Nonlinear vibrations and chaotic phenomena of functionally graded material truncated conical shell subject to aerodynami
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O R I G I NA L
S. W. Yang · Y. X. Hao · L. Yang · L. T. Liu
Nonlinear vibrations and chaotic phenomena of functionally graded material truncated conical shell subject to aerodynamic and in-plane loads under 1:2 internal resonance relation Received: 16 April 2020 / Accepted: 17 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper focuses on the nonlinear dynamic responses of a functionally graded material (FGM) truncated conical shell under 1:2 internal resonance relation. The FGM truncated conical shell is subjected to the in-plane load and the aerodynamic load along the meridian direction. According to a power-law distribution, the material properties are assumed to be modified along the thickness direction smoothly and continuously and the material properties are temperature dependent. The aerodynamic load is obtained by the first-order piston theory with the curvature correction term. According to von Karman type nonlinear geometric relations, first-order shear deformation shell theory, Hamilton principle, the nonlinear equations of motion for the FGM truncated conical shell are established. Furthermore, the nonlinear equations of motion are reduced into a system of the ordinary differential equations by utilizing Galerkin procedure. The multiple scales method is used to obtain the averaged equations for the FGM truncated conical shell under the relations of 1:2 internal resonance and 1/2 subharmonic resonance. The frequency–response curves, time history diagrams, phase portraits, Poincare maps and bifurcation diagrams with different parameters are yielded by employing numerical calculations. The influences of exponent of volume fraction, Mach number, damping coefficient and in-plane load on the nonlinear resonance behaviors of the FGM truncated conical shell are investigated. The chaotic and periodic motions of the FGM truncated conical shell have been discussed in detail. Keywords Nonlinear vibrations · Chaos · Functionally graded materials · 1:2 Internal resonance · Conical shell
1 Introduction Functionally graded materials (FGMs) are famous for their excellence thermomechanical performances, and the FGM structures inspired researchers to explore the dynamics which are often used in the high strength or high temperature environments [1,2]. Because of the continuous and smooth varies of the volume fractions of the constituent materials for the functionally graded materials, the material properties, for example the S. W. Yang · Y. X. Hao (B) · L. Yang · L. T. Liu College of Mechanical Engineering and Beijing Key Laboratory of Electromechanical System Measurement and Control, Beijing Information Science and Technology University, Beijing 100192, People’s Republic of China E-mail: [email protected] S. W. Yang E-mail: [email protected] L. Yang E-mail: [email protected] L. T. Liu E-mail: [email protected]
S. W. Yang et al.
Young’s modulus, the thermal expansion coefficient and the density, all change continuously. Therefore, the FGM structures can relieve the problems
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