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An approach to the dynamic analysis of rotating tire-wheel-suspension units
.Journal of Sound and Vibrufion (1992) 156(3), 505-519
AN APPROACH TO THE DYNAMIC ANALYSIS OF ROTATING TIRE-WHEEL-SUSPENSION UNITS S. C. HUANG
AND
B. S.
Hsu
Department of Mechanical Engineering, National Taiwan Institute of Technology, Taipei, Taiwan, 10722, Republic of China (Received 28 February 1990, and in final form 17 May 1991) An approach to the dynamic analysis of a rotating tire-wheel-suspension unit is developed. A rotating ring on an elastic foundation and a vertically movable center mass model the tire and the wheel respectively. The interaction of the wheel motion with the rotating tire results in a parametrically excited system with periodic coefficients. The flexural response of
the rotating tire is shown, however, to he independent of the movement of the wheel and hence can be solved for analytically. The system response due to a concentrated force provided by ground contacting is solved for as an example to illustrate the solution procedure. Force transmission as a function of rotation and damping ratio is illustrated. The results show that the damping constant of the suspension hardly affects the force transmission and stability, for a constant point load. The damping ratio of the n= 1 mode of the tire, that is presumably provided by the tire sidewall, is found dramatically to dominate the force transmission to the axle and the stability criteria.
1. INTRODUCTION
The vibration of tires has been of great interest to automotive engineers, since knowledge of the natural frequencies and modes of tires is needed to interpret or predict the riding comfort and noise transmission. Serious studies on tire vibration have been accomplished during the past two decades. The first analytic approach to the modeling of a pneumatic tire was to use a ring on an elastic foundation [l-3]. Soedel [4] presented an analytic solution for the dynamic response of a rolling inflated tire using a linear thin shell theory and three-dimensional dynamic Green functions. Potts et al. [5] used a rotating, thin elastic ring under tension to model the tire belt, and treated the tire sidewall as an elastic foundation. They also checked the analytic results experimentally and a rather good agreement was found. In 1986, Kung et al. [6] investigated the dynamic response of a pneumatic tire-wheel unit. In this paper, they modeled the tire as a stationary ring on an elastic foundation and the wheel as a vertically movable center mass. They discovered that the motion of wheel was simply coupled with the rigid body mode of the tire. They also employed a finite element model for the same problem and concluded that the ring model was acceptable for the dynamic analysis of a pneumatic tire. In a later paper [7], they extended the analysis to the steady state response and force transmission at the wheel axle of a pneumatic tiresuspension system due to a harmonic force or displacement input, in which the receptance method was utilized. In the papers of Kung et al. [6,7] account was taken of the centrifugal effects due to rotation but not the Coriolis accelerations. Yet it has been shown [8-lo] that the effects of rotation, especially the Coriolis terms, significantly change the dynamic characteristics of the ring. The current research is basically an extension of the analysis of Kung et al. 505 0022-460X/92/150505+15 $08.00/O
IQ 1992Academic Press Limited
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S. C. HUANG
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[6,7] but with emphasis on the rotational effects on the system behavior. The tire is modeled as a ring rotating at a constant speed instead of as a stationary one. The sidewall of the tire and the wheel are modeled as distributed springs and a vertically movable center mass respectively. This inclusion of rotation generates a few interesting features that were not found by Kung et al. First of all, it makes the modal expansion difficult since the so-called traveling modes appear. Secondly, the subsequent system will be shown to become a paremetrically excited one with periodic coefficients of circular frequency equal to that of the rotation. The force transmission to the wheel axis is also investigated. and it will be shown that the force transmission depends on the coupling of rotational speed and damping ratio of the ring. The instability resulting from parametric excitation is considered too, and the results show that the instability, although not occurring in a practical range, is strongly dominated by the sidewall damping. The parameters which affect the stability are then discussed, and stability diagrams are drawn.
2. EQUATIONS OF MOTION The model of the tire-wheel-suspension unit is shown in Figure 1, where the tire is represented by a rotating ring of distributed springs in radial (kZ) and tangential (ko) directions. The wheel is modeled as a rigid center mass M, which is free to move vertically but is restrained horizontally. 0 denotes the tire rotational speed and p. is the internal pressure. z and 8 are ring-fixed co-ordinates in the transverse and circumferential directions respectively. y denotes the vertical displacement of the center mass. uz and ue represent the deflections of the ring in the z-0 co-ordinates. k, and c, are the elastic constant and the damping coefficient of the suspension. Note that k, has dimensions different from those of k, and ks. 2.1. ENERGY EXPRESSIONS The strain energy of the thin ring is expressed as h/2
2n
[&E~ + a’,~~] d6 dz,
U=ab
s -h/2
(2.1)
s0
Figure 1. The model of the tire-wheel-suspension
unit.
DYNAMICS
OF
TIRE-WHEEL-
SUSPENSION
UNITS
507
where oe and se denote the normal stress and strain in the 8 direction. og is the initial stress induced by the centrifugal force and is approximately equal to pa20 2 + (a/h)po [8]. a, b, h and f, denote the mean radius, the width, the thickness and the density of the ring (tire) respectively. The strain-displacement relationship is [9, lo] 138= (l/a)(u;, + uz) + (1 /2a2)[(u;, + U,)2+ (ul- ue)'l
+ (zla2)(uk- G),
(2.2)
where the primes signify differentiations with respect to 0. Note that in equation (2.2) the non-linear terms are considered only in computing the stored energy caused by the initial stress a;, but dropped in the subsequent derivation [9, 111, to assure a linear system. Substituting equation (2.2) into (2.1) one obtains the strain energy of the ring as U=b
+;(p~hR~+po)[(u;,+u,)~+(u:-ZQ)~] de, I
(2.3)
where D and K denote the bending and the membrane stiffnesses of the ring. The kinetic energy of the system after neglecting the rotary inertia of the ring is expressed as
-4 2
T=~abh
2n[(k-
u&)~+
(tie+u,R
+d2)‘] d8+;Mj2,
(2.4)
0
where the superscript dot indicates differentiation with respect to time. The other contributions to the total energy are from the applied distributed forces on the ring (q5, qe), the deflections of the springs (k;, ke, k,) and the dashpot (cS). They are combined and expressed as: v=f
2* {k,[u,-ysin(8+Rt)]2+ke[ue-ycos(8+Rt)]2 s0 -
(q,u, + qeue)} de + $k,y2 + ;c~)‘~.
(2.5)
Note that the net deflections of the distributed springs in the z and 8 directions are u, - y sin (8 + fit) and ue - y cos (8 + at), respectively. 2.2. EQUATIONS OF MOTION With all the energy expressions thus obtained Hamilton’s principle (6 f:: (U+ I/- T) dt =0) yields the equations of motion in terms of linear operators as L,( U, , ue)
+
phii, - 2phR& - k, sin (8 + CIt)y = qz,
AC&, , ue) + phiie + 2phf2ziz - ke cos (8 + Sar)y = qe , L&z,
ue) + Mj; + c$ + [k, + ubn(k, + ke)]y = 0,
(2.6a) (2.6b) (2.6~)
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S. C. HUANG
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where L,, Lo and Ly are the following linear differential operators: L,(&, 240)= (fI/U”)(U”--r.$)
-t (K/a2)(uZ+u~) - (phR2 +&a)($-
2l4) + (k,+pl)/a)u:, (2.7a)
LB(z&, ue) = (D/u”)@:-u;)
- (K/u2)(2.4~-+ z&i)- (phR*+p~/u)(2u:
+ u$) + (ke +po/u)us,
(2.7b) 2n L,,(&)
240) = -ub
[k,
sin (8+0t)u,+k,
cos (@+nt)u,]
de.
(2.7~)
s0 Note that the underlined terms in equations (2.6a, b) and the equation (26c) are extra terms, in respect to those for a center-fixed rotating ring [9]. It should be emphasized here that these terms are time-varying and of harmonic types. Hence, equations (2.6) constitute a parametrically excited system with periodically time-varying coefficients. If one sets R=O, i.e., one neglects the rotation of the ring, equations (2.6) yield the ones derived by Kung et al. [6], in which a time-invariant system was obtained. 3. FORCED RESPONSE OF THE SYSTEM
3.1.
DERIVATION
OF THE
FORCED
SOLUTION
Due to the periodic nature of the displacement functions u, and ue in the space coordinate 8, it is legitimate to represent the functions as ~~(8, t) = f [o.(t) n=O
cos (ne)
+ p.(t)
sin
40,0
cos (ne)
+ fin(f)
sin (&I)],
= f k(f) n=O
(ne)],
(3.la) (3.1 b)
where n is the circumferential wavenumber, and a,, P,, , &j,,and nn are the generalized coordinates. Note that in equations (3.1), the generalized co-ordinates contain the components of both inextensional and extensional modes [9] ; hence equations (3.1) provide a complete solution. Upon substituting equations (3.1) into the equations of motion (2.6) and utilizing the equalities listed in the Appendix, one obtains, for the generalized co-ordinates associated with different n a set of linear second order differential equations, in matrix form, 2” + G,W, + K,x, = Qn,
(3.2)
where the generalized co-ordinates vector x,, the matrices G,, , K, and the generalized force vector Qn are of the following forms : (i) for n # 1,
xZ=b,v Pn, 5”, %I, G,=
(3.3a)
(3.3b, c)
(3.3d)
OF TIRE-WHEEL-SUSPENSION
DYNAMICS
UNITS
509
(ii) for n = 1,
X:=t~l,p1,51,
G,=
f71YYL
0
2R
0
&lph
0
’
(3.4a, b)
L
0
0 c”
0
--a”
E
K,=
ii 0 - ,ysz(O - r4t)
0
ii
-6
0
z
6 0 -vce(t>
Y&40
-sz( 0 -G(t) --e(t)
(3.4~ 4
se(t) ft
Jsxs
Here d= ( l/ph)[n3(D/a4)
+n(K/a2)] + 2n(lR 2+p,,/pha),
(3.5a) (3Sb)
(3Sc)
s=(t) =k, sin (LB),
c=(t) = k, cos (Cl?),
(3.6a, b)
se(t)=ke
co(t) =ke
(at),
(3.6c, d)
y = abrt/M.
(3.7a, b)
E=
sin (at),
y(ke + kZ) + k,/M,
cos
Note that in equations (3.3b) and (3.4b), an equivalent viscous modal damping d,, for the ring is introduced in the same manner as in reference [9]. a, is related to the damping ratio Cmby ;1, = 2phC, w,,, where CD,,is the corresponding natural frequency of the ring. In equations (3.3), one sees that the co-ordinate y is uncoupled from the other four co-ordinates a,, /I,, , &, and n, for n # 1. This means that the movement of the wheel (center mass) interacts with the motion of the tire (ring) only through the n= 1 mode of the ring. The same result was obtained by Kung et al. [6] ; however, they did not consider the rotation of the tire and hence obtained a time-invariant system instead of a timevarying one. 3.2.
INEXTENSIONAL
ASSUMPTION
Although equation (3.2) provides a way of solving for the complete solution, the inextensional vibration of the ring is the one generally of practical significance and is predominant in the forced response. The inextensional assumption u3= -uh is hence made and, subsequently, one obtains the following relationships : (3.8a, b)
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S. C. HUANG
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Thus
(3.9a)
u,(O, t)= f [a,(t) cos (r10)+/3~(t) sin (no)], n=I
us(e, t) = f
A Pn(f) cos (ne) - A o,(t) sin (ne) n
n=l [ n
(3.9b)
. I
Note that the summation is taken from n = 1 instead of n =0, since n = 0 corresponds to the so-called breathing mode [l l] and is an extensional mode. The simplification reduces the system order by two, and it makes an analytic solution feasible. In the following, since equations (3.2) are very different for n # 1 and n = 1, we will discuss them separately. (i) n # 1. The co-ordinate y is uncoupled from the others for n # 1. Upon employing the inextensional assumption, an equation in terms of two generalized co-ordinates, a. and /In, is obtained:
. 0
I
;+
2a
+
Al
G+m
_2R
ph
(c?+n6) CE,
(6+na)
J-”
(ii+&)
ph
n(6F2)
0
S”
J
o
n(6F--8)
0an=
.
Pn
(3.10)
(Z+n6)
Note that equation (3.10) is valid only for n 22 modes. To clarify the expression, new symbols are introduced and equation (3.10) is rewritten as
(3.11)
where
X= il,/ph,
g=n(6Z-ri2)/(Z+nQ,
d=2n(6-tna)&i+&j,
ii1 =n(Ql6-ZQdl(E+nh
jiz=n(Q26--bQ3)/(d+n6).
Equation (3.11) describes the flexural behavior (n22) can be solved for analytically.
(3.12a-c) (3.12d, e)
of the tire (ring), and its solutions
DYNAMICS OF TIRE-WHEEL-SUSPENSION
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UNITS
(ii) n = 1. Substitute equation (3.8) with n = 1 into equation (3.2), and after arrangement, one obtains the following equation :
a, 0;:..
+
20 ph
-2R 2
0 0
0
0
cs z
0 ;:.
i
j;
k+k, _---02
k,+k, --sin 2ph k,+ks --cos
0
2ph +
0
kz+k+z
(k,+ke) sin Rt
-$(k,+k,)cosRt
0 al
SZt
2ph
2ph -f$
Rt
$(k,+ke)+$
PI Y
(3.13)
It has been recognized that the n = 1 mode corresponds to the rigid body mode of the ring, and it can be proven that a1 and /3, in equation (3.13) are indeed the displacements of the ring’s center in the rotating co-ordinates x’- y’. Equation (3.13) represents a periodic time-varying system (or parametric system), and its solution can only be obtained via a numerical method. In addition to the solution, the stability of such a system is also of concern and will be discussed. In the following, an illustrative example is presented, for selected parameters that adequately resemble a tire, which are listed in Table 1. The geometric dimensions of the example ring correspond to TABLE 1
The specifications of the tire-wheel-suspension unit example : materials properties and geometric parameters Tire density bending stiffness membrane stiffness mean radius lateral width radial thickness elastic constants radial tangential internal pressure
p0 = 2.07 x IO’N/m* (30 psi)
Wheel mass
M=8.78
Suspension stiffness
k,=3.6Ox
p=1.59x103Ns2/m4 0=2*175x 10’Nm K= 1,618 x 106N/m a=0.305 m b=O.l52m h=0*0127 m kz= 5.80 x lo6 N/m3 ke = 4.03 x lo6 N/m3
kg 104N/m
512
S. C.
HUANG
AND
B. S. HSU
an actual tire provided by a tire company. The properties of D and K are first selected from Gill’s thesis [ 121, in which he employed the composite theory and experimentally determined them. The values of k, and ke are calculated according to reference [ 131. However, not all of the values properly reflect the tire at hand, and minor modifications are hence made such that the calculated frequencies are close to the actual ones.
4. EXAMPLE
The calculation of the dynamic response of the selected example of a rotating tirewheel-suspension to a constant transverse force is presented here. As stated previously, the flexural modes of tire are uncoupled with the wheel and hence solved for independently. The n= 1 mode of tire and the vertical motion of wheel are coupled together and their relation is expressed in terms of force transmission. 4.1. GENERALIZED FORCE VECTOR Let a constant point load be acting vertically on the bottom of the tire, as shown in Figure 2. The force vector (qz, qe) is qz=(F/ab)6[6-e*(t)]=-(F/ab)6[e-(e~-Rt)],
qs=o,
(4.1)
where F is the amplitude of the applied force and the minus sign indicates the direction opposite to that of the positive transverse displacement. O*(t) is the location of the load at any instant in the rotating co-ordinate 8, and &, denotes the initial point load location at a time equal to zero: e.g., &, = 3a/2 in the example. The generalized force vector Q* is
(4.2)
Note that circumferential loading qe exists in reality, and can be included without further difficulty. It is not considered in this paper, however, for the sake of simplicity of the results. Y
Figure 2. Rotating co-ordinates (x’, y’, 0) and stationary co-ordinates (x, y, y).
DYNAMICS
OF
TIRE--WHEEL-SUSPENSION
RESPONSE OF THE TIRE (n > 2) 4.2. FLEiURAL Upon substituting equation (4.2) into equation equation (3.10) are found to be
a,=-(E/,/X)
cos [n(Dt-&J-n&],
/&=(F/JX)
513
UNITS
(3.10), the steady state solutions of sin [n(f2t-0,)-n&
for na2, (4.3)
where F= dF/2phabr(a’+
n@,
A = (f + &d2 - n*R *)’+ (&inn )*,
(4.4a, b)
n& = tan-’ {%&/(g + 4nf2 - n*52*)} By substituting obtained as
equation
(4.3) into equation
(4.4c) (3.9) the flexural response of the tire is
-
-
u,( VI,t) = - “gz -$ cos [n(0 + &It - 8* - &)I = - “E2 $
cos [n( 1y- 6* - &)I,
~dr,r)=~~~$sin
[n(8+nl-B’-L)l=~~2~sin
My--e*-&)I,
(4.5)
where w= 8 +nt, as shown in Figure 2, is the angle with respect to the non-rotating frame. To extract the effects of the Coriolis acceleration, we consider a hybrid model in which the centrifugal forces but not the Coriolis terms are included [9] : i.e., drop out the Coriolis terms from equations (2.6a, b). The flexural responses are shown in Figure 3, in which (a) shows the response of the tire and (b) shows that of the hybrid model. This figure shows the flexural responses at various values of R with m=O.2. It is seen that the responses are symmetric about the forcing point as R = 0, since the symmetry, as seen from equation (4.4c), relies on the coupling effect of damping and rotational speed. As f2 enters, it introduces a phase lag. In Figure 4 are illustrated the amplitude ratios that are (b)
(a)
0
r.p.m
1600 r.p.m
0 0 3600
Figure 3. Skady
r p.m.
state flexural response of tire for different values of R for (a) the rotating tire with stationary point load and (b) the hybrid model, with <.=0.02.
s.
C. HUANG
AND
B, s.
HsU
(b)
a (r.p.m. ~10~1
Figure 4. Amplitude ratios and phase lags of flexural mod? of tire or (a) n = 2, (b) n = 3 and (c) n = 4 modes with [,=0.2: -, the rotating tire with stationary point load; - - -, the hybrid model.
normalized with the corresponding non-rotating cases and the phase lags for the (a) n= 2, (b) n = 3 and (c) n = 4 modes. The figure explicitly shows the effect of Coriolis acceleration. As expected, the difference in the results for the two models increases with the rotational speed. The differences are significant as 0 reaches 2000 r.p.m., although the total responses (sum of all modes) as shown in Figure 3 did not show so clearly. 4.3. FORCE TRANSMISSION The rigid body motion of the rotating tire and the vertical movement of the wheel are governed by the following equation : &
.. ;:
-20
0
2
0
ph +
20
0j 0
0
cs YG_
kz+ko --f-J2 2ph +
0
k+ke_nz
0
2ph
-f$(kz+ke)sinf2t -Fcos
-$(k,+k~)cosRt
kz+ke
--sin 2ph
Lit
kz+k, --cosRt 2ph $(k,+ke)+$
(L?t - 3w/2) .
(4.6)
DYNAMICS
41.0
,
,
,
OF
I
,
TIRE-WHEEL-SUSPENSION
I
,
I
UNITS
515
9.0
(b) -
Time (s)
Figure 5. Responses of (a) the wheel, (b) the ring’s center in the horizontal direction, and (c) the ring’s center in the vertical direction: -, R=O, c,=O.OS; -- -, R=2OOOr.p.m., c,=O.OS; ---, a=2000 r.p.m., c,= 0.5 with F=200 kg, and cl =0.05.
Since equation (4.6) is a set of second order differential equations with periodically timevarying coefficients, a numerical approach must be used. In Figure 5 are shown the transient responses of the wheel and the ring’s center for different cases of Sz and cS with n = 1 modal damping c, = O-05 and F= 200 kg,, where & is the damping ratio of the suspension and is equal to c,/2Jjc,M’, with M’ the total mass of the tire and the wheel. In this figure, y denotes the motion of the wheel, and x, and y, represent the displacement of the ring’s center with respect to the stationary co-ordinates x-y. For the case of L2= 0, it is seen that x, = 0 in Figure 5(b) : i.e., the ring moves only vertically. This is consistent with intuition. However, as the rotation enters, there is a small deflection in the horizontal direction, although it is not so significant. It is also seen that the displacements of y and yC decrease with the increase of rotation. This phenomenon implies that the rotation of the tire affects the amount of force transmitted to the suspension. The relation of the rotation R and the force transmission K for various damping values is illustrated in Figure 6. Note that the force transmission K is defined here as the ratio of the force transmitted to the suspension (F’) to the applied constant point load (F) : i.e., K =: F’/F. It is seen that K decreases with the increase of rotation. This is not surprising, since the coexistence of rotation and damping of the ring shifts the maximum deflection point away from the load location [lo], and only a portion of the force in this case is transmitted vertically to the suspension. The phase shift (lag) for n= 1 is proportional to rotation and damping, and this explains the results in Figure 6. However, the force transmission K is not affected by the damping ratio of the suspension &, for this case of a constant point load. An increase of cS affects only the settling of the response, as shown in Figure 5. 5. STABILITY
ANALYSIS
As mentioned previously, for a parametric system stability is of primary concern, and we will address this point in this section. It is convenient to represent equation (4.6) as
516
S. C. HUANG AND B. S. HSU
0.0
1
1
1
I
0
I 2
J2trpm
I
I
I
I
4 x10?
Figure 6. Steady state force transmissibility (K = P/F)
for various values of cl
first order equations in state variables. After dropping the forcing terms, the system equa-
tions are rewritten as i(t)=A(t)z(f),
x(to) ‘zo,
where the state vector zT(t)=[dl(t), /j,(,),+(t), a,(t), jI,(t),v(t)], the initial state, and A(t) is a periodic matrix of the form &
-2R
k,+k, -_
0
A=
fl=
2R
Al ph
0
0
0
0 C.l ii
kz+k, --
2pA
fp
2Ph
-$(k,+ke)sinfft
to is the initial time ~0 is kz+k, - ,-----sinOr
0
W
ph
(5.1)
-@-$k;+k,)cosRt
k +k --2Bcos 2ph
RI
$(k:+ke)+$
I
0
0
0
0
0
0
1
0
0
0
0
0
0
I
0
0
0
Note that A(t) =A(t+
T), where T=2a.
(5.2)
Thus one has the eigenvalue problem
[~I--@(&-,+ T, to)]z=O,
(5.3)
where @(t, to) is the fundamental matrix (actually the state transition matrix) of equation (5.1), I is the identity matrix and the p’s are the Floquet mutipliers. The stability criteria can be briefly described as follows [ 141: (i) Asymptotically stable when all eigenvalues of clp(to + T, to) have magnitudes smaller than one, (Pit<
l9
i=l,2,3
,...,
n;
(5.4a)
(ii) marginally stable when all eigenvalues of @D( to + T, to) have magnitudes not exceeding one; at least one eigenvalue has magnitude one and for each of eigenvalues pj with magnitude one, its algebraic multiplicity m equals the geometric multiplicity q, IclilG
l,
i=l,2,3
,...,
n,
at least one ]p,]= 1, with q=m;
(5.4b, c)
DYNAMICS
OF
TIRE-WHEEL-SUSPENSION
517
UNITS
(iii) unstable when at least one eigenvalue of @(to + T, to) has magnitude greater than one, or there is at least one multiple eigenvalue with magnitude one for which q < m, IPil>l
of
with q
I/Jjl=17
(5.W
The discrete transition matrix @(to+ T, to) is obtained by numerically following first order differential equation over one period: 60, to) = A(Y’(t,
integrating
cp( to, to) = I.
to),
the (5.5)
Among all the parameters in A(t), the suspension damping is found to cause hardly any stability problem. It is justified from numerical integration that among the parameters the rotational speed and the damping ratio of the n= 1 mode affect the stability most significantly. The damping of the n = 1 mode is presumably provided by the sidewall of the tire. The numerical process was hence carried out by varying one parameter once at a time.
;
(b) (0) o-5 7.. .. . .... . .... . .... . .... . .... . .... . . , , 1 1 I , .... , .... , .... , .... , .... ................................................... ::: , .................................................................. ........................... ...................................... ............................................................. _ x: _ _................................................................... ........................... ....................................... ::: ............................ ........................................ ............................. :::: : 0.4 s:;::::::::::::::::::::::::::::::::::::: .. _ _ _ ............................. ............................. ........................................... .............................. ................................................ :::: .i ...................................................................................... ....................................................................................... _ _. ............................................ _ _. ................................. ..................................................................................... ..................................................................................... .................................................................................... ..................................... 0.3;:. ........................................................ ......................................................................................... m _ ................................. .::::::::::::::::::::::::::. .................................................................................. : ::: : ........................... .......................... _................................................... ............................................. _ _ ........................................................ ................................................. .::::::::::::::::::::::::: ................................................................................. ....... 0.2 ,::::: ~:IiiIi. iiiiiiiiiiiiiiiiiiiiiiji: ...................................................................................... ................................................. ................................................. ................................................................................ . ................................................................................ .............................. _.. _................................ I:. ......................................................................................... ................................................................................. ......................................................... ........... .::::::::::::::::::::::::. 0.1 7............................................................................................. _ _...::::::::::::::::::::::::::::::::: ................................. .:.... ............................................... ................................................. ................................................. ::::::::::::: ::::::: ::::: :::::: :::: ::::: ... ................................................. _. ............................................ _ _. ............................................ .................................................................................................. ................................................. ........................................... ..~~~~ ............................................ ........................................ 0.0 ::::,::::,::::,::::,::::,::t:,::::,::::,::::, 0.0
1.2
2.4
3.6
4.8
6.0
0
12
24
36
48
60
a (r.p.m.x10’) Figure 7. Stability diagram of the tire-wheel-suspension for unstable regions).
unit (dotted area for stable regions and blank area
The results are shown on the damping-rotation plot in Figures 7(a) and 7(b). In the figures, the dotted area represents the asymptotically stable region and the white area corresponds to the unstable region. In Figure 7(a), it is seen that instability occurs only at a speed far above the practical rotational speed. This means that there should be no concerns about the instability problem at the present stage. The inclusion of rotation, however, indicates a possibility of instability that may need to be considered for vehicles of the future. In Figure 7(b) the stability diagram at very high speeds, for the current model, is exaggerated to indicate the stability regions.
6. CONCLUSIONS The authors have employed the rotating-ring model for the tire and developed an approach to the study of suspension dynamics. It was discovered that the inclusion of rotation resulted in a parametrically excited problem. The wheel was found to interact with the tire through the n = 1 mode of the tire. For n 3 2, the flexural response of the tire is entirely independent of the wheel. A hybrid model was also considered to extract the effects of Coriolis acceleration. The results showed that at higher rotation speeds the Coriolis acceleration imposed significant changes on the flexural response of the tire. The rotation of the tire was also found to affect the force transmission. The coupling of rotation and damping induced a phase lag in the response. The force transmitted to the suspension in the vertical direction was hence reduced by an increase of rotational speed
518
s. c. HUANG
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and of sidewall damping. As to the rigid body mode of the ring, it was found to couple strongly with the motion of the wheel. The resulting time-varying system of periodic coefficients was solved by numerical integration. The results showed that rotation would push the tire horizontally even if the applied load was only in the vertical direction. This effect was not found by Kung et al. [7]. As rotation was considered, the parametric system possessed a possibility for instability that is not present without the inclusion of rotation. The results showed that although in the practical operation range no stability problem arises, there exists a possibility of instability if the rotation speed were to be drastically increased. The rotation and damping ratio of the n = 1 mode, that is presumably provided by the sidewall of the tire, were found to dominate the stability of the unit. The research described in this paper showed that, for the present application (a below 1500 r.p.m.), the inclusion of rotation makes no significant change in the total response. The approach, however, may be of value in prospective studies of tires for which the rotation ought to be taken into account: e.g., racing cars and perhaps vehicles of the future. Moreover, it has been shown that the most crucial stability parameter is sidewall damping, and the results presented may be useful to tire engineers.
ACKNOWLEDGMENT The authors are grateful to the National Science Council (NSC) of The Republic of China for the support of this research under the Grant No. NSC-78-0401-EOl l-07.
REFERENCES 1. R. N. DODGE 1965 Society ofAutomotiue Engineers Paper No. 650491. The dynamic stiffness of a pneumatic tire model. 2. J. T. TIELKING 1965 Society of Automotive Engineers Paper No. 650492. Plane vibration characteristics of a pneumatic tire model. 3. S. K. CLARK 1965 Society of Automotive Engineers Paper No. 650493. The rolling tire under load. 4. W. SOEDEL 1975 Journal of Sound and Vibration 41, 233-246. On the dynamic response of rolling tires according to thin shell approximation. 5. G. R. POTTS, C. A. BELL, L. T. CHAREK and T. K. ROY 1977 Tire Science and Technology 5, 202-225. Tire vibrations. 6. L. E. KUNG, W. SOEDEL and T. Y. YANG 1986 Journal of Sound and Vibration 106, 181-194.
7. 8. 9.
10. 11. 12. 13. 14.
Free vibration of a pneumatic tire-wheel assembly unit using a ring on an elastic foundation and a finite element model. L. E. KUNG, W. SOEDEL and T. Y. YANG 1986 Journal of Sound and Vibration 107, 195-213. On the dynamic response at the wheel axle of a pneumatic tire. S. C. HUANG 1987 Ph.D. Dissertation, Purdue University. Effects of Coriolis acceleration on the vibrations of spinning structures. S. C. HUANG and W. SOEDEL 1987 Journal of Sound and Vibration 115, 253-274. Effects of Coriolis acceleration on the free and forced in-plane vibrations of rotating rings on elastic foundation. S. C. HUANG and W. SOEDEL 1987 Journal of Sound and Vibration 118, 253-270. Response of rotating rings to harmonic and periodic loading and comparison with the inverted problem. W. SOEDEL 1981 Vibrations of Shells and Plates. New York: Marcel Dekker. D. E. GILL 1975 Master’s thesis, Purdue University. The determination of the mechanical properties of automobile tires. H. B. PACUKA 1971 National Bureau of Standards Monograph 122. Tire in-plane dynamics. Mechanics of Pneumatic Tires (S. K. Clark, editor). P. C. MILLER and W. 0. SCHIEHLEN 1985 Linear Vibrations. Dordrecht, The Netherlands: Martinus Nijhoff.
DYNAMICS
OF
TIRE-WHEEL-SUSPENSION
519
UNITS
APPENDIX 2n
ab
kZsin (8 + f2r)u, d0 =
-ablrk, sin (Rt)a,,-ahk,cos
(LB)&,,
(A.1)
s0 .ab
-abnke
kOsin (6 + G!t)uo de = o
sin (f2t)~n-abake cos (Qt)q,,
3 G-4.2)
2n -
k,sin(tI+SZt)ycos(nO)dO=
(A-3)
k,sin (B+Rt)y
(A.4)
s 0
2n -
sin (no) dO=
s0
s 2r
-
ke COS(8+f&)yCOS(ne)
-nke cos (Qt)y, o
do= i
0
3
n= n#l
1 i ’
(A.9
2n -
ke cos s 0
(0 + l2t)y
sin (no) de =
(A.61
AN APPROACH TO THE DYNAMIC ANALYSIS OF ROTATING TIRE-WHEEL-SUSPENSION UNITS S. C. HUANG
AND
B. S.
Hsu
Department of Mechanical Engineering, National Taiwan Institute of Technology, Taipei, Taiwan, 10722, Republic of China (Received 28 February 1990, and in final form 17 May 1991) An approach to the dynamic analysis of a rotating tire-wheel-suspension unit is developed. A rotating ring on an elastic foundation and a vertically movable center mass model the tire and the wheel respectively. The interaction of the wheel motion with the rotating tire results in a parametrically excited system with periodic coefficients. The flexural response of
the rotating tire is shown, however, to he independent of the movement of the wheel and hence can be solved for analytically. The system response due to a concentrated force provided by ground contacting is solved for as an example to illustrate the solution procedure. Force transmission as a function of rotation and damping ratio is illustrated. The results show that the damping constant of the suspension hardly affects the force transmission and stability, for a constant point load. The damping ratio of the n= 1 mode of the tire, that is presumably provided by the tire sidewall, is found dramatically to dominate the force transmission to the axle and the stability criteria.
1. INTRODUCTION
The vibration of tires has been of great interest to automotive engineers, since knowledge of the natural frequencies and modes of tires is needed to interpret or predict the riding comfort and noise transmission. Serious studies on tire vibration have been accomplished during the past two decades. The first analytic approach to the modeling of a pneumatic tire was to use a ring on an elastic foundation [l-3]. Soedel [4] presented an analytic solution for the dynamic response of a rolling inflated tire using a linear thin shell theory and three-dimensional dynamic Green functions. Potts et al. [5] used a rotating, thin elastic ring under tension to model the tire belt, and treated the tire sidewall as an elastic foundation. They also checked the analytic results experimentally and a rather good agreement was found. In 1986, Kung et al. [6] investigated the dynamic response of a pneumatic tire-wheel unit. In this paper, they modeled the tire as a stationary ring on an elastic foundation and the wheel as a vertically movable center mass. They discovered that the motion of wheel was simply coupled with the rigid body mode of the tire. They also employed a finite element model for the same problem and concluded that the ring model was acceptable for the dynamic analysis of a pneumatic tire. In a later paper [7], they extended the analysis to the steady state response and force transmission at the wheel axle of a pneumatic tiresuspension system due to a harmonic force or displacement input, in which the receptance method was utilized. In the papers of Kung et al. [6,7] account was taken of the centrifugal effects due to rotation but not the Coriolis accelerations. Yet it has been shown [8-lo] that the effects of rotation, especially the Coriolis terms, significantly change the dynamic characteristics of the ring. The current research is basically an extension of the analysis of Kung et al. 505 0022-460X/92/150505+15 $08.00/O
IQ 1992Academic Press Limited
506
S. C. HUANG
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[6,7] but with emphasis on the rotational effects on the system behavior. The tire is modeled as a ring rotating at a constant speed instead of as a stationary one. The sidewall of the tire and the wheel are modeled as distributed springs and a vertically movable center mass respectively. This inclusion of rotation generates a few interesting features that were not found by Kung et al. First of all, it makes the modal expansion difficult since the so-called traveling modes appear. Secondly, the subsequent system will be shown to become a paremetrically excited one with periodic coefficients of circular frequency equal to that of the rotation. The force transmission to the wheel axis is also investigated. and it will be shown that the force transmission depends on the coupling of rotational speed and damping ratio of the ring. The instability resulting from parametric excitation is considered too, and the results show that the instability, although not occurring in a practical range, is strongly dominated by the sidewall damping. The parameters which affect the stability are then discussed, and stability diagrams are drawn.
2. EQUATIONS OF MOTION The model of the tire-wheel-suspension unit is shown in Figure 1, where the tire is represented by a rotating ring of distributed springs in radial (kZ) and tangential (ko) directions. The wheel is modeled as a rigid center mass M, which is free to move vertically but is restrained horizontally. 0 denotes the tire rotational speed and p. is the internal pressure. z and 8 are ring-fixed co-ordinates in the transverse and circumferential directions respectively. y denotes the vertical displacement of the center mass. uz and ue represent the deflections of the ring in the z-0 co-ordinates. k, and c, are the elastic constant and the damping coefficient of the suspension. Note that k, has dimensions different from those of k, and ks. 2.1. ENERGY EXPRESSIONS The strain energy of the thin ring is expressed as h/2
2n
[&E~ + a’,~~] d6 dz,
U=ab
s -h/2
(2.1)
s0
Figure 1. The model of the tire-wheel-suspension
unit.
DYNAMICS
OF
TIRE-WHEEL-
SUSPENSION
UNITS
507
where oe and se denote the normal stress and strain in the 8 direction. og is the initial stress induced by the centrifugal force and is approximately equal to pa20 2 + (a/h)po [8]. a, b, h and f, denote the mean radius, the width, the thickness and the density of the ring (tire) respectively. The strain-displacement relationship is [9, lo] 138= (l/a)(u;, + uz) + (1 /2a2)[(u;, + U,)2+ (ul- ue)'l
+ (zla2)(uk- G),
(2.2)
where the primes signify differentiations with respect to 0. Note that in equation (2.2) the non-linear terms are considered only in computing the stored energy caused by the initial stress a;, but dropped in the subsequent derivation [9, 111, to assure a linear system. Substituting equation (2.2) into (2.1) one obtains the strain energy of the ring as U=b
+;(p~hR~+po)[(u;,+u,)~+(u:-ZQ)~] de, I
(2.3)
where D and K denote the bending and the membrane stiffnesses of the ring. The kinetic energy of the system after neglecting the rotary inertia of the ring is expressed as
-4 2
T=~abh
2n[(k-
u&)~+
(tie+u,R
+d2)‘] d8+;Mj2,
(2.4)
0
where the superscript dot indicates differentiation with respect to time. The other contributions to the total energy are from the applied distributed forces on the ring (q5, qe), the deflections of the springs (k;, ke, k,) and the dashpot (cS). They are combined and expressed as: v=f
2* {k,[u,-ysin(8+Rt)]2+ke[ue-ycos(8+Rt)]2 s0 -
(q,u, + qeue)} de + $k,y2 + ;c~)‘~.
(2.5)
Note that the net deflections of the distributed springs in the z and 8 directions are u, - y sin (8 + fit) and ue - y cos (8 + at), respectively. 2.2. EQUATIONS OF MOTION With all the energy expressions thus obtained Hamilton’s principle (6 f:: (U+ I/- T) dt =0) yields the equations of motion in terms of linear operators as L,( U, , ue)
+
phii, - 2phR& - k, sin (8 + CIt)y = qz,
AC&, , ue) + phiie + 2phf2ziz - ke cos (8 + Sar)y = qe , L&z,
ue) + Mj; + c$ + [k, + ubn(k, + ke)]y = 0,
(2.6a) (2.6b) (2.6~)
508
S. C. HUANG
AND
B. S. HSU
where L,, Lo and Ly are the following linear differential operators: L,(&, 240)= (fI/U”)(U”--r.$)
-t (K/a2)(uZ+u~) - (phR2 +&a)($-
2l4) + (k,+pl)/a)u:, (2.7a)
LB(z&, ue) = (D/u”)@:-u;)
- (K/u2)(2.4~-+ z&i)- (phR*+p~/u)(2u:
+ u$) + (ke +po/u)us,
(2.7b) 2n L,,(&)
240) = -ub
[k,
sin (8+0t)u,+k,
cos (@+nt)u,]
de.
(2.7~)
s0 Note that the underlined terms in equations (2.6a, b) and the equation (26c) are extra terms, in respect to those for a center-fixed rotating ring [9]. It should be emphasized here that these terms are time-varying and of harmonic types. Hence, equations (2.6) constitute a parametrically excited system with periodically time-varying coefficients. If one sets R=O, i.e., one neglects the rotation of the ring, equations (2.6) yield the ones derived by Kung et al. [6], in which a time-invariant system was obtained. 3. FORCED RESPONSE OF THE SYSTEM
3.1.
DERIVATION
OF THE
FORCED
SOLUTION
Due to the periodic nature of the displacement functions u, and ue in the space coordinate 8, it is legitimate to represent the functions as ~~(8, t) = f [o.(t) n=O
cos (ne)
+ p.(t)
sin
40,0
cos (ne)
+ fin(f)
sin (&I)],
= f k(f) n=O
(ne)],
(3.la) (3.1 b)
where n is the circumferential wavenumber, and a,, P,, , &j,,and nn are the generalized coordinates. Note that in equations (3.1), the generalized co-ordinates contain the components of both inextensional and extensional modes [9] ; hence equations (3.1) provide a complete solution. Upon substituting equations (3.1) into the equations of motion (2.6) and utilizing the equalities listed in the Appendix, one obtains, for the generalized co-ordinates associated with different n a set of linear second order differential equations, in matrix form, 2” + G,W, + K,x, = Qn,
(3.2)
where the generalized co-ordinates vector x,, the matrices G,, , K, and the generalized force vector Qn are of the following forms : (i) for n # 1,
xZ=b,v Pn, 5”, %I, G,=
(3.3a)
(3.3b, c)
(3.3d)
OF TIRE-WHEEL-SUSPENSION
DYNAMICS
UNITS
509
(ii) for n = 1,
X:=t~l,p1,51,
G,=
f71YYL
0
2R
0
&lph
0
’
(3.4a, b)
L
0
0 c”
0
--a”
E
K,=
ii 0 - ,ysz(O - r4t)
0
ii
-6
0
z
6 0 -vce(t>
Y&40
-sz( 0 -G(t) --e(t)
(3.4~ 4
se(t) ft
Jsxs
Here d= ( l/ph)[n3(D/a4)
+n(K/a2)] + 2n(lR 2+p,,/pha),
(3.5a) (3Sb)
(3Sc)
s=(t) =k, sin (LB),
c=(t) = k, cos (Cl?),
(3.6a, b)
se(t)=ke
co(t) =ke
(at),
(3.6c, d)
y = abrt/M.
(3.7a, b)
E=
sin (at),
y(ke + kZ) + k,/M,
cos
Note that in equations (3.3b) and (3.4b), an equivalent viscous modal damping d,, for the ring is introduced in the same manner as in reference [9]. a, is related to the damping ratio Cmby ;1, = 2phC, w,,, where CD,,is the corresponding natural frequency of the ring. In equations (3.3), one sees that the co-ordinate y is uncoupled from the other four co-ordinates a,, /I,, , &, and n, for n # 1. This means that the movement of the wheel (center mass) interacts with the motion of the tire (ring) only through the n= 1 mode of the ring. The same result was obtained by Kung et al. [6] ; however, they did not consider the rotation of the tire and hence obtained a time-invariant system instead of a timevarying one. 3.2.
INEXTENSIONAL
ASSUMPTION
Although equation (3.2) provides a way of solving for the complete solution, the inextensional vibration of the ring is the one generally of practical significance and is predominant in the forced response. The inextensional assumption u3= -uh is hence made and, subsequently, one obtains the following relationships : (3.8a, b)
510
S. C. HUANG
AND
B. S. HSU
Thus
(3.9a)
u,(O, t)= f [a,(t) cos (r10)+/3~(t) sin (no)], n=I
us(e, t) = f
A Pn(f) cos (ne) - A o,(t) sin (ne) n
n=l [ n
(3.9b)
. I
Note that the summation is taken from n = 1 instead of n =0, since n = 0 corresponds to the so-called breathing mode [l l] and is an extensional mode. The simplification reduces the system order by two, and it makes an analytic solution feasible. In the following, since equations (3.2) are very different for n # 1 and n = 1, we will discuss them separately. (i) n # 1. The co-ordinate y is uncoupled from the others for n # 1. Upon employing the inextensional assumption, an equation in terms of two generalized co-ordinates, a. and /In, is obtained:
. 0
I
;+
2a
+
Al
G+m
_2R
ph
(c?+n6) CE,
(6+na)
J-”
(ii+&)
ph
n(6F2)
0
S”
J
o
n(6F--8)
0an=
.
Pn
(3.10)
(Z+n6)
Note that equation (3.10) is valid only for n 22 modes. To clarify the expression, new symbols are introduced and equation (3.10) is rewritten as
(3.11)
where
X= il,/ph,
g=n(6Z-ri2)/(Z+nQ,
d=2n(6-tna)&i+&j,
ii1 =n(Ql6-ZQdl(E+nh
jiz=n(Q26--bQ3)/(d+n6).
Equation (3.11) describes the flexural behavior (n22) can be solved for analytically.
(3.12a-c) (3.12d, e)
of the tire (ring), and its solutions
DYNAMICS OF TIRE-WHEEL-SUSPENSION
511
UNITS
(ii) n = 1. Substitute equation (3.8) with n = 1 into equation (3.2), and after arrangement, one obtains the following equation :
a, 0;:..
+
20 ph
-2R 2
0 0
0
0
cs z
0 ;:.
i
j;
k+k, _---02
k,+k, --sin 2ph k,+ks --cos
0
2ph +
0
kz+k+z
(k,+ke) sin Rt
-$(k,+k,)cosRt
0 al
SZt
2ph
2ph -f$
Rt
$(k,+ke)+$
PI Y
(3.13)
It has been recognized that the n = 1 mode corresponds to the rigid body mode of the ring, and it can be proven that a1 and /3, in equation (3.13) are indeed the displacements of the ring’s center in the rotating co-ordinates x’- y’. Equation (3.13) represents a periodic time-varying system (or parametric system), and its solution can only be obtained via a numerical method. In addition to the solution, the stability of such a system is also of concern and will be discussed. In the following, an illustrative example is presented, for selected parameters that adequately resemble a tire, which are listed in Table 1. The geometric dimensions of the example ring correspond to TABLE 1
The specifications of the tire-wheel-suspension unit example : materials properties and geometric parameters Tire density bending stiffness membrane stiffness mean radius lateral width radial thickness elastic constants radial tangential internal pressure
p0 = 2.07 x IO’N/m* (30 psi)
Wheel mass
M=8.78
Suspension stiffness
k,=3.6Ox
p=1.59x103Ns2/m4 0=2*175x 10’Nm K= 1,618 x 106N/m a=0.305 m b=O.l52m h=0*0127 m kz= 5.80 x lo6 N/m3 ke = 4.03 x lo6 N/m3
kg 104N/m
512
S. C.
HUANG
AND
B. S. HSU
an actual tire provided by a tire company. The properties of D and K are first selected from Gill’s thesis [ 121, in which he employed the composite theory and experimentally determined them. The values of k, and ke are calculated according to reference [ 131. However, not all of the values properly reflect the tire at hand, and minor modifications are hence made such that the calculated frequencies are close to the actual ones.
4. EXAMPLE
The calculation of the dynamic response of the selected example of a rotating tirewheel-suspension to a constant transverse force is presented here. As stated previously, the flexural modes of tire are uncoupled with the wheel and hence solved for independently. The n= 1 mode of tire and the vertical motion of wheel are coupled together and their relation is expressed in terms of force transmission. 4.1. GENERALIZED FORCE VECTOR Let a constant point load be acting vertically on the bottom of the tire, as shown in Figure 2. The force vector (qz, qe) is qz=(F/ab)6[6-e*(t)]=-(F/ab)6[e-(e~-Rt)],
qs=o,
(4.1)
where F is the amplitude of the applied force and the minus sign indicates the direction opposite to that of the positive transverse displacement. O*(t) is the location of the load at any instant in the rotating co-ordinate 8, and &, denotes the initial point load location at a time equal to zero: e.g., &, = 3a/2 in the example. The generalized force vector Q* is
(4.2)
Note that circumferential loading qe exists in reality, and can be included without further difficulty. It is not considered in this paper, however, for the sake of simplicity of the results. Y
Figure 2. Rotating co-ordinates (x’, y’, 0) and stationary co-ordinates (x, y, y).
DYNAMICS
OF
TIRE--WHEEL-SUSPENSION
RESPONSE OF THE TIRE (n > 2) 4.2. FLEiURAL Upon substituting equation (4.2) into equation equation (3.10) are found to be
a,=-(E/,/X)
cos [n(Dt-&J-n&],
/&=(F/JX)
513
UNITS
(3.10), the steady state solutions of sin [n(f2t-0,)-n&
for na2, (4.3)
where F= dF/2phabr(a’+
n@,
A = (f + &d2 - n*R *)’+ (&inn )*,
(4.4a, b)
n& = tan-’ {%&/(g + 4nf2 - n*52*)} By substituting obtained as
equation
(4.3) into equation
(4.4c) (3.9) the flexural response of the tire is
-
-
u,( VI,t) = - “gz -$ cos [n(0 + &It - 8* - &)I = - “E2 $
cos [n( 1y- 6* - &)I,
~dr,r)=~~~$sin
[n(8+nl-B’-L)l=~~2~sin
My--e*-&)I,
(4.5)
where w= 8 +nt, as shown in Figure 2, is the angle with respect to the non-rotating frame. To extract the effects of the Coriolis acceleration, we consider a hybrid model in which the centrifugal forces but not the Coriolis terms are included [9] : i.e., drop out the Coriolis terms from equations (2.6a, b). The flexural responses are shown in Figure 3, in which (a) shows the response of the tire and (b) shows that of the hybrid model. This figure shows the flexural responses at various values of R with m=O.2. It is seen that the responses are symmetric about the forcing point as R = 0, since the symmetry, as seen from equation (4.4c), relies on the coupling effect of damping and rotational speed. As f2 enters, it introduces a phase lag. In Figure 4 are illustrated the amplitude ratios that are (b)
(a)
0
r.p.m
1600 r.p.m
0 0 3600
Figure 3. Skady
r p.m.
state flexural response of tire for different values of R for (a) the rotating tire with stationary point load and (b) the hybrid model, with <.=0.02.
s.
C. HUANG
AND
B, s.
HsU
(b)
a (r.p.m. ~10~1
Figure 4. Amplitude ratios and phase lags of flexural mod? of tire or (a) n = 2, (b) n = 3 and (c) n = 4 modes with [,=0.2: -, the rotating tire with stationary point load; - - -, the hybrid model.
normalized with the corresponding non-rotating cases and the phase lags for the (a) n= 2, (b) n = 3 and (c) n = 4 modes. The figure explicitly shows the effect of Coriolis acceleration. As expected, the difference in the results for the two models increases with the rotational speed. The differences are significant as 0 reaches 2000 r.p.m., although the total responses (sum of all modes) as shown in Figure 3 did not show so clearly. 4.3. FORCE TRANSMISSION The rigid body motion of the rotating tire and the vertical movement of the wheel are governed by the following equation : &
.. ;:
-20
0
2
0
ph +
20
0j 0
0
cs YG_
kz+ko --f-J2 2ph +
0
k+ke_nz
0
2ph
-f$(kz+ke)sinf2t -Fcos
-$(k,+k~)cosRt
kz+ke
--sin 2ph
Lit
kz+k, --cosRt 2ph $(k,+ke)+$
(L?t - 3w/2) .
(4.6)
DYNAMICS
41.0
,
,
,
OF
I
,
TIRE-WHEEL-SUSPENSION
I
,
I
UNITS
515
9.0
(b) -
Time (s)
Figure 5. Responses of (a) the wheel, (b) the ring’s center in the horizontal direction, and (c) the ring’s center in the vertical direction: -, R=O, c,=O.OS; -- -, R=2OOOr.p.m., c,=O.OS; ---, a=2000 r.p.m., c,= 0.5 with F=200 kg, and cl =0.05.
Since equation (4.6) is a set of second order differential equations with periodically timevarying coefficients, a numerical approach must be used. In Figure 5 are shown the transient responses of the wheel and the ring’s center for different cases of Sz and cS with n = 1 modal damping c, = O-05 and F= 200 kg,, where & is the damping ratio of the suspension and is equal to c,/2Jjc,M’, with M’ the total mass of the tire and the wheel. In this figure, y denotes the motion of the wheel, and x, and y, represent the displacement of the ring’s center with respect to the stationary co-ordinates x-y. For the case of L2= 0, it is seen that x, = 0 in Figure 5(b) : i.e., the ring moves only vertically. This is consistent with intuition. However, as the rotation enters, there is a small deflection in the horizontal direction, although it is not so significant. It is also seen that the displacements of y and yC decrease with the increase of rotation. This phenomenon implies that the rotation of the tire affects the amount of force transmitted to the suspension. The relation of the rotation R and the force transmission K for various damping values is illustrated in Figure 6. Note that the force transmission K is defined here as the ratio of the force transmitted to the suspension (F’) to the applied constant point load (F) : i.e., K =: F’/F. It is seen that K decreases with the increase of rotation. This is not surprising, since the coexistence of rotation and damping of the ring shifts the maximum deflection point away from the load location [lo], and only a portion of the force in this case is transmitted vertically to the suspension. The phase shift (lag) for n= 1 is proportional to rotation and damping, and this explains the results in Figure 6. However, the force transmission K is not affected by the damping ratio of the suspension &, for this case of a constant point load. An increase of cS affects only the settling of the response, as shown in Figure 5. 5. STABILITY
ANALYSIS
As mentioned previously, for a parametric system stability is of primary concern, and we will address this point in this section. It is convenient to represent equation (4.6) as
516
S. C. HUANG AND B. S. HSU
0.0
1
1
1
I
0
I 2
J2trpm
I
I
I
I
4 x10?
Figure 6. Steady state force transmissibility (K = P/F)
for various values of cl
first order equations in state variables. After dropping the forcing terms, the system equa-
tions are rewritten as i(t)=A(t)z(f),
x(to) ‘zo,
where the state vector zT(t)=[dl(t), /j,(,),+(t), a,(t), jI,(t),v(t)], the initial state, and A(t) is a periodic matrix of the form &
-2R
k,+k, -_
0
A=
fl=
2R
Al ph
0
0
0
0 C.l ii
kz+k, --
2pA
fp
2Ph
-$(k,+ke)sinfft
to is the initial time ~0 is kz+k, - ,-----sinOr
0
W
ph
(5.1)
-@-$k;+k,)cosRt
k +k --2Bcos 2ph
RI
$(k:+ke)+$
I
0
0
0
0
0
0
1
0
0
0
0
0
0
I
0
0
0
Note that A(t) =A(t+
T), where T=2a.
(5.2)
Thus one has the eigenvalue problem
[~I--@(&-,+ T, to)]z=O,
(5.3)
where @(t, to) is the fundamental matrix (actually the state transition matrix) of equation (5.1), I is the identity matrix and the p’s are the Floquet mutipliers. The stability criteria can be briefly described as follows [ 141: (i) Asymptotically stable when all eigenvalues of clp(to + T, to) have magnitudes smaller than one, (Pit<
l9
i=l,2,3
,...,
n;
(5.4a)
(ii) marginally stable when all eigenvalues of @D( to + T, to) have magnitudes not exceeding one; at least one eigenvalue has magnitude one and for each of eigenvalues pj with magnitude one, its algebraic multiplicity m equals the geometric multiplicity q, IclilG
l,
i=l,2,3
,...,
n,
at least one ]p,]= 1, with q=m;
(5.4b, c)
DYNAMICS
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517
UNITS
(iii) unstable when at least one eigenvalue of @(to + T, to) has magnitude greater than one, or there is at least one multiple eigenvalue with magnitude one for which q < m, IPil>l
of
with q
I/Jjl=17
(5.W
The discrete transition matrix @(to+ T, to) is obtained by numerically following first order differential equation over one period: 60, to) = A(Y’(t,
integrating
cp( to, to) = I.
to),
the (5.5)
Among all the parameters in A(t), the suspension damping is found to cause hardly any stability problem. It is justified from numerical integration that among the parameters the rotational speed and the damping ratio of the n= 1 mode affect the stability most significantly. The damping of the n = 1 mode is presumably provided by the sidewall of the tire. The numerical process was hence carried out by varying one parameter once at a time.
;
(b) (0) o-5 7.. .. . .... . .... . .... . .... . .... . .... . . , , 1 1 I , .... , .... , .... , .... , .... ................................................... ::: , .................................................................. ........................... ...................................... ............................................................. _ x: _ _................................................................... ........................... ....................................... ::: ............................ ........................................ ............................. :::: : 0.4 s:;::::::::::::::::::::::::::::::::::::: .. _ _ _ ............................. ............................. ........................................... .............................. ................................................ :::: .i ...................................................................................... ....................................................................................... _ _. ............................................ _ _. ................................. ..................................................................................... ..................................................................................... .................................................................................... ..................................... 0.3;:. ........................................................ ......................................................................................... m _ ................................. .::::::::::::::::::::::::::. .................................................................................. : ::: : ........................... .......................... _................................................... ............................................. _ _ ........................................................ ................................................. .::::::::::::::::::::::::: ................................................................................. ....... 0.2 ,::::: ~:IiiIi. iiiiiiiiiiiiiiiiiiiiiiji: ...................................................................................... ................................................. ................................................. ................................................................................ . ................................................................................ .............................. _.. _................................ I:. ......................................................................................... ................................................................................. ......................................................... ........... .::::::::::::::::::::::::. 0.1 7............................................................................................. _ _...::::::::::::::::::::::::::::::::: ................................. .:.... ............................................... ................................................. ................................................. ::::::::::::: ::::::: ::::: :::::: :::: ::::: ... ................................................. _. ............................................ _ _. ............................................ .................................................................................................. ................................................. ........................................... ..~~~~ ............................................ ........................................ 0.0 ::::,::::,::::,::::,::::,::t:,::::,::::,::::, 0.0
1.2
2.4
3.6
4.8
6.0
0
12
24
36
48
60
a (r.p.m.x10’) Figure 7. Stability diagram of the tire-wheel-suspension for unstable regions).
unit (dotted area for stable regions and blank area
The results are shown on the damping-rotation plot in Figures 7(a) and 7(b). In the figures, the dotted area represents the asymptotically stable region and the white area corresponds to the unstable region. In Figure 7(a), it is seen that instability occurs only at a speed far above the practical rotational speed. This means that there should be no concerns about the instability problem at the present stage. The inclusion of rotation, however, indicates a possibility of instability that may need to be considered for vehicles of the future. In Figure 7(b) the stability diagram at very high speeds, for the current model, is exaggerated to indicate the stability regions.
6. CONCLUSIONS The authors have employed the rotating-ring model for the tire and developed an approach to the study of suspension dynamics. It was discovered that the inclusion of rotation resulted in a parametrically excited problem. The wheel was found to interact with the tire through the n = 1 mode of the tire. For n 3 2, the flexural response of the tire is entirely independent of the wheel. A hybrid model was also considered to extract the effects of Coriolis acceleration. The results showed that at higher rotation speeds the Coriolis acceleration imposed significant changes on the flexural response of the tire. The rotation of the tire was also found to affect the force transmission. The coupling of rotation and damping induced a phase lag in the response. The force transmitted to the suspension in the vertical direction was hence reduced by an increase of rotational speed
518
s. c. HUANG
AND
B. s. HSU
and of sidewall damping. As to the rigid body mode of the ring, it was found to couple strongly with the motion of the wheel. The resulting time-varying system of periodic coefficients was solved by numerical integration. The results showed that rotation would push the tire horizontally even if the applied load was only in the vertical direction. This effect was not found by Kung et al. [7]. As rotation was considered, the parametric system possessed a possibility for instability that is not present without the inclusion of rotation. The results showed that although in the practical operation range no stability problem arises, there exists a possibility of instability if the rotation speed were to be drastically increased. The rotation and damping ratio of the n = 1 mode, that is presumably provided by the sidewall of the tire, were found to dominate the stability of the unit. The research described in this paper showed that, for the present application (a below 1500 r.p.m.), the inclusion of rotation makes no significant change in the total response. The approach, however, may be of value in prospective studies of tires for which the rotation ought to be taken into account: e.g., racing cars and perhaps vehicles of the future. Moreover, it has been shown that the most crucial stability parameter is sidewall damping, and the results presented may be useful to tire engineers.
ACKNOWLEDGMENT The authors are grateful to the National Science Council (NSC) of The Republic of China for the support of this research under the Grant No. NSC-78-0401-EOl l-07.
REFERENCES 1. R. N. DODGE 1965 Society ofAutomotiue Engineers Paper No. 650491. The dynamic stiffness of a pneumatic tire model. 2. J. T. TIELKING 1965 Society of Automotive Engineers Paper No. 650492. Plane vibration characteristics of a pneumatic tire model. 3. S. K. CLARK 1965 Society of Automotive Engineers Paper No. 650493. The rolling tire under load. 4. W. SOEDEL 1975 Journal of Sound and Vibration 41, 233-246. On the dynamic response of rolling tires according to thin shell approximation. 5. G. R. POTTS, C. A. BELL, L. T. CHAREK and T. K. ROY 1977 Tire Science and Technology 5, 202-225. Tire vibrations. 6. L. E. KUNG, W. SOEDEL and T. Y. YANG 1986 Journal of Sound and Vibration 106, 181-194.
7. 8. 9.
10. 11. 12. 13. 14.
Free vibration of a pneumatic tire-wheel assembly unit using a ring on an elastic foundation and a finite element model. L. E. KUNG, W. SOEDEL and T. Y. YANG 1986 Journal of Sound and Vibration 107, 195-213. On the dynamic response at the wheel axle of a pneumatic tire. S. C. HUANG 1987 Ph.D. Dissertation, Purdue University. Effects of Coriolis acceleration on the vibrations of spinning structures. S. C. HUANG and W. SOEDEL 1987 Journal of Sound and Vibration 115, 253-274. Effects of Coriolis acceleration on the free and forced in-plane vibrations of rotating rings on elastic foundation. S. C. HUANG and W. SOEDEL 1987 Journal of Sound and Vibration 118, 253-270. Response of rotating rings to harmonic and periodic loading and comparison with the inverted problem. W. SOEDEL 1981 Vibrations of Shells and Plates. New York: Marcel Dekker. D. E. GILL 1975 Master’s thesis, Purdue University. The determination of the mechanical properties of automobile tires. H. B. PACUKA 1971 National Bureau of Standards Monograph 122. Tire in-plane dynamics. Mechanics of Pneumatic Tires (S. K. Clark, editor). P. C. MILLER and W. 0. SCHIEHLEN 1985 Linear Vibrations. Dordrecht, The Netherlands: Martinus Nijhoff.
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519
UNITS
APPENDIX 2n
ab
kZsin (8 + f2r)u, d0 =
-ablrk, sin (Rt)a,,-ahk,cos
(LB)&,,
(A.1)
s0 .ab
-abnke
kOsin (6 + G!t)uo de = o
sin (f2t)~n-abake cos (Qt)q,,
3 G-4.2)
2n -
k,sin(tI+SZt)ycos(nO)dO=
(A-3)
k,sin (B+Rt)y
(A.4)
s 0
2n -
sin (no) dO=
s0
s 2r
-
ke COS(8+f&)yCOS(ne)
-nke cos (Qt)y, o
do= i
0
3
n= n#l
1 i ’
(A.9
2n -
ke cos s 0
(0 + l2t)y
sin (no) de =
(A.61