VIII International Conference “Heavy Machinery-HM 2014”, Zlatibor, 25-28 June 2014, <Start page>-<End page>

Natural Frequencies of a Tapered Cantilever Beam of Constant

Thickness and Linearly Tapered Width
Aleksandar Nikolić*1, Slaviša Šalinić1
1
Faculty of Mechanical and Civil Engineering
University of Kragujevac, 36000 Kraljevo, Dositejeva 19 (Serbia),

A method for determination of natural frequencies of a tapered cantilever beam in free bending vibration by a rigid
multibody system is proposed. The considerations are performed in the frame of Euler-Bernoulli beam theory. The method
consists of two steps. In the first step, the tapered cantilever beam is approximated by n flexible straight beam, and after
that all of the n segments are divided into k segments. In the second step, all of the flexible straight beams are replaced by
three rigid beams connected through revolute and prismatic joints with the corresponding springs in them. The results of
the proposed method are compared with similar methods proposed in literature.

Keywords: Free bending vibration, tapered cantilever beam, rigid multibody system, natural frequencies
approaches will be performed on the example from [4],
1. INTRODUCTION
where exact values of frequencies of the stepped cantilever
beam are determined. Also, the results of the presented
Research on dynamic characteristics of flexible approach will be compared with the results from [5],
tapered cantilever beams is very important in different where the tapered cantilever beam of a rectangular cross
engineering fields. These types of beams appear most section, constant thickness and linearly tapered width is
frequently as the result of a need for saving in material, analysed.
reduction of weight, better utilization of material,
increased rigidity, etc. A significant number of papers
2. A RIGID MULTIBODY MODEL OF A TAPERED
dedicated to the solution of this problem have been
recently published. CANTILEVER BEAM
This paper presents a new approach to
approximative determination of natural frequencies of free Let us consider free vibration of the flexible tapered
vibration of this type of beams using the main ideas cantilever beam with the length L, where its end A is
presented in [6] and [7]. A short analysis and adaptation of clamped, and the end B is free, as shown in Fig. 1. The
approaches from [1], [2], and [3] will be carried out for the beam thickness h is constant, whereas its width changes
purpose of comparing the obtained results with the results linearly along the beam, starting from bA in the clamped

b b
fb  x   B A x  bA , 0  x  L,
from similar approaches so that the procedure of analysis end of the beam, up to bB at the free end:
of this type of beams could be feasible. Comparison
between the presented approach and the two mentioned (1)
L
.

Figure 1: Tapered cantilever beam

*Corresponding author: Faculty of Mechanical and Civil Engineering

University of Kragujevac, 36000 Kraljevo, Dositejeva 19 (Serbia), nikolic.a@mfkv.kg.ac.rs
 VIII International Conference “Heavy Machinery-HM 2014”, Zlatibor, 25-28 June 2014, <Start page>-<End page>

A rigid multibody model of the tapered cantilever The springs of corresponding stiffness are placed in
beam will be created in two steps. In the first step, the the joints. The approximative model of the flexible
exact shape of the cantilever beam is approximated with n tapered cantilever beam is thus obtained in the form of an
flexible segments of constant width. In the second step, opened kinematic chain without branching made of 2n·k
each n flexible segment is divided into k equal flexible rigid segments connected through the corresponding joints
segments. In further text, the division in the first step will and springs in them (see Fig. 4). Let us determine the
be called primary division, and the division in the second parameters of the observed mechanical system which are
step will be called secondary division. Primary division necessary for further considerations.
should approximate the exact shape of the cantilever beam The stiffness of springs in the joints of the i-th
in the best way, and secondary division should segment based on [7], for the case of bending of the beam

12 E  I zi
additionally increase accuracy. in one plane, are:
cr  k 3 , cs  k zi ,
The parameters which define each of the obtained EI
segments are: 3
(5)
Li Li

r  2 j  1  2k  i  1 ,
-the modulus of elasticity of the material E,
-the shear modulus of the material: where the indices r and s are:

G
2 1    s  2 j  2k  i  1 , i  1, n, j  1, k ,
E (6)
, (2)

-the Poisson coefficient  , The length of the rigid segments is:

-the density of the material , l
-the length of the segment after primary division: lr  i , (7)
2
Li  , i  1,..., n, 3 
 2 li , j  k ,
L
(3)

n

ls  li  li , j  k  i  n,
-the width of the segment after primary division:
 L1
 fb ( 2 ), i  1, 
(8)

li , j  k  i  n,
bi   i 1 
 fb ( Lk  Li ), 1  i  n,
(4)

 k 1 2 where
li  i , li  i 1 ,
-the area of the cross section of the segment after primary L L
(9)
division Ai , 2k 4k

mr   Ai lr ,

 
-the axial moment of inertia for the principal axis z of the The mass of the rigid segments is:
cross section of the beam after primary division I zi ,
  A l   A l  , j  k  i  n,

ms  
The approximative shape of the tapered beam after
i 1 i (10)
  Ai ls ,
i i
primary and secondary divisions is shown in Fig. 2 by
dashed lines.
The position of the centre of mass of each rigid
2.1. Our approach
segment is defined by the local position vector of the
centre of mass in relation to the beginning of the segment:
ρcu  cu c  c  ,
Each of n·k flexible segments is divided into three T
rigid segments, where the first and second rigid segments u u
(11)
are interconnected through a prismatic joint, and the
second and third segments through a revolute joint (see
Fig. 3a).

Figure 2: An approximation of the cantilever tapered beam by stepped beams

A.Nikolić, S.Šalinić
 Proceedings of VIII International Conference “Heavy Machinery-HM 2014”, Zlatibor, 25-28 June 2014, <Start Page>-<End Page>

a) b)

c)
Figure 3: The rigid multibody model of the i-th flexible beam segment: a) Presented approach, b) Ref. [1], c) Ref. [2], [3]

Figure 4: The rigid multibody model of the flexible beam

c  0.5  lr ,
J cr    ar 2  lr 2  ,
where

 
mr

 
12
 0.5 A l 2  A l  l   0.5  l      
r

  A l   1 a 2  l 2     li  

2

, j  k  i  n,  i i  12 i  cs 2  
i 1 i

cs     
i i i i

Ai  li  Ai 1li  
i

 
 
0.5  ls ,    
 l 
(14)
   Ai 1li  ai 1  li2   li  i   cs  ,
2

 12   
1 2
c  0,  c  0, u  1, 2kn,
J cs
    

(12) 2

 j  k  i  n,
u u

ρu  lu 0 0 , u  1, 2kn,
m
 s  ai2  lr 2  ,
The local vectors of the rigid segments are:
T

 12
(13)

the axis  perpendicular to the plane of rotation is:

The moment of inertia of the rigid segment in relation to

Natural frequencies of a tapered cantilever beam of constant thickness and linearly tapered width
 VIII International Conference “Heavy Machinery-HM 2014”, Zlatibor, 25-28 June 2014, <Start page>-<End page>

where: The local position vector of the centre of mass of

h, if the cross section is rectangular ,

rigid segments ρ cu , the mass of the rigid segments mu and
ai   3
 the axis  J c  ( u  1, 2k  n ) may be defined from the
(15) the moment of inertia of the rigid segment in relation to
 2
di , if the cross section is circular ,

expressions (10)-(14), where li i li are given in the

u

0 1 0T , if the u-th joint is prizmatic,

The unit vectors of the axis of the u-th joint are:

eu  
0 0 1 , if the u-th joint is revolute,
expression (21).

eu  0 0 1 .
(16)
T The unit vectors of the axis of the u-th joint are:

where u  1, 2kn.
T
(22)

The coefficients u and u represent identifiers of

u  1, u  1, 2n  k
All joints in the kinematic chain are revolute, so that:
(23)

1, if the u -th joint is prismatic,

the joint type, where it holds that:

u   2.3. Approach from [2], [3]

0, if the u -th joint is revolute,
(17)

as well as that u  1  u References [2] and [3] propose discretization of

each of nk flexible segments so that they are divided into
two equal rigid segments which are interconnected by one
2.2. Approach from [1]
cylindrical spring and one revolute spring with the
corresponding rigidity (see Fig. 3c). This division results
Each of n·k flexible segments is divided into three
rigid segments, where the first and second rigid segments, in an open kinematic chain without branching made of nk
as well as the third and fourth ones, are interconnected rigid segments connected by the corresponding springs.
with a revolute joint (see Fig. 3b). The springs of the The stiffnesses of springs in the joints of the u-th segment
corresponding rigidity are placed in those joints. Similarly based on [2] and [3] are:
cTs  k , cM s  zi ,
to our approach, the approximative model of the flexible GAi EI
(24)
tapered cantilever beam in the form of an open kinematic Li Li

s  j  k  i  1 , i  1, n, j  1, k ,
chain without branching made of 2n·k rigid segments where the index s is:
connected with the corresponding joints and springs in
(25)
them is obtained. Let us determine the parameters of the
The length of the rigid segments is:
2l  , j  k ,
observed mechanical system which are necessary for

 i
further considerations.

ls  li  li , j  k  i  n,
The stiffnesses of springs in the joints of the i-th


segment based on [1] are: (26)

cr  cs  2k zi , i  1, n, li , j  k  i  n,
EI
(18)
Li
where
li  i , li  i 1 ,
where the indices r and s are defined in the expression (6).
L L
1 2 p 
The length of the rigid segments is: (27)
lr 
2k 2k
li , (19) The local position vector of the centre of mass of
p
2l  , j  k ,
the rigid segments ρ cu , the local vectors of the rigid
 i

segments ρu , the mass of the rigid segments mu and the
ls  li  li , j  k  i  n,

(20)
relation to the axis 
moment of inertia of the rigid segment of constant width in
J c  ( u  1, 2k  n ) may be
li , j  k  i  n, u

determined from the expressions (10)-(14), where

li , li i ls are given in (26) and (27).
where
li  p
Li 
, li  p i 1 ,
L
(21)
k k
1 1 
and where p  1   is the coefficient of division of
3. EIGENVALUE PROBLEM
2 3 Reference [1] and our approach use relative
the beam. Reference [1] shows that, especially for this coordinates for description of the system, whereas [2] and
value of the coefficient, the assumed model of the beam is [3] use absolute coordinates. That is the reason why the
reduced to a simpler shape which contains springs only in formation of differential equations of motion will be
the joints (see Fig. 3b). In an opposite case, the model of presented for the cases of using relative coordinates (for
the beam also contains a spring which connects the first our approach and the approach in [1]) and absolute
and third rigid bodies, and then the process of modelling coordinates (for the approaches in [2] and [3]).
the flexible beam by this method becomes considerably
complicated.

A.Nikolić, S.Šalinić
 Proceedings of VIII International Conference “Heavy Machinery-HM 2014”, Zlatibor, 25-28 June 2014, <Start Page>-<End Page>

3.1. Relative coordinates where ∆φv and ∆yv are relative joint displacements which,

v  v  v 1 ,
expressed as a function of absolute coordinates, read:
The potential energy of the system of springs in the (35)
yv  yv  zlvv  yv 1  zrv 1v 1 , 0  0, y0  0. (36)
 c   cu qu2 ,
joints reads:
zlv  cv , zrv  lv  cv ,
1 2 kn
(28) (37)
The absolute coordinates yv and  v represent
2 u =1
where qu (u=1,…,2kn) are relative joint displacements.

T   m  q  q q ,
The kinetic energy of the system is transverse displacements of the centres of masses and
1 2 kn 2 kn rotation about that centres of the v-th rigid segment in
(29) relation to the horizontal position, respectively. Axial
2  1  1
displacements of the centres of masses of the v-th rigid
where an overdot denotes the derivative with respect to segment are neglected because of the assumption of small
time, q=[q1,q2,…,q2kn]T is the vector of generalized deformations of the beam.

 
coordinates and
2 kn  
T   mv yv 2  J cv v 2 ,
rcu rcu
m  q    mu
The kinetic energy of the system is

     J Cu e T e  , (30)
T
1 kn
 q q 
(38)
 
u = 2 v 1
where an overdot denotes the derivative with respect to
the metric tensor coefficient of the inertia matrix of the
time. By applying the Lagrange equations of the second
system. For more details see [9].
kind for the case of conservative systems,
d  L  L
In Equation (30), mu is the mass of the u-th rigid
   0,
segment in the chain, J Cu  is its axial moment of inertia
dt  yv  yv
d  L  L
relative to the principal axis which is perpendicular to the

   0,
plane of beam bending, rcu is the vector of position of the
dt  v  v
centre of masses of the rigid body (Vu) in relation to the (39)

in which qu  t   0, qu  t   0 where v  1, k  n, and L  T   is the Lagrange function,

inertial frame Axyz. The configuration
q0=[q1=0,…,q2kn=0] T

(u=1,...,2kn) corresponds to the equilibrium position of the differential equations of motion of the mechanical system

Mz  Kz  02kn1 ,
flexible beam shown in Fig. 4 in the absence of gravity are obtained in the form:
and force at the free end of the beam B. Linearized (40)
differential equations of motion of the considered system where 02kn1R2kn1, z=[z1, z2,…,zkn]T is the vector of
of rigid bodies in the surroundings of the equilibrium absolute coordinates, and it holds that zv=[yv, φv] T

Mq  Kq  02n1 ,
position read (see [8]): (v=1,…,kn).
(31) The mass matrix is:
where 02n1R2kn1, K  diag (c1 ,..., c2kn ) is the stiffness M = diag (M1,1 , M2,2 ,..., Mkn, kn ), (41)

 
matrix, and MR2kn2kn is the mass matrix, whose where:
members are: Mv ,v  diag mv , J cv ,
 rc   rc 
(42)

m  q 0    mu  u   u 
T


2 kn

 q q0  q K1,1 .. 0 

The stiffness matrix is:
u = q0
 .. .. : 
0 0 0 ..

     J Cu e  q 0  e   q 0 , ( ,   1, 2kn),
 : : : :
 0 .. K v 1,v 1 K v 1,v 0 
2 kn
T

u
 
0 ..
K   0 .. K v ,v 1 K v ,v 1 0  , (43)
 0 .. 0 
(32) K v ,v ..

 
The partial derivative of the position vector rcu relative to 0 K v 1,v K v 1,v 1 ..
 : :: : 
generalized coordinate q at the position q0 reads:

  e  q 0     ρ k 1  q 0   ρCu  q 0  
  u   0 .. K kn, kn 
: : : ::

 k  1 
0 0 0 ..

 rcu 
      0  
  e q ,   u,
cTv1 cTv1 zrv 1
where:
 
 q q0 
  e  q 0   ρCu  q 0    e  q 0  ,   u ,
K v,v 1    , (44)
 cTv1 zlv 1 cM v  cTv1 zrv 1 zlv 
0,   u
 cTv1  cTv cTv1 zlv  cTv zrv 
 
K ,   cM v1  cM v  , (45)
(33)
 cTv1 zlv  cTv zrv 2
3.2. Absolute coordinates
 cTv1 zlv  cTv zrv 
2

 cTv 
K v ,v 1    , (46)
The potential energy of the system of springs in the cTv zlv 1

 cTv zrv cM v  cTv zlv 1 zrv 


 c   cM v v2  cTv yv2 , 
joints reads:
1 kn
(34) For more details, see [2] or [3].
2 v =1

Natural frequencies of a tapered cantilever beam of constant thickness and linearly tapered width
 VIII International Conference “Heavy Machinery-HM 2014”, Zlatibor, 25-28 June 2014, <Start page>-<End page>

Finally, the eigenvalue problem, formulated on the basis of 4.1. Example 1

 K -  2M  v  02kn1 ,
(31) and (40), reads:
(47) Let us observe the flexible cantilever beam with three

where  is the natural frequency of free vibration of the

stepped changes of the circular cross section with the
following characteristics:
- Young’s modulus: E  2.068  1011 N / m2 ,
flexible tapered cantilever beam, and vR2kn1 represents

mass density:   7850 kg / m3 ,

the eigenvector which corresponds to the given frequency.

total length: L  2.0 m ,

Approximate values of natural frequencies of the -
considered cantilever beam are obtained by solving the -
diameter: d1  0.03 m ,
eigenvalue problem (47).
-

d2 / d1  0.8 d3 / d1  0.65 , d 4 / d1  0.25 ,

- diameters ratio:
4. NUMERICAL EXAMPLE AND VERIFICATION OF
THE METHOD

L1  0.25L , L2  0.3L , L3  0.25L , L4  0.2L,

- length of the segments:

Verification of the efficiency of the presented

- area of the cross section of the segment after
method will be performed through two examples. The first

 du2
primary division of the beam:
example will treat the problem of determination of natural
frequencies of the flexible cantilever beam with three Au  ,
stepped changes of the circular cross section. Thus, 4
primary division is carried out in advance, so that n=4, and - axial moment of inertia for the principal axis z of

 du4
the influence of secondary divisions of the beam on the the cross section of the beam:

I zu 
accuracy of our method will be analyzed. Exact values of
natural frequencies of such a beam are determined in [4], ,
64
so it is a good example for comparing the accuracy of the
In further considerations, for convenience of comparisons
proposed approach with the relevant approaches presented
with the results from paper [4], the non-dimensional
frequency coefficients L  4  2  A1 L4 /(EI1z ) are used.
in [1] and [2]. The second example analyzes the tapered
cantilever beam of a rectangular cross section, constant
thickness and linearly variable width. The influence of Using the above theory, the approximative numerical
primary division on the accuracy of our method will be values of the first three non-dimensional frequency
analyzed in this example. The results achieved by using coefficients are obtained. These frequency coefficients
our approach will be compared with the results from [5]. along with the corresponding relative errors are shown in
Table 1. The errors are calculated as:
 100  100 [%].
approximative value
exact value

Figure 5: The three-stepped cantilever beam

Table 1 gives the comparative results obtained by using all frequency where the approach from [2] is slightly better.
three presented methods of discretization depending on the Besides, the relative error in determination of the first
number of secondary divisions of each segment of the frequency with one division of the beam segment is 0.059
beam. It also shows relative errors of the obtained values %, i.e. the error is far smaller than 1%. This fact is
of frequencies in relation to the exact values of frequencies particularly important if it is taken into account that the
from [4]. It can be noticed that the values of obtained values of the first frequency are of most significance in
frequencies, at the increased number of secondary studying dynamic characteristics of various technical
divisions of beam segments, converge faster toward the objects.
exact values if our approach is used, exept for the third

A.Nikolić, S.Šalinić
 Proceedings of VIII International Conference “Heavy Machinery-HM 2014”, Zlatibor, 25-28 June 2014, <Start Page>-<End Page>

c  1
4.2. Example 2 bB
, (48)
bA
Let the tapered cantilever beam of constant
dimensional frequency i , which is connected with the i-th
thickness and linearly tapered width be given (see Fig. 1). Let us also introduce the concept of the i-th non-

example. The beam length is L  0.5 m , and the thickness frequency i  rad / s  in the following way:
The material of the beam is the same as in the previous

is h  0.005 m.
 A1 L4
i   i ,
E  I zi
The area of the cross section of the segment after primary (49)

Au  bu h,
division of the beam is:
Table 2 gives the values of the first three non-
dimensional frequencies, where the value of the parameter
The axial moment of inertia for the principal axis z
c changes from 0 to 1, with the step 0.1, and for n=10 and
of the cross section of the beam is:
n=20, respectively. It can be noticed that there is very
I zu  u ,
b h3 good agreement between our results and the results from
12 [5]. It is obvious that the convergence of frequency toward
The beam width at the beginning and the end of the the values from [5] is faster at smaller values of the
beam will be varied in order to obtain necessary relations parameter c, i.e. when the beam is less tapered (see Fig. 6
of these dimensions for the needs of comparison of results. and Fig. 7). In that case it is enough for the number of
That is why the parameter related to the degree of beam segments of constant width (primary divisions) to be n=10,
tapering is introduced: and to achieve the satisfactory accuracy.

Table 1: Natural frequencies of the cantilever beam – comparison of the present paper results and the results from [4]

Number
of Non-dimensional frequency coefficients
divisions 1L 2 L 3 L
Relative error [%] Relative error [%] Relative error [%]
Appr. Appr. Appr. Appr. Appr. Appr.
Our Our Our
from from from ref. from from from
appr. appr. appr.
ref. [1] ref. [2] [1] ref. [2] ref. [1] ref. [2]
2.51159 2.51000 2.56814 4.31315 4.43415 4.87846 5.47403 5.79999 6.10351
1
0.059 -0.004 2.312 -2.975 -0.254 9.741 -5.938 -0.337 4.878
2.51200 2.49390 2.52396 4.43100 4.43826 4.53171 5.74267 5.77220 5.86927
2
0.076 -0.645 0.552 -0.324 -0.161 1.941 -1.322 -0.815 0.853
2.51114 2.49524 2.51612 4.44160 4.43855 4.48271 5.78781 5.77304 5.84107
3
0.042 -0.592 0.240 -0.086 -0.155 0.839 -0.546 -0.800 0.369
2.51051 2.49909 2.51213 4.44470 4.44009 4.45832 5.80856 5.78317 5.82606
5
0.016 -0.439 0.081 -0.016 -0.120 0.290 -0.190 -0.626 0.111
2.51030 2.50156 2.51104 4.44514 4.44120 4.45167 5.81386 5.79064 5.82184
7
0.008 -0.340 0.037 -0.006 -0.095 0.141 -0.099 -0.498 0.038
2.51018 2.50375 2.51046 4.44527 4.44222 4.44815 5.81658 5.79763 5.81958
10
0.003 -0.253 0.014 -0.003 -0.072 0.061 -0.052 -0.378 -0.001
Exact
solution 2.5101 4.44542 5.81961
[4 ]

Natural frequencies of a tapered cantilever beam of constant thickness and linearly tapered width
 VIII International Conference “Heavy Machinery-HM 2014”, Zlatibor, 25-28 June 2014, <Start page>-<End page>

Figure 6. Absolute error of natural frequency of: straight beam (c=0) in comparison with [5]

Figure 7. Absolute error of natural frequency of: maximum tapered beam (c=1),
in comparison with [5]

Table 2: Natural frequencies of the cantilever beam – comparison of the present paper results and the results from [5]

1 2 3
Non-dimensional frequencies

c
Our results Ref. Our results Our results
Ref. [5] Ref. [5]
n=10 n=20 [5] n=10 n=20 n=10 n=20
0 3.5169 3.5162 3.5160 21.8890 21.9984 22.035 60.3636 61.3642 61.6970
0.1 3.6307 3.6309 3.6310 22.0992 22.2156 22.254 60.5474 61.5696 61.9100
0.2 3.7612 3.7624 3.7629 22.3361 22.4607 22.502 60.7573 61.8044 62.1530
0.3 3.9125 3.9152 3.9160 22.6073 22.7417 22.786 61.0019 62.0784 62.436
0.4 4.0913 4.0956 4.0970 22.9240 23.0704 24.021 61.2942 62.4068 62.776
0.5 4.3067 4.3130 4.3152 23.3039 23.4659 23.519 61.6556 62.8145 63.199
0.6 4.5728 4.5822 4.5853 23.7773 23.9606 24.021 62.1238 63.3458 63.751
0.7 4.9134 4.9271 4.9317 24.4012 24.6162 24.687 62.7728 64.0892 64.527
0.8 5.3703 5.3907 5.3976 25.2983 25.5668 25.656 63.7744 65.2537 65.747
0.9 6.0272 6.0595 6.0704 26.7946 27.1727 27.299 65.6527 67.4945 68.115
1 7.0805 7.1374 7.1422 30.1108 30.8063 30.970 71.1455 74.3753 75.653

A.Nikolić, S.Šalinić
 Proceedings of VIII International Conference “Heavy Machinery-HM 2014”, Zlatibor, 25-28 June 2014, <Start Page>-<End Page>

5. CONCLUSION ACKNOWLEDGEMENTS
This research was supported under grants no.
This paper presents a new method of approximative ON174016 and no. TR35006 by the Ministry of
determination of frequency of the tapered cantilever beam Education, Science and Technological Development of the
Republic of Serbia. This support is gratefully
which can serve as an alternative to relevant approaches
acknowledged.
from [1], [2] and [3]. In such discretization of the flexible
tapered cantilever beam, a well-developed methodology
for mechanics of a system of rigid bodies is used for the REFERENCES
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Natural frequencies of a tapered cantilever beam of constant thickness and linearly tapered width