Modelling of vibration damping in pneumatic tyres - Taylor & Francis ...

Vehicle System Dynamics Vol. 43, Supplement, 2005, 145–155

Modelling of vibration damping in pneumatic tyres ATANAS A. POPOV* and ZUNMIN GENG School of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, UK The paper deals with the measurement, identification and modelling of vibration damping in heavy vehicle tyres. Recent developments in vibration analysis are applied in order to extend the damping modelling to more general cases than viscous or hysteretic damping models based on a dissipation matrix. Both the general first-order state-space approach and the second-order small-damping method are critically reviewed. A general procedure for damping identification is developed and implemented. The only limitation to this procedure is that structural linearity and reciprocity should be satisfied; this has been adequately proven for pneumatic tyres. The best theoretical results have been achieved by assuming non-proportional viscous damping in the tyre and through the application of modal analysis techniques based on complex-valued modes. Keywords: Pneumatic tyres; Damping identification; Modal analysis; Non-proportional viscous damping; Complex modes; Tyre modelling

1.

Introduction

The interaction between a tyre and the road, together with the interaction between a tyre and the surrounding air, leads to a multitude of noises and vibrations [1]. Since tyre dynamics are central to the transmission and dissipation of vibrational and acoustical energy, one can reasonably argue that tyre damping is essential for the underlying physical phenomena. However, the role of tyre damping in vehicle dynamics is somewhat overshadowed by other factors which are thought to be of more importance. For example, the vehicle designer relies mostly on damping in the shock absorbers in the low-frequency range and on dissipation of sound energy in the vehicle body within the audible-frequency range. A good balance in model complexity between a tyre and a vehicle is needed to obtain efficient and accurate vehicle simulations. It is relatively easy to measure or calculate the inertia and stiffness properties of a tyre; however, the accurate determination of damping parameters currently presents an unsolved problem. Damping is the removal of energy from a vibrating system [2]. The energy lost is either transmitted away from the system by some mechanism of radiation or dissipated within the system. All structures exhibit vibration damping but, despite a large body of literature on the subject, damping remains one of the least well-understood aspects of vibration analysis. *Corresponding author. Email: [email protected]

Vehicle System Dynamics ISSN 0042-3114 print/ISSN 1744-5159 online © 2005 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00423110500140765

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A. A. Popov and Z. Geng

It so happens that damping forces are usually small in magnitude when compared with other interactions in a mechanical system but play an important role in the dynamic response; yet their mathematical description remains much more complicated. There is a fundamental problem because it is not in general clear which state variables govern the damping forces. A widely used damping model, originated by Lord Rayleigh [3], assumes that instantaneous generalized velocities are the only relevant state variables which determine the damping forces; this is the celebrated ‘viscous damping’ model. It leads to a description of damping behaviour by a dissipation matrix, directly analogous to the mass and stiffness matrices of structural mechanics. A step further in the idealization, also introduced by Rayleigh, is to assume the damping matrix to be a linear combination of the mass and stiffness matrices, the so-called proportional damping. Under this assumption the frequencies and modes of vibration for the system under investigation have real values. There is no obvious reason to expect physical systems to exhibit proportional damping, and complex modes of vibration should be regarded as the norm; complex modes of vibration arise even with viscous damping provided that it is non-proportional. There is a variety of tyre models in the literature with different levels of complexity, ranging from a simple spring–damper element (single point contact) through flexible-ring-type models to detailed finite-element simulations (see [4] for an account). In all these models, when damping effects were included, they were almost invariably of the equivalent proportional viscous damping type. Moreover, damping values as high as 10% of critical damping have been reported and used for vehicle simulations [1, 5]. On the other hand, in studies mainly concerned with the performance of automotive suspensions, the whole amount of chassis damping was considered to arise from the shock absorbers under the assumption that tyre damping is much smaller in magnitude than suspension damping (see, for example, [6]). Other researchers in the area [7] disagreed with this assumption and argued that a separate consideration of tyre damping is essential because tyre damping is responsible for the coupling between sprung and unsprung mass motions at the wheel-hop frequency. Assuming the damping matrix to be of non-proportional viscous type, then direct decoupling of the second-order differential equations of structural dynamics is impossible. Generally, the first-order state-space approach [8] becomes the only feasible method. Woodhouse [9] showed that, for small damping, models of damping can be obtained by using directly the second-order differential equations of structural dynamics, employing a first-order perturbation expansion based on the undamped modes and natural frequencies. A method was proposed to obtain damping models from complex mode shapes and frequencies of vibration [10], and the feasibility of this approach was demonstrated on idealized linear arrays of damped spring–mass oscillators. However, it was not clear whether this approach was applicable to experimental data obtained for complex real structures. This important issue is addressed here in an attempt to employ the new theoretical results to the pneumatic tyre.

2.

Experimental measurements

A rig for tyre testing was designed, manufactured and commissioned (figure 1). The wheel was rigidly attached to a seismic table. The pneumatic tyre was only in contact with the wheel and could perform free vibrations in the frequency range of interest, up to 200 Hz. The experimental techniques employed were as follows: excitation by a shaker with a chirpsweeping input [11], and measurement by accelerometer positioned at 16 equally spaced points along the central circumference of the tyre. During the measurements the tip of the drive rod and the accelerometer were glued to the surface of the tyre. The excitation was

Modelling of vibration damping in pneumatic tyres

Figure 1.

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Modal testing set-up for a Dunlop truck tyre SP341 295/80R22.5: PC, personal computer.

applied in the radial direction, since only in-plane modes of tyre vibration were of interest. Simultaneous generation of excitation and acquisition of vibration response were performed through one National Instruments NI 4551 board [12]. The data were subsequently analysed with purpose-written programs in MATLAB [13]. The tyre used was a conventional single truck tyre Dunlop SP341 295/80R22.5, tested under different inflation pressures. A large set of measurements was made to check for the linearity and reciprocity of tyre structure [14, 15]. Linearity was proven by exciting the tyre with harmonic loading of different magnitudes. Reciprocity was observed to a sufficient accuracy on the ensemble-averaged frequency response functions (FRFs). These initial checks for linearity and reciprocity justified a model with a symmetric damping matrix [9].

3.

Procedures of modal analysis

In modal testing and analysis (see, for example, [8, 11]), any linear system with damping proportional only to velocities can be represented by a finite number N of degrees of freedom and a second-order differential equation of motion ¨ + [C] {x(t)} ˙ + [K] {x(t)} = {f (t)} , [M] {x(t)}

(1)

where [M], [C] and [K] are the mass, dissipation (damping) and stiffness matrices respectively, {x (t)} is the vector of generalized coordinates (radial tyre displacements in this particular case) and {f (t)} is the vector of generalized forces driving the vibration (excitation force on the tyre by the shaker). As a preliminary step towards the general analysis it can be assumed that the damping matrix of the tyre can be simultaneously diagonalized with the mass and stiffness matrices, meaning that damping is taken as proportional. In this case, a standard procedure is to transform equation (1) into the frequency domain and to decouple the equations of motion through a transformation of the generalised coordinates into modal coordinates by the N × N modal matrix [φ] = [{φ1 } , {φ2 } , . . . , {φN }] . Here, the matrix columns represent the mode shapes of the undamped system scaled for unit mass, or {φn }T [M] {φn } = [I ] holds with [I ] as an N × N unit matrix. Under the assumption of proportional viscous damping, the FRFs commonly employed in modal testing and analysis [8, 11] are Xj (ω) φj n φkn , = 2 Fk (ω) ω − ω2 + i2ζn ωn ω n=1 n N

Hj k (ω) =

j, k = 1, . . . , N.

(2)

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Here, the index j refers to the measurement (response) point, while the index k refers to the excitation point. The summation is taken over all available modes N for each frequency ω of excitation. The aim is to measure a large set of FRFs (the left-hand side of equation (2)) in order to obtain the modal parameters accurately: the natural frequencies ωn , the damping ratios ζn and the elements in the modal matrix [φ] scaled for unit mass. The parameter identification is usefully conducted through the ‘curve-fitting’ approach and begins with the identification of residues and poles in the FRFs. Equation (2) can be also rewritten in its complex conjugate pair form in the Laplace domain: Hj k (s) =

N N Rj∗kn iφj n φkn Rj kn −iφj n φkn + , + = ∗ s − pn s − pn n (s + σn − in ) n (s + σn + in ) n=1 n=1

(3)

1/2 where s is the Laplace variable and the poles are pn = −ζn ωn + iωn 1 − ζn2 = −σn + in ∗ 2 1/2 and pn = ipn = −ζn ωn − iωn 1 − ζn = −σn − in respectively. It should be noted that under the proportional damping assumption the mode shapes in [φ] are always vectors of real numbers. Also, when plotting phase angles derived from the complex-valued FRFs Hj k (ω) against the frequency ω, the phase angle shifts at resonances are always 0◦ or ±180◦ . Supposing now that the damping matrix [C] in equation (1) is of non-proportional viscous type; then a direct decoupling of the second-order differential equation becomes impossible. In this case, the first-order state-space method becomes a feasible approach for the investigation [16–18]. By grouping together the vibration displacements {x(t)} and velocities {y(t)} as a system state vector, the set of N second-order equations is now represented by a set of 2N first-order equations, as follows:   dy(t)     0 M −M 0 y(t) 0 dt + = . (4) M C  0 K x(t) f (t)  dx(t)   dt This formulation leads to a generalized eigenvalue problem [8]

1 p p 0 I . = −1 −1 M −K C −K p

(5)

The solution consists of a set of 2N eigenvalues appearing in complex conjugate pairs for underdamped systems; this means that, if pn = −σn + in is an eigenvalue, then pn∗ = ipn = −σn − in also belongs to the spectrum. The imaginary parts ±n of pn and pn∗ represent the damped natural frequency of each mode, while the real part −σn is related to the half-power bandwidth of the mode (a measure of modal damping). The corresponding eigenvectors to the eigenvalues pn and pn∗ , are also complex conjugate and have the form {n }pn {n∗ }pn∗ {n } = and {N+n } = , n = 1, . . . , N, (6) {n } {n∗ } where {n } and {n∗ } are N -dimensional complex conjugate eigenvectors corresponding to the displacements {x}. In a similar way to the classical modal interpretation with real eigenvectors when decoupling the equations of motions, the matrix [] = [{1 } , . . . , {2N }] (without any particular scaling)


149

represents a transformation between modal and physical coordinates; that is,

y (t) q (t) = [] ∗ , x (t) q (t)

and equation (4) is in modal coordinates   dq(t)       M dt T 0 T −M + [] [] [] M C 0  dq ∗ (t)      dt

0 q(t) 0 T = [] . [] ∗ K q (t) f (t)

(7)

However, the pair consisting of the displacement vector {n } and the pole pn can also be called a complex mode without any loss of information, since this mode fully characterizes the complex eigenvectors {n } and {N+n } as shown by equation (6). Two orthogononality conditions are necessary and sufficient for the set of 2N vectors, formed by {n } and {N+n }, and 2N poles pn and pn∗ to be the complete set of complex modes for a model with N degrees of freedom: {n }

T

M {n } = {n }T [C] {n } + pn {n }T [M] {n } + {n }T [M] {n }pn = an , C (8a) −M 0 {n }T {n } = {n }T [K] {n } − pn {n }T [M] {n }pn = bn , (8b) 0 K

0 M

with an∗ and bn∗ calculated in the same way. For the decoupled first-order differential equation (7), the nth and (N + n)th modes can be solved from an

dqn + bn qn = {n }T {f (t)} , dt

an∗

dqn∗ + bn∗ qn∗ = {n∗ }T {f (t)} . dt

(9)

Thus, one ends up with a set of 2N decoupled equations, and this is equivalent to having a set of independent 2N single-degree-of-freedom systems. The frequencies and damping for these oscillators are governed by pn = −bn /an and pn∗ = −bn∗ /an∗ . By applying standard techniques of analysis, such as the Duhamel convolution integrals and/or Laplace transforms [11, 18], a full set of transfer functions can be derived and usefully written in the matrix form N Rn Rn∗ + [H (s)] = s − pn s − pn∗ n=1 =

N {n }{n }T n=1

=

an

N {ψn }{ψn }T n=1

s − pn

1 { ∗ }{ ∗ }T 1 + n ∗ n s − pn an s − pn∗ +

{ψn∗ }{ψn∗ }T . s − pn∗

(10)

In the particular case of proportional damping with the introduced scaling of complex modes, this general procedure leads to {ψn } = ψn∗ = {φn } with an = 2iωn and an∗ = −2iωn , and

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the FRFs are the same as those described by equation (2): [H (ω)] =

N n=1

{φn } {φn }T . ωn2 − ω2 + i2ζn ωn ω

(11)

Equation (10) can be implemented within an optimization procedure with experimentally measured FRF data in order to derive a full set of modal parameters. The derivation obeys the orthogonality conditions of the complex modal theory and corresponds to a particular situation: the unit-mass normalization of real modes. Further considering equation (1) with general viscous damping, its second-order eigenproblem is det −λ2 [M] + iλ [C] + [K] = 0. (12) Under a small-damping assumption (e.g. a damping ratio of less than 5%), the roots of equation (12), namely the complex eigenvalues λn , are close to the undamped eigenvalues ωn , and the complex eigenvectors {ψn } are also close to the undamped modal vectors {φn }. Thus, a solution for the eigenvalues can be obtained in the form [3, 9] {ψn } =

N

αk(n) {φk } ,

αn(n) = 1

and

(n) αk 1

∀ k = n.

(13)

k=1

One should note that the scaling chosen for the eigenvectors is of particular importance here; this fact has not been given sufficient attention in earlier work [9]. Substituting the complex normal mode {ψn } and λn back into the equation of motion and applying the orthogonality properties of the undamped modes {φn }, one has −λ2n αk(n) + iλn

N

Ckj = {φk }T [C] φj .

αj(n) Ckj + ωk2 αk(n) = 0,

(14)

j =1

For k = n, equation (14) yields for the eigenvalues −λ2n + iλn Cnn + ωn2 ≈ 0,

λn ≈ ±ωn +

iCnn , 2

(15)

while, for k = n, it gives for the complex modes αk(n) ≈

iωn Ckn , ωn2 − ωk2

{ψn } ≈ {φn } + i

N ωn Ckn {φk }. ωn2 − ωk2 k=1

(16)

k=n

Equation (14) can be also rewritten as [14]  −λ2n αk(n) + iλn αn(n) Ckn + αk(n) Ckk +

N

  + ωk2 αk(n) = 0. αj(n) Ckj

(17)

j =k=n

This leads for the modes to αk(n) ≈ −

iλn Ckn iλn Ckn ≈ , (λn − λk )(λn + λ∗k ) ωk2 − λ2n + iλn Ckk

{ψn } ≈ {φn } + i

N k=1 k =n

λn Ckn {φk }. (λn − λk )(λn + λ∗k )

(18)


151

The relationship between undamped (real) and complex modes within the small nonproportional viscous damping assumption originated by Rayleigh [3] is known as Rayleigh’s small-damping assumption. Some important conclusions can be drawn from equations (15) and (18). (i) The eigenvalues λn employed in the second-order approximation are different from the eigenvalues pn obtained by the first-order derivation. However, based upon Rayleigh’s small-damping assumption, an approximate relationship between the two sets of eigenvalues holds, namely pn ≈ iλn . Equation (15) shows that the damped natural frequencies depend, to this order of approximation, on the diagonal terms of the damping matrix in normal coordinates. (ii) As shown by equations (18), the real part {ψn } of the complex eigenvectors equals the undamped modes {φn }, while the imaginary part is purely caused by the arbitrarily distributed damping. As a complex-valued contribution to the eigenvector, the off-diagonal terms of the transformed damping matrix C exert a dominant influence upon the imaginary part of the eigenvector.

4.

Results and discussion

By employing equation (3) in the pole-pair optimization procedure [11, 19], the FRFs and in-plane mode shapes of tyre vibration are identified. Figure 2 shows the real and imaginary parts, the amplitudes and the phases of measured data (solid curves) and a fitted FRF (dotted curves) at a tyre inflation pressure of 7.5 bar. All tyre resonances involving in-plane bending of the tread band are revealed within the frequency range from 0 to 200 Hz. There are important discrepancies between measured and fitted data in the range between 140 and 200 Hz. Moreover, deviations in the phase–frequency characteristics of the FRF from the proportional damping assumption are observed at resonances (see [14, 15] for more details), namely for phase angles between 0◦ and ±180◦ . This proves the need for more general procedures of damping identification, based upon complex modes. Figure 3 demonstrates the same experimental data as those shown in figure 2, together with the fitted FRF through a procedure based now on the pole-pair optimization of

Figure 2. Measured and identified real and imaginary parts, amplitudes and phases by the real modal analysis for an FRF of tyre vibration at an inflation pressure of 7.5 bar.

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Figure 3. Measured and identified real and imaginary parts, amplitudes and phases by the complex modal analysis for an FRF of tyre vibration at an inflation pressure of 7.5 bar.

equation (10). The theoretical results in figure 3 (dotted curves) generally show much better agreement with the experimental data for the whole frequency range, in comparison with the identification results in figure 2. Figure 4 illustrates the discrepancy between the fitted and measured FRFs; the solid curve represents the real modal analysis, while the double-dotdashed curve represents the complex modal anlysis. Obviously, the curve fitting with complex modes yields a significant improvement on the results obtained in the higher-frequency range between 150 and 200 Hz. An objective measure of the fit is provided by the rms value of the discrepancy between theoretical and experimental results, which is denoted by RMSD and

Figure 4.

FRF discrepancies and RMSD values for the results obtained through real and complex modal analyses.


153

given by 1/2 m 2 1 th exp H (i) − H (i) , RMSD = m i=1

(19)

where the summation is over all available discrete points within the range from 0 to 200 Hz, and m is the total number of points. Equation (19) gives RMSD = 0.0011 for real modes and a lower value of 0.000 53 for the procedure based upon complex modes. Obviously, the application of complex modes within the non-proportional viscous damping context offers a better description of the system damping than the real modal analysis approach does. Further analysis, not presented here but given in [15], reveals that the real and imaginary parts of the complex modal matrix [], obtained from the identification with tyre vibration data, are nearly proportional to each other; this means that the complex modes can be approximately transformed into equivalent real modes by changing the phases in the solution. This demonstrates that the tyre damping is not far from proportional; however, for an accurate description of the damping properties, one should apply the general procedure based on complex modes, as justified by the presented analysis. The application of Rayleigh’s small-damping approximation, as derived in equations (15) and (18), proves to be a useful tool for the implementation of a step-by-step procedure for the identification of damping in tyres and the derivation of the corresponding dissipation matrix [15]. One problem that needed special attention here concerned the incompleteness of modal data from experiments. In modal analysis, there is a transformation between the damping matrix in modal coordinates and the damping matrix in physical coordinates. If complete modes were obtained from experiments, then the modal matrix would be a square matrix, and the damping matrix in physical coordinate could be easily obtained by inverse transformation. However, in practice the experimental modes are often incomplete, and as a result the modal matrix is not a square matrix. In this case, the modes are truncated and a sort of power-leakage phenomenon is unavoidable. Based upon this consideration, a pseudo-inverse transform was introduced in order to evaluate the damping matrix in physical coordinates and at the same time to improve the effects from the modal truncation. In figure 5, the identified viscous damping matrices in modal and physical coordinates are shown at a tyre pressure of 7.5 bar. The obtained damping matrices are symmetric and positive definite, even without the application of any special symmetry-preserving steps in the algorithm. This gives futrher confidence in the accuracy achieved through the identification

Figure 5. Identified general viscous damping matrix in pneumatic tyres at inflation pressure of 7.5 bar: (a) modal coordinates; (b) physical coordinates.

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procedure. The negative damping distribution in some of the physical coordinates can be attributed to the incompleteness of the modal information. Physically, the phenomenon can be identified as power leakage.

5.

Conclusions

The following conclusions can be drawn. (i) A test rig facility for measuring the vibration of truck tyres has been developed. Extensive set of measurements of vibration in truck tyres has been made. (ii) Both the general first-order state-space approach and the second-order small-damping method have been critically reviewed and applied. (iii) An appropriate method for normalization (scaling) of complex eigenvectors has been used for damping identification in pneumatic tyres. This method is of importance for further studies into general viscous and non-viscous damping models. (iv) Based upon the interpretation of complex mode shapes and the normalization procedure, an improved expression for the FRFs in the small-damping case (second-order systems of differential equations) has been proposed. It was validated through the experimental data for the tyre. (v) A new general procedure (algorithm) for damping identification has been developed and implemented. The only limitation to this procedure is that linearity and reciprocity should be verified to sufficient accuracy. Acknowledgements This work was supported by a grant from the Engineering and Physical Sciences Research Council of the UK. The authors are grateful to Dr D.J. Cole from the University of Cambridge for valuable advice and discussions. References [1] Pottinger, M.G., Marshall, K.D., Lawther, J.M. and Thrasher, D.B., 1986, A review of tyre/pavement interaction induced noise and vibration. In: M.G. Pottinger and T.J.Yager (Eds) The Tyre Pavement Interface, ASTP Special Technical Publication 929 (Philadelphia, PA: American Society for Testing and Materials), pp. 183–287. [2] Crandall, S.H., 1970, The role of damping in vibration theory. Journal of Sound and Vibration, 11, 3–18. [3] Lord Rayleigh, 1945, Theory of Sound, 2nd edition, reissue (New York: Dover Publications). [4] Clark, S.K., 1981, Mechanics of Pneumatic Tires, 2nd edition (Washington, DC: US Department of Transport). [5] Stutts, D.S. and Soedel, W., 1992, A simplified dynamic model of the effect of internal damping on the rolling resistance in pneumatic tires. Journal of Sound and Vibration, 155, 153–164. [6] Williams, R.A., 1997, Automotive active suspensions. Part 1: basic principles. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 211, 415–426. [7] Zaremba, A., Hampo, R. and Hrovat, D., 1997, Optimal active suspension design using constrained optimization. Journal of Sound and Vibration, 207, 351–364. [8] Maia, N.M.M. and de Silva, J.M.M. (Eds), 1997, Theoretical and Experimental Modal Analysis (Baldock, Hertfordshire: Research Studies Press). [9] Woodhouse, J., 1998, Linear damping models for structural vibration. Journal of Sound and Vibration, 215, 547–569. [10] Adhikari, S. and Woodhouse, J., 2001, Identification of damping. Part 1: viscous damping. Journal of Sound and Vibration, 243, 43–61. [11] Ewins, D.J., 2000, Modal Testing: Theory, Practice and Application, 2nd edition (Baldock, Hetrfordshire: Research Studies Press). [12] National Instruments, 2002, The Measurement and Automation Catalog (Austin, TX: National Instruments). [13] 2002, MATLAB Version 6, The Language of Technical Computing (Natick, MA: The MathWorks Inc.).


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[14] Geng, Z., Popov, A.A. and Cole, D.J., 2002, Modelling of vibration damping in pneumatic tyres: appropriate interpretation of complex modes. Proceedings of the International Conference on Noise and Vibration Engineering, Leuven, Belgium, 16–18 September, pp. 485–494. [15] Geng, Z., Popov, A.A. and Cole, D.J., 2004, Measurement, identification and modelling of damping in pneumatic tyres. (submitted). [16] Balmès, E., 1997, New results on the identification of normal modes from experimental complex modes. Mechanical Systems and Signal Processing, 11, 229–243. [17] Garvey, S.D., Penny, J.E.T. and Friswell, M.I., 1998, The relationship between the real and imaginary parts of complex modes. Journal of Sound and Vibration, 212, 75–83. [18] Ibrahim, S.R. and Sestieri, A., 1995, Existence and normalization of complex modes in post experimental use in modal analysis. Proceedings of the 13th International Modal Analysis Conference, Vol. 1, Nashville, TN, pp. 483–489. [19] Lang, G.F., 1989, Demystifying complex modes. Sound and Vibration, 36–40.