Parametric excitation of tire-wheel assemblies by a stiffness non-uniformity

Journal of Sound and Vibration (1995) 179(3), 499–512

PARAMETRIC EXCITATION OF TIRE–WHEEL ASSEMBLIES BY A STIFFNESS NON-UNIFORMITY D. S. S Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of Missouri–Rolla, Rolla, Missouri 65401–0249, U.S.A.

C. M. K  W. S School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, U.S.A. (Received 2 October 1992, and in final form 1 October 1993) A simple model of the effect of a concentrated radial stiffness non-uniformity in a passenger car tire is presented. The model treats the tread band of the tire as a rigid ring supported on a viscoelastic foundation. The distributed radial stiffness is lumped into equivalent horizontal (fore-and-aft) and vertical stiffnesses. The concentrated radial stiffness non-uniformity is modeled by treating the tread band as fixed, and the stiffness non-uniformity as rotating around it at the nominal angular velocity of the wheel. Due to loading, the center of mass of the tread band ring model is displaced upward with respect to the wheel spindle and, therefore, the rotating stiffness non-uniformity is alternately compressed and stretched through one complete rotation. This stretching and compressing of the stiffness non-uniformity results in force transmission to the wheel spindle at twice the nominal angular velocity in frequency, and therefore, would excite a given resonance at one-half the nominal angular wheel velocity that a mass unbalance would. The forcing produced by the stiffness non-uniformity is parametric in nature, thus creating the possibility of parametric resonance. The basic theory of the parametric resonance is explained, and a parameter study using derived lumped parameters based on a typical passenger car tire is performed. This study revealed that parametric resonance in passenger car tires, although possible, is unlikely at normal highway speeds as predicted by this model unless the tire is partially deflated.

1. INTRODUCTION

In an earlier paper [1], the authors proposed a simple three-degree-of-freedom discrete model to demonstrate that the dynamic fore-and-aft (horizontal) forces measured on a tire test stand (Figure 1) could originate from a rotating mass unbalance located on the tread band. The authors also explained the influence of eccentricity or radial run-out. Although this model proved that the fore-and-aft phenomena could arise from mass unbalance, it did not eliminate other possible causes. In this paper an attempt is made to address the question of what role stiffness non-uniformities might play in the tire response. Stiffness non-uniformity may be defined as a circumferential variation in the radial stiffness of the tire. The effect of this variation is to cause the resultant forces measured at the axle to vary as the tire rolls with a constant deflection. This is usually the case when a tire is on a test fixture, where the radial displacement of the contact region can be set to the desired amount to simulate vertical loading. The main purpose of this paper is to show how radial stiffness non-uniformities can contribute to the dynamic fore-and-aft 499 0022–460X/95/030499 + 14 $08.00/0

7 1995 Academic Press Limited

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force by exciting an oscillatory variation on top of the nominal rotation rate V0 , and that the primary resultant forcing will occur at a frequency of 2V0 . This forcing appears in terms of both external and parametric excitation in the equations of motion. An additional purpose is to demonstrate that parametric excitation can lead to multiple resonances at certain combinations of system parameters such as angular velocity and magnitude of stiffness non-uniformity. 2. MATHEMATICAL MODEL

Although stiffness non-uniformity in an actual tire is distributed continuously, this model will treat it as a point-wise lumped parameter. This type of non-uniformity is equivalent to the presence of a discrete point-spring of stiffness ks between the tread band and the hub. A further simplification is made by assuming that the tread band consists of a rigid ring which is supported by a radial viscoelastic foundation of stiffness k and viscosity c. In addition, the ring is assumed to remain in rolling contact with the ground or test wheel with no slipping and, thus with a fixed vertical position of the center of mass Y0 . Such simplifications are very helpful in stripping the problem down to the features necessary to describe certain response phenomena of an actual tire. Earlier research has shown [1, 2] that very simple models can yield fundamental information about the physics of complicated systems, provided that the limitations of the simple models are understood. If the surface velocity of the drum is v0 , the average angular speed is V0 = v0 /R. The rolling radius is taken to be approximately equal to the average tread band radius. This oscillating rolling motion superimposed on the average spinning of the tire causes a horizontal shift in the location of the geometric center of the ring, described by the displacement u. The geometrical relationships are shown in Figure 2. A moving reference frame xyz is attached to the tread ring (with origin at the ring mass center O') and rotates with angular velocity v = Vk relative to the stationary reference frame XYZ (with origin at O). Here, k denotes the right-hand system unit vector normal to the plane of the tire. Note that V = V0 − f , because the ring will execute a dynamic rolling oscillation which is superimposed on the average rolling speed V0 . The assumption of rolling contact relates the velocity of the horizontal displacement of the tread band mass center to the variation in the rolling frequency. This relationship is simply u˙ = Rf , which implies that u = Rf. The constant angle u* locates the spring non-uniformity on the ring, measured s counterclockwise from the x-axis of the moving reference frame. The tread ring will resist

Figure 1. A schematic of the non-uniformity test set-up.

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Figure 2. A schematic of the stiffness non-uniformity model.

pure horizontal motion without rotation because of the elastic foundation between ring and hub. This distributed radial stiffness k can be related to a lumped equivalent transverse linear stiffness K0 , by the relation k = K0 /pR [1]. The free body diagram of the tread ring is shown in Figure 3. Summing moments about the contact point C yields (a list of notation is given in the Appendix) IC u¨ = −K0 uR + ks [Y0 cos (V0 t + u* s ) − u sin (V0 t + u* s )] sin (V0 t + u* s )R, R

(1)

where IC is the moment of inertia about the contact point and is given by IC = IO' + MT R 2, MT is the mass of the tread band, and IO' is the moment of inertia about the center of mass O'.

Figure 3. A free body diagram of a tread ring.

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After some simplification, equation (1) becomes u¨ + cˆu˙ + [K − k cos 2 (V0 t + u* s )]u = k Y0 sin 2(V0 t + u* s ).

(2)

where K = (1/MT )(K0 + ks /2)(R/rc )2,

k = (1/2MT )(R/rc )2ks ,

cˆ = (c/MT )(R/rc )2,

and rc is the radius of gyration of the ring about the contact point C. Summing the forces in the horizontal and vertical directions yields the fore-and-aft vertical hub bearing forces Fx = (ks Y0 /2) sin 2(V0 t + u* s ) − [K0 + (ks /2) cos 2(V0 t + u* s )]u

(3)

Fy = (ks Y0 /2)[1 + cos 2(V0 t + u* s )] − (ks /2) sin 2(V0 t + u* s )u + K0 Y0 − MT g,

(4)

and

respectively. The last two terms in equation (4) are static terms due to the initial vertical deflection (which is set to simulate static loading), Y0 , and the weight of the tread band, MT g. Equation (2) is in the form of a forced, damped Mathieu-type equation [3]. The most significant information that can be obtained from equation (2) is that both the external forcing term (the right side of equation (2)) and the parametric forcing term (on the left side of (2)) have a frequency of twice the nominal rolling frequency. Note that the magnitude of the external forcing differs from that of the parametric forcing by a factor of Y0 : that is, a reduction in the hub preload reduces the external excitation while leaving the parametric excitation unchanged. The literature [4] regarding the fore-and-aft force generally agrees that the first harmonic component (excitation once per revolution) of the measured force is the most dominant in magnitude. This kind of excitation is consistent with the presence of a mass unbalance [1]. However, the second harmonic force (excitation twice per revolution) is often of appreciable magnitude as well, as measured by Schuring [5], who found that the maximum second harmonic amplitude of the experimentally measured fore-and-aft force was approximately 14% of the maximum first harmonic in magnitude. He also found that the peak second harmonic fore-and-aft force occurred at approximately one-half of the rolling rate at which the peak first harmonic was observed. This observation supports the theory that a radial stiffness non-uniformity may contribute to the overall fore-and-aft force. In fact, Schuring’s experiments seem to indicate that there is usually a fairly localized region (i.e., of small arc length) of the tire which exhibits a greater radial stiffness. As stated earlier, such a stiffness non-uniformity would excite the wheel at a frequency of twice the rolling rate. It follows that this stiffness non-uniformity would excite the same resonance that a mass unbalance would excite, but at one-half of the rolling rate. There seems to be no (to the best knowledge of the authors) experimental correlation in the literature between measured stiffness non-uniformity and the magnitude of the second harmonic fore-and-aft response. Such experimental data would be required to test this simple model of stiffness non-uniformity. The non-existence of appropriate experimental results notwithstanding, there are practical questions to be asked regarding the existence of a stiffness non-uniformity induced parametric excitation. Perhaps the most interesting question is the possibility of parametric instability. Because of the parametric forcing, there is theoretically an infinite number of resonances, instead of only one as is the case for a non-parametrically forced single-degree-of-freedom system [6]. Of course, a continuous model would exhibit an infinite number of parametric resonances for each mode of vibration [3, 7]. Also, these parametric resonances occur over a continuous range of system

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Figure 4. The numerical solution of equation (2) with V0 = 100 rad/s and ks = 2·6% k: ----, u(t); · · · ·, ua (t).

parameters, rather than at the discrete combinations for which resonances occur in non-parametrically excited systems. In the following sections, solutions of the equation of motion in both the time and the frequency domain will be discussed, as well as the resulting fore-and-aft force response. 3. SOLUTION OF EQUATION OF MOTION

The homogeneous form of equation (2) has a long history and has been very thoroughly studied [8]. The homogeneous, undamped form of equation (2) has solutions in terms of the so-called Mathieu’s functions [9, 10]. Due to the fact that the appropriate set of Mathieu’s functions is dependent upon the system parameters K and k , and that they must be represented as an infinite series, the analytical evaluation of the forced system is prohibitively complex [9]. Therefore, a numerical integration procedure due to Bulrich and Sto¨r [11] and implemented by Press et al. [12] has been used here to evaluate equation (2) directly. The values K0 =1·9×106 N/m, Y0 =0·015 m, R=0·272 m and IC =0·9672 kg-m2 were chosen for this numerical simulation [1]. Viscous damping was chosen to be 10% [13] of the critical damping value of 769 N s/m when ks = 50 000 N/m. Since there seems to be little published data regarding the magnitude of ks , two values were chosen for the following numerical simulations which were equal to 2·6% and 26% of the nominal lumped horizontal stiffness, K0 . The actual range of stiffness variation that might be found in a given set of tires would have to be found by experimentation and is probably closer to 2·6% or less. However, the larger stiffness nonuniformity was included in this paper to accentuate the parametric excitation effects. 3.1.   The temporal solution to equation (2) over 25 forcing periods (p/V0 ) at V0 = 100 radians per second (rad/s) (corresponding to roughly 100 km/h vehicle speed), and with a parametric stiffness (stiffness non-uniformity) ks = 0·026K0 , is plotted in Figure 4 along with an analytical solution of the same equation omitting the parametric forcing term (i.e., keeping only the K term in equation (2)). Comparison of the two solutions (the solution of equation (2) is denoted u, and is plotted with a solid line, and the solution of the approximate equation is denoted ua and is plotted with a bolder dotted line) reveals that at this level of non-uniformity, the two solutions are virtually identical. The same two solutions are plotted again in Figure 5, but for ks = 0·26K0 . At this level of non-uniformity,

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Figure 5. The numerical solution of equation (2) with V0 = 100 rad/s and ks = 26% k: ----, u(t); · · · ·, ua (t).

a significant difference between the two solutions can be seen. This result indicates that under stable parametric combinations for this example, the parametric forcing can be neglected provided that the magnitude of the parametric forcing is low enough. However, even though the system is not operating near any resonances, but with large parametric forcing, omitting the parametric excitation term can result in a significant underestimation of maximum amplitude. The solution to equation (2) is plotted in Figure 6 when an unstable combination of parameters (ks = 9·5 × (105) N/m and V0 = 214 rad/s) is present. The result is clearly an exponentially growing solution. This type of instability is different from that which occurs in non-parametrically forced systems, in that the unbounded growth is exponential rather than polynomial in nature. This will be discussed in further detail in section 4. The natural frequency v0 of the undamped non-parametrically forced system with response ua previously referred to is dependent upon the parametric stiffness ks as well as K0 . For example, when ks = 0·026K0 or 50 000 N/m, v0 = 385 rad/s. The damping value chosen is dependent upon the critical damping value given by cˆcr = 2zK .

(5)

The damped natural frequency (vd ), given by vd = 12 z4k − cˆ 2 ,

Figure 6. An unstable solution at V0 = 214 rad/s and ks = 9·5 × 105 N/m.

(6)

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Figure 7. The frequency response of equation (2) for Ks = 2·6% k and 1% damping.

of the non-parametrically forced system with ks = 0 is 380 rad/s for 10% critical viscous damping. When ks is 0·026K0 , vd = 383 rad/s for 10% damping. 3.2.   A simple numerical mapping procedure was used to determine the frequency response of the parametrically forced equation of motion. The method consisted of integrating equation (2) over 100 periods—well past the attenuation of the transient response—and then searching over the second half of the resultant discrete series of n amplitude data points for the maximum value of =u(ti )=. The frequency response of equation (2) where ks = 2·6% K0 and with light damping (1%) is as shown in Figure 7. The largest spike at V0 = 385 rad/s corresponds to the system natural frequency. The next largest resonance at V0 1 192 rad/s corresponds to the first (n = 1) type I parametric resonance as described by Streit et al. [14] and presented in some detail in section 4.1. The smaller spike at V0 1 96 rad/s is also a type I resonance. The largest spike, corresponding to the natural frequency of the system, is termed a type II resonance. The resultant fore-and-aft force for this level of parametric forcing is shown in Figure 8. The system response for ks = 26% K0 is shown in Figure 9. Here the unbounded nature of the response at V0 = 385 rad/s is clearly evident. The response at V0 = 192 rad/s, although large, remains bounded.

Figure 8. The resultant fore-and-aft force for Ks = 2·6% k and 1% damping.

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Figure 9. The frequency response of equation (2) for Ks = 26% K0 and 1% damping.

4. FLOQUET THEORY

The Floquet theory [15] supplies the principal tool from which the stability of a parametrically forced linear system can be determined. The theory elegantly and readily lends itself to computer implementation. The following is a brief and necessarily incomplete (see Yakubovitch and Starzhinskii [16] for a more thorough explanation) introduction to Floquet theory, and the ancillary method of transition point determination. Since the occurrence of parametric instability is solely a function of the parametric forcing, it is sufficient to investigate the homogeneous system. The homogeneous form of equation (2) can be written q˙(t) = A(t)q(t),

(7)

where q=

67

u , u˙

(8)

u and u˙ are the state variables for the system, and A=

$

1 −(K − k sin 2(V0 t + u* s ))

%

0 . −cˆ

(9)

The solution of equation (8) can be written in terms of its principal matrix solution F(t) [15] as q(t) = F(t)q(0),

(10)

where F(0) = I. Since A(t) = A(t + T), it follows that F(t + T) is also a fundamental matrix solution [17]. Furthermore, from Floquet theory [3] it is known that F(t + T) = CF(t),

(11)

where C is a constant, non-singular matrix (usually referred to as the ‘‘monodromy matrix’’ for F(t)), given by C = F(T). From equation (11) it follows that F(t + nT) = CnF(t),

n = 0, 1, 2, . . . .

(12)

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If the monodromy matrix C has linearly independent eigenvectors, it is well known [18] that it can be written as C=V

$ %

(13)

$ %

(14)

l1 0 V−1 , 0 l2

where V is the eigenvector matrix for C, and l1 and l2 are the eigenvalues for C. From equations (10), (11) and (12) we have q(nT) = V

l1n 0 V−1q(0). 0 l2n

If C does not have linearly independent eigenvectors (as can occur when l1 = l2 = l), it can be shown that equation (14) becomes q(nT) = V

$

%

l n nl n − 1

q(0), V−1 0 ln

(15)

where V is the so-called generalized eigenvector matrix for C. From either equation (14) or (15) it is clear that 0 q(nT): 1 a

for

=lmax = Q 1 (asymptotically stable), =lmax = = 1 (bounded free response), =lmax = q 1 (unstable).

(16)

The results described by equations (7)–(16) are due to Floquet [3, 14], and provide an elegant means of determining the stability of solutions to linear systems of equations with periodic coefficients in the form of equation (7). From equation (12), it is seen that stability can be determined from knowledge of the solution of equation (7) over just one period of the excitation. The major difficulty in this determination is the construction of F(t), and the subsequent formation of C. There are at least three main approaches to this problem. (1) The appropriate Mathieu’s functions, the choice of which is dependent upon the values of the system parameters [9] (K and k in the current study), can be used. (2) The actual periodic coefficient (cos 2(V0 t + u* s ) in this study) can be approximated by a series of step functions and the principal matrix solution can be constructed as the product of the principal matrix solutions for each constant step [18–20]. This method has the advantage that it is applicable to systems with general periodic coefficients for which no analytical solution exists, and it may also be used to approximate the complete forced solution to such general equations [22]. (3) Numerical integration may be used to construct F(T) [13]. Here, equation (7) is solved numerically once using q1 (0) = (1, 0)T as initial conditions and a second time using q2 (0) = (0, 1)T as initial conditions. With these initial conditions, the fundamental matrix solution at t = T becomes the monodromy matrix; i.e., F(T) = C.

(17)

The third method above was chosen for determination of the monodromy matrix. 4.1.    The points at which regions of instability come in contact with the V-axis (ks = 0) are termed ‘‘transition points’’ [14, 16]. Knowledge of the location of these points is important, because in a lightly damped system an unbounded response can occur even when the parametric forcing is small, provided that the forcing frequency is close to a transition frequency.

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In the case of a single-degree-of-freedom system such as the one described by equation (2), the location of transition points may be determined by investigating the undamped, homogeneous, non-parametrically forced case. Thus, equation (2) may be written u¨ + v02 u = 0,

(18)

v02 = K .

(19)

where

The solution for equation (18) may be written u(t) = u0 cos v0 t + (u˙0 /v0 ) sin v0 t.

(20)

u˙ (t) = −u0 v0 sin v0 t + u˙0 cos v0 t,

(21)

Thus,

where u0 = u(0) and u˙0 = u˙ (0). Equations (20) and (21) may be combined in matrix form, and written as

6 7

u(t) = u˙ (t)

&

cos v0 t −v0 sin v0 t

sin v0 t v0 cos v0 t

'6 7

u0 = F0 (t)u(t). u˙0

(22)

'

(23)

From equation (17), then, we have C = F0 (T) =

&

cos v0 T

−v0 sin v0 T

sin v0 T v0 . cos v0 T

The eigenvalues of C may be determined from the characteristic equation (l − l1 )(l − l2 ) = l 2 − 2l cos v0 T + 1 = 0.

(24)

From equation (24) it is seen that l1 l2 = 1

(25)

l1 + l1 = −2 cos v0 T.

(26)

and

This implies that either (a) l1 and l2 are real with =l1 = = 1/=l2 = = real, or (b) l1 and l2 are complex with =l1 = = =l2 = = 1. From this we conclude that the Floquet multipliers must be real when a multiplier lies outside of the unit circle. Therefore, the Floquet multipliers must exit the unit circle along the real-axis leaving the following two possibilities for changes in stability. (a) l1 = l2 = +1. This situation is as shown in Figure 10, where the Floquet multiplier leaves the unit circle at +1 on the real axis. The loss of stability will be referred to as a type I instability. Returning to either equation (14) or equation (15), we see that the type I loss of stability results in a divergent growth in the solution at a rate of =lmax = n. Substitution of l1 = l2 = 1 into equations (25) and (26) gives: V0 = v0 /2n,

n = 1, 2, . . . .

(27)

(b) l1 = l2 = −1. This situation is as shown in Figure 11, where the Floquet multiplier leaves the unit circle at −1 on the real axis. The loss of stability will be referred to as a

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Figure 10. The movement of type I multipliers.

type II instability. Returning again to either equation (14) or equation (15), we see that the type II loss of stability results in an oscillatory growth in the solution as (−1)n =lmax = n. Substitution of l1 = l2 = −1 into equations (25) and (26) gives V0 = v0 /(2n + 1),

n = 0, 1, 2, . . . .

(28)

Recall that both the parametric and external excitation for this problem are at a frequency of 2V0 . Therefore, a rotation rate of V0 = v0 will correspond to both the standard externally excited resonance and a type II (n = 0) parametric resonance. In addition, an infinity of parametric resonances (both types I and II) will exist. In the next section, we will explore the ranges of V0 for which unstable behavior is expected. 4.2.     The type of frequency analysis described in section 3.2 leaves a major question unanswered: What combinations of system parameters (in this case angular speed and parametric stiffness) will lead to an unbounded response? This question is very important to tire manufacturers, who need to know the maximum limit of tire non-uniformity allowable for safe operation. In the present example, it is desirable to know the maximum allowable parametric stiffness for safe operation at normal driving speeds. The Floquet theory enables this type of determination. A numerical algorithm based upon the Floquet theory was used to determine the regions of parametric instability in the (V0 , ks )-plane. The algorithm iterated over the (V0 , ks ) parameter-plane, testing a discrete number of parametric points to determine stability. Due to the time-intensive nature of this procedure, the resolution of the parameter-plane search was of necessity rather coarse. The results of this analysis are displayed in

Figure 11. The movement of type II multipliers.

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Figure 12. The parametric stability plot for equation (2) with (a) zero damping, (b) 1% damping and (c) 2% damping.

Figures 12(a)–(c), where unstable parameter combinations are denoted by a dot (·). In the cases shown, damping was varied from 0% to 2% of critical. The results demonstrate that damping causes the regions of instability to move away from the frequency axis. In fact, the thin bands of instability at lower frequencies vanish altogether. Examination of equation (2) helps to explain why the regions of instability are slanting to the right. As the stiffness non-uniformity parameter, ks , is increased, K increases and as a result, the stationary (non-parametrically excited) natural frequency increases. The large branch of unstable points originating from the V0-axis at approximately 385 rad/s corresponds to the type II region of instability which originates at the natural frequency of the system. The thinner streaks of unstable points correspond to type I regions of instability. It should be noted that, due to the fact that the regions of type I instability decrease in width as the angular velocity V0 approaches zero (in fact, they become infinitesimally thin), only a few of them can be captured using the numerical method described above. 5. CONCLUSIONS

Even while the very simplified model of a tire which was investigated here is limited in scope, the results should be of interest to the tire industry. It has been shown here that a localized stiffness non-uniformity contributes to the fore-and-aft force not at the frequency of the rolling speed, but at a frequency of twice the rolling speed. This means that for the commonly observed fore-and-aft force effect at tire rolling frequency, the explanation of tread band mass non-uniformity offered earlier by the authors [1] remains plausible. However, the present model provides insight into the second-harmonic, lower level, fore-and-aft forces measured by Schuring [5]. The strongest resonance appears to occur at a traveling speed corresponding to half of the fundamental natural frequency of the tire, which means that parametric resonances are unlikely to occur at normal highway speeds unless the tire is partially deflated.

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A seecond conclusion is that stiffness non-uniformity may lead to a parametric forcing and, hence, the possibility of multiple parametric resonances. It has been shown that the strongest regions of instability occur at rates of rotation of the tire corresponding to speeds above those commonly encountered in present-day automobile travel. While the instability regions seen at lower rotation rates are readily diminished in size through the introduction of even small amounts of damping, at higher stiffness non-uniformity levels (say, due to belt separation), typical damping will not be sufficient to suppress the instability and parametrically excited resonances may dominate the response. Future theoretical efforts should expand the model to allow structural ring flexibility (the present model corresponds to a rigid body mode approximation). In addition, while the theoretical model described here represents a plausible explanation of the effect of a localized stiffness non-uniformity in the tire, more experimental results are required to test the theory.

ACKNOWLEDGMENT

The authors would like to acknowledge the support of the Bridgestone/Firestone Tire and Rubber Company, which made this study possible.

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APPENDIX: LIST OF SYMBOLS R u ua V0 f u* s Y0 O O' k ks K0 c MT IC rc q(t) A(t) F(t) C T l V V

v0

nominal rolling radius (m) horizontal (fore-and-aft) displacement of center of mass (m) horizontal (fore-and-aft) displacement of center of mass neglecting parametric forcing (m) nominal rolling rate (rad/s) variation from the nominal rolling rate (rad/s) angle of application of the pointwise spring nonuniformity (rad) maximum constant vertical deflection of mass center (m) inertial origin tread band fixed origin distributed radial stiffness (N/m/m) pointwise radial stiffness nonuniformity (N/m) nominal lumped equivalent stiffness (N/m) equivalent viscous damping constant (N s/m) mass of tread band (kg) mass moment of inertia of tread band with respect to contact point (kg m2) radius of gyration of the ring about the contact point (m) state space vector for system periodic coefficient matrix for state space representation principal matrix solution of state space system monondromy matrix period of coefficient matrix A(t) eigenvalue of C eigenvector of C generalized eigenvector of C natural frequency of equivalent non-parametrically forced system