will yield this value of load.
eter of %-inch will come close to it. In this size the spring will
spring with G = 11.5 X 10" pounds per square inch. If, say,
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158
MECHANICAL SPRINGS
100,000 pounds per square inch stress to the net solid height of
the active coils in the spring. It is clear that springs with a large
solid height and vice versa. Thus a spring of .100-inch wire and
100,000 pounds per square inch, Fig. 79. This means that the
equal to the solid height. At 60,000 pounds per square inch the
however, exceed 3 per cent and will usually be within 2 per cent.
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Chapter VIII considers these variables. This means that the charts
CHAPTER VIII
COMPRESSION SPRINGS
springs are shown in Fig. 80. The most common typeends set
stress for a given load) than would be the case where the ends
are simply squared and closed, while in Fig. 80c, the ends are
left plain without any grinding. This type of spring would give
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Fig. 80d is the same as that at c except that the ends have been
sion springs requires that the effect of the end turns be esti-
alytical work by Vogt1 indicates that for the usual design of end
coil with ends squared and ground, Fig. 80a, the number of active
Page 468.
157
158
MECHANICAL SPRINGS
pression spring has 10 free coils and 12 total coils (tip to tip of
bar) then on this basis the number of active coils would be 10J/2,
with the load, and this increases the number of inactive turns.
load the number of active turns was equal to n' -f- % where n'
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figure for average active turns varying from n' to n' + Vt. Be-
cause the total number of turns is n' + 2 for the usual type of
end turn. Fig. 80a, this means that the inactive turns found in
solid height divided by bar or wire diameter. Since for the usual
shape of end coil, Fig. 80a, the "solid turns" are equal to the
"total turns" measured from tip to tip of bar, minus V-i turn,
2"The Effect of Overstrain on Closely Coiled Helical Springs and the Variation
of the Number of Active Coils with Load", Virginia Polytechnic Institute, Engineering
pears that for the usual design of end coil the number of in-
tion from total turns. Probably a mean value of P/i inactive coils
while a lower figure may be used for lower loads. The seating
associates2.
usual type of end turn. Test results concerning the other types
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of inactive coils are as follows: For plain ends, Fig. 80c, active
Fig. 80d, active turns are n 1. // 2% turns at each end are set
roughly as n 5.
ECCENTRICITY OF LOADING
160
MECHANICAL SPRINGS
the other.
here. However, the final results of the analysis are given in the
eccentricity e and coil radius r and the abscissas being the num-
ber of turns n' between tip contact points. The total number of
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e =1.123(2-1) (128)
\Vz turns greater than the number of coils n' between tip contact
that where the spring index is fairly large the stress will be in-
This is borne out also by experiments made by Pletta and Maher"Helix Warping
0.40
Q30
0.20
QI0
30
as
Fig. 81Ratio e/r between eccentricity and coil radius as a funtion of n'
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end of the spring were not parallel. The loads were then ad-
be calculated.
and load being given. Springs tested had ground end coils of
the usual form. In the last column the values of the ratio e/r
tions 128 and 129 are given. For comparison the test values of
162
MECHANICAL SPRINGS
Table XIX
Load Eccentricity _ e
Turns
Coil Radius
Spring
Outside
Wire
Between Tip
Total
Calcu-
No.
Diameter
Diameter
Contact Points
Load
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Test
lated
(in.)
(in.)
H'
(lb.)
2%
.177
34
.12
.12
2%
.177
34
.04
.12
2%
.177
4U
31
.09
.11
r-.
.177
1',
29
.14
.11
2-i
.177
38
.19o
.23
2*i
.177
38
.13
.19 av.
.23
may be excessive.
Table XX
A.S.T.M.
Specification
Material (No.)
Wire
Permissible
Diameter
Variation
(in.)
(in.)
.028 to .072
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.001
.073 to .375
.002
.003
.0003
.027 to .063
.0005
.001
.093 to .148
+ .001
.149 to .177
.0015
.178 to .250
.002
eter of the spring after winding will be slightly greater than the
mean diameter D.
is
64 PrVi
5=-
Gd<
^Manual of Spring Engineering published American Steel and Wire Co., Page 97.
lfi-4
MECHANICAL SPRINGS
where e and A are small quantities, relative to unity, the true de-
64Pr>(l+c)'n
64Pr3n
8. (1+3.-4X) (130)
(ja,
radius is one per cent greater than the nominal, i.e., e=.01,
while at the same time the true wire diameter is one per cent
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, 64Pr>n
values the actual deflection will be 1.07 times the nominal de-
cent, i.e., A=.02. Assuming the true mean coil diameter of this
been made about 2 per cent less than the nominal, which meant
that e was .02 Using this value in Equation 130 the true de-
actual coil diameter had been made slightly smaller than the
variation in the stress; a one per cent change in the coil diameter,
a one per cent change in stress. Usually, the stress does not have
Table XXI
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3 to 4 .0550 .0730
4 to 5 .0725 .095
5 to 8 .125
6 to 7 .165
7 to 8 .210
requires not only that the effective turns be known, but also
average figure for modulus of rigidity for carbon and alloy steel*
166
MECHANICAL SPRINGS
pounds per square inch. On the basis of the test data given in
Table XXII
Modulus of Rigidity
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the coil pitch such that when the spring is compressed solid, no
with a coil pitch sufficiently great so that the elastic limit of the
shown in Fig. 82b for a spring of large index7. At low loads be-
also been decreased. If the initial free length of the spring is made
the final free length may be held to the specified value. At the
yond a certain initial free length, the final length after the set-
ting operation will be the same. The reason for this is that the
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168
MECHANICAL SPRINGS
Table XXIII
Helical Springso
Stress
at Solid Compres-
sion up to which
it is not necessary
to remove set
(lb./sq. in.)
Maximum Stress
at Solid Compression
Diameter
Material
(in.)
(lb./sq. in.)
Music Wire
up to .032
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.032 to .062
.062 to .125
130,000
110,000
100,000
90,000
180,000
170,000
160.000
150,000
up to .032
.032 to .062
.062 to .125
120,000
100,000
90.000
80.000
170,000
160.000
150,000
140,000
Oil-tempered wire
80,000
140,000
up to .125
over.125
85,000
75,000
140,000
120,000
Phosphor bronze
General sizes
40,000
70,000
for some time, however, if the settage stress is too high, the free
length will increase slightly. This again will change the load-
CHAPTER IX
lar to the buckling of a long slender column when the load ex-
BUCKLING
the case of the usual steel column, on the other hand, this de-
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crease is small and may be neglected. The reason for this lies in
usually less than .1 per cent. This is not true, however, for a
also given here. For addit-'onal references on the buckling of springs see articles
by E. Hurlbrink, Zei(. Vrr. d. Inj.. V. 54, Paee 138, 1910; by R. Grammel. Zeit Aneru;.
Math. Mech., V. 4, Page 384, 1924; and by Biezeno and Koch, Zett Angew Math. Mcch.
169
170
MECHANICAL SPRINGS
the spring in its unstressed condition- and a, fi, " are the same
P.
1+-"P<
(131)
AG
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ber of coils per unit length (Equations 135, 141, and 144). Hence
III
p.- !. ()
^By compressive rigidity is me(int the ratio of load to deflection per unit of length
for the case of a bar under direct compression. For a bar of cross-sectional area A
and modulus of elasticity E the compressive rigidity is equal to AE. Likewise the
flexural rigidity is the ratio of bending moment to curvature for a beam in pure
bending and is equal to modulus of elasticity times moment of inertia of the cross-
section. The shearing rigidity is equal to the ratio of shearing force to shearing
deflection per unit of length and for a beam is equal to modulus of rigidity times cross-
scctional area GA, multiplied by a constant depending on the shape of the section.
Gd<h
(135)
ltzL= 64P^n
h Gd'L
L-l P
'-(-T>
(137)
Equating values of Prr given by Equations 134 and 137, the fol-
(/0-/) V2?. 1
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dB EI + 0T (139)
172
MECHANICAL SPRINGS
In this case EI and GIP are the flexural and torsional rigidities of
the wire cross section, respectively. The twist due to the bending
along the axis yy of the moment; that due to M< must be multi-
rM rM
El GIp
'--/'-(
rM rM
EI
)d<t>
irMr
EI
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Integrating this,
/ EI \
(1+g7t)
(140)
one inch axial length will be n6/l. This will be also equal to the
I.
nir Mr f E \
17 ~eT\ +"2g)
From this the flexural rigidity /?, which is the ratio of bend-
173
00 =
2LEIG
nirr(2G+E)
.(141)
Qr
EI
.(142)
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de=_-sin <t>ds
17-4
MECHANICAL SPRINGS
EI
turn of spring,
Since there will be n/l turns per inch axial length, the total
ny irn Qr3
~TT=~l7 EI
LEI
7o ,- (144)
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son's ratio,
z3-z2+(3+2r)mz-m = 0 (145)
where
m = (146)
It will be found that this equation has one real positive rool
Equation 137,
P_*^-_a#(1_,) (147)
P-CbLCk (148)
I0|IIIIII 1I 1IIIII I
5 0 10
15
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that the buckling load factor Ca is now to be taken from the upper
'See article by Biezeno and Koch, loc. cit. for a further discussion of this problem.
Tests carried out by these investigators show good agreement with the analysis
nrovided that the number of coils is not too small and that the coils do not touch
before buckling occurs. Also comments in Machine Design, July 1943, Page 144.
176
MECHANICAL SPRINGS
Gd,
P=CL - - (149)
64r3n
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the spring constant CK = 142 pounds per inch. From Fig. 86 the
tion; hence curve b of Fig. 86 for fixed ends may be used. Then
stress of 60,000 pounds per square inch the actual load on the
190 pounds, taking 60 per cent of the value for 100,000 pounds
able margin between the working load and the buckling load.
stand not only axial loads, but also transverse loads as indicated
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178
MECHANICAL SPRINGS
latter are called upon to absorb lateral forces due to the un-
such conditions the combined effect of the axial load P and the
axial load relative to the buckling load the larger the effect of
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from Equation 148 for the case of built-in ends using curve b
of Fig. 86. If P is the axial load acting on the spring, the ratio
approximately by"
=C, (151)
load were acting the results of beam theory may be used. The
'A more exact method of determining this factor is given in the reference of
Footnote 5. This shows that the approximate expression is sufficiently accurate for
practical use.
179
S'-j2El
QP
* (152)
12(5
Ql
S. (153)
Ql3 Ql
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2Qnr
.=-(.204/'+1.06r=) (155)
lateral deflection 8.
torsion moment due to the axial load P will be Pr. The effective
180
MECHANICAL SPRINGS
M-Krt 2 )1 T (156)
16M, 4c-1
irrf3 4c-4
_ 16Pr/ .615 \
Equation 16). The direct shear stress at the inside of the coil due
stress is
t-t. + t. (158)
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ratio:
approximately (159)
2r 2Pr
The ratio J/r= 9.5/2.13 =-4.48 and from Fig. 86 the buckling
2(200)(8)X2W r 1
(for Cl=2) is
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Thus it is seen that in this case the actual deflection with axial
s Ol
pounds per square inch may be expected due to the lateral load.
182
MECHANICAL SPRINGS
TEST DATA
This steel plate carried the axial load while the transverse load
155. The range in deviation in the test points was from about
60 per cent to 125 per cent of the theoretical values, most of the
values.
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tween test and theory lies in (1) imperfect clamping of the end
Equation 151.
CHAPTER X
SINGLE SPRINGS
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August 15, 1941, Page 134). However, the method used by the author differs from
that used by Jennings in that a distinction is made between static and variable
loading. Also the usual deflection formula is used instead of the Wood formula used
183
184
MECHANICAL SPRINGS
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LT=-i-P (160)
comes
"-Tr ,,
Page 84. In this discussion a number of curves based on free-height volume and a
MAXIMUM SPACE-EFFICIENCY
185
t" = K ^ (162)
ira3
* rndtrj
8 GK'
44
4V
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or
nd3= (164)
ir (c+1)3
4Cr
C--S--(cT1)2- (166)
cupied and a given peak stress, the energy stored depends only
180
MECHANICAL SPRINGS
in the lower curve of Fig. 89. This curve shows that for variable
.40
> 36
a.
.28
.24
.2or
.10
C"v
/(3-SPI
*ING NES
ci /
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(2-SPR
NG NEST^
c,
(single
spring)
6 8 10 12
SPRING INDEX C ~ J
14
peated only a few times during the service life of the spring as
MAXIMUM SPACE-EFFICIENCY
187
16Pr
K,
3/
SPRING r>
iest)
s/
j\
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(2-SPRI
NG NEST'
4 (single
spring)\
V-
ir
6 8 10
SPRING INDEX C = *r
12
14
188
MECHANICAL SPRINGS
t,2V
where
C.- - (168)
K> (c+l)" v;
around three).
SPRING NESTS
helical springs
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MAXIMUM SPACE-EFFICIENCY
189
n^' - (169)
tion, it will be assumed that the free lengths of the springs com-
prising the nest are also the same. This means that the total
deflection of each spring is the same at any given load, i.e., that
Ji-a, (170)
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GdiKi Gd2K2
be written
GKi GK;
that the spring indexes c, and c-. (and hence also the curvature
the spring indexes are made the same, the energy coefficients
C,- (which depend only on the indexes) will be the same for
both springs. Using Equation 165 this means that the total
where V, and V-. are the volumes enclosed by the outer and
AG
(-( (1 ~) d73)
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V,-i,d,(c+l)'
44
tained:
v:-c;:)' >
MAXIMUM SPACE-EFFICIENCY
191
U= C
.(175)
where
(176)
spring index in Fig. 89. From this it is seen that for a two-spring
they are also higher than would be the case if the free-height
volume had been taken as a basis. Thus for example the analysis
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is used as a basis.
(177)
4G
where
(178)
192
MECHANICAL SPRINGS
becomes
AG
where
U=>C."' (181)
where
Fig. 90. From these it appears that for static loads the maximum
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the two kinds of loading will be small. In any case, the results
index should be used, say around 3 for a single spring and about
this chapter, the minimum amount of space required for the spring
CHAPTER XI
TENSION SPRINGS
considered.
curvature of the wire or bar at the point where the hook joins
the body of the spring, at A, Fig. 93b, may occur. This curvature
tension springs occur at such points and this is one reason why
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that an initial load must be applied before the coils will begin to
193
194
MECHANICAL SPRINGS
estimate for the case shown in Fig. 93 (which has a sharp bend
at A' (where the sharp bend begins) due to the load P is Pr ap-
(b)
radius r2 being the radius at the start of the bend in the plane
obtained from the curve of Fig. 180, Chapter XVII, for torsion
ing stress must also be added the direct tension stress 4P/Vd-'.
32Pr 4P
(183)
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-K, +
At the point A, Fig. 93b near the point where the bend joins
tion 14, Chapter II, for the case of pure torsion acting on a
TENSION SPRINGS
195
will then be
the axial load. This direct shear, however, does not act at the
as large as possible.
(184)
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196
MECHANICAL SPRINGS
bend as well as on the radii r, and r2. Since the peaks of these
tion effect smaller, than is the case with the spring shown in
mined first. To this is added the deflection due to the end coils.
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tween points where the loops start, the total active turns would
where in this case r is the mean radius of the loop (taken equal
given by Equation 7.
For the full coil turned up, Fig. 94, the experimental work
Pr'
TENSION SPRINGS
197
2r/d, the higher the index the lower the initial tension values.
p.=-:f- use)
16r
Table XXIV
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T 25,000
* 22,500
Jj 20,000
18,000
'16,200
"14,500
"13,000
.. 11,600
10,600
,., 9.700
8,800
, _ 7.900
3 7,000
mum initial tension the stress, from Table XXIV, due to initial
198
MECHANICAL SPRINGS
loop designs, some of which are shown in Fig. 95, may be used.
FULL LOOP
(a)
SMALL EYE
FLAM SQUARE
CUT ENDS
(d)
HOOK
V HOOK
(9)
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EXTENDED EYE
<h)
SWIVEL BOLT
til
tion 18. Thus, if the loop is at the side, the moment arm of the
the usual type of end loop is used, the line of action of the load
Often tension springs are made with plain ends, Fig. 95d.
dicated in Fig. 96a. When using these, the spring is close wound
and the ends of the spring are spread apart by screwing the
TENSION SPRINGS
199
raw ID
the spreading apart of the turns near the ends is set up. This
the end coils are spread. A second type of spring end is shown
in Fig. 96b. This is screwed into the ends of the spring coil.
the end loop is bent up, the moment arm of the load at the point
a design, while more expensive than the usual form of end loop,
use a U-shape piece having hooks at each end to fit over the
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200
MECHANICAL SPRINGS
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Fig. 98Method of
supporting end
loops to avoid
to whipping action
TENSION-COMPRESSION SPRINGS
TENSION SPRINGS
201
In this case nine springs having the shape shown in Fig. 99 are
Fig. 100. One of these plates is moved back and forth by a crank
taken so that the end coils have a gradual transition between the
body of the spring and the straight portion of the end. In this
manner the curvature of the end turns may be reduced and the
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Fig. 100Assembly of
tension - compression
202
MECHANICAL SPRINGS
ance limit is 70,000 pounds per square inch for springs tested in a
centration effects near the end turns would tend to reduce the
endurance range below this value. On the other hand, the en-
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CHAPTER XII
SPRINGS
tion is that more material may be packed into a given space for
great. Where static loads are involved and springs are cold-set,
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the axial load and r the mean coil radius. To calculate the tor-
'For a further discussion of this point together with theoretical results see
paper by Sayre and de Forest"New Spring Formulas and New Materials in Precision
Spring Scale Design" presented at the Annual A.S.M.E. Meeting, December, 1934.
aIn this case, the index may be considered as the ratio 2r/a between mean di-
203
204
MECHANICAL SPRINGS
ing machine. Rates vary from 1/10 to 1xk pounds per inch deflection
+.
fly'
(186)
the coordinates x and tj, q is the pressure per unit of area of the
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Enwx
bn cos . Y
.(187)
sults become
Sir> tln? \
n-l 3 & \
cosh
cosh
nirb
2a
\cOs -
. (188)
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206
MECHANICAL SPRINGS
er
(189)
rm = k(2Gda) (190)
Table XXV
b/
k,
k.
1.
.875
.1406
.208
1.2
.739
.219
1.S
.848
.196
.231
2.
.930
.229
.246
2.5
.968
.249
. .258
a.
.985
.263
.287
4.
.997
.281
.282
5.
.999
.291
.291
10.
1.000
.312
.312
1.000
.333
.333
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.166
brane. the ratio q/S being again replaced by 2GH. Since twice
V=2 / / zdxdy
t/0/2 /-6/2
by using Equation 188 for z and taking q/S = 2Gh the expres-
1 / 192a 1 nwb \
3 I *-b ni 2a
M,=kiGea?b
Solving for 6,
where fc, depends on the ratio b/a and may be obtained from
Table XXV.
coil radius, Fig. 102. The maximum shearing stress then becomes
i.e., the stress increase due to curvature and direct shear is neg-
M,
k-sOrb
(192)
>>>
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duces to
208
MECHANICAL SPRINGS
,,-^ (194)
2*Pr>n
i=^6G (195)
duces to
44.6 Pr3n
(196)
Ga'
theory5 shows that for an index greater than four the error will
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than 4.
194. For large spring indexes this value of stress will be approxi-
mately correct, but for small or moderate indexes the error will
Page 269.
Chapter II. In this case the analysis" shows that for small pitch
angles and indexes greater than three an expression for the factor
c c2 c3
.(197)
A.&Pr
(198)
curve for round wire (Fig. 30 Chapter II) shows that the values
of K' are somewhat under the K values for round wire, the differ-
ence being about 7 per cent for an index of 3, and 4 per cent for
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an index of 4.
210
MECHANICAL SPRINGS
equal to
and taking the spring index equal to 2r/a, for finding K'. The
torsion will give results correct to within 4 per cent for spring
per cent less than that figured from the usual formula in Equa-
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cent less.
i.e., one having the same outside coil diameter, number of turns
cross section. The loads thus found are multiplied by the factor
1.06 and the deflections by .738 to find those for the square wire
tions 198 and 200 should be used instead of the charts mentioned.
6,+&.+2a,
(199)
(200)
211
angle a.
4.8Pr cos a
t = X
.62( )tan:a
/ a \ / a V 1 / a \3 V 2p /
2p J
(201)
. (202)
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7These equations were derived by Goehner, loc. cit.. Page 271, using methods
212
MECHANICAL SPRINGS
a3 1 c c2 J
was done for round wire springs, Equation 51. The analysis
= ^' (204)
COS a G
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.31 cos* a E
1 + ^T~
means that the deflection will be about 3 per cent lower than that
RECTANGULAR-WIRE SPRINGS
lion is parallel to the spring axis and where the ratio b/a is
r^K'Pr^Sa) (206)
equation will give results accurate to within a few per cent for in-
3 P(2r+a)
2- ^.63*," (207)
rectangle. This will also be true where the bar is coiled in the
the maximum stress tends to occur at the inside of the coil be-
effects here tend to come into play. Where the spring is coiled
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flatwise, Fig. 106, the peak stress may occur either on the short
ratio b/a.
abVab
be taken from Fig. 107 depending on the ratio a/'b or b/a and
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the side of the section parallel to the axis of the spring; hence,
=6, load P = 300 pounds, from Fig. 107 for a/b=2 and c=6,
-0 Pr -
abVab
5.88(300)1.5_
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a=-
19.&Pr>n
Ga' (b -.56a)
(209)
216
MECHANICAL SPRINGS
2*Pr*n
where
C=aJ(6/3-.21a) (212)
spring axis as in Fig. 102, Equation 210 will yield results ac-
210 will also give results accurate to within a few per cent, for
for the smaller indexes, and for larger indexes where higher
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2*Pr'n
5 . (213)
GC"
a factor C2 where
C21
'"Goehner, loc. cit., Page 271. It is assumed that the pitch angle is under 12
degrees.
the results given by Equation 210 for such cases may be around
for small pitch angles may be simply carried out by the use of
is given by
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218
MECHANICAL SPRINGS
where the constant 7 depends on the ratio a/b or b/a, Fig. 109.
in testing machine
inch (steel). From Fig. 109 the constant 7=6.7 for a/b = 2,
8Pr>n 6.7X8X300X3.37X5 . ,
S = y = = 1.51 inch.
a-b'G MX^X11.5X10
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these formulas used in practice with the more exact theory for
that the formulas given here will yield more satisfactory results
for the calculation of such springs than will the empirical form-
by using the chart of Fig. 107 and neglecting the pitch angle.
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count'":
l3See author's article in Machine Design, July, 1930 for further details of this com-
parison.
220
MECHANICAL SPRINGS
(216)
the wire cross section about an axis parallel to the spring axis.
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against load are represented by the full lines in Figs. Ill and
112 which show the results of two tests on two coils, one of in-
dex 3.07 and the other of index 4.14. For comparison, dashed
formula, Equation 198, are also shown. It may be seen that the
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this chapter for square and rectangular bar springs are based
this is, however, beyond the scope of this book. In the absence
is concerned.
CHAPTER XIII
JBy a slow rate is meant one in which the time of application of the load is
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222
223
against time is shown in Fig. 114b. This latter curve also rep-
(a)
spring which will be reflected from the end, the time for the
wave to travel from one end of the spring to the other being
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224
MECHANICAL SPRINGS
extension wave will be seen to travel back and forth along the
DESIGN CONSIDERATIONS
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crank radius and connecting rod length is not too large, the ex-
225
per minute).
crankshaft.
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follows:
Fig. 11.5Helical
spring subject to
reciprocating mo-
tion
226
MECHANICAL SPRINGS
due to resonance effects and this may increase the stress range
springs several methods are open. In the first place the natural
resonance will occur only for the higher order harmonics (which
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these end coils will close up thus changing the natural frequency
227
pressing against the center coils of the spring have also been
tance x from the left end of the spring. The active length of
small element A from its mean position (or position when the
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3Dte Federn, by Gross and Lehr, Page 115. published by V.D.I., Berlin, 1938.
'.The Surging of Engine Valve Springs, by Swan and Savage, Sp. Rep. No. 10,
228
MECHANICAL SPRINGS
ird-yds ff-y
K= - ,i (219)
This follows from the equation force equals mass times accelera-
tion since the acceleration of this element is d2y dt2. The par-
Gd'Ay Gd' dy
P -= :i2r (220)
64r3 64r' ds
and this will be the net force F(, acting to accelerate the ele-
comes
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ds 32 r2 as"
2. Air damping
force per unit length of the wire per unit of velocity, the damp-
ing force Fa is
Fd=c-y-ds (222)
dt
229
ing by ds
= c (223,)
Ag dt2 32 r- ds2 ds
dy I dy d-y I- a2y
: = and =
dt2 dt dx'
where
Gd,
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64r"n
='\ (227)
b= w (228)
= o"-^- (229)
dt' dx2
'See for example article by C. H. Kent, Machine Design, October, 1935, for a
230
MECHANICAL SPRINGS
NATURAL FREQUENCY
where </> (x) and i/<(f) are functions of x and t respectively. Then
dt2 dx'
or
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1 rfV _ a2 d2*
~+V(i)-0 (231)
Cp<t> u>J*Cx)
+=-^ = 0 (232)
dx2 a-
o>X tax
aa
Nostrand.
231
(wX <ttX \
A&in .+ B,cos J
aa/
value of f,
second requires that sin wl/a=0. This means that the following
relations hold:
or
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for the natural frequency of the spring (in cycles per second)
becomes
w = m kM
'2t 2 II W
frequency, m= 1, becomes
232
MECHANICAL SPRINGS
2*r2/i y 32y
d J Gg
(236)
and the number of active coils. For the usual steel springs where
One Spring End FreeFor a spring with one end free and
spring.
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hanging on its end as shown in Fig. 117, the lowest natural fre-
3510d
(237)
3510X.3
TT):x6
(238)
233
and k the spring constant in pounds per inch deflection, the fre-
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Fig. 117Weight
on helical spring
y0=csin wj
(240)
234
MECHANICAL SPRINGS
in Fig. 118. Here the dot-dash line represents the stress due to
sumed given by
y = F(x)sin aj (241)
frequency.
The boundary conditions are: For x=0 Fig. 116, y=0 re-
in space. For x=l, y = csin wt since the other end of the spring
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natural frequency.
y-=^sin(2*U+4>) (242)
OA
<f> is a given phase angle. From this solution, for small damp-
r = r.,^ (243)
235
i.e., t, may be around 100 to 300 times t8c It has also been
TO VALVE LIFT.
TIME t
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ing due to clashing and lifting of the end turns from the sup-
ports.
cycles per second and that the camshaft speed is 1200 revolu-
tions per minute, this means that resonance between this natural
236
MECHANICAL SPRINGS
frequency of the spring and the tenth harmonic of the valve lift
2*X200r
t, = = Zalr,,
equal to the valve lift, and if the alternating stress due to the
tenth harmonic of the lift curve is, for example, .002t (as found
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rr-r.+2r, (244)
the valve in the closed position, the range in stress will be from
237
DESIGN EXPEDIENTS
helpful:
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of practical importance.
CHAPTER XIV
* /
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238
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239
240
MECHANICAL SPRINGS
y}
STACKED N PARALLEL
STACKED IN SERIES
of this type.
cone height and thickness greater than about 1.3 should not
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THEORY
,See for example, Theory of Plates and ShellsS. Timoshenko, McGraw Hill,
241
shown by the dotted outline in Fig. 124c. If the ratio r/r, be-
tween outer radius and inner radius is not too large, tests show
central holes have also been made on the basis of this assump-
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and these have yielded good results when compared with those
242
MECHANICAL SPRINGS
the ratio r/ri between outer and inner radii is not over 3 (which
using this method is not over about 5 per cent, while the error in
The solution for the initially coned disk spring which fol-
out by two radial planes subtending a small angle d6, Fig. 124a,
under the action of the external load, a radial cross section ro-
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ter being the total rotation of the cross section about point O
distance from the axis of the spring between the upper and
and the angle <t> produce moments about point O which resist
becomes
3A. M. Wahl and G. Lobo, Jr."Stresses and Deflections in Flat Circular Plates
with Central Holes", Transactions A.S.M.E., 1930. A.P.M. 52-3. Also S. Tiinoshenko
"The Uniform Section Disk Spring", Transactions A.S.M.E.. 1936, Page 305. A
similar solution for radially tapered springs is given by W. A. Brecht and the writer
"The Radially Tapered Disc Spring", Transactions A.S.M.E., 1930, A.P.M. 52-4.
243
It=[cx cos(#-4>)\d9
l,-h=de[x cos(p-<t>)-cos 0|
or
It will be further assumed that the angles fi and <f> are small
<t>"
The last expression for 1cos <f> is obtained by using the cosine
h-k=d8x<t>(p - (246)
(-t)
. - V- (2>
Z, cx
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ratio. Thus the stress due only to the radial motion dr becomes
(E 1' (248)
244
MECHANICAL SPRINGS
tion 248:
EtdHtf-*) (p - ) x'dx
(1-V)(c-*)
r. -I
Etdeaf}-*)^ - -f-)r . ,
1 ir I i
set up by the rotation <f> of the element dx, Fig. 12Ab, it is neces-
Et'
D - (250)
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12(1-ms)
Et3
12(1/*-)
vature is then
2= =
cx cx
"Timoshenko, loc. eit., Part II, Page 120, gives a further discussion of plate rigidity.
Et^dx
ofA/2= (252
6rff-" (253)
fdx 2(1-m')(c-*)
Ed>y
(1-m')(c-x)
2M," = 2rfM2 = -
2 12(1-M')(c-x)
M," due only to the angular motion of the elements of the sec-
tion becomes
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M,=Aft'+M1'
or
246
MECHANICAL SPRINGS
found from the condition that the sum of all forces over the
force acting in the plane of the disk. Since the bending stresses
a" have no force resultant, only the stress a/ due to the radial
Xc-r,
o,'tdx = 0
r-r' xdx
c = (257)
fog,
. 2 r , r X
(I- ri r i -I
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Ml 2r
Solving for P,
P=^L_ (259)
(r-r,)d9
Taking
247
where
"^7[-('-T> + ']
. (263)
in Fig. 125. From Equation 263 it may be seen that the load P
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248
MECHANICAL SPRINGS
on Page 249.
from O will be the sum of the stresses a,' and a". Hence, using
will occur when xcr, and t/=t/2. Taking these values to-
"-o^K-tM (265)
where
ft'-Jfcp" (267)
The stress a., at the lower inner edge of the spring is ob-
- o^K'-t)-*'] (268)
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PRACTICAL DESIGN
249
P=C,Cr
where
c'-7i-^[(T-T)(7-i)+1]
. (2691
. (270)
The factor Cl thus depends on the ratios h/t and S/t while
- "DISC THICKNESS
and 126. The curve of Fig. 126 may be used to obtain greater
It should be noted that the curves of Figs. 122 and 126 also
ous ratios h/t between initial cone height and thickness. This is
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250
MECHANICAL SPRINGS
-20'
and for a considerable range the spring rate is very low. For
When h/t reaches a value of about 2.8 the load drops below zero
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(271)
251
*-^K-f-i)H >
If the positive sign is used before the constant C..' the stress in
the upper inner edge of the spring is obtained, while using the
thus seen that the stress is a function of r/r(, h/t and h/t.
Figs. 127 and 128. For ratios a = r/r( equal to 1.5, the curves
that within the range shown by the curves, for values of h/t
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252
MECHANICAL SPRINGS
upper inner edge where 8<2/j. For deflection 8 equal to 2h, the
upper and for 8>2/i, the tension in the lower edge becomes the
stresses at the two edges become equal, and for 8/<>2 the upper
curve yields higher values. For most practical cases where h/t
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deflection curve desired is that for h/t 1.5, Fig. 121. The
h/t1.5, this being the maximum value. Solving this for t and
From Fig. 126 for h/t=1.5, C, = 1.68 on the flat part of the
curve and from Fig. 125 for a = 2, C2=1.45. From Equation 269
P=C,CV
= 1300 lb.
253
STRESS ON
mate distribution of
constant-load type of
disk spring
STRESS ON
LOWER SURFACE
(TENSION)
.25/.134 or about 2. From Fig. 128 for 8/f=2 the factor Kt will
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pounds the thickness of the disk may be reduced about 2 per cent.
the spring may deflect %-inch before coming against the stop.
K, =12.5 for 8/<=3, h/t = 2.5. From Equation 271 on Page 250,
254
MECHANICAL SPRINGS
Fig. 125 for a=2, C2 = 1.45 and from Fig. 122 the maximum
Ef
To
This load is too high since 520 pounds were desired. To get a
lower peak load, since the latter from Equation 269 increases
inch. For the same shape of curve h/t must be kept the same
from Fig. 128 for h/f=2.5 and 8/f=3.2, the factor K, is practic-
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ally the same as for 8/f=3. From Fig. 122 for 8/f=3.2, h/t=
2.5, C=1.3. Since C,=4.6 at the peak load, the load when the
pounds.
of Fig. 121 and the curve for 7i/f=1.5 of Fig. 126 such that the
the outside diameter of the spring, then the constant load P and
P=CZ)' (273)
25S
(274)
stress and on the ratio D/d or (r/r() between outer and inner
in constant - load
Belleville springs.
ness t is chosen
in accordance with
is 30x10 pounds
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curves of Figs. 130 and 131. It should be noted that the thick-
MECHANICAL SPRINGS
Table XXVI
Spring
Thickness
Maximum
Deflection
Constant
Load
Maximum Stress
200,000
(D/d)
f 1.25
\ 1.5
D/80
D/67.4
D/63.8
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(0
(5 = 2.25<)
D/35.5
D/29.9
D/28.3
(P)
14.5 D2
18.5 D3
17.4 D2
I 2.0 to 2.5
f 1.25
1.5
D/92.5
D/77.8
D/73.7
D/41
8.15 D!
10.4 D3
9.8 D2
150,000
D/34.6
12.0 to 2.5
D/32.7
100,000
f 1.25
1.5
D/113
D/95.4
D/90.2
D/50.2
D/42.3
D/40
3.62 D3
4.62 DJ
4.35 D3
2.0 to 2.5
the springs are steel for which the modulus of elasticity E may
taking r = D/2:
D \aC
'2~\ K,E ~ K
- (275)
where
257
P= i=CD2
K>
where
.(277)
4CIC,E
X4
I2Q
III!
D=OUTSIDE DIAMETER
60L
00000
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part of the curve for h/t=1.5, Fig. 126. Using these values the
258
MECHANICAL SPRINGS
ever, the final equations are not exact and, in practice, devia-
disks have a ratio h/t of about 1.45 and show clearly the constant-
J-
11
a*
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DEFLECTION IN INCHES
of springs.
259
Laszlo' with the results of some tests carried out by Lehr and
Table XXVII
Distance from
Measured
Calculated Stress
Inner Edge
Stress
(Eq. 271)
(mm)
(lb./$q. in.>
(Ib.Isq. in.)
70,500
63,900
66,700
56,000
62,200
45,900
54,500
10
41,400
49,000
20
29,200
35,300
30
22,100
25,200
40
16,450
17,450
50
11,750
11,350
60
8,450
6,400
TO
5,630
2,520
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69,200
tion was not large. Such good agreement as Table XXVII indi-
WORKING STRESSES
Equation 271 may be used even though the yield point of the
steel from which the spring is made is only 120,000 pounds per
260
MECHANICAL SPRINGS
which may occur will redistribute the stress and allow the re-
tion which also makes for a more favorable condition. Also due
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values. Hence
. Af = (r-r.) (278)
hole in the spring, the expression for maximum stress >r becomes
GM 3 P (279)
2(r-/\)t2 a- t2
2(rr,)t76.
per square inch figured from Equation 271 and has an outside
261
t(.0297)2 'H
80,000 pounds per square inch figure is well below the tension
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residual stresses present in the usual case the range in the upper
the presetting operation, with the result that tension stresses will
262
MECHANICAL SPRINGS
of the material.
Since most initially coned disk springs are heat treated after
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CHAPTER XV
rection of load application while at the same time high loads are
linear for small deflections and concave upward for large ones.
In the latter case the spring becomes stiffer as the load increases.
section as shown in Fig. 1341. As- will be shown later the use of
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RADIALLY-TAPERED SPRINGS
ing the commutator bars together, while at the same time allow-
the v-rings or commutator bars. For this purpose, the disk spring
is well suited.
t See paper by W. A. Brecht and the author, "The Radially Tapered Disk Spring",
263
261
MECHANICAL SPRINGS
.P
and inner radii is not over 3. For stress, the agreement between
>
r,
\t
rt
A^
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(t2Kr)
method is used.
complete spring being shown in Fig. 134. Under the load P the
(c-x)de
approximately. Thus
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initial length Zl or
_ ly-U _ 2.
l, CX
angle <f> is small, i.e., if the spring deflection is small, the term
E<t,(kr)
--E<t>k
(281)
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spring.
267
dM
aydxdydd (282)
The total moment acting on the slice cut out of the disk will
M'=de J j t
(283)
M'=de
P(r-r<)
(284)
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2*
= <t>(r<,-ri) (286)
Equation 285,
p-^r[T-+Tw+v<+r<1)] (287)
-^T (288)
=C'P (289)
Eh'
where
a'+a+l
5.73 / a-1 \
and a=r/n.
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Eh'S r 1.5i2 -i ,.
P= H (292)
C'rJ L h-(a2+a + l) J
or
289
m1+d"1''m'+--f =0 (294)
dr Ar
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m2=D(^+M-^-) (296)
In these equations:
per inch
n = Poisson's ratio
t = 2*r.
Materials, 2nd Edition, Vol. 2, Page 135, Van Nostrand; also by A. NadaiKtastische
expression:
dr* r dr r3 2T*,r
6irk,(n-\)r2
where
*Vt-*
radial moments act along the edge; hence from Equation 295,
\ dr r /r-r
(+"f)-
ofa'
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6x*,(l-) (- |- + s + M)
c- ^
6xA,(l-,i) (- - -s + )
where
a,-oT*
271
The maximum stress in the plate will occur at the inner edge
6(m,)r_ri
*= (301)
Using Equations 296, 298, 299, and 300, and the derivative
stress becomes
where
Km 12-MA-B) 1-2,
and
B-(l--|--^)(a--a-W.) + (M-8-_)
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0= - dW- (304)
dr
or, integrating
w=-f<t>dr+C3
272
MECHANICAL SPRINGS
C,r(-"2+'> C,r(-"3-') P
w - ^TT-TT + Cl(305)
1 1 Gwkl(n\)r
that the deflection w is zero at the outer edge of the plate where
CV(-"!+ Cr (-'"-> P
C + : + T-. - -..(306)
.1 1 6rk,(fil)r
+ss
22
Pr2
i = C (307)
Etbs
where
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(---+,)(--)
means that for small deflections (say, less than about half the
273
most cases for diameter ratios a less than 3. However, the agree-
and K' will be over 10 per cent for values of a greater than about
1.4, the values given by the exact theory being somewhat higher
Table XXVIII
a K K' C V
1 985 .954 .0 .0
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for deflection, derived from the more exact plate theory by the
5=0" - *~ I (309)
[- 1 -
+ f2 ('++1) -
or
unity. For large deflections, the equation corrects for the effect
Equation 293.
that for a given load, outside diameter and stress the deflection
274
MECHANICAL SPRINGS
tions for heat treating and forging than disks of larger ratios.
figure the spring as though the load were applied exactly at the
spring with respect to the outer will also be reduced in the ratio
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275
0.28
Fig. 139Deflection
KP
S=C
Pr.'
Etb''
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.157- inch
sign the point of load application is Vfe-inch inside the edge as in-
dicated in Fig. 140. Thus the distance d=23A inches and rr4 =
101,000 pounds per square inch and the deflection at the points
276
MECHANICAL SPRINGS
Since these deflections are much smaller than half the thick-
greatly reduced.
FT
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Fig. 141Arrangement
A = steel cylinder, B
F = cylinder, E = exten-
ance was allowed between the edges of the disk spring, the cylin-
the same time it was possible to read the extensometers while the
277
or not the load was central. For measuring deflections a dial gage
was used.
disk springs having outer diameters varying from 3.8 to 4Y* inches,
using Equations 302 and 309 (correction being made for the
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In all cases, good agreement was found between test and cal-
culated values for stress. At the lower loads for deflection in all
test and theory, due probably to the fact that the point of appli-
278
MECHANICAL SPRINGS
bOOO
4000
?3O00
2000
S 1000
(Fig. 140) became less, thus making the spring slightly stiffer
test and theory was sufficiently good for most practical purposes.
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axis and at a distance y from the middle surface of the disk will
be
(311)
an angle d6, Fig. 137a. The total moment M" will be the integral
Nostrand, 1941.
J.
JES7
Thus
f" f
cx
deE+tHog,a ,'
12
- 6P{r~ri) (312a)
TiEtHog,a
1V
6Pr,
<-t)
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PrJ
a=c'; (3i4)
where
C (315)
v log,a
mum when i/=/2 and r=r,. Using these values and the value of
(316)
280
MECHANICAL SPRINGS
where
3 a-1
K'~ (317)
7T lOgrat
cular plate symmetrically loaded, then from plate theory the fol-
dr1 r dr r- D
where
M = Poisson's ratio
For the case shown in Fig. 133 where the load P is distrib-
2xr
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<?=-
Pr / , \ C,r C2
281
\ dr r Jr.,.
unit length is
ing expressions for </> and d<t>/dr, substituting in Equation 321 the
(322)
where
a2 1
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where
cW^+^y (325)
\ a- / or21
282
MECHANICAL SPRINGS
Table XXIX
1.2S
1.5
:j
.955
1.10
1.26
1.48
1.88
2.17
2.34
.955
1.07
1.38
1.74
2.07
2.37
.341
.519
.672
.734
.784
.704
.343
.524
.689
.773
.775
.760
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1.18
20 25 10 35
r. _ OUTER RADIUS
* rc INNER RADIUS
* Paper on "Stresses nnd Deflections in Flat Circular Plates with Central Holes"
by G. Lobo, Jr. and the writer, Transactions ASME, 1930, A.P.M. 52-3 gives a fur-
ther discussion of flat circular plates with various loading and edge conditions.
283
LARGE DEFLECTIONS
tions are small, say, not over half the spring thickness, for rea-
269 and 270. This gives the following expression for the load P:
Et>
P=C,C, (326)
where
C,- (327)
(l-M^t \2f- J
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284
MECHANICAL SPRINGS
shows how the deflection curve deviates from a straight line after
Ef-
tr-ifi- (328)
r-
where
*'-7i^(Cl'a+c0 (329)
tions 266 and 267. Values of K, are also given by the curves for
SIMPLIFIED CALCULATION
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S/ DEFLECTION
4 "SPRING THICKNESS
285
The deflection at any other load less than the maximum load
Table XXX
Maximum
Diameter
Spring
Maximum
Load
Stress. (jm
Ratio
Thickness
Deflection
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(lb./sq. in.) (
tt = D/d)
()
(5 = 1.75()
(8 = 1.75f)
f 1.25
D/80
D/45.7
42D3
200,000 J
1.5
D/67.4
D/38.5
53.8D'
2 lo 2.5
D/63.8
D/36.4
50.5D'
1.25
D/92.5
D/52.8
23.7D'
150,000
1.5
D/77.8
D/44.4
30.1D'
2 to 2.5
D/73.7
D/42.2
28.4D'
1.25
D/113
D/64.5
10.5D:
100,000
1.5
D/95.4
D/54.5
13.4D2
2 to 2.5
U/90.2
D/51.5
12.6D2
CHAPTER XVI
riety of forms. Because of the shapes which are possible for this
the flat cantilever spring (Figs. 147 and 148) and their applica-
spring is that the end of the spring may be guided along a defi-
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road shocks, but also to carry lateral loads and, in some cases to
CANTILEVER SPRINGS
the spring built in at one end and loaded at the other, the maxi-
P/3
s=TeT(330)
286
287
T~
b,
_I
PI3
3EI
d-M2) (331)
.3, the deflection given by this equation is about 10 per cent less
than that given by Equation 350. The reason for this difference
lies in the fact that for a spring of relatively great width compared
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4A
bo ,
1;
j.
.2 .4
RAtIO ^
288
MECHANICAL SPRINGS
by2
6PI
(332)
bohT-
given later.
of the usual shape as shown in Fig. 149 may, for practical pur-
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Fig. 147. For a given load P the maximum stress is again given
depending on the ratio b/b between width at the free and built-
PP s
where
3 r1 6 / * \7 3 , *M
pends on b/bo and may be taken from the curve of Fig, 148. It
, The nominal stress is obtained by dividing the bending moment by the section
289
equal this gives the maximum deflection for a given value of load.
2P
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amount S1 the ordinary theory will hold. However, when the de-
distance from the built-in end O of the beam (Fig. 150) and 8
290
MECHANICAL SPRINGS
dx'
M-f)T
P(x.-x)
Eh
.(334)
'-'[' -f(-r)]
expression results:
dx-
P(x-x)
(335)
tions which require that at x0, y=0, and dy/dx = 0, the reduc-
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SMALL DEFLECTION
at end of spring and width at built-in edge. In Fig. 151 are given
J See Die Fcdern, by Gross and Lehr, published by V.D.I., Berlin, 1938, Page 133,
291
UI4
a.
to.?
TRIANGULAR
RECTANGULAR
! PROFILE.
EI.
and 1. This means that, for example, if c=l and the stress is
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cent less than this. For springs of rectangular profile the varia-
for the triangular profile (b/b = 0). It is clear that the correc-
considered.
292
MECHANICAL SPRINGS
PP
200X(30)JX12
= .77
EIa 30X10"X6X(M)3
6PI
boh2
However, from Fig. 151, for c=.77, b/b = .2, there is a 6V2
20
18
16
1-
-J
12
7-
+-
a.
UJ
cr
UJ
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14
TRIANGULAR
PROFILE
^ fay
R E C TANGUL AR
PROFILE
per square inch. From Fig. 148 for b/b = .2, K,=1.31 and the
PI'
"3EIa
= 10.1 inches
on the curve of Fig. 152 for c=.77 and fo/fo=.2 which indicates
293
since in this case the width b is large compared with the thickness
tion 333.
of n leaves, then from Fig. 149, b = nib1 and b=nbi- The ratio
b/b will be njn and the curves of Figs. 148, 151, and 152 may
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tice is the cantilever spring with one end rigidly built in and the
'For more complicated cases of elliptic leaf springs and those supported by links
or shackles, the reader is referred to the book by Gross and Lehr, loc. cit.
2<)l
MECHANICAL SPRINGS
12EI
36Eh
(336)
(337)
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the clamped edge. Actual test data relative to the values of such
Where the axial load P, Fig. 154 is not small compared to the
In such cases a more accurate analysis shows that the stress and
C, = -
(338)
1-
A>1-.178 (339)
* CT
3sEh
o = K! - - (340)
OP
a=c'W/" (341)
*^F\Tv -m 111 i Y-
tion )
EI r< / dy y EW
(342)
8 A more exact method is applied to the case of a simply supported beam in Theory
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296
MECHANICAL SPRINGS
higher order. The lateral force Q does the work @A8 and it may
. Q/J / i \ _ c_W_
Ql Ps
<r= A
2Z 2Z
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297
I 00
b96
z 94
5 .92
_l
9: .90
V)
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u' 86
tr
l-
w no
0 I 2 .3 .4 .5 .6 .7 .8
RATIO *
P,r EI Ij*
The curve of Fig. 156 shows that where the axial load is half
the critical buckling load, the factor C, = 2, i.e., twice as much de-
beam theory were used. On the other hand for ratios P/Pr, equal
to .5 or less, the curve of Fig. 157 indicates that the stress formula
PLATE SPRING
298
MECHANICAL SPRINGS
GENERATOR TRAMS:
/^///////////A
ZEE
ZEE
ZCH
'//)//- ^
dashed line of Fig. 159. From the beam theory, the deflection
Pl,H,
(343)
-A
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2EI
T^T^r A - -
the load at each end. For very wide springs in relation to the
6PI,
(344)
299
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av. If omax and amln are the maximum and minimum nominal
300
MECHANICAL SPRINGS
of actual test data the value of Kf may be taken equal to the theo-
but in other cases the two factors may be nearly equal. Fatigue
sections probably will be satisfactory for use until more test data
are available.
hole in a flat spring for holding the spring in place or for manu-
3.0i
<
Ld
1.8
Ld
DC
Ld
1.4
SM
all d/h-
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STRIP THICKNESS
HOLE DIAMETER
LAR
GE d/h (PF
r.i
10BA
RVE)
3LE SHAP
E OF
RATIO $
.4 .6
HOLE DIAMETER
STRIP WIDTH
1.0
center. A bolt through this hole holds the leaves together and
301
()i .
MOMENT
ratio d/w and d/h, factor is 3 compared to 1.85 for small d/w
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of ratio d/w between hole diameter and plate width. Thus where
On the other hand, for thin springs (for example, those made
Goodier7 shows that for a small hole in a wide strip under pure
a thicker plate where d/h is small. Since when the hole diam-
"Transactions ASME, Aug., 1934, Page 617, and Mechanical Engineering, Aug.,
1936, Page 485 discuss descriptions of such tests, together with theoretical stress con-
centration factors.
2 Philosophical Magazine, V. 22, 1936, Page 69. This work has also been checked
See N.A.C.A. Technical Note No. 740. Values of K( = 1.59 were found for rf/t=.145
302
MECHANICAL SPRINGS
the probable shape of this curve is that shown dashed in Fig. 161
the full lines of Fig. 163. Thus the stress range for complete re-
small hole is put into the strip, assuming that d/'w is small and
d/h large (Fig. 162) the factor K, 1.85. Since this material
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mean stress line <r and either the upper or lower full line are also
for this type of stress application has been reduced by the pres-
ence of the hole from 180,000 to 112,000 pounds per square inch
is concerned.
somewhat higher endurance limit for the strip with a hole than
the figures determined in this way, due to "size effect" for such
303
should be on the safe side for design. Also the assumption that
and polished. For the thicker sections, such as are used in leaf
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and plate springs, with the surface in the condition left by rolling
MEAN STRESS
and not ground after heat treatment on the basis of available test
Fig. 164. The upper and lower curves A and A' represent the
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305
>
)w
1]
wJ
1 r\
11
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>
306
MECHANICAL SPRINGS
thick so that the ratio d/h between notch diameter and spring
NOTCH DIAMETER
Wll
that the factor Kt would be practically the same as that for a strip
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with a hole of the same diameter d and the same width w. It is,
therefore, suggested that the lower curve of Fig. 161 may also
307
of Kt thus found for various values of the ratio r/h between mean
obvious.
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is tested in the form shown by the dashed line b in Fig. 169 which
308
MECHANICAL SPRINGS
pletely reversed bending stress for a .48 per cent carbon steel
I.61 1 1 1 - 1
:.0r
TT SPRING THICKNESS
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materials (where the surface has been ground after heat treat-
309
__y b
rn
^UNIFORM WIDTH
may be expected13.
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These are of the order of .001-inch total; to obtain say one per
t4 Machine Design, Nov., 1939, presents a more complete description of these ex-
tensometers.
"If a square element is imagined as cut out of a flat specimen under load and
if this element is imagined to be rotated until no shearing stresses act on its sides, the
The two rounded points P and P' are pressed lightly against the
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311
causing lateral slippage of the points P and P' (Fig. 171) resting
on the specimen.
the test specimen will cause a relative motion of points P and P'
tubes attached to two knife edges B and B'. These knife edges arc
held together by the spring steel strips S (Fig. 172a) which are
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at 1943 Annual A.S.M.E. meeting gives design formulas for elastic hinges. These may
312
MECHANICAL SPRINGS
tion of the specimen occurs point B' moves to say B". This causes
result is that the whole assembly pivots about the point O, and
tion takes place (about 35 in this case). Thus any relative mo-
make them stiff enough so that buckling due to the clamping load
will not occur, while at the same time sufficient flexibility must
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the U-shaped magnet shown. The flat springs D and D' allow
Fig. 173Sketch of
tensometer
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CHAPTER XVII
about the coil axis. Because of the mode of stressing such springs
typical shapes of ends for torsion springs are shown in Figs. 175
and 176. The design of spring end is made primarily from the
around a rod, one end of it being fastened to the rod while the
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314
315
as to wind the spring, then the moment tending to twist the spring
the spring and guide, the actual moment may decrease along
load them in such a way that the spring tends to wind up as the
load is applied. The reason for this is that the residual stresses
tion as to subtract from the peak stress due to the loading, pro-
vided that the load is in the same direction as that in which the
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316
MECHANICAL SPRINGS
spring, the reaction will be against the arbor and the peak bend-
spring are clamped, or if special ends are used, some stress con-
arbor or rod, about which the spring is wound, and the inner
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diameter of the spring. If this is not done, the spring may bind
or wrap around the arbor and high stresses may be set up. The
if the spring end deflects 90 degrees or V4-turn and the spring has
eter of the spring and the inside diameter of the tube. This
317
the case for circular wire. Consequently, for the same peak stress,
axis of the coil. For the usual small pitch angles, the spring may
ture (or the spring axis) and including a small angle dj>, the
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radius of the center line of the bar, V>, the mean coil diameter, is
THEORY
CENTER LINE
318
MECHANICAL SPRINGS
sumed that plane cross sections normal to the center line remain
(r.-y)d*
and the stress o> acting will be this elongation times the modulus
This equation shows that die stress distribution across the sec-
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1. The sum of the normal forces acting over a radial cross section
must be zero since no net external force (but only an external mo-
ment) acts
2. The sum of the moments of the elementary forces about the neutral
y(Ad<t>)
(346)
319
d<t>
Since
J r-
-y
ray
2A = 0
EAd* r
drJydA=
M (347).
JydA = Ae
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E(Ad<t>) M
d<t> Ae
becomes
My (348)
Ae(r-y)
where i/ = /i, and ry=ru the inside radius of the bar, Fig. 178.
maximum stress am.r at the inside of the coil, Fig. 179, becomes
Aer,
-Mh,
Aen
320
MECHANICAL SPRINGS
f'^-O (349)
Letting
/"--mA (3511
J r-y,
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J ) y, r J L ry, J r
e(l+m)A
mA = 0
or
-i--i(i+^+^-+4+..+(^y",+..)
ryi r \ r r- r3 \ r / /
for in,
1 r y&l ,M + + y, + _ ,
n y_/s r>i n \ r r1 r3 /
m = 1 1 h H h (354
A,-- rm (355)
21+m
V 2 1 + m/
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lar section,
"~bhr\ c-1 7
(356)
This gives
322
MECHANICAL SPRINGS
1/1
3c5 - 1.8
becomes
Om0z Kl
6M
where
. (357)
3c--c-.8
3c(c-l)~
distribution of stress in
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K2 drops with increase in index, since the spring bar then ap-
which means that the peak stress is about 30 per cent greater in
this case than that figured by the usual formula in which the
323
_ Mds _ \2Mds
* EI Ebh'
r \2Mds _ 12MI
where l=2wnr is the active length of the spring wire and n the
1.6
LS
I.I
'0,
(CIRCULAR
WIRE)
J. . i
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(RECTANGULAR -
WIRE)
4 6 8 10
"Comparison of the usual beam equation with more exact results calculated from
curved-bar theory shows that the difference is negligible for practical purposes. This is
analogous to helical compression springs, where the usual equation for deflection is
accurate enough for most practical purposes, although derived in an elementary way.
324
MECHANICAL SPRINGS
24irMrn
0= radians (358)
Eon3
The angular twist of the spring in degrees will be 57.3 times this
value.
24wPR-rn ,nrn
-SST-(359)
wire are more frequently used for reasons of economy. The stress
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(t-)
*m; (360)
Ae
(-!)
erm/) l-\-m),
V2l+m/
ffm0
Taking the spring index c==2r, d and A - *(/- 4 for circular wire,
(361)
325
(362)
T 4 V 2c
11
ing for m,
111
m=111
4c' 8c 16c*
For practical springs where the index c>3, this series con-
verges rapidly and two terms are sufficient for practical use. Thus
11-1(1\
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(363)
4c-
326
MECHANICAL SPRINGS
4c"c-1
g- , , (365)
4c(c 1)
ured from the usual formula for a straight circular bar, i.e., stress
Fig. 180 that the values of K, do not differ much from those of K.,
circular wire the same expression may be used as that used for
Mds G4MI
EI ird,E
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12SMrn ,. ,.,,
0 = radians (366)
Ed,
64Mrn , ,
rEd'
the end of a lever arm of radius R, Fig. 177, then M = PR and the
l28PR"rn (368)
Ed1
327
GRAVITY
/AXIS
"S NEUTRAL
1 AXIS
end of the lever arm is lV* pounds. From Fig. 180 the factor K,
for c=9.4 is 1.08. Using Equation 364 the maximum stress is,
taking M = PR,
tZ3 xX(.04)3
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xX30X10X(.04)
Table XXXI
<lb./sq. In.)
328
MECHANICAL SPRINGS
XXXI.
Plain
carbon
steels
Table XXXII
".004-.009
.010-.020
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.021-.040
.041-.060
.061-.080
.081-.100
.101-.ISO
.151-.225
.226-.400
.401-.625
Monel metal
Brass
Hard Drawn
160,000
160,000
160,000
140,000
130,000
110,000
-Kind of Wire-
Tempered
180,000
180,000
180,000
185,000
145,000
135,000
125,000
140,000
60,000
30,000
Music
280,000
270.000
240,000
220,000
210,000
200.000
185.000
165,000
wire, as indicated. Values are also given for stainless steel, monel
using Equations 357 or 364 which take into account the stress
CHAPTER XVIII
SPIRAL SPRINGS
come in contact, the analysis for the spring may be carried out
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329
330
MECHANICAL SPRINGS
indicated in Fig. 184. It also will be assumed that the spring has
M.-Pr+M, (369)
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librium,
M=P(r+y)+Mi-Rx (370)
SPIRAL SPRINGS
331
tion 370,
In this the integral is taken over the total length of the spiral.
(371)
(372
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332
MECHANICAL SPRINGS
dU dU = 0: =0
dR dM,
/'M dM ,
/"* (374)
J dM,
0 (375)
(0 376)
dM y dM
dM, r' dR X
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tion 376,
SPIRAL SPRINGS
333
the angular deflection due to this moment. Thus the angular de-
tion 372)
d>= = / ds (379)
v dM, Jo EI 6Mo'
dM y
= 1 + (380)
aAf. r
Using Equations 371 and 380 in 379 angular deflection <f> becomes
tical purposes:
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D t/o t/0
This means that the first three integrals of Equation 382 are
zero. Hence
RxHs=0
s:
184, the radial load R will be zero. For a small number of turns,
334
MECHANICAL SPRINGS
P=0 which means that the tangential force at the end A, Fig.
184, also vanishes for the condition assumed. Since R is also zero,
-tt (385)
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6M
(383)
(384)
SPIRAL SPRINGS
335
turing reasons the outer end of a spiral spring may be held with
will act at the pinned end A and the loading conditions will be
will be
M,-Pr (387)
Assuming that the coils do not touch each other, the mo-
becomes
M=P(r+y)-Rx (388)
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(389)
or
336
MECHANICAL SPRINGS
Rx'ds=0 (390)
Since / x-ds is different from zero this means that the radial
force R is also equal to zero for the pin-ended case. Fig. 185.
-ir/K1+7)-**](1+7)*
M. p'/. . 2y
* EI
r ^
J r; 4
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*=1.25~ (392)
will have about 25 per cent more angular deflection than the cor-
SPIRAL SPRINGS
337
- *g Z' (~>
outer end. This means that the stress at this end will also be the
point, it may still happen that in some cases this is the limiting
will tend to reduce the stress given by Equation 393. For a more
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and the total length 15 inches. Required the stress and the de-
inch,
, bh' .5(.06)' ,
'=12- 12-=9X10'
'nrrMJ 1.25X25X15 , J.
v EI 30X10X9X10-
grees.
12M 12X25 , .
bh' .5(.06)'
338
MECHANICAL SPRINGS
occurs will be about half this or 84,000 pounds per square inch
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to move along a circular arc about O, the work done by the force
Using this and Equation 394 in Equation 375, the following ex-
pression is obtained:
dM
dR
(395)
SPIRAL SPRINGS
339
<f>, the moment M, will also move through the same angle. From
dU 1 /' 8M
%J 0
(386)
to be directed along the arc of motion of the end A, Fig. 186, this
dU
dP EI
dM
Mds
. (398)
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dp
and from this the actual value of Fig iseSpiral spring with
340
MECHANICAL SPRINGS
spring with clamped ends and a large number of turns) the ratio
IB,
14
12
sty
III 1 1 1 . . .
360 440 520 600 680 760 840 920 1000 1080
for various values of the function A=(r2 r1)/ri have been ob-
angle swept out by the radius vector in traveling from one end
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smaller when the total coil angle 6 is near 360, 720 or 1080 de-
grees, i.e. for 1, 2, or 3 full turns. This is shown by the dips in the
ber of turns if possible. From Fig. 187 it is also seen that as the
.6, the ratio M,/M is still equal to almost 1.2 which means that
maximum stress will still be almost 20 per cent higher than that
SPIRAL SPRINGS
341
(400)
J. MJ
on the total angle 6 of the spiral and on the ratio A. Values of this
51 1
^V
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y-
! k. f
3,
440
Fig. 188Stiffness factor p for spiral spring with small number of turns
and .6 and for more than two turns of the spiral this factor differs
from unity by less than 4 per cent and may usually be neg-
342
MECHANICAL SPRINGS
w 80 160 240 320 400 480 560 640 720 800 880 960 1040
have been worked out4 as functions of the angle along the spiral
for various numbers of turns and for various values of the ratio A;
the results are plotted in Fig. 189. The ordinates represent radial
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the spring due to the external moment while \p is the angle meas-
SPIRAL SPRINGS
343
ured along the spiral from the outer end. By using these curves
The curves of Figs. 187 and 188 apply only to spiral springs
with clamped outer ends. Where the outer end is pin connected
and few turns are involved, an analysis may be carried out using
WORKING STRESSES
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clock spring during its life may be subject to less than 5000 cycles
and hence may be stressed much higher than would be the case
are present (as for example in the spiral spring for the balance
Chapter XXIII.
344
MECHANICAL SPRINCS
190. Here the spring is shown wound up on the arbor. When the
strip and h is the thickness, the total sectional area of the wound
spring will be Ih. But from Fig. 190 this is also equal to
(tt/4' (d,2d,2) assuming that the coils are wound tightly so that
cU-^^-lh+dS
(401)
turns n becomes
(402)
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2h
-Ih+df -d,
1403)
2h
will be separated by the thickness of the oil film and this will in-
SPIRAL SPRIXGS
345
(404)
equal to hi
(405)
From this
Dl = ^D,2-hl
d,' - hi
(406)
2h
-(407)
SPRING
CASE
SPRING
CASE
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ARBOR
Fig. 191Unwound
N=n-n'
4h -(D,+d,)
2h
(408)
<
with the arbor or case are neglected in this derivation the results
tion factor k less than unity. Values of this correction factor are
m (409)
In
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wide, .015-inch thick and 100 inches long. The arbor diameter
Table XXXIII
m 5 4 3 2 1.5
is %-inch and the inner diameter of the case 2V4 inches. The
I[(2.25)'-(.375)"]
m = -=2.58
.015X100
SPIRAL SPRINGS
347
the number of turns is small and the outer end is clamped the
curves given.
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CHAPTER XIX
RING SPRINGS
m Fi<is. 192 and 193. When an axial load is applied, sliding oc-
curs along the conical surfaces with the result that the inner
rings are compressed and the outer rings extended. In this man-
139 and "Characteristics of the Ring Spring", American Machinist. Feb. 14, 1924.
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348
RING SPRINGS
349
stresses.
since the ring behaves like a thick cylinder under internal or ex-
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350
MECHANICAL SPRINGS
STRESS CALCULATIONS
is shown in Fig. 194. From this it may be seen that on the com-
per inch deflection) is obtained than for the return stroke. This
is due to the friction forces on the conical faces of the rings which,
in
in
<
<
y/
//
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12 3 4 5
DEFLECTION, INCHES
RING SPRINGS
351
of friction). This latter force acts in the direction shown when the
be neglected'.
acts in the direction shown, the total radial force acting will be
tial center line of the ring will be the total radial force divided
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by InrTi, where r( is the mean radius of the ring. Hence this load
may be expressed as
2 (N cos a F sin a)
Nlcosa-psina)
p = (410)
Page 164.
352
MECHANICAL SPRINGS
N(cos a-iisma)
ac= -.
P COSa-VLSilta
Oc = ; :; (413).
irAi sina+fiCOSa
Ptana
a'=-VA~K (414)
where
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tanafa+tana)
K= (415)
1 it tan a
tion 415.
Ptana I AIRS
TrAK
RING SPRINGS
353
by Fig. 196. It should be noted that Equations 414 and 416 give
load P only.
lent stress in the outer rings, the compressive stresses due to the
P=
N- (417)
formula:
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N- "A". (418)
COS a fi Sin a
load P and stress at, Fig. 192 (which length may be obtained from
tions), then the average compression stress <' in the contact re-
gion is
N cos a
(419)
2irrb
This holds since the total area over which the force N acts is
354
MECHANICAL SPRINGS
'Jill (420)
2rmb(ltitan a)
DEFLECTION
flections of the rings must first be found. For the inner ring the
due to each inner ring will be two times the radial value acrm/E
divided by tan a. (The factor two is used since there are two
ner rings in the spring (a ring of half the full section being con-
a,-^ (421)
E tan a
i = ^^ (422)
Etana
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E tan a
Prmn / . A
wEA
sidering that in this case the direction of the friction forces F, Fig.
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where
g_to(tona-M) (426)
The ratio between the load P, (return stroke) and the load
JO. (427)
PK
Hence to find the ratio of the spring constants for the return and
ratio Kx/K for the given values of /i and a. This is true since the
given deflection.
DESIGN CALCULATION
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100000X4.58X18X2
= = 3.26 in.
tX29X10X.584X.095
RING SPRINGS
357
.095
P 100000
i 3.26
= 30,700 lb/in.
100000X.25 ,AnnM.. ,
= 143,000 lb/sq n.
tjX.584X.095
smaller than A., since it has been found from experience that
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sion. For example, draft gear springs have been designed for
000 pounds per square inch in inner rings when spring is solid".
143000X.584 .
"2X4.58X.79X.97
358
MECHANICAL SPRINGS
Table XXXIV
Inches
1.1).
Pound*
O.D.
B.
tin
P>
3.750
3.093
1.521
2.124
.980
3.020
2.240
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.750
250
.778
1135
1420
3000
3250
3500
382
150
1630
1465
1430
.540
152
.480
.1750
-0265
.0286
i.h:,(,
.230
041
.0378
.724
.1785
.0164
.0179
1.813
1.906
4.305
1.917
2.242
1.498
1.530
3.982
1.530
1.785
.204
.0366
.0522
.0602
.1436
.0625
.0961
5000
5000
5760
60011
6800
1880
CHAPTER XX
VOLUTE SPRINGS
and relatively thin bar or blade, which has been wound to form
the blade has the shape shown in Fig. 199. After winding but
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359
-360
MECHANICAL SPRINGS
certain load. This means that beyond a certain load some of the
of the bar is frequently tapered near the inner end of the coil'.
volute springs, each element of the coil may, for practical pur-
cult to compute. Some of these stresses arise from the fact that
the resultant load P, Fig. 197 in general will not be axial as as-
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of tapering the inner end thickness sec article by B. Sterne, "Characteristics of the
Volute Spring," Journal S.A.E., June 1942, Page 221. See also paper by H. O. Fuchs,
"Notes on Secondary Stresses in Volute Springs," Transactions ASME, July 1943, Page
543; and "A Design Method for Volute Springs", Journal S.A.E., Sept. 1943, Page 317.
Results of fatigue tests arc given in article by B. Sterne, Transactions A.S.M.E., July
Vol. 1, Page 94, gives results of strain measurements and eccentricity determinations
on volute springs.
VOLUTE SPRINGS
361
spring at zero load is indicated by the line AB, a being the free
the height of the blade center line, and the abscissa the distance
from outer end A. At moderate loads before the outer end starts
ACD.
toming load) at which the outer coil just starts to bottom, the
in Fig. 201. Above this load, as the coils bottom, the spring be-
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curves upward.
tl
UJ
<
362
MECHANICAL SPRINGS
that the coil radius r at any angle 6 from the built-in outer end A
where
H= r~r<- (429)
active coils.
tangular cross section, where the long side of the section is paral-
lel to the spring axis and where the width b (Fig. 197a) is greater
6*Pr -. (430)
Gbh?
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dB 3Pr>d8
dJ = s - (431)
2t G6*(l-.63A)
to start for constant, free helix angle when the slope dS/ds at the
dS .
= tan a
()
VOLUTE SPRINGS
363
becomes
(432).
432, the initial bottoming load P, for constant free helix angle
becomes
Gbh'a^l - .63y
by 57.3).
two conditions, namely, where the loads are less than initial bot-
DEFLECTION
are less than the initial bottoming load P Equation 428 and
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3'^,^^('-^)'^'-
Gbh?
(434)
364
MECHANICAL SPRINGS
n is the number of active coils, the total deflection (for loads un-
3 s3
design of the end coils (Fig. 197). Where these latter are tapered
-2
0 2 A .6 .8 W
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read from the curve of Fig. 202. It should be noted that Equation
VOLUTE STW.VCS
365
435 will also apply for the case of a variable free helix angle pro-
sidered as composed of two parts, e.g., a part 8' (Fig. 200) due to
a part 8" due to the deflection of the free portion CD. Assum-
ing that the coils have bottomed to a radius r* and angle 6' as in-
tained:
VaGbh^l
'- f)
(437)
2irn / c' \
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r- (l--) (438)
by
\ 4irn /
^ 7Tar,
"0 (441)
mi
MECHANICAL SPR1NGS
J,' GbhHX-.&Zh/b)
-r(&+-5r-') >
using this curve and that of Fig. 202, the deflection at any load
c r. J P
. (445)
P=P2. Using Equation 445 and taking /?= (r r, )/> the final
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'-(Af-(~)''
446)
Equation 428. This also gives the difference between free and
solid height:
^jy'('--^)4,mU^('--r) (447)
VOLUTE SPRINGS
367
. (448)
Since the free and solid heights of the spring are known,
the helix angle a (in radians) may be calculated from this equa-
, = 2irnra/fi (449)
I.0
.80
.70
.60
.50
.40
.30
.25
.20
.15
.10
.07
.06
.05
.04
I-
.03
.025
if
Oi
.019
.01
.008
.007
.006
.005
.004
.0035
.003
iff
>
jy
<
1.2
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.08
1.6
1.7
1.8
368
MECHANICAL SPRINGS
to calculate P P2, 8, and 8, from Equations 433, 446, 447 and 449.
When P<P,: Where the load P is less than the initial bot-
toming load P the peak stress will occur at the maximum radius
spring with b>3h and with the long side of the rectangle paral-
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i.e., where the blade is wide compared to the thickness, the term
JP(c^l)-
2hb
When P>P1: Where the load is greater than the initial bot-
3P(c+l)
when P?P,
(450)
(452)
VOLUTE SPRINGS
30!)
this in Equation 452, the stress at any load P (for P>P,) becomes
3p(c.y^+i)
( - -4)
(453)
2hb
When final bottoming occurs, the load P = P,. Using the value
3P.(c, + l)
(' - <)
. (454)
2hb
DEFLECTION, INCHES
-200O
8000
6000
'/
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oooo
AUUU
2000
000 40000 GO
and the value of Pt given by Equation 433, the stress r2 for final
2Ga(a+l)
.(455)
370
MECHANICAL SPRINGS
.5.
free and solid heights; thus 8., = 2%-inch. From Equation 448
St 2.5
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= .0531 radians
, . ,. , 2irX4X2.5X.75
2irnr,
2X11.5X(10)X.0531X11 1L , .
(10)=
11.5X(10yX5X(.25)3X.0531X.969
1= 3(2.5)2
3X2460X21 ,L ,
2X.25X5X.969
2460
P2 - = 9840 lb
(.5)2
371
io
.6
.2
0 .2 4 .6 .8 1.0
r.
Equation 449, using the value of K, = .47 given by Fig. 202 for
The value 82 for the final bottoming load P., will be the difference
between the free and solid height, i.e., 8., = 7.5 5 = 2.5-inch.
stress curve may be plotted, since the stress will vary linearly with
load up to initial bottoming load P,. The stress at any load be-
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as given in Fig. 204 are obtained for this case. From these dia-
>>*
from the inner to the outer radius has been carried out by
aa ai r r,-
tively (Fig. 197). The relative variation of the helix angle may
(457)
6 S.A.E. Journal, Sept. 1943, Puge 317. This also discusses design of presetting
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VOLUTE SPRINGS
373
against r,'r in Fig. 205, for the various values of z. This gives an
idea of the relative variation in free helix angle with radius, for
different values of z.
deflection diagrams will be obtained for all springs with given val-
been computed by Fuchs" and are given in Figs. 206, 207, 208, 209,
and 210. On each figure curves are drawn for r jr equal to .3, .4,
i-
a,
<'
Ml
II,
I1
'/I
,"
77;
JL
(//
'ffl
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<
//'A
OEFLECTION RATIOf
^DEFLECTION RAtIOj
374
MECHANICAL SPRINGS
Sh = nraaiK3 (458)
K _ 2i r \
zd-q3) , (1-z)(1-9')-
(459)
may be taken from the curve of Fig. 211 for various values of z.
where
(460)
Values of K> are plotted in Fig. 212 against <7 = r,/r. For
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VOLUTE SPRINGS
375
Fig. 211Constant K,
plotted as a function of
3_ P,(c+1)
2 hb
(462)
becomes:
2r'
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(463)
376
MECHANICAL SPRINGS
formula
Equation 433 taking a=a( and using r, instead of r,.. This gives
GbVa, / h \
p'=-3^ V ~ -63t)
(465)
Ghat ^ c.+l \
(466)
expression:
r^-ai (467)
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(7 = .515, z = .434 from Equation 457. From Fig. 211, for q .515,
11X10"X7.5X(.4)3X.060X.053X5.5
Pm _ = 6550 lb
S, = 4X3.75X.060X5.5 = 4.95
z=y* and z = ^, (Figs. 207 and 208) for q = .5 does not amount
to the peak deflection. Hence either curve may be used for con-
VOLUTE SPRINGS
377
11X10"X.4X.076
3/75"
11X10'X.4X.06
1.93
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inch.
CHAPTER XXI
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378
RUBBER SPRINGS
379
the predicted and actual behavior of such springs are the fol-
lowing:
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COMPRESSION SPRINGS
Fig. 214Compression
380
MECHANICAL SPRINGS
pected for short slabs than would be the case if dry surfaces
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Applied Mechanics, March, 1938, Page A13; and "Rubber SpringsShear Loading"
by the same author, journal of Applied Mcch., Dec., 1939, Page A159. Other articles
ASME, 1933, Page 45: "Rubber Cushioning Devices"Hirschfield and Piron, Trans-
actions ASME, Aug. 1937, Page 471; "The Mechanical Characteristics of Rubber"
F. L. Haushalter, Transactions ASME. Feb., 1939, Page 149. "Use of Rubber in Vi-
bration Isolation"E. H. Hull, Journal of Applied Mechanics, Sept. 1937, Page 109.
RUBBER SPRINGS
381
n.Eg (hfi)'
(468)
4 3 6 7 89 10 IS 20 30 40 30 60 60 100
from J. F. D. Smith
the ratio f} will be 8/4=2. From Fig. 215 the modulus of elas-
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382
MECHANICAL SPRINGS
becomes:
57X310 (1X2)"
430
V32
= 11.6%
may be constructed.
the form shown in Fig. 217 consisting of two rubber pads bonded
to steel plates are widely used for vibration isolation and machine
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-J
RUBBER
.STEEL
2AG
radians (469)
RUBBER SPRINGS
383
since in this case there are two pads si/bjected to the load P.
(To obtain the angle in degrees the value given by Equation 469
B4
I01 i i i i i i i i i i i i i
from J. F. D. Smith
h-htany (470)
Equation 469.
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384
MECHANICAL SPRINGS
as indicated by the full lines of Fig. 219, the position of the rub-
between the rubber and the steel. If the final deflection leaves
the case, and if S/h is not too great, it may be shown that Equa-
tion 471 can be used with enough accuracy for practical purposes2.
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radius r will be
dy/dr will be equal to the negative value of the shear angle. The
tive is equal to the tangent of the shear angle, using Equation 472,
8 becomes
. , r ^/l 1\ 6s / 1 1 \
1890
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386
MECHANICAL SPRINGS
. (476)
P , r
Fig. 221 the sheaf stress t will be constant and better utilization
proportional to r
for stress:
2*rh
2>r(
2ttKo 2irrf/h9
= Const (477)
t is constant in this case, the shear angle v = t/G will also be con-
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= (r-r,)tan-
-(r-,)tan(--) (478)
RUBBER SPRINGS
387
, P(r.-r,)
o ;
(479)
(480)
In this case the maximum shear stress will occur when r=r( and
is
2xi\'A
(481)
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dr tan y
(482)
where 1 is the shear angle t/G. Dividing this by r yields the ele-
tion 480,
, dr / M \
de= tan ( )
r \ 2wr'hG /
flection becomes
rr / c 1 c3 2 c* \ ,
M/11\
9= ( 1 (485
4*hG \ rr r.' J
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that the thickness h, Fig. 223 varies inversely as the square of the
h=h~ (486)
M Const (487)
the elemental ring shown shaded in Fig. 223b will be, as before,
dr tan y
d8= -
or
de= tan ( J
9-[to..Gd!b)H^ (488)
(a) (b)
Mlog,
(489)
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390
MECHANICAL SPRINGS
M 10000 , .
rm = = = 80 lb/sq in.
ity is 125 pounds per square inch. Using the first term of the
10003
4,r(5)(125)
If the spring were of the constant stress type (Fig. 223) with
hi=5 inches, 7i=2.22 inches from Equation 487 the stress is the
moment is
10000X.405
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2tX4X5X125
Equation 489 .
ALLOWABLE STRESSES
RUBBER SPRINGS
891
These values agree roughly with those of Keys4 who states that
of a flat shear spring, Fig. 217, at 50 pounds per square inch shear
shear angle in 100 days. Other tests on 1-inch thick rubber sand-
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392
MECHANICAL SPRINGS
1. Steady-state vibration.
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ing support, (as shown on Fig. 117 of Chapter XIII). This is the
cal VibrationsJ. P. Den Hartog, McGraw-Hill, Second Edition 1940, and Vibration
a more comprehensive discussion of rubber mountings for aircraft and military equip-
ment.
RUBBER SPRINGS
393
tion for the relative motion y between the mass and the support
T^s^-rio'"" . .
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where a>2=k/m
y+a sin iat or using Equation 492 for y and taking <,>/<=///n
(490)
(492)
a, sin at
(493)
394
MECHANICAL SPRINGS
does not apply since in such cases the effect of damping (which
seen that the amplitude of motion of the mass has been reduced
the mass is subjected will also be reduced in the same ratio. Thus
The chart of Fig. 224, based on Equations 490 and 493, and
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RUBBER SPRINGS
395
disturbing frequency is, say, 800 cycles per minute and the static
pected.
the chart of Fig. 224 are based on the assumption that damping
ever, for best accuracy, damping must be taken into account. The
tions in the usual way6. Actual tests, however, show that the in-
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396
MECHANICAL SPRINGS
P,=
P.
(494)
This equation shows that the reduction in vibration for the no-
to .08 have been observed but this may vary considerably for dif-
stant damping ratio c/cc. This curve also indicates that, for con-
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this statement does not hold true in all cases particularly where
/// equal to 1.6 and higher have been used in tank and aircraft
(495)
RUBBER SPRINGS
397
//
V1
V|
|/
//
III
1 CUIrt
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//
'INDlCA
TO V
"\
OK \
ftllT
fi
IWK1
DAMPING
AHIMIU
MS
vs
J, 1
l\
\N
1.0
VT
so
tudes and to design the mountings so that the motion across the
398
MECHANICAL SPRINGS
methods.
the effective stiffness also increases and with it the natural fre-
quency. This will tend to throw the system out of resonance and
load.
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that excessive stresses in the rubber are avoided under the anti-
CHAPTER XXII
since in most cases, load and deflection are given, which means
that the spring must store a given amount of energy. This is the
case, for example, in the design of landing gear springs for air-
ous types of springs, such as helical, leaf, cantilever, etc., from the
rial, assuming a given maximum stress. This will give the de-
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load P at its end1. Since the bar is loaded axially, the stress dis-
tribution across the section is uniform and for this reason this
the length and A the cross-section area, the stress a will be P/A.
The energy stored will be equal to the area under the load-deflec-
1 This will be called a "tension-bar" spring to distinguish it from the helical ten-
399
400
MECHANICAL SPRINGS
17=
2E 2E
a similar analysis shows that for variable loads the energy stored
u."5e%> (497)
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CANTILEVER SPRINGS
ness (Fig. 147, Chapter XVI). For small deflections, the deflec-
tion from beam theory is, using the notation of Chapter XVI,
=W;(498)
ENERGY-STORAGE CAPACITY
401
6Pi obh>
~l*rmP- 5 (499)
1 2PVJ
u-irPs--m (500)
Using Equation 499 in this, and taking the volume V of the ma-
r5V
U= IBE(501)
when the stress is just equal to the yield stress o- will be, from
Equation 501,
"-w (502)
is only 1/9 the value of energy which may be stored in the ideal
to the yield point, the cantilever spring will still have a consider-
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able margin before general yielding over the cross section occurs.
one as shown later. Since the tension-bar spring does not have
ing complete yielding over the section at the built-in end of the
over the cross section for complete yielding. This means that,
yield stress o-, while on the compression side, the stress3 is equal
a Actually, for most spring steels the stress will tend to rise after the yield point
402
MECHANICAL SPRINGS
(503)
This is one-fourth the value for the ideal case. (Equation 496).
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be only 1/9 that of the latter1. The reason for "this may be found
this respect, the cantilever spring still finds a field of use particu-
(504)
U.-
(505)
18Kf'E
ENERGY-STORAGE CAPACITY
403
6Pl3
sEb;hT (506)
1 3P"-P
U Pi (507)
2 Eboh'
Using Equation 506 in Equation 507 and taking the volume V==
u-He (508)
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(509)
ideal spring at the yield point, Equation 496. For complete yield-
ing over the section of the spring, the load will be about 50 per
cent above the value given by Equation 509 and for this condi-
U> (510)
a.'V
6M
bh2
(w>
24irMm
*=n^-rad,ans
comes
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the moment and the angle. Using this equation the energy be-
1 l2*M2rn
2 * Ebh?
u= bek7 (513)
ENERGY-STORAGE CAPACITY
405
will then be
be
is higher than the curvature factor K2, Fig. 180, the former should
32M
u = Ar
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irtP
From this
"=32*7 (516)
128Afrn
*~ Ed,
tt 1 64M2rn
Um --M*=--- (517)
406
MECHANICAL SPRINGS
<72V
u-1kJe- (518)
stress reaches the yield point <tv is, neglecting the curvature factor
Kt as before,
yielding,
u-im (520)
ideal spring.
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u-~i0E (521)
Assuming the same value of Kf for both cases. Equation 521 thus
SPIRAL SPRINGS
tangular wire, provided that the ends are clamped and that there
Pr=Mt
16M,
T=
TCP
TCP
M, = r (522)
I under a torque Mt is
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32MJ
<t>=
Td'G
1 16M.H
t'V
408
MECHANICAL SPRINGS
4G
(524)
G=
2(1+m)
Using these values in Equation 524, for static loading the energy
and that for complete yielding over the section. This margin
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the basis of Equations 519 and 526 is probably too favorable for
tion of the round bar (as would occur after complete yielding
Comparing this with Equation 522 this means that, for com-
plete yielding, the moment is equal to 4/3 times the value when
(525)
V3
"**-t(-it-)
(527)
the stress in the outer fiber just reaches the yield point in torsion.
u-TiE (528)
This value is only about 23 per cent lower than the stored
energy in the ideal case and shows that the helical compression
limit in torsion, from Equation 523 the energy stored thus be-
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comes
"-w (529)
pression is obtained:
U"^K? (530)
410
MECHANICAL SPRINGS
Equation 7) is
64PHn
Gd,
"-i"-^ <->
Equation 89
16Pr
or
P. "d3"
16rK.
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with the exception of the factor K0. Again taking t, equal to the
equation becomes
4.62K.'E
To get the energy storage for complete yielding over the sec-
j Chapters V and VI give a more complete discussion of helical springs under static
ENERGY-STORAGE CAPACITY
411
t/.=
(534)
2.6 K.'E
From Equations 533 and 535 it is seen that the larger the
spring index, the larger the energy storage per unit volume of
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section, it appears that the ideal spring is not a great deal more
efficient than other types. Thus, for example on this basis (from
torsion all have about 75 per cent of the capacity of the ideal
K, = l.l and from the last column of Table XXXV, the energy-
(535)
4.62K2
412
MECHANICAL SPRINGS
are assumed, the ideal spring is far more efficient than the other
Table XXXV
Torsion
Spring
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Energy stored at first or complete yielding (static loads) for ideal case a 2V/2E-
Energy stored at endurance limit (variable loads) for ideal case ce* V/2JS.
t Values of fatigue strength reduction factors Kf in this row will vary among the
siderably and for this reason the figures given should be consid-
types.
CHAPTER XXIII
SPRING MATERIALS
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1 Static and endurance properties of helical springs were discussed in Chapter IV.
'Article by C. T. Eakin, Iron Age, August 16, 1934, Page 18, "Mechanical
Springs", published by Wallace Barnes Co., 1944, and "Manual on Design and Ap-
plication of Helical and Spiral Springs for Ordnance", published by SA.E. War En-
413
414
MECHANICAL SPRINGS
Table XXXVI
ASTM
Material
Music wire
Oil-tempered
over A
(camp. A
wire
dia.
Oil-tempered wire
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under A dia.
(comp. B). . . .
Hard-drawn spring
Hot-wound carbon
steel he 1 i c a 1
springs"
Chrome -vanadium
Speci-
Man-
Phos-
Sul-
fication
Carbon
ganese phorus
phur
Silicon
(max.
(max.
(%)
(%)
%>
%)
(%)
A228-41
.70 to
.20 to
.03
.03
.12 to
.60
.30
A229-41
.55 to
.80 to
.045
.050
.10 to
.75
1.20
.30
A229-41
.55 to
.60 to
.045
.050
.10 to
.75
.90
SPRING MATERIALS
415
the basis of these data, the endurance diagram of Fig. 226 has
been drawn up. This represents what may be expected for good
inch. Again, it may be seen that for ground and polished speci-
than for the others. Higher values may also be expected for high-
lower'1 than those shown in Fig. 226. This is shown by the tests
Table, the limiting range of stress in the master leaf of the spring
was only about 30,000 to 40,000 pounds per square inch. For
Table XXXVII
Ultimate
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Elastic
Modulus
Elongation
Modulus
tensile
limit in
of
of
Material
strength
tension
elasticity
2 inches
Rigidity"
Hard-drawn spring
(Ib./sq. in.)
(Ib./sq. in.)
(lb./sq. in.)
<%)
(Ib./sq. in.)
wire
160.000 to
60%
30 x 10"
11.4 x 10"
Oil-tempered spring
310,000t
of T.S.t
wire
170 000 to
310,000t
70 to 853
of T.S.
30 x 10"
11.4 x 10"
Music wire
255,000 to
440,000t
60 to 75r;
of T.S.
30 x 10"
11.5x10"
Annealed, high-
carbon wire
250,000 to
200.000 to
416
MECHANICAL SPRINGS
160000, , 1 1 1
u 1200001 1 1 1 1
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SPRING MATERIALS
417
the upper and lower curves of Fig. 227. As will be seen from these
Table XXXVIII
Elonga- Endur-
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curves the tensile strength of music wire may vary from 255,000
pounds per square inch for the larger wire sizes to 440,000 pounds
per square inch for the smaller. The carbon content of this ma-
terial usually will vary with the wire size, the smaller sizes hav-
range within .1 per cent in carbon content for a given wire size.
Usually music wire is not used for springs larger than about %-
special order.
stresses. This bluing treatment may call for heating the springs
the larger sizes and for 15 to 30 minutes for the smaller sizes.
6c
Il I
VI
11
In
0" .
da
: 23
II
.07
0- .
del
II
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MECHANICAL SPRINGS
>- gg
O o
3 *-
S3
II
111
: U Eg
Sin SH
3;.
.2 -J
II
Jll
r-'r
ce ,e
. g_
SS
3 ft
SPRING MATERIALS
419
can steels are being used to make this material with satisfactory
440000/
01 02 03 04 OS 06 07 08 09 10 II J2 .13 J4 15 16
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fications are plotted against wire size in Fig. 228. Further limita-
of not more than 30,000 pounds per square inch in a single lot in
sizes below .120-inch, and not more than 25,000 pounds per
tensile strengths of this wire are somewhat below the values for
r>
a:
i/1
>-
F>
"8
Table XL
(Round Specimens)
Endurance
Limit in
Reve nasi
Bending
Investi-
gator
HankitLs1
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Hankins0
Shelton, Swanger"
Weibel-
Johnson4
Elongation
Par Cent
0 000
* 200
* 0600
+ 02600
* 2">00~)
-2000
0 000
0 200
* 8000
2000
=2800
0"
0*
40.5,
2.-}
2.-J
408
0|
Yield
Point
(Tension)
lb./on. In.
06200
42800
020
020
0000
0200
2200
42400
UK.
Strength
1 Tension)
lb.,sq. in.
0000
0702
220
2200
200000
2000
2002
0000
SPRING MATERIALS
421
wire are usually wound cold and then given a thermal treatment
g 320000
#300000
IS 20 25
that the carbon in any one lot of material shall not vary by more
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than .20 per cent and the manganese by not more than .30 per
cent. Usually the higher carbon contents are used for the larger
sizes.
strength in a single lot shall not vary more than 40,000 pounds
per square inch for sizes below .072-inch, nor by more than 30,-
000 pounds per square inch for sizes above .072-inch. Winding is
422
MECHANICAL SPRINGS
such as, for example, in torsion springs with certain shapes of end
Table XLI
bid. Lis*
TNcknta
of
*>
III
Condition
Surtae
CtSti)
./*, In.
ML
Rant* at
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5Shai*
fc./a*.1n.
.6% Commercial
Carbon Spring
Steel
Hardened and
A. Heed.
350-370
0 to
itatson and
Tempered
42000
Bradley
. o% Commercial
Carbon Spring
Steel
Hardened and
.062 inch
machined
from surface
after hen I
treatment
350-370
0 to
128000
a*
Silico-
o.y. 900-c
T. 540C
As Heed.
390-400
0 to
63000
Manganeac
Steel
Silico-
O.Q. 9000C
T. 540*C
.062 inch
machined
from surface
0 to
110000
Mnnganesc
SPRING MATERIALS
-123
cases, the springs may be wound hot from either carbon or alloy-
steel bars and then heat-treated. For carbon steel bars, the com-
Table XLII
Limiting
Found by Batson and Bradlev, Dept. of Sci. & Ind. Research (British) Special
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5 In master leaf.
Stress concentration effects act to reduce strength. These are due to clamps used
Fahr. in a salt bath. This will give a hardness around 375 to 425
on Table XXXVII.
concerned.
424
MECHANICAL SPRINGS
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ing and may vary from 160,000 to 320,000 pounds per square
Springs of 18-8 stainless steel wire are wound cold and may
750 degrees Fahr. for 15 minutes to an hour, the shorter time be-
Carbon, max
Chromium
Nickel
.15%
16.00-20.00$
8.00-12.00%
26%
SPRING MATERIALS
425
about 2 per cent beryllium and the rest copper together with
Table XLIII
Ultimate
Wire Size
Tensile Streng
(to.)
(lb/sq in.)
.0104
320,000
.0135
313,000
.0173
306,000
.0258
.0410
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288,000
269,000
.0625
251,000
.0915
234,000
.1480
207,000
.207
185,000
.263
171,000
.307
162,000
quenched from 1475 degrees Fahr. and then cold drawn to in-
XXXVIII.
copper and 30 per cent zinc which is cold rolled to give it high
Copper 29 per cent; nickel 66 per cent; aluminum 2.75 per cent.
cold drawn. After winding, springs are given a final heat treat-
1942, and "New Alloys for Springs," Product Engineering, June 1938, give additional
426
MECHANICAL SPRINGS
per cent nickel, this material also has good mechanical proper-
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things being equal, this means that the glass spring will usually
the spring so that stresses may be kept to low values, this ma-
"Article by Betty, et a!.. Transactions ASME, July 1942, Page 465 gives data
7 Article by Colin Carmiehael, Machine Design, August 1942, Page 85, gives fur-
ther details on the use of ulas, as well as article bv T. J. Thompson, Product Engineer-
INDEX
Composition 414
Angular deflection
Composition 414
Description 422
Belleville springs
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Theory 240
Theory 169
Simple 286
storage capacity)
Composition 414
Charts
Helical springs
428
MECHANICAL SPRINGS
cal springs)
Constant-load springs
Belleville 254
Disk 254
Fatigue tests 22
cies 224
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Creep
Correction factor
springs 99
Damping
Decarburization
Deflection
loading 295
Deflection (continued)
Ordinary formula 29
act theory 48
Charts 217
INDEX
429
Dimensions
Effect of variations in 23
lus of rigidity 82
ville springs)
Constant-thickness type
stresses 285
Radially-tapered type
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Tests 162
springs 183
Deflection 196
Shape 198
Stress in 193
Types 198
ance limits)
Elliptical law 16
Simplification of 15
Straight-line law 15
Endurance limits
430
MECHANICAL SPRINGS
Functions of springs , 2
Glass springs
426
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Composition 414
Manufacture 421
Advantages 25
Buckling 169
Charts
Cold-setting 167
Combined stress
Shear-energy theory 45
Maximum-shear theory 44
Deflection
Elementary theory 29
Exact theory 47
Ordinary formula 29
Small index 48
Heat-treatment 423
Natural frequency
INDEX
431
Stress
Charts 214
Deflection
Charts 217
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Stress
Charts 214
due to 300
Composition 414
Heat-treatment 423
Manufacture 423
Composition 414
Index, spring
Effect on deflection
Round-wire springs 48
Effect on stress
ville springs)
Large deflections
432
MECHANICAL SPRINGS
Modulus of rigidity
Effect of decarburization 77
Overstraining, effect of 76
Phosphor bronze 85
Temperature coefficient of 80
Temperature effects 79
termination 83
Music wire
Composition 414
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Manufacture 416
Physical properties
In bending 420
In torsion 418
Presetting
ment 224
Calculation 230
bon steel 17
Pulsating load 17
Oil-tempered wire
Composition 414
Manufacture 419
Deflection 56, 62
Stress 51, 66
INDEX
433
Rubber springs
Advantages 378
Deflection 381
Damping 395
Shear spring
Simple 382
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gravity 398
Scale springs 2
Helical springs 91
Size of shot 92
intensity 92
Temperature effect 94
Spiral springs
Stress 334
Few turns
Many turns
Spring ends
Spring materials
Composition 414
434
MECHANICAL SPRINGS
352
Stress (continued)
Ring springs
Rubber springs
Shear 382
334, 337
Static component of 12
Volute spring
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Stress concentration
Notched bars 17
300, 306
Stress concentration
Flat springs
Clamped ends
Due to holes
Due to notches
urements )
Stress cycles
Constant amplitude 12
Few 18, 94
Variable amplitude 13
reducing 226
Surface decarburization
Belleville springs
299
307
300
306
120
125
236
237
233
15
262
Tension springs
INDEX
435
Valve springs
226, 237
Valve-spring wire
Definition 12
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ance range) 10
range) 12
Variations in dimensions
Vibration
Volute springs
wire, etc.)
Working stresses
ticity)
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1 64 04 3 ... 10
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U. C. BERKELEY LIBRARIES
C077D73517
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