Analysis and Comparison of Nonlinear Dynamic Behavior of Viscoelastic and Elastic Rectangular Plates under Supersonic Aerodynamic Flow
Authors
Abstract
Complex nonlinear behaviors such as chaotic motion have devastating effects on dynamic systems. In this study, nonlinear behavior of simply supported rectangular viscoelastic plates was examined during supersonic aerodynamics and compared with the nonlinear elastic plate. Classical plate theory was used to obtain the plate equations, and Von- Kármán strain-displacement relations were used to consider the nonlinear geometric effects. The Kelvin Voigt model was also used to describe the viscoelastic properties and the “first-order piston theory" was used for supersonic aerodynamic flow. The equations of motion of the rectangular plate were extracted using the Lagrangian method and then, discretized by the Rayleigh-Ritz method. Solution of the equations was performed using fourth order Runge Kutta method. To investigate the dynamic behavior of the plates, the eigenvalues of the system, time history curves, phase portraits, Poincaré maps, and bifurcation diagrams were studied and analyzed. The results show that in some aspect ratios, the threshold for the occurrence of the flutter in the viscoelastic plate will be lower than that in the elastic plate. On the other hand, when the control parameter increases, complex nonlinear behavior such as chaos in the elastic plate goes simpler in the viscoelastic plate, such as periodic motion.
Keywords
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4. Xie, D., Xu, M., Dai, H. and Dowell, E. H., “Observation and Evolution of Chaos for a Cantilever Plate in Supersonic Flow”, Journal of Fluids and Structures, Vol, 50, pp. 271-291, 2014.
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6. Lukasiewicz, S. and Xia, Z. Q., “Nonlinear Damped Vibrations of Simply- Supported Sandwich Plates in a Rapidly Changing Temperature Field”, Nonlinear Dynamics, Vol. 9, pp. 369–389, 1996.
7. Sun, Y. X. and Zhang, S. Y., “Chaotic Dynamic Analysis of Viscoelastic Plates”, International Journal of Mechanical Sciences, Vol. 43, No. 5, pp. 1195-1208, 2001.
8. Pourtakdoust, S. H. and Fazelzadeh, S. A., “Chaotic Analysis Of Nonlinear Viscoelastic Panel Flutter in Supersonic Flow:, Nonlinear Dynamics, Vol. 32, No. 4, pp. 387-404, 2003.
9. Khudayarov, B., “Flutter of a Viscoelastic Plate in a Supersonic Gas Flow”, International Applied Mechanics, Vol. 46, No. 4, pp. 455-460, 2010.
10. Merrett, C. G. and Hilton, H., “Elastic and Viscoelastic Panel Flutter in Incompressible, Subsonic and Supersonic Flows”, ASDJournal, Vol. 2, No. 1, pp, 53-80, 2010.
11. Saksa, T., Banichuk, N., Jeronen, J., Kurki, M. and Tuovinen, T., “Dynamic Analysis for Axially Moving Viscoelastic Panels”, International Journal of Solids and Structures,Vol. 49, pp. 3355-3366, 2012.
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13. Yang, X. D., Yu, T. J., Zhang, W., Qian, Y. J. and Yao, M. H.,“ Damping Effect on Supersonic Panel Flutter of Composite Plate with Viscoelastic Mid- Layer”, Composite Structures, Vol. 137, pp. 105-113, 2016.
14. Cunha- Filho, A. G., De Lima, A. M. G., Donadon, M. V., and Leão, L. S., “Flutter Suppression of Plates Using Passive Constrained Viscoelastic Layers”, Mechanical Systems and Signal Processing, Vol 79, pp. 99-111, 2016.
15. Usmonov, B. Sh., “Dynamic Instability of Viscoelastic Plate in Supersonic Flow”, International Journal of Advanced Engineering, Management and Science (IJAEMS), Vol. 3, No. 2, pp. 35-39, 2017.
16. Wang, X., Yang, Z., Wang, W. and Tian, W. Nonlinear Viscoelastic Heated Panel Flutter with Aerodynamic Loading Exerted on Both Surfaces”, Journal of Sound and Vibration, Vol. 409, pp. 306-317, 2017.
17. Rade, D. A., Deü, J.- F., Castello, D. A., de Lima, A. M. G. and Rouleau, L., Nonlinear Structural Dynamics and Damping chapter 5: Passive Vibration Control Using Viscoelastic Materials, Mechanisms and Machine Science, Vol. 69, pp. 119-168, Springer Nature Switzerland AG, 2019.
18. Sherov, A.G., Khudayarov, B.A., Ruzmetov, K.Sh. and Aliyarov, J., “Numerical Investigation of the Effects Angles of Attack on the Flutter of a Viscoelastic Plate”, Advances in Aircraft and Spacecraft Science, Vol. 3, pp. 215-228, 2020.
19. Khudayarov, B., Turayev, F., Zhuvonov, Q., Vahobov, V., Kucharov, O., and Kholturaev, Kh, “Oscillation Modeling of Viscoelastic Elements of Thin- Walled Structures”, In IOP Conference Series: Materials Science and Engineering, Vol. 883, No. 1, p. 012188. IOP Publishing, 2020.
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21. Ashley, H., Zartarian, G., “Piston Theory a New Aerodynamic Tools for the Aeroelastician”, Vol. 23, No. 12, pp. 1109-1118, 1956.
22. Dowell, E. H., Aeroelasticity of Plates and Shells, Noordhoff, Leyden, 1975.
23. Xue, D. Y., “Finite Element Frequency Domain Solution of Nonlinear Plate Flutter WithTemperature Effects and Fatigue Life Analysis”, PhD dissertation, Engineering Mechanics, OldDominion University, Norfolk, VA; 1991.
24. Rao, S. S., Vibration of Continuous Systems, John Wiley and Sons, Hoboken, New Jersey 2007.
25. Valizadeh, N., Natarajan, S., Gonzalez- Estrada, O. A., Rabczuk, T., Bui, T. Q. and Bordas, S. P., “NURBS- Based Finite Element Analysis of Functionally Graded Plates: Static Bending, Vibration, Buckling and Flutter”, Composite Structures, Vol. 99, pp.309-326, 2013.
26. Grover, N., Maiti; N. K. and Singh, B. N., “Flutter Characteristics of Laminated Composite Plates Subjected to Yawed Supersonic Flow Using Inverse Hyperbolic Shear Deformation Theory”, Journal of Aerospace Engineering, Vol. 29, No. 2, p. 04015038, 2016.
27. Xie, D., Xu, M., Dai, H., and Dowell, E. H., “Proper Orthogonal Decomposition Method for Analysis of Nonlinear Panel Flutter with Thermal Effects in Supersonic Flow”, Journal of Sound and Vibration, 337, pp. 263-283, 2015.
28. Abdel- Motaglay, K., Chen, R., Mei, C., “Nonlinear Flutter of Composite Oanels under Yawed Supersonic Flow Using Finite Elements”, AIAA Journal, Vol. 37, No. 9, pp. 1025-1032, 1999.
29. AsadiGorgi, H., Dardel, M., and Pashaei, M .H., Effects of All- Over Part- Through Cracks on the Aeroelastic Characteristics of Rectangular Panels”, Applied Mathematical Modelling, Vol. 39, No. 23-24, pp. 7513-7536, 2015.
2. Dowell, E. H., “Nonlinear Oscillations of a Fluttering Plate II”, AIAA Journal, Vol. 5, No. 10, pp 1856-1862,1967.
3. Weiliang, Y. and Dowell, E. H., “Limit Cycle Oscillation of a Fluttering Cantilever Plate”, AIAA Journal, Vol. 29, No. 11, pp. 1929-1936. 1991.
4. Xie, D., Xu, M., Dai, H. and Dowell, E. H., “Observation and Evolution of Chaos for a Cantilever Plate in Supersonic Flow”, Journal of Fluids and Structures, Vol, 50, pp. 271-291, 2014.
5. Xia, Z.Q. and Lukasiewicz, S., “Non- Linear, Free, Damped Vibrations of Sandwich Plates”, Journal of Sound and Vibration, Vol, 175, No. 2, pp.219-232, 1994.
6. Lukasiewicz, S. and Xia, Z. Q., “Nonlinear Damped Vibrations of Simply- Supported Sandwich Plates in a Rapidly Changing Temperature Field”, Nonlinear Dynamics, Vol. 9, pp. 369–389, 1996.
7. Sun, Y. X. and Zhang, S. Y., “Chaotic Dynamic Analysis of Viscoelastic Plates”, International Journal of Mechanical Sciences, Vol. 43, No. 5, pp. 1195-1208, 2001.
8. Pourtakdoust, S. H. and Fazelzadeh, S. A., “Chaotic Analysis Of Nonlinear Viscoelastic Panel Flutter in Supersonic Flow:, Nonlinear Dynamics, Vol. 32, No. 4, pp. 387-404, 2003.
9. Khudayarov, B., “Flutter of a Viscoelastic Plate in a Supersonic Gas Flow”, International Applied Mechanics, Vol. 46, No. 4, pp. 455-460, 2010.
10. Merrett, C. G. and Hilton, H., “Elastic and Viscoelastic Panel Flutter in Incompressible, Subsonic and Supersonic Flows”, ASDJournal, Vol. 2, No. 1, pp, 53-80, 2010.
11. Saksa, T., Banichuk, N., Jeronen, J., Kurki, M. and Tuovinen, T., “Dynamic Analysis for Axially Moving Viscoelastic Panels”, International Journal of Solids and Structures,Vol. 49, pp. 3355-3366, 2012.
12. Amabili, M., “Nonlinear Vibrations of Viscoelastic Rectangular Plates”, Journal of Sound and Vibration,Vol 362, pp. 142-156 2016.
13. Yang, X. D., Yu, T. J., Zhang, W., Qian, Y. J. and Yao, M. H.,“ Damping Effect on Supersonic Panel Flutter of Composite Plate with Viscoelastic Mid- Layer”, Composite Structures, Vol. 137, pp. 105-113, 2016.
14. Cunha- Filho, A. G., De Lima, A. M. G., Donadon, M. V., and Leão, L. S., “Flutter Suppression of Plates Using Passive Constrained Viscoelastic Layers”, Mechanical Systems and Signal Processing, Vol 79, pp. 99-111, 2016.
15. Usmonov, B. Sh., “Dynamic Instability of Viscoelastic Plate in Supersonic Flow”, International Journal of Advanced Engineering, Management and Science (IJAEMS), Vol. 3, No. 2, pp. 35-39, 2017.
16. Wang, X., Yang, Z., Wang, W. and Tian, W. Nonlinear Viscoelastic Heated Panel Flutter with Aerodynamic Loading Exerted on Both Surfaces”, Journal of Sound and Vibration, Vol. 409, pp. 306-317, 2017.
17. Rade, D. A., Deü, J.- F., Castello, D. A., de Lima, A. M. G. and Rouleau, L., Nonlinear Structural Dynamics and Damping chapter 5: Passive Vibration Control Using Viscoelastic Materials, Mechanisms and Machine Science, Vol. 69, pp. 119-168, Springer Nature Switzerland AG, 2019.
18. Sherov, A.G., Khudayarov, B.A., Ruzmetov, K.Sh. and Aliyarov, J., “Numerical Investigation of the Effects Angles of Attack on the Flutter of a Viscoelastic Plate”, Advances in Aircraft and Spacecraft Science, Vol. 3, pp. 215-228, 2020.
19. Khudayarov, B., Turayev, F., Zhuvonov, Q., Vahobov, V., Kucharov, O., and Kholturaev, Kh, “Oscillation Modeling of Viscoelastic Elements of Thin- Walled Structures”, In IOP Conference Series: Materials Science and Engineering, Vol. 883, No. 1, p. 012188. IOP Publishing, 2020.
20. Lakes, R., Viscoelastic Materials, Cambridge University Press, NewYork, USA, 2009.
21. Ashley, H., Zartarian, G., “Piston Theory a New Aerodynamic Tools for the Aeroelastician”, Vol. 23, No. 12, pp. 1109-1118, 1956.
22. Dowell, E. H., Aeroelasticity of Plates and Shells, Noordhoff, Leyden, 1975.
23. Xue, D. Y., “Finite Element Frequency Domain Solution of Nonlinear Plate Flutter WithTemperature Effects and Fatigue Life Analysis”, PhD dissertation, Engineering Mechanics, OldDominion University, Norfolk, VA; 1991.
24. Rao, S. S., Vibration of Continuous Systems, John Wiley and Sons, Hoboken, New Jersey 2007.
25. Valizadeh, N., Natarajan, S., Gonzalez- Estrada, O. A., Rabczuk, T., Bui, T. Q. and Bordas, S. P., “NURBS- Based Finite Element Analysis of Functionally Graded Plates: Static Bending, Vibration, Buckling and Flutter”, Composite Structures, Vol. 99, pp.309-326, 2013.
26. Grover, N., Maiti; N. K. and Singh, B. N., “Flutter Characteristics of Laminated Composite Plates Subjected to Yawed Supersonic Flow Using Inverse Hyperbolic Shear Deformation Theory”, Journal of Aerospace Engineering, Vol. 29, No. 2, p. 04015038, 2016.
27. Xie, D., Xu, M., Dai, H., and Dowell, E. H., “Proper Orthogonal Decomposition Method for Analysis of Nonlinear Panel Flutter with Thermal Effects in Supersonic Flow”, Journal of Sound and Vibration, 337, pp. 263-283, 2015.
28. Abdel- Motaglay, K., Chen, R., Mei, C., “Nonlinear Flutter of Composite Oanels under Yawed Supersonic Flow Using Finite Elements”, AIAA Journal, Vol. 37, No. 9, pp. 1025-1032, 1999.
29. AsadiGorgi, H., Dardel, M., and Pashaei, M .H., Effects of All- Over Part- Through Cracks on the Aeroelastic Characteristics of Rectangular Panels”, Applied Mathematical Modelling, Vol. 39, No. 23-24, pp. 7513-7536, 2015.
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