Large Vibration Amplitude of Circular Functionally Graded Plates Resting on Pasternak Foundations
In the present study, the problem of geometrically nonlinear free vibrations of functionally graded circular plates (FGCP) resting on Pasternak elastic foundation with immovable ends was studied. The material properties of the functionally graded composites examined were assumed to be graded in the thickness direction and estimated through the rule of mixture. The theoretical model is based on the classical Plate theory and the Von Kármán geometrical nonlinearity assumptions. Hamilton’s principle is applied and a multimode approach is derived to calculate the fundamental nonlinear frequency parameters, which are found to be in a good agreement with the published results dealing with the problem of functionally graded plates. On the other hand, the influence of the foundation parameters on the nonlinear frequency to the linear frequency ratio of the FGCP has been studied. The effect of the linear and shearing foundations is to decrease the frequency ratio, where it increases with the effect of the nonlinear foundation stiffness.
[1] M. Koizumi, The concept of FGM. Ceram Trans Func Grad Mater
1993;34:3–10.W.-K. Chen, Linear Networks and Systems (Book
style).Belmont, CA: Wadsworth, 1993, pp. 123–135.
[2] M. Koizumi, FGM activities in Japan. Composite B 1997; 28:1–4.
[3] S .Suresh, Mortensen A. Fundamentals of Functionally Graded
Materials: Processing and Thermomechanical Behavior of Graded
Metals and Metal-Ceramic Composites. London, UK: IOM
Communications Ltd, 1998.
[4] Y .Miyamoto, Kaysser W A, Rabin B H, et al. Functionally Graded
Materials: Design, Processing and Applications. Boston, UK: Kluwer
Academic Publishers, 1999.
[5] M. Haterbouch, R. Benamar, The effects of large vibration amplitudes
on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates, part I: iterative and explicit analytical
solution for non-linear transverse vibrations, Journal of Sound and
Vibration 265 (2003) 123–154.
[6] A. Allahverdizadeh, M.H. Naei, M. Nikkhah Bahrami, Nonlinear free
and forced vibration analysis of thin circular functionally graded plates,
Journal of Sound and Vibration 310 (2008) 966–984.
[7] Z. H. Zhou, K. W. Wong, X. S. Xu, A. Y. T. Leung, Natural vibration of
circular and annular thin plates by Hamiltonian approach, Journal of
Sound and Vibration 330(2011) 1055-1017.
[8] J. N. Reddy , Jessica Berry , Nonlinear theories of axisymmetric bending
of functionally graded circular plates with modified couple stress,
Composite Structures 94 (2012) 3664-3668.
[9] T.Y. Wu, G.R. Liu, A differential quadrature as a numerical method to
solve differential equations, Comput. Mech. 24 (2) (1999) 197–205.
[10] J. Yang, H.J. Xiang, Thermo-electro-mechanical characteristics of
functionally graded piezoelectric actuator, Smart Mater. Struct. 16 (3)
(2007) 784–797.
[11] O. Civalek, Nonlinear analysis of thin rectangular plates on Winkler–
Pasternak elastic foundations by DSC–HDQ methods, Appl. Math.
Model. 31 (2) (2007) 606–624.
[12] A. Zerkane, K. EL Bikri and R. Benamar, "A homogenization procedure
for nonlinear free vibration analysis of functionally graded beams resting
on nonlinear elastic foundations”, Applied Mechanics and Materials
Journal (accepted).
[13] C.Y. Chia, Nonlinear Analysis of Plates, McGraw-Hill, New York,
1980.
[14] R. Benamar, M.M.K. Bennouna, R.G. White, The effects of large
vibration amplitudes on the mode shapes and natural frequencies of thin
elastic structures, part II: fully clamped rectangular isotropic plates,
Journal of Sound and Vibration 164 (1991) 399–424.
[15] M. El Kadiri, R. Benamar, R.G. White, The non-linear free vibration of
fully clamped rectangular plates: second non-linear mode for various
plate aspect ratios, Journal of Sound and Vibration 228 (2) (1999) 333–
358.
[16] R. Benamar, M.M.K. Bennouna, R.G. White, The effects of large
vibration amplitudes on the mode shapes and natural frequencies of thin
elastic structures, part III: fully clamped rectangular isotropic platesmeasurements
of the mode shape amplitude dependence and the spatial
distribution of harmonic distortion, Journal of Sound and Vibration 175
(1994) 377–395.
[1] M. Koizumi, The concept of FGM. Ceram Trans Func Grad Mater
1993;34:3–10.W.-K. Chen, Linear Networks and Systems (Book
style).Belmont, CA: Wadsworth, 1993, pp. 123–135.
[2] M. Koizumi, FGM activities in Japan. Composite B 1997; 28:1–4.
[3] S .Suresh, Mortensen A. Fundamentals of Functionally Graded
Materials: Processing and Thermomechanical Behavior of Graded
Metals and Metal-Ceramic Composites. London, UK: IOM
Communications Ltd, 1998.
[4] Y .Miyamoto, Kaysser W A, Rabin B H, et al. Functionally Graded
Materials: Design, Processing and Applications. Boston, UK: Kluwer
Academic Publishers, 1999.
[5] M. Haterbouch, R. Benamar, The effects of large vibration amplitudes
on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates, part I: iterative and explicit analytical
solution for non-linear transverse vibrations, Journal of Sound and
Vibration 265 (2003) 123–154.
[6] A. Allahverdizadeh, M.H. Naei, M. Nikkhah Bahrami, Nonlinear free
and forced vibration analysis of thin circular functionally graded plates,
Journal of Sound and Vibration 310 (2008) 966–984.
[7] Z. H. Zhou, K. W. Wong, X. S. Xu, A. Y. T. Leung, Natural vibration of
circular and annular thin plates by Hamiltonian approach, Journal of
Sound and Vibration 330(2011) 1055-1017.
[8] J. N. Reddy , Jessica Berry , Nonlinear theories of axisymmetric bending
of functionally graded circular plates with modified couple stress,
Composite Structures 94 (2012) 3664-3668.
[9] T.Y. Wu, G.R. Liu, A differential quadrature as a numerical method to
solve differential equations, Comput. Mech. 24 (2) (1999) 197–205.
[10] J. Yang, H.J. Xiang, Thermo-electro-mechanical characteristics of
functionally graded piezoelectric actuator, Smart Mater. Struct. 16 (3)
(2007) 784–797.
[11] O. Civalek, Nonlinear analysis of thin rectangular plates on Winkler–
Pasternak elastic foundations by DSC–HDQ methods, Appl. Math.
Model. 31 (2) (2007) 606–624.
[12] A. Zerkane, K. EL Bikri and R. Benamar, "A homogenization procedure
for nonlinear free vibration analysis of functionally graded beams resting
on nonlinear elastic foundations”, Applied Mechanics and Materials
Journal (accepted).
[13] C.Y. Chia, Nonlinear Analysis of Plates, McGraw-Hill, New York,
1980.
[14] R. Benamar, M.M.K. Bennouna, R.G. White, The effects of large
vibration amplitudes on the mode shapes and natural frequencies of thin
elastic structures, part II: fully clamped rectangular isotropic plates,
Journal of Sound and Vibration 164 (1991) 399–424.
[15] M. El Kadiri, R. Benamar, R.G. White, The non-linear free vibration of
fully clamped rectangular plates: second non-linear mode for various
plate aspect ratios, Journal of Sound and Vibration 228 (2) (1999) 333–
358.
[16] R. Benamar, M.M.K. Bennouna, R.G. White, The effects of large
vibration amplitudes on the mode shapes and natural frequencies of thin
elastic structures, part III: fully clamped rectangular isotropic platesmeasurements
of the mode shape amplitude dependence and the spatial
distribution of harmonic distortion, Journal of Sound and Vibration 175
(1994) 377–395.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:65101", author = "El Kaak Rachid and El Bikri Khalid and Benamar Rhali", title = "Large Vibration Amplitude of Circular Functionally Graded Plates Resting on Pasternak Foundations", abstract = "In the present study, the problem of geometrically nonlinear free vibrations of functionally graded circular plates (FGCP) resting on Pasternak elastic foundation with immovable ends was studied. The material properties of the functionally graded composites examined were assumed to be graded in the thickness direction and estimated through the rule of mixture. The theoretical model is based on the classical Plate theory and the Von Kármán geometrical nonlinearity assumptions. Hamilton’s principle is applied and a multimode approach is derived to calculate the fundamental nonlinear frequency parameters, which are found to be in a good agreement with the published results dealing with the problem of functionally graded plates. On the other hand, the influence of the foundation parameters on the nonlinear frequency to the linear frequency ratio of the FGCP has been studied. The effect of the linear and shearing foundations is to decrease the frequency ratio, where it increases with the effect of the nonlinear foundation stiffness.
", keywords = "Non-linear vibrations, Circular plates, Pasternak foundation, functionally graded materials.", volume = "7", number = "9", pages = "1816-5", }
