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Free vibration of a pneumatic tire-wheel unit using a ring on an elastic foundation and a finite element model
Journal
of Sound and
FREE USING
Vibration (1986) 107(2), 181-194
VIBRATION A RING
OF A PNEUMATIC ON
AN ELASTIC
FINITE
TIRE-WHEEL
FOUNDATION
ELEMENT
UNIT AND
A
MODEL
L. E. KuNcit AND W. SOEDEL Ray W. Hemck Laboratories,
School of Mechanical Engineering AND
T. Y. YANG School of Aeronautics
and Astronautics,
Purdue University,
(Received
23 November
West Lafayette,
Indiana 47907, U.S.A.
1984)
Natural frequencies and mode shapes of a pneumatic tire without suspension are investigated using a lZd.o.f., geometrically non-linear, doubly curved, thin shell finite element of revolution with laminate composite materials. The wheel is assumed to be free to move within its own plane. The results of the free vibration analysis indicate that only the radial modes of n = 1 are affected by the wheel’s freedom to move. To evaluate the finite element modeling, a simplified elastic ring-spring model is studied. The tire is modeled as a circular, elastic ring supported by distributed spring in both radial and circumferential directions. The wheel is modeled as a rigid mass to which the distributed spring is attached. The two models are found to agree and complement each other. While the simplified ring-spring model is easy and practical to use to obtain preliminary results, the complex finite element model can give more detailed and accurate results for both free vibration and dynamic response analyses.
1. INTRODUCTION The first approach to the modeling of a pneumatic tire was to use a ring supported by an elastic foundation [l-4]. The elastic foundation was modeled as a distributed spring in radial direction. The model was then extended [SJ to include the tangential stiffness effect between tread band and wheel rim and the pretension of the tread band due to internal pressure and/or rotational speed. Recently the ring model was used to investigate
the vibration response of fixed axle rotating tires [6]. This shows that even though there are numerical techniques available now for analyzing the structural behavior of tires, the relatively simple mathematical model of a ring is still a useful tool for understanding tire behavior. It should be noted that in all the above studies a fixed-wheel tire was presumed. Since the long range objective of tire dynamics research is to predict the dynamic response at the wheel axle of a rolling tire, in order to study the vibration transmitted to the vehicle body through the suspension, the assumption of a fixed-wheel tire can no longer be used. The wheel has to be allowed to have at least one degree of freedom in vertical direction. Consequently the dynamic characteristics of the tire-wheel unit will be different from those of a fixed wheel tire. How are these dynamic characteristics different from each other? Two approaches, one using the ring model and the other using the finite element procedure developed in the present research, have been used to study this question. t
Now with Central Research
Laboratory,
The Firestone
Tire and Rubber Company
181 0022460X/86/110181
+ 14 $03.00/O
@ 1986 Academic
Press Inc. (London)
Limited
182
1.. 1.. KUNG,
W. SOEDEL
AND
.I‘. Y. YANG
The reason for using a ring model is that the finite element program results needed to be interpreted. Mathematical derivations for the natural frequencies and mode shapes of a circular ring supported by a distributed spring attached to a movable center mass are presented first. The analysis will show that only the n = 1 modes are affected by the freedom of the center mass. The finite element procedure is described next. A numerical example and discussions follow. The conclusions drawn from the ring model support the finite element results.
2. A RING Figure tangential
MODEL WITH MOVABLE FOUNDATION
1 shows an elastic ring supported by distributed directions. The distributed springs are attached
springs, in both radial and to the rigid center mass M
h
Figure
1. A ring on elastic foundation
with one d.o.f. center mass.
After dspacement
Figure
2. Displacements
of a differential
ring element
and the center
mass.
VIBRATION
OF
PNEUMATIC
TIRE-WHEEL
UNIT
183
which is free to move in y-direction. The radial and tangential displacements of the ring are w and u respectively. The radial and tangential stiffness constants (in N/m* or lbf/in*) of the distributed spring are k, and k, respectively. The symbol 0 is the angular co-ordinate measured counterclockwise from the x-axis. The radius, width and thickness of the ring are R, b and h, respectively (a list of nomenclature is given in the Appendix). Figure 2 illustrates the relationship between a differential ring element and its corresponding point on the edge of the center mass after deflection from its equilibrium position. The elongations of the distributed spring in radial and tangential directions are (w - y sin 0) and (II - y cos 0) respectively. The equations of motion of a ring are [7] (D/R4)[~3~/~63-~4w/~~4]-(K/R2)[~u/~~+w]+q,=ph~2~/~t2,
(o/R”)[a*~/ae*
- a3w/de3] + (zc/R*)[a*tgae*+
(2.1)
aw/ae] + 4. = pha*~/at*.
(2.2)
Note that the inextensional assumption is not used here since it is not a permissible simplification for all loading situations a tire may encounter. The equation of motion for the center mass is 27
2% k(w-y
I0
sin e) (sin B)bR de+
cos 0) (cos 8)bR de = Mj,
k,(v-y I0 277
or
Mf+~Rb(k,,,+k,)y=Rb
(k,,,w sin e + k,V
cos
e)
de.
(2.3)
I 0
The
term q,,, in equation (2.1) is replaced by sin e)
qw=(N;/R2)(a2w/a0*)-k,(w-y
(2.4)
to account for the foundation stiffness k, and the pretension NL due to internal pressure and/or rotation [8]. The term q. in equation (2.2) is replaced by
e)
qv=-ku(u-ycos to account for the foundation
(2.5)
stiffness K, in the tangential direction. For the eigenvalue
analysis, let
w = w,(e)
e'+,
u =
V,(e) e’“n’,
y =
y,
e'"o'.
(2.6a-c)
Equations (2.1), (2.2) and (2.3) with substitution of equation (2.6) to eliminate the time variable become (D/R4)[d3Vn/de3-d4Wn/d0’]-(K/R2)[dVn/de+ + (N;/R*)
W,,]
d* W,,/de* - k,,,( W,, - Y, sin 0) + phd, W,, = 0,
(2.7)
(D/R4)[d2Vn/d~2-d2Wn/d~3]+(K/R2)[d2Vn/d~2+dWn/d~] - k”( V,, - Y, cos 0) + phw2, V,, = 0,
(2.8)
27r
(kw,
Y&rRb(k,,,+k,)-o:M]=Rb
sin 0+ k,V, cos e) de.
(2.9)
I0
Let the mode shape functions
w,(e) = A,
W,,( 0) and V,,( 0) be cos
(ne - b),
v,(e)=R,sin(ne+),
(2.10a, b)
L. E. KUNG,
184
W. SOEDEL
AND
‘I-. Y. YANG
where (b is an arbitrary angle. Substitution of equations (2.10) into equation (2.9) results in
[k&,cos(nf+~)sinO+k,B,sin(nf+-Q,)
Y,,=Rb
xcos e]df3/[~Rb(k,,,+k,)-&Ml, or
Y,=U~Rb(kJ,,-k,B,)sin
(2.11)
~]/[~Rb(k,+k,)-w2,M]}6,,,
where 6,, is the Kronecker delta &,={A
;;;;;}.
For the cases of n # 1, equation (2.11) vanishes and equations (2.7) and (2.8) are identical to the equations of motion for the ring with a fixed center mass: (D/R4)[d3VJd03-d4WJd04]-(K/R2)[dV,/dfI+ +(N;/df12)-k,,,W,+pho;
W,]
(2.12)
W,, =O,
(D/R4)d2V,/d82-d3W,/d83]+(K/R2)[d2Vn/dO2+dW,/dB]-k,V,+pho2,V,=0. (2.13) Substitution equation
of equation
(2.11) into equations
n4D K phw: -F-Rz+L----n3D R4
n*N’
(2.12) and (2.13) results in the matrix n3D
kw
R2
nK
-F-R2
n*D
nK ph+F-z-
R2
= (0).
n*K
(2.14)
k,
The characteristic equation is obtained by setting the determinant of the coefficient matrix of (2.14) to zero: (2.15) w4,+clwz,+c2=o, c,=(-l/ph)[(n2+l){(n2D/R4)+(K/R2)}+(n2N~/R2)+k,+k,],
(2.16)
c2=(l/p2h2)[n2(n2-1)2(DK/R6)+n2(k,+n2N~/R2){(D/R4)+(K/R2)} + k,{(n4D/R4)+(K The eigenvalues,
+ n*NL)/R*}+
(2.17)
k,k,].
or natural frequencies squared, w;, for all values of n are the roots of
equation (2.15): i = 1,2.
&=f[-c*+(-I)‘_], The amplitude
ratio of the corresponding equations (2.14) as
Ynif
&i
(2.18)
mode shape is determined from either one of
=
0,
i=l,2.
(2.19b)
For the case of n = 1, equation (2.11) becomes Y, = [ mRb( k,,A, - kJ.3,) sin +I/[ rRb( k,,, + k) - o:M]. Since the value of C#Jis arbitrary, the substitutions
(2.20)
of C#J = 0 and C$= ~12 result in two
VIBRATION
OF PNEUMATIC
TIRE-WHEEL
185
UNIT
orthogonal modes for n = 1. With the substitution of C#J = 0, equation (2.20) vanishes and equations (2.7) and (2.8), with n = 1, have the same form as equations (2.12) and (2.13). The characteristic equation becomes &+p,&+pz=
(2.21)
0,
(2.22)
P~=(-~/~~)[~{(~IR~)+(KIR~)}+(N~IR~)+~,+~I, p2= (lip2h2)[(k,+ko+N~iR2){(DIR4)+(KIR2))+(~N;iR2)+~~].
(2.23)
The second subscript 0 of the variable o in equation (2.21) denotes that 4 = 0. The squares of the natural frequencies are the roots of equation (2.21): UJ:,i=$-pr
i=l,2.
+ (-l)im],
(2.24)
The ratios of mode shape amplitudes are
= phmfoi - (D/&l - (K/R21- k
WR4~+WIR2~
A,,i -=
B,oi Pho:,i-(DIR4)-{(K+N~)lR2}-k, ylOi/
BIOi
=
(D/R4)+(K/R2)
O9
’
(2’25a)
(2.25b)
i= 1,2.
The substitution of 4 = 1r/2 into equation (2.20) gives (2.26)
Yl=aA,-/3&,
a = rRbk,,,/ rRb( k,,, + k&o:, M,
/3 = mRbk,/[ ?rRb(k
+ k,,) - w;, M].
(2.27,2.28)
The second subscript 1 of the variable w in both equations (2.27) and (2.28) denotes that
C#J = 7r/2. Equations (2.7) and (2.8), with n = 1, become (D/R4)[d3VI/d&h4W,/de4]-(K/R’)[dVr/de+ + ( N’,/R2)
WJ
(2.29)
d2 W/de2 - k( W, - Yl sin 0) + PO:, WI = 0,
(D/R4)[d2V,/de2-d3W,/d63]+K/R2[d2VI/de2+dWr/df?] (2.30)
- k,( VI - Yl cos 0) + ph W:, V, = 0. Equations (2.10) become W,(e) = A,
cos (8 -
VI = B, sin (6 - r/2)
m/2) = A, sin 0,
= -RI cos 8. (2.31a, b)
Substitution of equations (2.31) and (2.26) into equations (2.29) and (2.30) results in the matrix equation
= 0. D K -m-w_ R4 R2
The characteristic
(2.32)
kfP
equation for equations (2.31) is a third-order UT, + 4*& +
ql= (-llph)[(2D/R4)+{(2R+
92w:,+
q3 =
polynomial in o:r:
0,
N~)IR’}+k,+k,l-(~RblM)[k,+%l,
(2.33)
(2.34)
e= (lIp2h2)[~(DIR4)+(~IR2)+(~‘,IR2)+k,+~~~(DIR4)+(~IR2)~ +~{(N~lR2)+~,)1+(~RblMph) x[{(2D/R4)+((2K+N~)IR2)+k,+k,}(k,+k,)-(k2,+kt)l,
(2.35)
186
L. E. KUNG.
W. SOEDEL
AND
I‘.Y. YANG
qD=(~Rb/M~Zh2)[{(D/R4)+(K/R2)+k,}k~.+2~k,{(D/R4)+(K/R’)} +k:{(K/R4)+((K+N;)/R2)+k,}-(k,,+k,) x[{(D/R4)+(K/RZ)+(N;/R2)+kk,+k,.}{(D/R4)+(K/RZ)} (2.36)
+k~(J%lR2)+kw>ll. Three natural frequencies, w:,~, i = 1,2,3, can be obtained amplitude ratios of the corresponding mode shapes are
from equation
(2.33) and the
(DIR4)+(KIR2)+kvP
A,,i -= ElIi
(2.37a)
ph&-(D/R4)-(K/R*)-(N;/R*)-kk,(l-a) =
[phlJ& - ~~I~4~-~~I~2~--k,~~-~~l/~~~I~4~+~~I~2~+~,P1,
YllilBt,r = a(AtlilBlli)+P, 3.
A FINITE
ELEMENT
MODEL
(2.37~)
i = 1,2,3.
FOR TIRE-WHEEL
(2.37b)
SIMULATION
Figure 3(a) shows a thin shell of revolution having a contour similar to that of a pneumatic tire. The angular co-ordninate 0 is measured using the right-hand rule about the z-axis. The local displacements u, u, and w of a point on the shell surface are indicated.
(b) Cb)
(0)
Figure
3. (a) A thin shell of revolution
and (b) the geometry
of an element.
A circular piece of the shell of revolution, shown in Figure 3(b), is modeled by a 12-d.o.f., doubly curved, geometrically non-linear thin shell finite element of revolution with smeared properties of laminate composite materials. The polar, meridional and circumferential radii are r, RI and R, respectively. The meridional length of the kth element is C,. The local co-ordinate x is measured in the meridional direction from the medium of the element. The six degrees of freedom assumed at each end circle of the finite element are w, u, functions for w, u, and u are assumed u, p(aw/ax - u/R,), au/ax and au/ax. Displacement to be composed of the products of a sinusoidal variation in the circumferential direction and a cubic polynomial variation in the meridional direction: w( x, e) = (a, + a,x + u3x2 + &$x3) cos (ne - 4), ~(~,e)=(a,+a,~+a,~*+a~~3)~0~(ne-~),
u(x, e)=(~,+u,,x+a1,x2+u,2~3) Here a,,~,,...,
a,, are polynomial
coefficients
sin (no-4). and 4 is an arbitrary
(3.1) angle.
VIBRATION
OF
PNEUMATIC
TIRE-WHEEL
UNIT
187
The stiffness matrix of the finite element is derived by using the principle of minimum potential energy [9]. The strain energy of the system is formulated by using the thin shell assumption [lo] and the constituent properties of laminated composite materials. The strain energy is then expressed in terms of nodal displacements through several transfonnations. The stiffness matrix, which is obtained by performing the second derivatives of the strain energy with respect to the nodal displacements, consists of three parts: linear, first order non-linear, and second order non-linear terms. It is necessary that the non-linear parts of the stiffness matrix be included in order to account for the prestressed effect due to inflation pressure and/or tire rotation. The consistent mass matrix and consistent force vector [ 1l] were used. The non-linear equilibrium state of the tire due to inflation pressure is calculated by using either an iterative or self-correcting incremental procedure. Then the free vibration analysis is performed about the established equilibrium state. As annular steel plate is used to model the wheel to which a tire is mounted, as illustrated in Figure 4. The steel plate is divided equally in two halves and they are placed
Figure 4. Modeling
the wheel by a pair of annular plate elements.
at the end of the tire finite element mesh. The wheel in the present model is considered as a rigid mass and the steel plate, which has the same mass as that of a wheel, is made fairly rigid compared with the tire material. Other criteria of modeling a wheel may be used as well. 4.
NUMERICAL EXAMPLE AND DISCUSSIONS
A P185/80R13, automotive tire-wheel unit is used. The tire cross-sectional profile is obtained by placing an unstretched strip of the tire on a grid paper in the following manner. The outer rim-to-rim distance is O-152 m (6 in) and the radius of the crown point is O-311 m (12.25 in). The material properties of the tire is grouped into four zones: belt,
188
L. E. KUNG,
W. SOEDEL
AND
T. Y. YANG
symmetry
I.
0
1
I.
1
2 And
Figure 5. Cross-sectional zones.
1.
”
3
4
1
5
coordmate z (in1
profile and finite element mesh of a P185/80R13
tire with four material property
upper sidewall, mid-sidewall and lower sidewall. The constituent properties of each material zone are given in the form of the [A], [B], [D] matrices for laminated composites. Figure 5 illustrates the cross-sectional profile, the material property zones and the finite element mesh for the half tire. A convergence study has shown that 10 finite elements are adequate for modeling the tire and wheel in this study. The wheel is modeled by an annular steel plate with outer and inner diameters of 0.343 m (13.5 in) and O-038 m (1.5 in) respectively. The weight of the wheel is 86 N (19.3 lbr). The inflation pressure is 206,700 N/m* (30 psi). 4.1. RING MODEL There is no intention to have the ring model predict precisely natural frequencies of the actual tire. The purpose of the ring model is to explain the curious effect of selective wheel coupling first observed with the finite element model. The only reason to select parameters for the ring model close to those of the actual tire is cosmetic; the general effect is the same, whatever the ring parameters are. In order to calculate natural frequencies and mode shapes close to the realistic tire by using equations (2.18), (2.19), (2.24), (2.25), (2.33) and (2.37), the equivalent values of ph, k, k, D/R4, Kl R2 and NfJR2 of the tire need to be evaluated. The value of ph is the sum of the ph for the belt and a third ph of the sidewall. The values of the membrane rigidity K and the bending rigidity D are replaced by the belt membrane and bending rigidities, respectively, in the circumferential direction. The radial and tangential stiffness constants of the tire sidewall are [8] (but see also reference [5]) k, = (2p/b tan CY~) (N/m3 k, = (2pL cos ao/baoho(l - ho/4R)
or
lbr/in3),
(N/m3 or lbr/in’).
(4.1) (4.2)
VIBRATION
OF PNEUMATIC
TIRE-WHEEL
UNIT
189
Here p is the inflation pressure of the tire (N/m’ or IbJin’), a0 is the circular angle of the tire sidewall (rad), L is the arc length of the tire sidewall (m or in), ho is the vertical distance of the tire sidewall (m or in), and b is the width of the tread band. The prestressed force resultant NL due to inflation pressure is [8] Ni =pR[(l
+phRfi2/P)/(1
tan (Ye)] (N/m or IbJin),
+2PR2/Kb
(4.3)
where a is the rotational speed of the tire (rad/s). The calculated natural frequencies of the tire with a fixed wheel are shown in Figure 6 by solid circles for values of n from 0 to 14. The lower curve represents modes dominated by transverse or radial mode components and the upper curve represents the tangentially or circumferentially dominated modes. The natural frequencies of the tangential modes beyond n = 3 are too high to be plotted.
I
4
I
1
1
8
I
12
Crctriferentdmodeno n
Figure 6. Natural frequencies obtained by the ring for various circumferential modes. (1) Modes dominated by radial displacement; (2) modes dominated by tangential displacement. 0, Fixed wheel; 0, one d.o.f. wheel.
The natural frequencies of the same tire with a one degree of freedom wheel are also presented in the same figure by hollow circles. The natural frequencies for both cases are listed in Table 1. For n f 1, the resulting natural frequencies are the same as those of the fixed wheel case. For n = 1, four natural frequencies, instead of two in the fixed wheel case, are obtained. Among the four frequencies, two agree well with those obtained for the fixed wheel case both in frequency (see those for 80 and 350 Hz in Figure 6) and in mode shape (see Figure 7). The other two natural frequencies result from the interaction between the wheel and the missing orthogonal mode in Figure 8. As a result of the interaction, the missing n = 1 transverse mode splits into two new modes (see Figure 9). In the first mode the wheel mass moves along with the tire in the form of a rigid body mode and a zero frequency results. In the second mode the wheel moves against the tire at a natural frequency of 141 Hz.
190
L. E. KUNG,
w. S~EDEL TABLE
Natural frequencies
of a P185/80R13 -foundation
0
1 2 3 4 5 6 I 8 9 10 11 12 13 14
r Fixed wheel 258 19 85 91 120 154 198 253 318 393 477 571 673 785 906
Movable
tire calculated from the ring on elastic model (Hz)
wheel
258 0,79,141 85 97 120 154 198 253 318 393 417 571 613 785 906
r. Y. YANC;
1
Natural frequency of mode shape with dominant transverse displacement n
ANI)
Natural frequency of mode shape with dominant tangential displacement ,Fixed wheel 17 356 555 781 1016 1255 1497 1739 1982 2226 2471 2715 2960 3205 3450
Movable
wheel
77 356 555 781 1016 1255 1497 1739 1982 2226 2411 2715 2960 3205 3450
Figure 7. Orthogonal mode shapes dominated by tangential displacement for n = 1. Natural frequency 356 Hz.
VIBRATION
Figure
8. Mode shape
dominated
OF PNEUMATIC
by radial
TIRE-WHEEL
displacement
for n = 1. Natural
Figure 9. Two mode shapes dominated by radial displacement for n = 1, resulting one d.o.f. center mass. Natural frequencies: (a) 0 Hz; (b) 141 Hz.
4.2.
FINITE
ELEMENT
191
UNIT
frequency
79 Hz.
from the inclusion
of the
RESULTS
As mentioned earlier, the wheel is modeled with an annular steel plate of the same amount of mass. The method of controlling the freedom of the wheel is achieved by setting appropriate boundary conditions at the wheel. The boundary conditions are set such that the wheel is free to move within its own plane. The wheel can interact in two orthogonal directions with the tire. The calculated natural frequencies of the tire with a fixed and a movable wheel are presented in Figure 10. The natural frequencies obtained for the fixed wheel tire and those for the movable wheel tire are shown in Figure 10 by solid and hollow circles, respectively, for eight different circumferential wave numbers. Only for the n = 1 transverse modes are the
192
L. E. KUNG,
0
W. SOEDEL
4
2
Clrcumferentlol
Figure 10. Natural
frequencies,
obtained
AND
r. Y. YANG
6 mode
8
10
no n
by the finite element model, for the first four meridionally symmetric
and eight circumferential modes. 0, Fixed wheel; 0, one d.o.f. wheel. natural frequencies affected by the inclusion of wheel mass. This behavior is the same as that concluded from the ring model. The n = 1 modes dominated by tangential motion are unaffected by the wheel. The bottom curve, which is marked as mw = 3, is the equivalent of the lower curve in Figure 6. The curve marked as mu = 1 is the equivalent of the upper curve in Figure 6. The other two curves represent mw = 5 and mu = 3 modes, which are next to the two fundamental modes mw = 3 and mv = 1, respectively. A list of the natural frequencies of mw = 3 and mu = 1 for both fixed and movable wheel tires is presented in Table 2. TABLE
Natural frequencies
n 1 2 3 4 5 6 7 8
Natural frequency of mode shape with dominant transverse displacement (mw = 3) \ r Fixed wheel Movable wheel 78 103 124 145 165 185 207 233
2
of a P185/80R13 tire calculated from thejnite model (Hz)
0,86 103 124 145 165 185 207 233
element
Natural frequency of mode shape with dominant tangential displacement (mu = 1) r Fixed wheel Movable wheel 209 222 245 272 319 350 386 415
209 222 245 272 319 350 386 415
A comparison between the mw = 3 curve in Figure 10 obtained from the finite element model and the lower curve in Figure 6 obtained from the ring model reveals that the ring model provides reasonably accurate natural frequencies for n = 0 to 8. The discrepancy increases as n increases. It must be noted that it is not the intention that the ring model results should be as accurate as those of the finite element model. The purpose of the ring model is to provide some results to check on the validity of the finite element results
VIBRATION
OF PNEUMATIC
TIRE-WHEEL
UNIT
193
and to help understand the complex vibration behavior of tires in a simple manner. The conclusions drawn from the present study about the interaction between the wheel and the n = 1 modes of the tire thus have been independently verified by two mathematical modeling approaches.
5. SUMMARY The free vibration of an automotive tire with movable wheel was analyzed by using a 12 d.o.f., doubly curved, geometrically non-linear, thin shell finite element of revolution. The finite element procedure took advantage of the axisymmetry of tires so as to achieve computation efficiency by using relatively fewer number of d.o.f. (66) as compared with other existing types of finite elements. The wheel was modeled by a pair of annular steel plates (see Figure 4), which was represented by one finite element. The calculated natural frequencies indicated some interesting characteristics of a tire with movable wheel. In order to understand better and to explain the findings obtained from the rather complex finite element method, a simple analytical model of a circular ring on an elastic foundation was derived. The two models, one simple and one complex, complement each other. The tire with a movable wheel was modeled by using a circular ring supported by an elastic foundation with a movable center mass. The movable center mass (or wheel) was allowed only one degree of freedom along its y-axis (see Figure 1). The results of the derivation based on thin shell theory indicated that only the n = 1 modes dominated by transverse motion were affected by the freedom of the wheel. The affected mode shape was an in-plane, pseudo-rigid body mode (the tire belt moves as a rigid body while the wheel holds still). Two new modes resulted from the interaction between the movable wheel and the effected mode shape (see Figure 9). In the first mode the wheel moves with the tire and the natural frequency is always 0 Hz. In the second mode the wheel moves against the tire and the natural frequency is usually higher than that of the pseudo-rigid body mode of the fixed wheel. The results obtained by the finite element method showed that indeed only one of the n = 1 modes was affected by the freedom of the wheel. The rest of the mode shapes remained unchanged from those of the fixed wheel tire. The physical explanation is, as guided by the ring model, that all other mode shapes, with n # 1, have a force distribution between the wheel rim and the tire that varies sinusoidally around the rim in such a way that any wheel motion effects integrate out.
ACKNOWLEDGMENT The authors gratefully acknowledge the continuing financial support from the Firestone Tire and Rubber Company and the permission to publish this paper. Valuable technical assistance and advice were received from Dr J. D. Walter and Dr S. K. Jha.
REFERENCES 1. F. BOHM 1966 Ingenieur Archiu 35, 82-101. Mechanik des Gurtelreifens. 2. R N. DODGE 1965 Sociery of Automotive Engineers Paper No. 650491. The dynamic stiffness of a penumatic tire model. 3. J. T. TIELKING 1965 Society of Automotive Engineers Paper No. 650492. Plane vibration characteristics of a pneumatic tire model. 4. S. K. CLARK 1965 Society ofAutomotive Engineers Paper No. 650493. The rolling tire under load. 5. H. B. PACEJKA 1971 National Bureau Standards Monogruph 122. Tire in-plane dynamics. Mechanics of pneumatic tires (editor S. K. Clark).
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AND
T. Y. YANG
G. R. POTTS, C. A. BELL, L. T. CHAREK and T. K. ROY 1977 Tire Science and Technology, TSTCA 5, 202-225. Tirevibrations. W. SOEDEL 1981 Vibrations of Shells and Plates. New York: Marcel Dekker. W. SOEDEL 1975 Shorr Course, Firestone Tire and Rubber Company. Notes on tire vibrations using a ring model. L.E.KuNG,T.Y.YANG, W.SOEDEL~~~ L.T.CHAREK 1985JournalofSoundand Vibration 102, 329-346. Natural frequencies and mode shapes of an automotive tire with interpretation and classification using 3-D computer graphics. 10. J. L. SANDERS JR. 1963 Quarterly of Applied Mathematics 21, 21-36. Nonlinear theories for thin shells. 11. H. M. ADELMAN, D. S. CATHERINES and W. C. WALTON JR 1969 NASA TND-4972. A method for computation of vibration modes and frequencies of orthotropic thin shells of revolution. APPENDIX:
An,An, al,a2,...,a12
Bn, bn, b ck Cl,
c2
D h h, i K L mv mw N;, n P P23P2 qt,q2 4" 9w R2
k, r t VII 4
0
WI?
W x Y", Y", Y Z a,
P
QO
P s "1 i R 010, "101 91, "11: ""
NOMENCLATURE
amplitude of W, displacement coefficients of element displacement functions amplitude of V,, displacement width of ring meridional length of kth finite element of revolution coefficients of polynomial equation flexural rigidity thickness of ring vertical distance of tire sidewall imaginary unit (=Gi) membrane rigidity arc length of tire sidewall meridional half-wave number in v-displacement meridional half-wave number in w-displacement prestressed force resultant circumferential full-wave number inflation pressure coefficients of polynomial equation coefficients of polynomial equation distributed tangential load distributed radial load radius of ring radii of curvature along meridional and circumferential directions, respectively radial co-ordinate time nth mode shape in v displacement component along meridional and circumferential directions, respectively nth mode shape in w displacement component normal to the surface of element of revolution meridional co-ordinate amplitude of nth mode shape for center mass displacement of center mass axial co-ordinate dimensionless constants circular angle of tire sidewall density of ring Kronecker delta angular co-ordinate phase angle rotating speed of tire natural frequency of n = 1 and 4 = 0 natural frequency of n = 1 and 4 = ?r/2 natural frequency (rad/s)
of Sound and
FREE USING
Vibration (1986) 107(2), 181-194
VIBRATION A RING
OF A PNEUMATIC ON
AN ELASTIC
FINITE
TIRE-WHEEL
FOUNDATION
ELEMENT
UNIT AND
A
MODEL
L. E. KuNcit AND W. SOEDEL Ray W. Hemck Laboratories,
School of Mechanical Engineering AND
T. Y. YANG School of Aeronautics
and Astronautics,
Purdue University,
(Received
23 November
West Lafayette,
Indiana 47907, U.S.A.
1984)
Natural frequencies and mode shapes of a pneumatic tire without suspension are investigated using a lZd.o.f., geometrically non-linear, doubly curved, thin shell finite element of revolution with laminate composite materials. The wheel is assumed to be free to move within its own plane. The results of the free vibration analysis indicate that only the radial modes of n = 1 are affected by the wheel’s freedom to move. To evaluate the finite element modeling, a simplified elastic ring-spring model is studied. The tire is modeled as a circular, elastic ring supported by distributed spring in both radial and circumferential directions. The wheel is modeled as a rigid mass to which the distributed spring is attached. The two models are found to agree and complement each other. While the simplified ring-spring model is easy and practical to use to obtain preliminary results, the complex finite element model can give more detailed and accurate results for both free vibration and dynamic response analyses.
1. INTRODUCTION The first approach to the modeling of a pneumatic tire was to use a ring supported by an elastic foundation [l-4]. The elastic foundation was modeled as a distributed spring in radial direction. The model was then extended [SJ to include the tangential stiffness effect between tread band and wheel rim and the pretension of the tread band due to internal pressure and/or rotational speed. Recently the ring model was used to investigate
the vibration response of fixed axle rotating tires [6]. This shows that even though there are numerical techniques available now for analyzing the structural behavior of tires, the relatively simple mathematical model of a ring is still a useful tool for understanding tire behavior. It should be noted that in all the above studies a fixed-wheel tire was presumed. Since the long range objective of tire dynamics research is to predict the dynamic response at the wheel axle of a rolling tire, in order to study the vibration transmitted to the vehicle body through the suspension, the assumption of a fixed-wheel tire can no longer be used. The wheel has to be allowed to have at least one degree of freedom in vertical direction. Consequently the dynamic characteristics of the tire-wheel unit will be different from those of a fixed wheel tire. How are these dynamic characteristics different from each other? Two approaches, one using the ring model and the other using the finite element procedure developed in the present research, have been used to study this question. t
Now with Central Research
Laboratory,
The Firestone
Tire and Rubber Company
181 0022460X/86/110181
+ 14 $03.00/O
@ 1986 Academic
Press Inc. (London)
Limited
182
1.. 1.. KUNG,
W. SOEDEL
AND
.I‘. Y. YANG
The reason for using a ring model is that the finite element program results needed to be interpreted. Mathematical derivations for the natural frequencies and mode shapes of a circular ring supported by a distributed spring attached to a movable center mass are presented first. The analysis will show that only the n = 1 modes are affected by the freedom of the center mass. The finite element procedure is described next. A numerical example and discussions follow. The conclusions drawn from the ring model support the finite element results.
2. A RING Figure tangential
MODEL WITH MOVABLE FOUNDATION
1 shows an elastic ring supported by distributed directions. The distributed springs are attached
springs, in both radial and to the rigid center mass M
h
Figure
1. A ring on elastic foundation
with one d.o.f. center mass.
After dspacement
Figure
2. Displacements
of a differential
ring element
and the center
mass.
VIBRATION
OF
PNEUMATIC
TIRE-WHEEL
UNIT
183
which is free to move in y-direction. The radial and tangential displacements of the ring are w and u respectively. The radial and tangential stiffness constants (in N/m* or lbf/in*) of the distributed spring are k, and k, respectively. The symbol 0 is the angular co-ordinate measured counterclockwise from the x-axis. The radius, width and thickness of the ring are R, b and h, respectively (a list of nomenclature is given in the Appendix). Figure 2 illustrates the relationship between a differential ring element and its corresponding point on the edge of the center mass after deflection from its equilibrium position. The elongations of the distributed spring in radial and tangential directions are (w - y sin 0) and (II - y cos 0) respectively. The equations of motion of a ring are [7] (D/R4)[~3~/~63-~4w/~~4]-(K/R2)[~u/~~+w]+q,=ph~2~/~t2,
(o/R”)[a*~/ae*
- a3w/de3] + (zc/R*)[a*tgae*+
(2.1)
aw/ae] + 4. = pha*~/at*.
(2.2)
Note that the inextensional assumption is not used here since it is not a permissible simplification for all loading situations a tire may encounter. The equation of motion for the center mass is 27
2% k(w-y
I0
sin e) (sin B)bR de+
cos 0) (cos 8)bR de = Mj,
k,(v-y I0 277
or
Mf+~Rb(k,,,+k,)y=Rb
(k,,,w sin e + k,V
cos
e)
de.
(2.3)
I 0
The
term q,,, in equation (2.1) is replaced by sin e)
qw=(N;/R2)(a2w/a0*)-k,(w-y
(2.4)
to account for the foundation stiffness k, and the pretension NL due to internal pressure and/or rotation [8]. The term q. in equation (2.2) is replaced by
e)
qv=-ku(u-ycos to account for the foundation
(2.5)
stiffness K, in the tangential direction. For the eigenvalue
analysis, let
w = w,(e)
e'+,
u =
V,(e) e’“n’,
y =
y,
e'"o'.
(2.6a-c)
Equations (2.1), (2.2) and (2.3) with substitution of equation (2.6) to eliminate the time variable become (D/R4)[d3Vn/de3-d4Wn/d0’]-(K/R2)[dVn/de+ + (N;/R*)
W,,]
d* W,,/de* - k,,,( W,, - Y, sin 0) + phd, W,, = 0,
(2.7)
(D/R4)[d2Vn/d~2-d2Wn/d~3]+(K/R2)[d2Vn/d~2+dWn/d~] - k”( V,, - Y, cos 0) + phw2, V,, = 0,
(2.8)
27r
(kw,
Y&rRb(k,,,+k,)-o:M]=Rb
sin 0+ k,V, cos e) de.
(2.9)
I0
Let the mode shape functions
w,(e) = A,
W,,( 0) and V,,( 0) be cos
(ne - b),
v,(e)=R,sin(ne+),
(2.10a, b)
L. E. KUNG,
184
W. SOEDEL
AND
‘I-. Y. YANG
where (b is an arbitrary angle. Substitution of equations (2.10) into equation (2.9) results in
[k&,cos(nf+~)sinO+k,B,sin(nf+-Q,)
Y,,=Rb
xcos e]df3/[~Rb(k,,,+k,)-&Ml, or
Y,=U~Rb(kJ,,-k,B,)sin
(2.11)
~]/[~Rb(k,+k,)-w2,M]}6,,,
where 6,, is the Kronecker delta &,={A
;;;;;}.
For the cases of n # 1, equation (2.11) vanishes and equations (2.7) and (2.8) are identical to the equations of motion for the ring with a fixed center mass: (D/R4)[d3VJd03-d4WJd04]-(K/R2)[dV,/dfI+ +(N;/df12)-k,,,W,+pho;
W,]
(2.12)
W,, =O,
(D/R4)d2V,/d82-d3W,/d83]+(K/R2)[d2Vn/dO2+dW,/dB]-k,V,+pho2,V,=0. (2.13) Substitution equation
of equation
(2.11) into equations
n4D K phw: -F-Rz+L----n3D R4
n*N’
(2.12) and (2.13) results in the matrix n3D
kw
R2
nK
-F-R2
n*D
nK ph+F-z-
R2
= (0).
n*K
(2.14)
k,
The characteristic equation is obtained by setting the determinant of the coefficient matrix of (2.14) to zero: (2.15) w4,+clwz,+c2=o, c,=(-l/ph)[(n2+l){(n2D/R4)+(K/R2)}+(n2N~/R2)+k,+k,],
(2.16)
c2=(l/p2h2)[n2(n2-1)2(DK/R6)+n2(k,+n2N~/R2){(D/R4)+(K/R2)} + k,{(n4D/R4)+(K The eigenvalues,
+ n*NL)/R*}+
(2.17)
k,k,].
or natural frequencies squared, w;, for all values of n are the roots of
equation (2.15): i = 1,2.
&=f[-c*+(-I)‘_], The amplitude
ratio of the corresponding equations (2.14) as
Ynif
&i
(2.18)
mode shape is determined from either one of
=
0,
i=l,2.
(2.19b)
For the case of n = 1, equation (2.11) becomes Y, = [ mRb( k,,A, - kJ.3,) sin +I/[ rRb( k,,, + k) - o:M]. Since the value of C#Jis arbitrary, the substitutions
(2.20)
of C#J = 0 and C$= ~12 result in two
VIBRATION
OF PNEUMATIC
TIRE-WHEEL
185
UNIT
orthogonal modes for n = 1. With the substitution of C#J = 0, equation (2.20) vanishes and equations (2.7) and (2.8), with n = 1, have the same form as equations (2.12) and (2.13). The characteristic equation becomes &+p,&+pz=
(2.21)
0,
(2.22)
P~=(-~/~~)[~{(~IR~)+(KIR~)}+(N~IR~)+~,+~I, p2= (lip2h2)[(k,+ko+N~iR2){(DIR4)+(KIR2))+(~N;iR2)+~~].
(2.23)
The second subscript 0 of the variable o in equation (2.21) denotes that 4 = 0. The squares of the natural frequencies are the roots of equation (2.21): UJ:,i=$-pr
i=l,2.
+ (-l)im],
(2.24)
The ratios of mode shape amplitudes are
= phmfoi - (D/&l - (K/R21- k
WR4~+WIR2~
A,,i -=
B,oi Pho:,i-(DIR4)-{(K+N~)lR2}-k, ylOi/
BIOi
=
(D/R4)+(K/R2)
O9
’
(2’25a)
(2.25b)
i= 1,2.
The substitution of 4 = 1r/2 into equation (2.20) gives (2.26)
Yl=aA,-/3&,
a = rRbk,,,/ rRb( k,,, + k&o:, M,
/3 = mRbk,/[ ?rRb(k
+ k,,) - w;, M].
(2.27,2.28)
The second subscript 1 of the variable w in both equations (2.27) and (2.28) denotes that
C#J = 7r/2. Equations (2.7) and (2.8), with n = 1, become (D/R4)[d3VI/d&h4W,/de4]-(K/R’)[dVr/de+ + ( N’,/R2)
WJ
(2.29)
d2 W/de2 - k( W, - Yl sin 0) + PO:, WI = 0,
(D/R4)[d2V,/de2-d3W,/d63]+K/R2[d2VI/de2+dWr/df?] (2.30)
- k,( VI - Yl cos 0) + ph W:, V, = 0. Equations (2.10) become W,(e) = A,
cos (8 -
VI = B, sin (6 - r/2)
m/2) = A, sin 0,
= -RI cos 8. (2.31a, b)
Substitution of equations (2.31) and (2.26) into equations (2.29) and (2.30) results in the matrix equation
= 0. D K -m-w_ R4 R2
The characteristic
(2.32)
kfP
equation for equations (2.31) is a third-order UT, + 4*& +
ql= (-llph)[(2D/R4)+{(2R+
92w:,+
q3 =
polynomial in o:r:
0,
N~)IR’}+k,+k,l-(~RblM)[k,+%l,
(2.33)
(2.34)
e= (lIp2h2)[~(DIR4)+(~IR2)+(~‘,IR2)+k,+~~~(DIR4)+(~IR2)~ +~{(N~lR2)+~,)1+(~RblMph) x[{(2D/R4)+((2K+N~)IR2)+k,+k,}(k,+k,)-(k2,+kt)l,
(2.35)
186
L. E. KUNG.
W. SOEDEL
AND
I‘.Y. YANG
qD=(~Rb/M~Zh2)[{(D/R4)+(K/R2)+k,}k~.+2~k,{(D/R4)+(K/R’)} +k:{(K/R4)+((K+N;)/R2)+k,}-(k,,+k,) x[{(D/R4)+(K/RZ)+(N;/R2)+kk,+k,.}{(D/R4)+(K/RZ)} (2.36)
+k~(J%lR2)+kw>ll. Three natural frequencies, w:,~, i = 1,2,3, can be obtained amplitude ratios of the corresponding mode shapes are
from equation
(2.33) and the
(DIR4)+(KIR2)+kvP
A,,i -= ElIi
(2.37a)
ph&-(D/R4)-(K/R*)-(N;/R*)-kk,(l-a) =
[phlJ& - ~~I~4~-~~I~2~--k,~~-~~l/~~~I~4~+~~I~2~+~,P1,
YllilBt,r = a(AtlilBlli)+P, 3.
A FINITE
ELEMENT
MODEL
(2.37~)
i = 1,2,3.
FOR TIRE-WHEEL
(2.37b)
SIMULATION
Figure 3(a) shows a thin shell of revolution having a contour similar to that of a pneumatic tire. The angular co-ordninate 0 is measured using the right-hand rule about the z-axis. The local displacements u, u, and w of a point on the shell surface are indicated.
(b) Cb)
(0)
Figure
3. (a) A thin shell of revolution
and (b) the geometry
of an element.
A circular piece of the shell of revolution, shown in Figure 3(b), is modeled by a 12-d.o.f., doubly curved, geometrically non-linear thin shell finite element of revolution with smeared properties of laminate composite materials. The polar, meridional and circumferential radii are r, RI and R, respectively. The meridional length of the kth element is C,. The local co-ordinate x is measured in the meridional direction from the medium of the element. The six degrees of freedom assumed at each end circle of the finite element are w, u, functions for w, u, and u are assumed u, p(aw/ax - u/R,), au/ax and au/ax. Displacement to be composed of the products of a sinusoidal variation in the circumferential direction and a cubic polynomial variation in the meridional direction: w( x, e) = (a, + a,x + u3x2 + &$x3) cos (ne - 4), ~(~,e)=(a,+a,~+a,~*+a~~3)~0~(ne-~),
u(x, e)=(~,+u,,x+a1,x2+u,2~3) Here a,,~,,...,
a,, are polynomial
coefficients
sin (no-4). and 4 is an arbitrary
(3.1) angle.
VIBRATION
OF
PNEUMATIC
TIRE-WHEEL
UNIT
187
The stiffness matrix of the finite element is derived by using the principle of minimum potential energy [9]. The strain energy of the system is formulated by using the thin shell assumption [lo] and the constituent properties of laminated composite materials. The strain energy is then expressed in terms of nodal displacements through several transfonnations. The stiffness matrix, which is obtained by performing the second derivatives of the strain energy with respect to the nodal displacements, consists of three parts: linear, first order non-linear, and second order non-linear terms. It is necessary that the non-linear parts of the stiffness matrix be included in order to account for the prestressed effect due to inflation pressure and/or tire rotation. The consistent mass matrix and consistent force vector [ 1l] were used. The non-linear equilibrium state of the tire due to inflation pressure is calculated by using either an iterative or self-correcting incremental procedure. Then the free vibration analysis is performed about the established equilibrium state. As annular steel plate is used to model the wheel to which a tire is mounted, as illustrated in Figure 4. The steel plate is divided equally in two halves and they are placed
Figure 4. Modeling
the wheel by a pair of annular plate elements.
at the end of the tire finite element mesh. The wheel in the present model is considered as a rigid mass and the steel plate, which has the same mass as that of a wheel, is made fairly rigid compared with the tire material. Other criteria of modeling a wheel may be used as well. 4.
NUMERICAL EXAMPLE AND DISCUSSIONS
A P185/80R13, automotive tire-wheel unit is used. The tire cross-sectional profile is obtained by placing an unstretched strip of the tire on a grid paper in the following manner. The outer rim-to-rim distance is O-152 m (6 in) and the radius of the crown point is O-311 m (12.25 in). The material properties of the tire is grouped into four zones: belt,
188
L. E. KUNG,
W. SOEDEL
AND
T. Y. YANG
symmetry
I.
0
1
I.
1
2 And
Figure 5. Cross-sectional zones.
1.
”
3
4
1
5
coordmate z (in1
profile and finite element mesh of a P185/80R13
tire with four material property
upper sidewall, mid-sidewall and lower sidewall. The constituent properties of each material zone are given in the form of the [A], [B], [D] matrices for laminated composites. Figure 5 illustrates the cross-sectional profile, the material property zones and the finite element mesh for the half tire. A convergence study has shown that 10 finite elements are adequate for modeling the tire and wheel in this study. The wheel is modeled by an annular steel plate with outer and inner diameters of 0.343 m (13.5 in) and O-038 m (1.5 in) respectively. The weight of the wheel is 86 N (19.3 lbr). The inflation pressure is 206,700 N/m* (30 psi). 4.1. RING MODEL There is no intention to have the ring model predict precisely natural frequencies of the actual tire. The purpose of the ring model is to explain the curious effect of selective wheel coupling first observed with the finite element model. The only reason to select parameters for the ring model close to those of the actual tire is cosmetic; the general effect is the same, whatever the ring parameters are. In order to calculate natural frequencies and mode shapes close to the realistic tire by using equations (2.18), (2.19), (2.24), (2.25), (2.33) and (2.37), the equivalent values of ph, k, k, D/R4, Kl R2 and NfJR2 of the tire need to be evaluated. The value of ph is the sum of the ph for the belt and a third ph of the sidewall. The values of the membrane rigidity K and the bending rigidity D are replaced by the belt membrane and bending rigidities, respectively, in the circumferential direction. The radial and tangential stiffness constants of the tire sidewall are [8] (but see also reference [5]) k, = (2p/b tan CY~) (N/m3 k, = (2pL cos ao/baoho(l - ho/4R)
or
lbr/in3),
(N/m3 or lbr/in’).
(4.1) (4.2)
VIBRATION
OF PNEUMATIC
TIRE-WHEEL
UNIT
189
Here p is the inflation pressure of the tire (N/m’ or IbJin’), a0 is the circular angle of the tire sidewall (rad), L is the arc length of the tire sidewall (m or in), ho is the vertical distance of the tire sidewall (m or in), and b is the width of the tread band. The prestressed force resultant NL due to inflation pressure is [8] Ni =pR[(l
+phRfi2/P)/(1
tan (Ye)] (N/m or IbJin),
+2PR2/Kb
(4.3)
where a is the rotational speed of the tire (rad/s). The calculated natural frequencies of the tire with a fixed wheel are shown in Figure 6 by solid circles for values of n from 0 to 14. The lower curve represents modes dominated by transverse or radial mode components and the upper curve represents the tangentially or circumferentially dominated modes. The natural frequencies of the tangential modes beyond n = 3 are too high to be plotted.
I
4
I
1
1
8
I
12
Crctriferentdmodeno n
Figure 6. Natural frequencies obtained by the ring for various circumferential modes. (1) Modes dominated by radial displacement; (2) modes dominated by tangential displacement. 0, Fixed wheel; 0, one d.o.f. wheel.
The natural frequencies of the same tire with a one degree of freedom wheel are also presented in the same figure by hollow circles. The natural frequencies for both cases are listed in Table 1. For n f 1, the resulting natural frequencies are the same as those of the fixed wheel case. For n = 1, four natural frequencies, instead of two in the fixed wheel case, are obtained. Among the four frequencies, two agree well with those obtained for the fixed wheel case both in frequency (see those for 80 and 350 Hz in Figure 6) and in mode shape (see Figure 7). The other two natural frequencies result from the interaction between the wheel and the missing orthogonal mode in Figure 8. As a result of the interaction, the missing n = 1 transverse mode splits into two new modes (see Figure 9). In the first mode the wheel mass moves along with the tire in the form of a rigid body mode and a zero frequency results. In the second mode the wheel moves against the tire at a natural frequency of 141 Hz.
190
L. E. KUNG,
w. S~EDEL TABLE
Natural frequencies
of a P185/80R13 -foundation
0
1 2 3 4 5 6 I 8 9 10 11 12 13 14
r Fixed wheel 258 19 85 91 120 154 198 253 318 393 477 571 673 785 906
Movable
tire calculated from the ring on elastic model (Hz)
wheel
258 0,79,141 85 97 120 154 198 253 318 393 417 571 613 785 906
r. Y. YANC;
1
Natural frequency of mode shape with dominant transverse displacement n
ANI)
Natural frequency of mode shape with dominant tangential displacement ,Fixed wheel 17 356 555 781 1016 1255 1497 1739 1982 2226 2471 2715 2960 3205 3450
Movable
wheel
77 356 555 781 1016 1255 1497 1739 1982 2226 2411 2715 2960 3205 3450
Figure 7. Orthogonal mode shapes dominated by tangential displacement for n = 1. Natural frequency 356 Hz.
VIBRATION
Figure
8. Mode shape
dominated
OF PNEUMATIC
by radial
TIRE-WHEEL
displacement
for n = 1. Natural
Figure 9. Two mode shapes dominated by radial displacement for n = 1, resulting one d.o.f. center mass. Natural frequencies: (a) 0 Hz; (b) 141 Hz.
4.2.
FINITE
ELEMENT
191
UNIT
frequency
79 Hz.
from the inclusion
of the
RESULTS
As mentioned earlier, the wheel is modeled with an annular steel plate of the same amount of mass. The method of controlling the freedom of the wheel is achieved by setting appropriate boundary conditions at the wheel. The boundary conditions are set such that the wheel is free to move within its own plane. The wheel can interact in two orthogonal directions with the tire. The calculated natural frequencies of the tire with a fixed and a movable wheel are presented in Figure 10. The natural frequencies obtained for the fixed wheel tire and those for the movable wheel tire are shown in Figure 10 by solid and hollow circles, respectively, for eight different circumferential wave numbers. Only for the n = 1 transverse modes are the
192
L. E. KUNG,
0
W. SOEDEL
4
2
Clrcumferentlol
Figure 10. Natural
frequencies,
obtained
AND
r. Y. YANG
6 mode
8
10
no n
by the finite element model, for the first four meridionally symmetric
and eight circumferential modes. 0, Fixed wheel; 0, one d.o.f. wheel. natural frequencies affected by the inclusion of wheel mass. This behavior is the same as that concluded from the ring model. The n = 1 modes dominated by tangential motion are unaffected by the wheel. The bottom curve, which is marked as mw = 3, is the equivalent of the lower curve in Figure 6. The curve marked as mu = 1 is the equivalent of the upper curve in Figure 6. The other two curves represent mw = 5 and mu = 3 modes, which are next to the two fundamental modes mw = 3 and mv = 1, respectively. A list of the natural frequencies of mw = 3 and mu = 1 for both fixed and movable wheel tires is presented in Table 2. TABLE
Natural frequencies
n 1 2 3 4 5 6 7 8
Natural frequency of mode shape with dominant transverse displacement (mw = 3) \ r Fixed wheel Movable wheel 78 103 124 145 165 185 207 233
2
of a P185/80R13 tire calculated from thejnite model (Hz)
0,86 103 124 145 165 185 207 233
element
Natural frequency of mode shape with dominant tangential displacement (mu = 1) r Fixed wheel Movable wheel 209 222 245 272 319 350 386 415
209 222 245 272 319 350 386 415
A comparison between the mw = 3 curve in Figure 10 obtained from the finite element model and the lower curve in Figure 6 obtained from the ring model reveals that the ring model provides reasonably accurate natural frequencies for n = 0 to 8. The discrepancy increases as n increases. It must be noted that it is not the intention that the ring model results should be as accurate as those of the finite element model. The purpose of the ring model is to provide some results to check on the validity of the finite element results
VIBRATION
OF PNEUMATIC
TIRE-WHEEL
UNIT
193
and to help understand the complex vibration behavior of tires in a simple manner. The conclusions drawn from the present study about the interaction between the wheel and the n = 1 modes of the tire thus have been independently verified by two mathematical modeling approaches.
5. SUMMARY The free vibration of an automotive tire with movable wheel was analyzed by using a 12 d.o.f., doubly curved, geometrically non-linear, thin shell finite element of revolution. The finite element procedure took advantage of the axisymmetry of tires so as to achieve computation efficiency by using relatively fewer number of d.o.f. (66) as compared with other existing types of finite elements. The wheel was modeled by a pair of annular steel plates (see Figure 4), which was represented by one finite element. The calculated natural frequencies indicated some interesting characteristics of a tire with movable wheel. In order to understand better and to explain the findings obtained from the rather complex finite element method, a simple analytical model of a circular ring on an elastic foundation was derived. The two models, one simple and one complex, complement each other. The tire with a movable wheel was modeled by using a circular ring supported by an elastic foundation with a movable center mass. The movable center mass (or wheel) was allowed only one degree of freedom along its y-axis (see Figure 1). The results of the derivation based on thin shell theory indicated that only the n = 1 modes dominated by transverse motion were affected by the freedom of the wheel. The affected mode shape was an in-plane, pseudo-rigid body mode (the tire belt moves as a rigid body while the wheel holds still). Two new modes resulted from the interaction between the movable wheel and the effected mode shape (see Figure 9). In the first mode the wheel moves with the tire and the natural frequency is always 0 Hz. In the second mode the wheel moves against the tire and the natural frequency is usually higher than that of the pseudo-rigid body mode of the fixed wheel. The results obtained by the finite element method showed that indeed only one of the n = 1 modes was affected by the freedom of the wheel. The rest of the mode shapes remained unchanged from those of the fixed wheel tire. The physical explanation is, as guided by the ring model, that all other mode shapes, with n # 1, have a force distribution between the wheel rim and the tire that varies sinusoidally around the rim in such a way that any wheel motion effects integrate out.
ACKNOWLEDGMENT The authors gratefully acknowledge the continuing financial support from the Firestone Tire and Rubber Company and the permission to publish this paper. Valuable technical assistance and advice were received from Dr J. D. Walter and Dr S. K. Jha.
REFERENCES 1. F. BOHM 1966 Ingenieur Archiu 35, 82-101. Mechanik des Gurtelreifens. 2. R N. DODGE 1965 Sociery of Automotive Engineers Paper No. 650491. The dynamic stiffness of a penumatic tire model. 3. J. T. TIELKING 1965 Society of Automotive Engineers Paper No. 650492. Plane vibration characteristics of a pneumatic tire model. 4. S. K. CLARK 1965 Society ofAutomotive Engineers Paper No. 650493. The rolling tire under load. 5. H. B. PACEJKA 1971 National Bureau Standards Monogruph 122. Tire in-plane dynamics. Mechanics of pneumatic tires (editor S. K. Clark).
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G. R. POTTS, C. A. BELL, L. T. CHAREK and T. K. ROY 1977 Tire Science and Technology, TSTCA 5, 202-225. Tirevibrations. W. SOEDEL 1981 Vibrations of Shells and Plates. New York: Marcel Dekker. W. SOEDEL 1975 Shorr Course, Firestone Tire and Rubber Company. Notes on tire vibrations using a ring model. L.E.KuNG,T.Y.YANG, W.SOEDEL~~~ L.T.CHAREK 1985JournalofSoundand Vibration 102, 329-346. Natural frequencies and mode shapes of an automotive tire with interpretation and classification using 3-D computer graphics. 10. J. L. SANDERS JR. 1963 Quarterly of Applied Mathematics 21, 21-36. Nonlinear theories for thin shells. 11. H. M. ADELMAN, D. S. CATHERINES and W. C. WALTON JR 1969 NASA TND-4972. A method for computation of vibration modes and frequencies of orthotropic thin shells of revolution. APPENDIX:
An,An, al,a2,...,a12
Bn, bn, b ck Cl,
c2
D h h, i K L mv mw N;, n P P23P2 qt,q2 4" 9w R2
k, r t VII 4
0
WI?
W x Y", Y", Y Z a,
P
QO
P s "1 i R 010, "101 91, "11: ""
NOMENCLATURE
amplitude of W, displacement coefficients of element displacement functions amplitude of V,, displacement width of ring meridional length of kth finite element of revolution coefficients of polynomial equation flexural rigidity thickness of ring vertical distance of tire sidewall imaginary unit (=Gi) membrane rigidity arc length of tire sidewall meridional half-wave number in v-displacement meridional half-wave number in w-displacement prestressed force resultant circumferential full-wave number inflation pressure coefficients of polynomial equation coefficients of polynomial equation distributed tangential load distributed radial load radius of ring radii of curvature along meridional and circumferential directions, respectively radial co-ordinate time nth mode shape in v displacement component along meridional and circumferential directions, respectively nth mode shape in w displacement component normal to the surface of element of revolution meridional co-ordinate amplitude of nth mode shape for center mass displacement of center mass axial co-ordinate dimensionless constants circular angle of tire sidewall density of ring Kronecker delta angular co-ordinate phase angle rotating speed of tire natural frequency of n = 1 and 4 = 0 natural frequency of n = 1 and 4 = ?r/2 natural frequency (rad/s)