The exact solutions of the nonlinear vibration systems are extremely complicated to be received, so it is crucial to analyze their approximate solutions. This paper employs the spreading residue harmonic balance method (SRHBM) to derive analytical approximate solutions for the fifth-order nonlinear problem, which corresponds to the strongly nonlinear vibration of an elastically restrained beam with a lumped mass. When the SRHBM is used, the residual terms are added to improve the accuracy of approximate solutions. Illustrative examples are provided along with verifying the accuracy of the present method and are compared with the HAM solutions, the EBM solutions, and exact solutions in tables. At the same time, the phase diagrams and time history curves are drawn by the mathematical software. Through analysis and discussion, the results obtained here demonstrate that the SRHBM is an effective and robust technique for nonlinear dynamical systems. In addition, the SRHBM can be widely applied to a variety of nonlinear dynamic systems.
This paper investigates the stability behavior in three-degree-freedom composite laminated piezoelectric rectangular plate system based on the Routh-Hurwitz criterion and the normal form theory. Firstly, the stability behavior of the composite laminated piezoelectric rectangular plate is determined by applying the Routh-Hurwitz criterion. Then, the normal forms of the composite laminated piezoelectric rectangular plate systems for the case of one non-semi-simple double zero and two pairs of pure imaginary eigenvalues are obtained by using the normal form theory. Finally, numerical simulations are carried out to illustrate the main results.
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