How to cite this paper
Jandaghian, A., Jafari, A & Rahmani, O. (2014). Vibrational response of functionally graded circular plate integrated with piezoelectric layers: An exact solution.Engineering Solid Mechanics, 2(2), 119-130.
Refrences
Abramowitz, M., & Stegun, I. A. (Eds.). (1964). Handbook of Mathematical Functions: With Formulars, Graphs, and Mathematical Tables (Vol. 55). DoverPublications.com.
Batra, R. C., & Geng, T. S. (2002). Comparison of active constrained layer damping by using extension and shear mode piezoceramic actuators. Journal of Intelligent Material Systems and Structures, 13(6), 349-367.
Batra, R. C., & Liang, X. Q. (1997). The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators. Computers & structures, 63(2), 203-216.
Behjat, B., & Khoshravan, M. R. (2012). Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates. Composite Structures, 94(3), 874-882.
Bian, Z. G., Lim, C. W., & Chen, W. Q. (2006). On functionally graded beams with integrated surface piezoelectric layers. Composite structures, 72(3), 339-351.
Ebrahimi, F., & Rastgo, A. (2008a). An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory. Thin-Walled Structures, 46(12), 1402-1408.
Ebrahimi, F., & Rastgoo, A. (2008b). Free vibration analysis of smart annular FGM plates integrated with piezoelectric layers. Smart Materials and Structures, 17(1), 015044.
Ebrahimi, F., Rastgoo, A., & Atai, A. A. (2009). A theoretical analysis of smart moderately thick shear deformable annular functionally graded plate. European Journal of Mechanics-A/Solids, 28(5), 962-973.
Es & apos; haghi, M., Hashemi, S. H., & Fadaee, M. (2011). Vibration analysis of piezoelectric FGM sensors using an accurate method. International Journal of Mechanical Sciences, 53(8), 585-594.
Fakhari, V., Ohadi, A., & Yousefian, P. (2011). Nonlinear free and forced vibration behavior of functionally graded plate with piezoelectric layers in thermal environment. Composite Structures, 93(9), 2310-2321.
Hashemi, S. H., Khorshidi, K., Es’haghi, M., Fadaee, M., & Karimi, M. (2012). On the effects of coupling between in-plane and out-of-plane vibrating modes of smart functionally graded circular/annular plates. Applied Mathematical Modelling, 36(3), 1132-1147.
He, X. Q., Ng, T. Y., Sivashanker, S., & Liew, K. M. (2001). Active control of FGM plates with integrated piezoelectric sensors and actuators. International Journal of Solids and Structures, 38(9), 1641-1655.
Huang, X. L., & Shen, H. S. (2006). Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments. Journal of Sound and Vibration, 289(1), 25-53.
Jandaghian, A. A., Jafari, A. A., & Rahmani, O. (2013). Transient Bending of a Circular Plate Integrated with Piezoelectric layers. Applied Mathematical Modelling,37(12-13), 7154-7163.
Kargarnovin, M. H., Najafizadeh, M. M., & Viliani, N. S. (2007). Vibration control of a functionally graded material plate patched with piezoelectric actuators and sensors under a constant electric charge. Smart materials and structures, 16(4), 1252.
Liew, K. M., He, X. Q., Ng, T. Y., & Kitipornchai, S. (2002). Active control of FGM shells subjected to a temperature gradient via piezoelectric sensor/actuator patches. International journal for numerical methods in engineering, 55(6), 653-668.
Liew, K. M., He, X. Q., Ng, T. Y., & Kitipornchai, S. (2003). Finite element piezothermoelasticity analysis and the active control of FGM plates with integrated piezoelectric sensors and actuators. Computational Mechanics, 31(3-4), 350-358.
Loja, M. A. R., Soares, M., & Barbosa, J. I. (2013). Analysis of functionally graded sandwich plate structures with piezoelectric skins, using B-spline finite strip method. Composite Structures,96, 606-615.
Ootao, Y., & Tanigawa, Y. (2000). Three-dimensional transient piezothermoelasticity in functionally graded rectangular plate bonded to a piezoelectric plate. International Journal of Solids and Structures, 37(32), 4377-4401.
Rahmani, O., Khalili, S. M. R., Malekzadeh, K., & Hadavinia, H. (2009). Free vibration analysis of sandwich structures with a flexible functionally graded syntactic core. Composite Structures, 91(2), 229-235.
Ray, M. C., & Batra, R. C. (2007). Vertically reinforced 1-3 piezoelectric composites for active damping of functionally graded plates. AIAA journal, 45(7), 1779-1784.
Tanigawa, Y. (2001). Control of the transient thermoelastic displacement of a functionally graded rectangular plate bonded to a piezoelectric plate due to nonuniform heating. Acta mechanica, 148(1-4), 17-33.
Tiersten, H. F. (1969). Linear piezoelectric plate vibrations: Elements of the linear theory of piezoelectricity and the vibrations of piezoelectric plates. Plenum Press.
Timoshenko, S., & Woinowsky-Krieger, S. (1959). Theory of plates and shells (Vol. 2, pp. 240-246). New York: McGraw-hill.
Vel, S. S., & Batra, R. C. (2001). Exact solution for rectangular sandwich plates with embedded piezoelectric shear actuators. AIAA journal, 39(7), 1363-1373.
Wang, Q., Quek, S. T., Sun, C. T., & Liu, X. (2001). Analysis of piezoelectric coupled circular plate. Smart Materials and Structures, 10(2), 229.
Yapeng, W.X.S. (2003) Three-dimensional solution for the free vibration of functionally gradient piezoelectric plates, Acta Mechanica Solida Sinica 1:010
Batra, R. C., & Geng, T. S. (2002). Comparison of active constrained layer damping by using extension and shear mode piezoceramic actuators. Journal of Intelligent Material Systems and Structures, 13(6), 349-367.
Batra, R. C., & Liang, X. Q. (1997). The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators. Computers & structures, 63(2), 203-216.
Behjat, B., & Khoshravan, M. R. (2012). Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates. Composite Structures, 94(3), 874-882.
Bian, Z. G., Lim, C. W., & Chen, W. Q. (2006). On functionally graded beams with integrated surface piezoelectric layers. Composite structures, 72(3), 339-351.
Ebrahimi, F., & Rastgo, A. (2008a). An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory. Thin-Walled Structures, 46(12), 1402-1408.
Ebrahimi, F., & Rastgoo, A. (2008b). Free vibration analysis of smart annular FGM plates integrated with piezoelectric layers. Smart Materials and Structures, 17(1), 015044.
Ebrahimi, F., Rastgoo, A., & Atai, A. A. (2009). A theoretical analysis of smart moderately thick shear deformable annular functionally graded plate. European Journal of Mechanics-A/Solids, 28(5), 962-973.
Es & apos; haghi, M., Hashemi, S. H., & Fadaee, M. (2011). Vibration analysis of piezoelectric FGM sensors using an accurate method. International Journal of Mechanical Sciences, 53(8), 585-594.
Fakhari, V., Ohadi, A., & Yousefian, P. (2011). Nonlinear free and forced vibration behavior of functionally graded plate with piezoelectric layers in thermal environment. Composite Structures, 93(9), 2310-2321.
Hashemi, S. H., Khorshidi, K., Es’haghi, M., Fadaee, M., & Karimi, M. (2012). On the effects of coupling between in-plane and out-of-plane vibrating modes of smart functionally graded circular/annular plates. Applied Mathematical Modelling, 36(3), 1132-1147.
He, X. Q., Ng, T. Y., Sivashanker, S., & Liew, K. M. (2001). Active control of FGM plates with integrated piezoelectric sensors and actuators. International Journal of Solids and Structures, 38(9), 1641-1655.
Huang, X. L., & Shen, H. S. (2006). Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments. Journal of Sound and Vibration, 289(1), 25-53.
Jandaghian, A. A., Jafari, A. A., & Rahmani, O. (2013). Transient Bending of a Circular Plate Integrated with Piezoelectric layers. Applied Mathematical Modelling,37(12-13), 7154-7163.
Kargarnovin, M. H., Najafizadeh, M. M., & Viliani, N. S. (2007). Vibration control of a functionally graded material plate patched with piezoelectric actuators and sensors under a constant electric charge. Smart materials and structures, 16(4), 1252.
Liew, K. M., He, X. Q., Ng, T. Y., & Kitipornchai, S. (2002). Active control of FGM shells subjected to a temperature gradient via piezoelectric sensor/actuator patches. International journal for numerical methods in engineering, 55(6), 653-668.
Liew, K. M., He, X. Q., Ng, T. Y., & Kitipornchai, S. (2003). Finite element piezothermoelasticity analysis and the active control of FGM plates with integrated piezoelectric sensors and actuators. Computational Mechanics, 31(3-4), 350-358.
Loja, M. A. R., Soares, M., & Barbosa, J. I. (2013). Analysis of functionally graded sandwich plate structures with piezoelectric skins, using B-spline finite strip method. Composite Structures,96, 606-615.
Ootao, Y., & Tanigawa, Y. (2000). Three-dimensional transient piezothermoelasticity in functionally graded rectangular plate bonded to a piezoelectric plate. International Journal of Solids and Structures, 37(32), 4377-4401.
Rahmani, O., Khalili, S. M. R., Malekzadeh, K., & Hadavinia, H. (2009). Free vibration analysis of sandwich structures with a flexible functionally graded syntactic core. Composite Structures, 91(2), 229-235.
Ray, M. C., & Batra, R. C. (2007). Vertically reinforced 1-3 piezoelectric composites for active damping of functionally graded plates. AIAA journal, 45(7), 1779-1784.
Tanigawa, Y. (2001). Control of the transient thermoelastic displacement of a functionally graded rectangular plate bonded to a piezoelectric plate due to nonuniform heating. Acta mechanica, 148(1-4), 17-33.
Tiersten, H. F. (1969). Linear piezoelectric plate vibrations: Elements of the linear theory of piezoelectricity and the vibrations of piezoelectric plates. Plenum Press.
Timoshenko, S., & Woinowsky-Krieger, S. (1959). Theory of plates and shells (Vol. 2, pp. 240-246). New York: McGraw-hill.
Vel, S. S., & Batra, R. C. (2001). Exact solution for rectangular sandwich plates with embedded piezoelectric shear actuators. AIAA journal, 39(7), 1363-1373.
Wang, Q., Quek, S. T., Sun, C. T., & Liu, X. (2001). Analysis of piezoelectric coupled circular plate. Smart Materials and Structures, 10(2), 229.
Yapeng, W.X.S. (2003) Three-dimensional solution for the free vibration of functionally gradient piezoelectric plates, Acta Mechanica Solida Sinica 1:010