Multi-level residue harmonic balance method for nonlinear vibration of the beam

Hasan, Abu; Rahman, Mohammad Shaifur

doi:10.1177/14613484211038403

Cited by 2 publications

(1 citation statement)

References 27 publications

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“…The result from the proposed HBM agrees well with the result from Runge–Kutta’s numerical integration method. Hasan and Rahman (2022) used the multi-level residue HBM for solving the geometrically cubic nonlinear vibration of the simply supported beam. The results were evaluated using a numerical integration method.…”

Section: Introductionmentioning

confidence: 99%

A harmonic balance solution for the intrinsic 1D nonlinear equations of the beams

Siami

Nitzsche

2023

Journal of Vibration and Control

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In this paper, a harmonic balance method is introduced to solve the intrinsic 1D nonlinear equations presented by Hodges for analyzing initially curved and twisted anisotropic rotating beams. First, the nonlinear first-order partial differential equations are discretized in space domain using the Galerkin approach. Then, the time response of the system of equations is approximated by using first and the second harmonic terms under the harmonic excitations applied to the structure. The harmonic balance algorithm has been developed for the linearized system of equations about the steady-state solution and for fully nonlinear equations. The differences between these two systems have been addressed under consideration of different levels of external loads. Fully analytical formulae have been presented for the HB solution of the linearized equations in this work. In the proposed approach for the nonlinear system, the Jacobian matrix utilized in the numerical solution part is obtained analytically to increase the time efficiency of the solution method. The results of the proposed approach for an initially twisted and curved blade are compared with a different algorithm based on a finite difference approach to discretize the equations and a time marching method to solve the system of equations in time domain. The comparison between the outputs of the harmonic balance method and the time marching method demonstrate high correlations between these distinct methods. It should be noted that the accuracy of the time marching method solution has been evaluated in another publication by the authors using the output of FLIGHTLAB for a blade with different types of boundary conditions. The proposed harmonic balance-based method delivers a very time-efficient approach for time analysis of twisted/curved beams and blades under harmonic loads in comparison with other employed methods. The developed approach can be a powerful computational tool for design optimization of composite blades.

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Section: Introductionmentioning

confidence: 99%

A harmonic balance solution for the intrinsic 1D nonlinear equations of the beams

Siami

Nitzsche

2023

Journal of Vibration and Control

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Chaotic control of a simply supported beam in a multidimensional system

Liu,

Xun,

2024

Sci Rep

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In this work, a control strategy based on fuzzy sliding mode control (FSMC) is applied to effectively manage the large-amplitude chaotic vibrations exhibited by a simply supported beam. The analysis considers the nonlinear beam structure, which is subjected to an external load. Hamilton’s principle is employed, and the equations of motion are derived for the studied structure. The 3rd Galerkin discretization is employed to derive the ordinary differential governing equation of the structure. A control strategy is presented to decrease the response of the obtained multidimensional system. The vibration of a one-dimensional system is compared with that of the derived multidimensional system. As shown throughout the study, a multidimensional nonlinear system of the structure must be considered for accurate dynamic estimation. By using the recurrent neural network (RNN) model, we can accurately predict chaotic motion and effectively apply control strategies to suppress chaotic motion. The efficacy and suitability of the employed control strategy have been demonstrated by controlling chaos in the beam’s multidimensional system.

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Multi-level residue harmonic balance method for nonlinear vibration of the beam

Cited by 2 publications

References 27 publications

A harmonic balance solution for the intrinsic 1D nonlinear equations of the beams

A harmonic balance solution for the intrinsic 1D nonlinear equations of the beams

Chaotic control of a simply supported beam in a multidimensional system

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