Nguyen, Hong-Hai (2008) A new approach for periodic structures. Application to the calculation of tire vibrations. PhD thesis Structures et matériaux, UR NAVIER, Equipe dynamique des structures et identification, ENPC p.142.

Full text available as: - These_NGUYEN_Hong-Hai.pdf ( 11680 Kb )

Licence: Copyright

Abstract

This work gives a dynamical approach for periodic structures and the application to tire modellings. The static and dynamic properties of the constitutive materials of the tire have been measured. Some specific models of homogenization have been used to determinate the mechanical equivalent characteristics. The frequency response functions have been measured at several points on the tire in two cases of excitation force: on the tread belt and on the sidewalls. The dynamical theory of periodic structures is presented. A transformation into the cartesian reference system of the structures having the periodicity in the non-cartesian reference system has been suggested. The utilization of transformation in the case of tire ensures periodic conditions in the exploitation of the matrices of cells. The mobilities at several points of the tire have been calculated and compared with the measurement. This work allows in one side the dynamical studies of tire in the high frequency range and in the other side a parametrical study of the influence of the material properties and the tire inflation pressure.

References

[1] J. F. Hamet. Propagating wave formulation of tire vibrations using the orthotropic plate model. Inter-Noise, 2002.

[2] J. M. Muggleton, B. R. Mace, and M. J. Brennan. Vibrational response prediction of a pneumatic tyre using an orthotropic two-plate wave model. Journal of Sound and Vibration, pages 929-950, 2003.

[3] L. Jia, Y. Xu, and J. Zhang. Free vibration analysis of radial pneumatic tires using Bezier function. Journal of Sound and Vibration, 285 :887{903, 2005.

[4] R. Merzouki, B. Ould-Bouamama, M.A. Djeziri, and M. Bouteldja. Modelling and estimation of tire-road longitudinal impact efforts using bond graph approach. Mechatronics, 17(2-3) :93-108, 2007.

[5] R. J. Pinnington. A wave model of a circular tyre. Part 1 : belt modelling. Journal of Sound and Vibration., 290 :101-132, 2006.

[6] R. J. Pinnington. A wave model of a circular tyre. Part 2 : side-wall and force transmission modelling. Journal of Sound and Vibration., 290 :133{168, 2006.

[7] J. Lemaitre and J.-L. Chaboche. Mécanique des matériaux solides. Dunod, 2004.

[8] C. CARROT. Viscoélasticité linéaire des polymères fondus. Techniques de l'Ingénieur, AM4 - Plastiques et composites(AM3620), 1999.

[9] Kenneth G. McConnell. Vibration Testing : Theory and Practice. Wiley-Interscience, 1995.

[10] R. Caracciolo, A. Gasparetto, and M. Giovagnoni. An experimental technique for complete dynamic characterization of a viscoelastic material. Journal of Sound and Vibration, 272 :1013-1032, 2004.

[11] B. Garnier. Isolation antivibratoire et antichoc - définitions. Principes physiques. Techniques de l'Ingénieur, BR2-Bruit et vibrations(B5140), 1994.

[12] D. Backström and A. C. Nilsson. Modelling the vibration of sandwich beams using frequency-dependent parameters. Journal of Sound and Vibration, 300 :589-611, 2007.

[13] J. Lee and K. Kim. Characterization of complex modulus of viscoelastic material subject to static compression. Mechanics of Time-Dependent Materials, 5 :255-271, 2001.

[14] G. Kergourlay, E. Balmès, and G. Legal. A characterization of frequency-temperature-prestress effects in viscoelastic films. Journal of Sound and Vibration, 297 :391-407, 2006.

[15] Jean-Marie Berthelot, Mustapha Assarar, Youssef Sefrani, and Abderrahim El Mahi. Damping analysis of composite materials and structures. Composite Structures, In Press, Corrected Proof :-.

[16] Jean Courbon. Vibrations des poutres. Techniques de l'Ingénieur, traité Construction, 1984.

[17] K. Miller. Testing and analysis : Mesuring the dynamic properties of elastomers for analysis. Axel Product, Inc, DynamicRev1, 2000.

[18] H.H. Nguyen, J. Cesbron, F. Anfosso-Ledee, H.P. Yin, S. Erlicher, and D. Duhamel. Dependence of the contact area on the velocity of a rolling tire. In Acoustics08 - Paris, 2008.

[19] James E. Mark and Burak Erman. Science and Technology of Rubber, Third Edition. Elsevier - Academic Press, 4 2005.

[20] Hans B Pacejka. Tire and Vehicle Dynamics. SAE International, 12 2005.

[21] P. Andersson and K. Larsson. Validation of a high frequency three-dimensional tyre model. Acta Acustica united with Acustica, 91 :121-131, 2005.

[22] K. Larsson and W. Kropp. A high frequency three-dimensional tyre model based on two coupled elastic layers. Journal of Sound and Vibration, 203 :889-908, 2002.

[23] N. M. M. Maia and J. M. M. E Silva. Theoretical and Experimental Modal Analysis. Research Studies Press ; 1st edition, 1998.

[24] R. J. Pinnington and A. R. Briscoe. A wave model for a pneumatic tyre belt. Journal of Sound and Vibration., 253 :969{987, 2001.

[25] J. F. Hamet. Tire/road noise : time domain green's function for the orthotropic plate model. Inter-Noise, 2002.

[26] R. J. Pinnington. Radial force transmission to the hub from an unloaded stationary tyre. Journal of Sound and Vibration, 253(5) :961-983, June 2002.

[27] K. Larsson, S. Barrelet, and W. Kropp. The modelling of the dynamic behavior of tyre tread blocks. Applied Acoustics, 63 :659-677, 2002.

[28] Y. Waki, B. Mace, and M. Brennan. Vibration analysis of a tyre using the wave finite element method. In Inter-Noise, 2007.

[29] D. Duhamel, B. R. Mace, and M. J. Brennan. Finite element analysis of the vibrations of waveguides and periodic structures. Journal of Sound and Vibration, 2005.

[30] I. Lopez, R.E.A. Blom, N.B. Roozen, and H. Nijmeijer. Modelling vibrations on deformed rolling tyres-a modal approach. Journal of Sound and Vibration, 307(3-5) :481-494, November 2007.

[31] S. C. Huang. The vibration of rolling tires in ground contact. Int. Journal of Vehicle Design, 13, 1992.

[32] Werner Soedel. Vibrations of shells and plates. Marcel Dekker, Inc. New York, 2004.

[33] S. C. Huang and W. Soedel. Response of rotating rings to harmonic and periodic loading and comparison with the inverted problem. Journal of Sound and Vibration, 118 :253-270, 1987.

[34] S. C. Huang and W. Soedel. Effects of coriolis acceleration on the free and forced in-plane vibrations of rotating rings on elastic foundation. Journal of Sound and Vibration, 115 :253-274, 1987.

[35] L. Gry. Dynamic modelling of railway track based on wave propagation. Journal of Sound and Vibration, 195 :477-505, 1997.

[36] A. Houmat. Three-dimensional free vibration analysis of plates using the h-p version of the finite element method. Journal of Sound and Vibration, 290(3-5) :690-704, March 2006.

[37] Y. K. Koh and R. G.White. Analysis and control of vibrational power transmission to machinery supporting structures subjected to a multi-excitation system, part i : Driving point mobility matrix of beams and rectangular plates. Journal of Sound and Vibration, 196(4) :469-493, October 1996.

[38] A. Bocquillet, M. N. Ichchou, and L. Jezequel. Energetics of axisymmetric fluid-filled pipes up to high frequencies. Journal of Fluids and Structures, 17 :491-510, 2003.

[39] S. Finnveden. Spectral finite element analysis of the vibration of straight fluid-filled pipes with flanges. Journal of Sound and Vibration, 199 :125-154, 1997.

[40] N. R. Harland, B.R.Mace, and R.W.Jones. Wave propagation, reflection and transmission in tunable fluid-filled beams. Journal of Sound and Vibration, 241 :735-754, 2001.

[41] L. Houillon, M. N. Ichchouh, and L. Jezequel. Wave motion in thin-wallde structures. Journal of Sound and Vibration, 281 :483-507, 2005.

[42] Marcelo T. Piovan, Carlos P. Filipich, and Victor H. Cortinez. Exact solutions for coupled free vibrations of tapered shear-flexible thin-walled composite beams. Journal of Sound and Vibration, 316(1-5) :298-316, September 2008.

[43] L.S. Beale and M.L. Accorsi. Power flow un two- and three- dimensional frame structures. Journal of Sound and Vibration, 185 :685-702, 1995.

[44] Von Flotow. Disturbance propagation in structural networks. Journal of Sound and Vivration, 106 :433-450, 1986.

[45] L. Brillouin. Wave propagation in periodic structures. New York, Dover, 1953.

[46] D. J. Mead. A general theory of harmonic wave propagation in linear periodic systems with multiple coupling. Journal of Sound and Vibration, 27 :235-260, 1973.

[47] D. J. Mead. Wave propagation and natural modes in periodic systems : I. mono-coupled systems. Journal of Sound and Vibration, 40 :1-18, 1975.

[48] D. J. Mead. Wave propagation and natural modes in periodic systems : Ii. multi-coupled systems, with and without damping. Journal of Sound and Vibration, 40 :19-39, 1975.

[49] D. J. Mead. Wave propagation in continuous periodic structures : research contributions from southampton. Journal of Sound and Vibration, 190 :495-524, 1996.

[50] D. Wang, C. Zhou, and J. Rong. Free and forced vibration of repetitive structures. International Journal of Solids and Structures, 40 :5477-5494, 2003.

[51] J.-M. Mencik and M.N. Ichchou. Multi-mode propagation and diffusion in structures through finite elements. European Journal of Mechanics - A/Solids, 24(5) :877-898, 2005.

[52] M. Géradin. Théorie des vibrations. Application à la dynamique des structures. Masson, Paris, 1993.

[53] S. Gopalakrishnan, A. Chakraborty, and D. Roy Mahapatra. Spectral Finite Element Method. Springer, 2008.

[54] S. Finnveden. Finite element techniques for the evaluation of energy flow parameters. Proc. Novem, pages 235-260, 2000.

[55] S. Finnveden. Formulas for modal density and for input power from mechanical and fluid point sources in fluid filles pipes. Journal of Sound and Vibration, 169 :43-53, 1997.

[56] B. Aalami. Waves in prismatic guides of arbitrary cross section. Journal of Applied Mechanics, 40 :1067-1072, 1973.

[57] L. Gavric. Finite element computation of dispersion properties of thin-walled waveguides. Journal of Sound and Vibration, 173 :113-124, 1994.

[58] L. Gavric. Computation of propagative waves in free rail using a finite element technique. Journal of Sound and Vibration, 184 :531-543, 1995.

[59] S. Finnveden. Evaluation of modal densit and group velocity by o finite element method. Journal of Sound and Vibration, 273 :51-75, 2004.

[60] F. Birgersson, S. Finnveden, and C. M. Nilsson. A spectral super element for modelling of plate vibration. Part 1 : general theory. Journal of Sound and Vibration, 287 :297-314, 2005.

[61] F. Birgersson and S. Finnveden. A spectral super element for modelling of plate vibration. Part 2 : turbulence excitation. Journal of Sound and Vibration, 287 :315-328, 2005.

[62] P. J. Shorter. Wave propagation and damping in linear viscoelastic laminates. Journal of the Acoustical Society of America, 115 :1917-1925, 2004.

[63] D. Duhamel, B. R. Mace, and M. J. Brennan. Finite element prediction of wave motion in structural waveguides. Rapport interne, University of Southampton.

[64] A. Bocquillet. Méthode Energétique de Caractérisations Vibroacoustiques des Réseaux Complexes. PhD thesis, Ecole Centrale de Lyon, février 2000.

[65] L. Gry and C. Gontier. Dynamic modelling of railway track : a periodic model on a generalized beam formulation. Journal of Sound and Vivration, 199 :531-558, 1997.

[66] W. X. Zhong and F. W. Williams. On the direct solution of wave propagation for repetitive structures. Journal of Sound and Vivration, 181 :485-501, 1995.

[67] A. Luongo and F. Romeo. Real wave vectors for dynamic analysis of periodic structures. Journal of Sound and Vibration, 279 (1-2) :309-325, January 2005.

[68] R. J. Guyan. Reduction of stiffness and mass matrices. AIAA Journal, 3 :380-387, 1965.

[69] M. Petyt. Introduction to finite element vibration analysis. Cambridge University Press, New York USA, 1990.

[70] Jean Salençon. Mécanique des milieux continus : Tome 1 : Concepts généraux. Ecole Polytechnique, 2005.

[71] R. S. Langley, N. S. Bardel, and H. M. Ruivo. The response od two-dimensional periodic structures to harmonic point loading : A theorical and experimental study of a beam grillage. Journal of Sound and Vibration, 207 :521-535, 1997.

[72] R. S. Langley. The response of two-dimensional periodic structures to impulsive point loading. Journal of Sound and Vibration, 201 :235-253, 1997.

[73] R. S. Langley. The response of two-dimensional periodic structures to point harmonic forcing. Journal of Sound and Vibration, 197 :447-469, 1996.

[74] R. S. Langley. On the viration conductivity approach to high frequency dynamics for two-dimensional structural components. Journal of Sound and Vibration, 182 :637-657, 1995.

[75] D. Duhamel. Finite element computation of green's functions. Engineering Analysis with Boundary Elements, 31(11) :919-930, November 2007.

[76] I. Kogevnikov. Modelling mechanical system with an infinite number of degrees of freedom - Application to the tyre dynamics. PhD thesis, Ecole Nationale des Ponts et Chaussées, 2006.

[77] A. Sameur. Modèle de contact pneumatique-chaussée pour la prévision du bruit de roulement. PhD thesis, Ecole nationale des Ponts et Chaussées, 2004.

[78] K. L. Johnson. Contact Mechanics. Cambridge University Press, 1987.

[79] R. Ghoeishi, P. Cartraud, T. Messager, and P. Davies. Modélisation du comportement de câbles synthétiques. In Journée AUM/AFM, 2004.

[80] Anne Nawrocki and Michel Labrosse. A finite element model for simple straight wire rope strands. Computers & Structures, 77(4) :345-359, July 2000.

[81] Taylor & Francis Group, editor. Viscoelastic material property characterization for tyre finite element analysis. Haris Shankar Sighania Elastomer & Tyre Reseach Institute, India, 2007.

[82] Company KYOWA. Gages for general stress measurement kfg. http ://www.kyowa-ei.co.jp/english/products/gages/pdf/kfg-01.pdf.

[83] J.M. Berthelot. Matériaux composites : Comportement mécanique et analyse des structures. 2005.

[84] Z. Hashin. Analysis of composite materials - a survey. Journal of Applied Mechanics, 450 :481-505, 1983.

[85] Antonio Miravete. 3-D Textile Reinforcements In Composite Materials. CRC, 1999.

[86] Senthil S. Vel and R. C. Batra. Exact solution for thermoelastic deformations of functionally graded thick rectangular plates. AIIAA Journal, 40(7) :1421-1433, 2002.

[87] S. Li. Introduction to Micromechanics and Nanomechanics. University of California Berkeley, California, 2006.

[88] Amit K. Naskar, A. K. Mukherjee, and R. Mukhopadhyay. Studies on tyre cords : degradation of polyester due to fatigue. Polymer Degradation and Stability, 83(1) :173-180, January 2004.

[89] C. J. M. van den Heuvel and E. A. Klop. Relations between spinning, molecular structure and end-use properties of polyethylene naphthalate tyre yarns. Polymer, 41(11) :4249-4266, May 2000.

[90] H. Barguet, B. Chauvin, A. Domingo, and T. Pottier. Câble composite pour le pneumatique. Brevet WO/2007/090603, Société de Technologie Michelin, 8 2007.

[91] Y. Chevalier. Comportements élastique et viscoélastique des composites. Techniques de l'Ingénieur, traité Plastiques et Composites, ARCH1(A7750), 5 1988.

[92] R. M. Christensen. Theory of Viscoelasticity. Dover Publications, Inc, 2003.

[93] Jean Salençon. Viscoélasticité. 1983.

[94] Z. Sobotka. Rheology of Materials and Engineering Structures. Elsevier Science Publishing Company, 1984.

[95] Alex Huynh Kim Long. Analysis of elastomer's dynamic behaviour : modeling and identification. PhD thesis, Ecole Nationale des Ponts et Chaussées, 2005.

[96] Roger Brown. Physical Testing of Rubber. Springer, 2005.

[97] A. Ya. Malkin. Rheology Fundamentals. ChemTec Publishing, 1994.

[98] J. Plusquellec. Vibrations. Techniques de l'Ingénieur, Bruit et vibrations :BR200, 2004.

[99] G. W. Laird and H. B. Kingsbury. A methode of determining complex moduli of viscoelastic materials. Experimental Mechanics, 13(3) :126-131, March 1973.

[100] D. J. Ewins. Modal Testing : Theory, Practice and Application. Taylor & Francis Group, 2 edition, 8 2001.

[101] S. N. Goyanes, A. Roncaglia, F. Saavedra, and G. H. Rubiolo. About the measurement of dynamic mechanical properties of bi-layer systems. Materials Science and Engineering A, 370(1-2) :431-434, April 2004.

[102] Hans Georg Bossemeyer. Evaluation technique for dynamic moduli. Mechanics of Time-Dependent Materials, 5(3) :273-291, 9 2001.

[103] J. Betten. Creep Mechanics. Springer, 2005.

[104] A.V. Shenoy. Rheology of Filled Polymer Systems. Springer, 1999.

[105] Etienne Balmès. Méthodes de conception et de validation en vibration. Laboratoire de Mécanique Sols, Structures et Matériaux, Ecole Centrale Paris, 2 2005.

[106] E. Balmès, Corus M., and Germès S. Model validation for heavily damped structures. Application to a windshield joint. SDTools.

[107] R. F. Gibson and R. Plunkett. A forced-vibration technique for measurement of material damping. Experimental Mechanics, 7(8) :297-302, 1977.

[108] Liming Yu, Yue Ma, Chungen Zhou, and Huibin Xu. Damping efficiency of the coating structure. International Journal of Solids and Structures, 42(11-12) :3045-3058, June 2005.

[109] Jean-Marie Berthelot and Youssef Sefrani. Damping analysis of unidirectional glass and Kevlar fibre composites. Composites Science and Technology, 64(9) :1261-1278, July 2004.

[110] Jean-Marie Berthelot. Damping analysis of laminated beams and plates using the ritz method. Composite Structures, 74(2) :186-201, July 2006.

[111] Jean-Marie Berthelot and Youssef Sefrani. Longitudinal and transverse damping of unidirectional fibre composites. Composite Structures, 79(3) :423-431, July 2007.

[112] N.W. Tschoegl, Wolfgang G. Knauss, and Igor Emri. Poisson's ratio in linear viscoelasticity – a critical review. Mechanics of Time-Dependent Materials, 6(1) :3-51, 2002.

[113] ABAQUS Theory Manual. ABAQUS, Inc., 2006.

[114] S. Gade and H. Herlufsen. Digital Filter Techniques vs. FFT Techniques for Damping Measurements. Bruel & Kjaer Technical Review, 1994.

[115] T. M. Nguyen. Dynamique non linéaire des systèmes mécaniques couplés : Réduction de modèle et Identification. PhD thesis, Ecole Nationale des Ponts et Chaussées, 2007.

[116] Y. Chevalier. Essais dynamiques sur composites – caractérisation aux basses fréquences. Techniques de l'Ingénieur, traité Plastiques et Composites, AM6(AM5400), 7 2002.

[117] J. Cesbron. Influence de la texture de chaussée sur le bruit de contact pneumatique-chaussée. PhD thesis, Ecole Centrale de Nantes, 2007.

[118] NF E90-150. Etalonnage des accéléromètres ou chaînes accélérmétriques dans la gamme des moyennes fréquences. Association Française de Normalisation, 1981.

[119] NF E90-151. Etalonnage des accéléromètres. Etalonnage en continu. Association Française de Normalisation, 1981.

[120] NF E90-152. Vibrations et chocs mécaniques : Fixation mécanique des accéléromètres. Association Française de Normalisation, 1993.

[121] J. D. Achenbach. Wave Propagation in Elastic Solids. Elsevier Science Ltd, 1999.

[122] Karl F. Graff. Wave motion in elastics solids. New York, Dover, 1973.

[123] A. Ehrlacher. Modèle de Matériaux Composites. Ecole Nationale des Ponts et Chaussées, 1999.

[124] T. Lachand-Robert. Analyse harmonique, distributions, convolution. Technique de l'Ingénieur, A142 - traité Mathématiques pour l'ingénieur, 1993.

[125] M. Lalanne, P. Berthier, and J. Der Hagopian. Mécanique des vibrations linéaires. Masson, 1995.

Table of content

Introduction générale - ix

I. Etudes bibliographiques - 1

1. Description des pneumatiques : Géométrie et matériaux - 2

2. Modélisation du pneumatique. Méthodes et résultats existants - 5

3. Structure guide d'onde et structure périodique - 10

4. Conclusion et perspectives - 15

II. Propriétés statiques des matériaux constituants du pneumatique - 17

1. Généralités - 19

2. Caoutchouc - 20

3. Couche élémentaire dans le pneu - 23

4. Revêtement intérieur - 31

5. Flanc - 33

6. Bande de roulement - 33

7. Conclusion - 36

III. Propriétés dynamiques des matériaux constituants du pneumatique - 37

1. Comportement dynamique des matériaux viscoélastiques - 39

2. Principe d'analyse dynamique - 50

3. Essais de relaxation - 54

4. Vibration d'une poutre - 61

5. Une application dans le calcul de l'aire de contact en fonction de la vitesse - 70

6. Synthèse sur les matériaux - 71

IV. Mesure des fonctions de réponse en fréquence d'un pneumatique - 73

1. Objectifs de la mesure - 75

2. Dispositif de mesure - 75

3. Erreurs et approximations de la chaîne de mesure - 79

4. Premier essai - 80

5. Mesures avec le pot vibrant : Pneu sans pression interne - 85

6. Mesures avec le pot vibrant : Pneu avec pression interne - 87

7. Identification sommaire des types de modes : bande de roulement et section - 92

8. Conclusion - 96

V. Structures périodiques - 99

1. Généralités sur la vibration des structures - 100

2. Théorie de la vibration des structures périodiques unidimensionnelles - 104

3. Transformation des matrices dans un repère spécifié - 110

4. Application à une poutre sollicitée par une force ponctuelle - 115

5. Conclusion - 117

VI. Modèle numérique périodique pour le pneumatique - 119

1. Pneumatique de section homogène - 120

2. Pneumatique de section réelle - 123

3. Comparaison avec des mesures - 128

4. Etudes paramétriques : Sensibilité aux propriétés des matériaux - 130

5. Conclusions - 133

Conclusion générale et Perspectives - 135

Annexe - 137

A. Traction d'une couche composite. Modules d'Young et coefficients de Poisson apparents - 137

B. Transformation de Laplace et transformation de Fourier - 141

C. Estimation des erreurs en fonction de la discrétisation en fréquence - 141

Bibliographie - 143

Statistiques de consultation

Repository Staff Only: edit this item