Vehicle Vibration and Ride 1
Vehicle Vibration and Ride 1
Vehicle Vibration and Ride 1
Brief discussion on suspension concepts 1 DOF -car ride model Terrain descriptions for vibration Random excitation of -car model 2DOF -car ride model
Each wheel should primarily undergo vertical motion; a single degree of freedom.
The figure shown (from Matschinsky) shows how this degree of freedom can be realized. For an independent wheel suspension, the design might have: a. strictly vertical travel b. a combination of vertical and lateral displacement and a rotation (camber change) c. a general non-linear coupler movement that exhibits constrained motion With two wheels mounted together on a single wheel carrier, as on a rigid beam axle (d), the suspension requires two degrees of freedom so that each each wheel will have one degree of freedom. The necessary design will permit the axle both parallel travel and rolling motion relative to the vehicle body.
Suspension design objectives may require both kinematic and vibration analysis
1. 2. 3. Motional requirements are met through kinematic considerations (kinematics, multibody dynamics) Isolation is achieved by including elastic and dissipative elements (vibration/ride modeling) Since the kinematic, elastic, and dissipative design will impact how the tire interacts with the road, as well as how the body reacts, it is essential to understand any available knowledge about the vehicles dynamics in developing a suspension system. A balance between handling and ride is almost always necessary.
Achieving these functional objectives may be attempted through geometric suspension design and/or through active suspension methods.
1 Wheel and tire 6 track rod, driven by 7 7 steering link/gearbox
4.
2 Wheel carrier, maintains the wheel bearings, brake caliper, and overall attitude of the wheel 3 Wishbone link or A-arm 4 transverse link from vehicle body to wheel carrier 5 tension link (compliant)
8 spring 9 damper
10 drive shaft
The remaining slides focus on models commonly used for ground vehicle ride analysis. These models can help build insight into how/why certain suspension designs are adopted.
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin
Noise refers to
aural vibrations frequency range 25 to 20,000 Hz (high)
Ride and noise are perceived differently by humans, so there is a need to adopt methods that help quantify and control. See Appendix B for more on human response (ref. Wong, Ch. 7).
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin
Key analysis can begin with even simpler system models illustrating isolation function
You can show that:
The -car model can provide insight into several key measures of vehicle ride
These measures are discussed in Wong (2001) and summarized in notes provided on Vehicle Ride. 10
1 Transmissibility Ratio TR ( 2 f , 0.05 ) 2
2 TR ( 2 f , 0.10 )
k s2 + (b)2 Z Z = = Y (k m 2 )2 + (b )2 Y
At higher frequencies, you may feel more with lighter unsprung mass. A lighter unsprung mass provides better vibration isolation in the midfrequency range.
0.1
Z Y
The frequency response tells us how the amplitude or phase of the response will depend on the frequency of the forcing function, y(t) (which we can relate to the terrain profile, y(x), and the forward velocity, V).
mass ratio =
Good isolation
mus ms
1 .10 4
The 1/4 car model is also helpful in introducing the role that controllable or active elements can play in vehicle suspensions.
Department of Mechanical Engineering The University of Texas at Austin ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin
Z Y
7 DOF 2 DOF
y=
2 DOF for pitch and bounce 15 DOF
dy dy dx dy = = V dt dx dt dx
Z k s2 + (b)2 = Y (k m 2 )2 + (b )2
pz = mz = F mg F = Fs + Fc = ks xs + b( y z ) x s = v = y z
y ( x ) = given road profile
Note: this variable represents the compression/extension of the spring element (not its total length).
Department of Mechanical Engineering The University of Texas at Austin
tan = z (t ) =
mb 3 (k m 2 )2 + (b )2
b bc
Z Yo sin(t ) Y y (t ) = Yo sin(t )
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control
These curves show the effect of damping, although all curves go through,
n = 2
See Appendix A.
= XV
The wavenumber of the road, , is a measure of the rate of change with respect to distance or length. In time, we relate period, T, to frequency, . In space, we relate wavenumber, , to wavelength, .
A road can then be described by a spectrum that is a function of wavenumber.
T = V
2 2 V 2 = = V = V T
T = 2
Units:
= 2
Units:
cycles/distance
Department of Mechanical Engineering The University of Texas at Austin
So you can relate frequency, , to a forward vehicle velocity using the wavenumber-based description of a road profile.
[ ] = rad sec
[ ] = rad m
You can develop basic functions to quantify these spectra. For example, S g ( ) = Csp N For vehicle vibration, you can convert this to units of frequency by the relation,
Wong (2001)
Sg ( f ) =
1 S g ( ) V
= 2
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control
Z Y
cycles/m
2 rad/m
Sv ( f ) = H ( f ) S g ( f )
The vibration spectrum Your linear vibration model
The input
Then, you can use measures, such as:
100 2
= 15.915
10 S g ( ) 10
6
constant
S g ( ) =
In this case,
0 0 1 2
1 S g ( = ) V V
100 1 mm 2 2 V rad/s
c k ; = ; n = n m 2 km
n = 2 (1.5) rad/sec = 0.1
H ( )
Hmag2( r)
rms vibration =
f2 f1
Let,
S ( f )df
wavenumber
Department of Mechanical Engineering The University of Texas at Austin ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control
= 2
S g ( ) =
2 r
10
S z ( ) = H ( ) S g ( )
100
5
100
10 S y ( )
10
10 S g ( ) 10
6
10 10 Sv ( )
6
Hmag2( r)
10 S v ( ) 1
= S z ( )d
2 z 0
0 10 20
Units: rad/sec
20 Hza
0.1
10
0.1
0 2 r 4
10
20
0. 1
S g ( )
H ( )
S z ( )
100
S z ( )
100
Assume:
V = 30 km/hr
v2 := Sv( ) d 0 Hza
100 10 1
10 10 Sv ( ) 1
6
10 10 Sv ( ) 1
6
v2 = 150.
RMS
mm
v2 = 4.609 mm
10 Sv ( )
0.1 0.01
0.1
10
20
0.1
v2 = 12.25 mm
Department of Mechanical Engineering The University of Texas at Austin
for V = 30 km/hr
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control
1 .10
10
20
10
20
Summary
The base-excited model should be used to understand the basic vibration problem in a -car vehicle model. The transmissibility ratio illustrates the frequency response of the base-excited mass-spring-damper system (see also Appendix B). Road profiles can be transformed into input forcing power spectral densities which drive the system. Basic functions provide a way to estimate the response spectrum and critical values such as the rms velocity or acceleration. Vehicle ride models can become more complex as you add the effect of additional masses, etc.
ms 1 + ks ( z1 z2 ) = 0 z
mus 2 + ks ( z2 z1 ) + ktr z2 = 0 z
Assume the response of each variable will take form,
Wong, Fig. 7.5
z1 = Z1 cos nt z2 = Z 2 cos nt
Plug into the equations above leads to two equations valid for any Z1 and Z2 so long as,
2 ms n + k s
ks
2 ms n + k s + ktr
ks
=0
f ns =
Two solutions:
n21 = n22 =
1 2
1 k s ktr 1 = ms k s + ktr 2
RR = ride rate
RR can be an order of magnitude larger than ms unsprung mass, while the suspension stiffness
is an order of magnitude lower than the equivalent tire stiffness.
f n us =
1 2
k s + ktr mus
Case 1: Neglect mus, find equivalent stiffness (RR) Case 2: Assume vehicle acts like big inertia and mus bounces between ground and inertia
n1 = 6.563 rad/sec
f n1 = 1.045 Hz
f n1,2 =
n 2 = 66.19 rad/sec
f n 2 = 10.54 Hz
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control
n1,2 2
=
f n s = 1.045 Hz f n us = 10.53 Hz
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin
c 2 km
d = n 1 2
NOTE: With a computer, easy enough to solve these problems, but sometimes it is good to get feel for magnitudes.
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin
ktr >> ks
f ns =
1 2
1 k s ktr 1 ms k s + ktr 2
ks ms
fn_us = 10.372 Hz
1 m
NOTE: In 4th ed., these equation numbers are 7.20 and 7.21, respectively.
Gillespie (1992)
f ns
f n2 s 10 =
1 2
k 10 g s W
= mg k
= static deflection
From Gillespie (1992)
f n us
1 2
ks + ktr mus
Department of Mechanical Engineering The University of Texas at Austin ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin
Transfer Functions
for Car Model - Vibration Isolation 1
10 1 Transmissibility Ratio TR ( 2 f , 0.05 ) 2
2 TR ( 2 f , 0.10 )
You may feel more higher frequency vibration with a lighter unsprung mass. A lighter unsprung mass provides better vibration isolation in the midfrequency range.
0.1
NOTE: In 4th ed., these equation numbers are 7.23 and 7.25, respectively.
mass ratio =
mus ms
1 .10 4
This is a measure of vibration isolation, or the response of the sprung mass to the excitation from the ground. Here we look at the effect of the ratio of unsprung to sprung mass (0.05, 0.1, 0.2, 0.75).
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin
Transfer Functions
for Car Model - Vibration Isolation 2
stiffness ratio = ktr ks
In this region, you get better isolation with a stiffer tire.
Transfer Functions
for Car Model - Vibration Isolation 3
Varying damping
Higher damping is better in the vicinity of the natural frequency of the sprung mass. In this region, you get better isolation with lower damping ratio.
Transfer Functions
for Car Model - Suspension Travel
( z2 z1 )max
z0
At frequencies below the natural frequency of the sprung mass, a softer suspension leads to higher suspension travel.
As expected, a stiffer tire (relative to suspension) transmits more force to sprung mass.
A higher stiffness ratio corresponds to a softer suspension spring stiffness. Softer suspension provides better overall isolation, except in mid region.
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control
This is measured by deflection of the suspension spring or by relative displacement of the sprung and unsprung masses.
Varying stiffness ratio
Transfer Functions
for Car Model - Dynamic Tire Deflection
Not good!
Transfer Functions
for Car Model - Dynamic Tire Deflection
( z0 z2 )max
Light damping Bad shock absorber?
Varying damping ratio See Problem 7.6 Varying stiffness ratio
( z0 z2 )max
z0
z0
Wong (2001) This is a measure of road holding, since the dynamic tire deflection ratio shown is a measure of the normal force on the ground contact.
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin
Wong (2001) Better vibration isolation with softer suspension, but to get better roadholding at a frequency of excitation close to the unsprung mass natural frequency, a stiffer suspension spring should be used.
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin
2 2 V 2 = = V = V T
V = f
Department of Mechanical Engineering The University of Texas at Austin
Result/Usage
Understanding of the influence of sprung and unsprung masses, etc., on suspension performance.
Insight into vibration isolation, suspension travel, and road holding capabilities.
Time-domain simulations, allow nonlinear effects, active system integration, transient evaluation.
Department of Mechanical Engineering The University of Texas at Austin
Summary
Basic 1 and 2 DOF models provide good tools for studying the dependence of ride performance on component parameter values. Ride analysis focuses on vibrational response of a vehicle to road excitation, allowing study of the dependence on the distribution of mass, stiffness, and damping. The transfer function models can also show how some objectives can be at odds with others (introducing the need for controls). It can take time and experience to use basic models effectively, so nonlinear simulation ends up being a strong tool that can help overcome difficulties with building insight. Later we will examine how active elements and feedback principles are used in controlled suspensions.
ME 360/390 Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin
References
1. 2. 3. 4. 5. W.T. Thomson, Theory of Vibration with Applications, Prentice-Hall, 1993. Gillespie, T.D., Fundamentals of Vehicle Dynamics, SAE, Warrendale, PA, 1992. Liljedahl, et al, Tractors and their power units, ASAE, St. Joseph, MI, 1996. Wong, J.Y., Theory of Ground Vehicles, John Wiley and Sons, Inc., New York, 2001. Karnopp, D. and G. Heess, Electonically Controllable Vehicle Suspensions, Vehicle System Dynamics, Vol. 20, No. 3-4, pp. 207-217, 1991.
Appendix A
Base-excited model and analysis
Appendix A 1
Appendix A 2
Appendix A 3
b bc
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Appendix B
Human response to vibration
Wong (2001)
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ISO 2631
Transverse vibration
Octave Bands
Octave bands are geometrically related by the recursive relation,
f n +1 = 2k fn
Where the two frequencies, fn and fn+1 are successive band limits (lower and upper), and the index k is a positive integer or a fraction according to a whole octave or fractional octave. For example, if k = 1, the ratio between successive bands is 2. If k = 1/3, then the ratio between the upper and lower band limits is 1.26. Associated with each band is a center frequency, fc, which is given by the geometric mean, f = f f
c n +1 n
An octave bandwidth is, f n+1 f n = BW (bandwidth) Contrast this with a decade. If you advance a decade, it is a 10-times increase.
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