Volume 43 Issue 8
Aug.  2022
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ZHANG Yuhang, LIU Wenguang, LIU Chao, LÜ Zhipeng. Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells[J]. Applied Mathematics and Mechanics, 2022, 43(8): 857-868. doi: 10.21656/1000-0887.420273
Citation: ZHANG Yuhang, LIU Wenguang, LIU Chao, LÜ Zhipeng. Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells[J]. Applied Mathematics and Mechanics, 2022, 43(8): 857-868. doi: 10.21656/1000-0887.420273

Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells

doi: 10.21656/1000-0887.420273
  • Received Date: 2021-09-09
  • Accepted Date: 2021-11-03
  • Rev Recd Date: 2021-10-26
  • Available Online: 2022-07-01
  • Publish Date: 2022-08-01
  • The nonlinear vibration responses of functionally graded materials (FGMs) shells with different cone angles under external loads were studied. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the 1st-order shear deformation theory, the von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations were simplified into a 1DOF nonlinear vibration differential equation under Volmir’s assumption. Finally, the nonlinear motion equations were solved with the harmonic balance method and the Runge-Kutta method, and the amplitude frequency response characteristic curves of the FGMs conical shells were obtained. The effects of different material distribution functions and different ceramic volume fraction exponents on the amplitude frequency response curves of conical shells were discussed. The bifurcation diagrams of conical shells with different cone angles, as well as time process diagrams and phase diagrams for different excitation amplitudes, were described. The motion characteristics were characterized by Poincaré maps. The results show that, the FGMs conical shells present the nonlinear characteristics of hardening springs. The chaotic motions of the FGMs conical shells are restrained and not prone to motion instability with the increase of the cone angle. The FGMs conical shell present a process from the periodic motion to the multi-periodic motion and then to chaos with the increase of the excitation amplitude.

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