An enriched multiple scales method for harmonically forced nonlinear systems

Cacan, Martin R.; Leadenham, Stephen; Leamy, Michael J.

doi:10.1007/s11071-014-1508-9

Cited by 16 publications

(5 citation statements)

References 40 publications

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“…It is worth mentioning that the homotopy method does not need to describe the nearness of excitation frequency to natural frequency. It regards that the final vibration frequency is equal to the excitation frequency [ 26 ]. By introducing an embedding parameter, this method can effectively describe the transition process between an ideal linear system and a practical nonlinear system.…”

Section: Primary Frequency Responsementioning

confidence: 99%

“…In Section 2 , the governing equation of motion of a folded-MEMS comb drive resonator is introduced. In Section 3 , the MMS combined with the homotopy concept is applied to deduce the primary resonance solution [ 25 , 26 ]. A case study is carried out to reveal the existence of low and high-energy branches and intersection properties under primary resonance conditions.…”

Section: Introductionmentioning

confidence: 99%

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Vibration Identification of Folded-MEMS Comb Drive Resonators

Han

Jin

et al. 2018

Micromachines

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Natural frequency and frequency response are two important indicators for the performances of resonant microelectromechanical systems (MEMS) devices. This paper analytically and numerically investigates the vibration identification of the primary resonance of one type of folded-MEMS comb drive resonator. The governing equation of motion, considering structure and electrostatic nonlinearities, is firstly introduced. To overcome the shortcoming of frequency assumption in the literature, an improved theoretical solution procedure combined with the method of multiple scales and the homotopy concept is applied for primary resonance solutions in which frequency shift due to DC voltage is thoroughly considered. Through theoretical predictions and numerical results via the finite difference method and fourth-order Runge-Kutta simulation, we find that the primary frequency response actually includes low and high-energy branches when AC excitation is small enough. As AC excitation increases to a certain value, both branches intersect with each other. Then, based on the variation properties of frequency response branches, hardening and softening bending, and the ideal estimation of dynamic pull-in instability, a zoning diagram depicting extreme vibration amplitude versus DC voltage is then obtained that separates the dynamic response into five regions. Excellent agreements between the theoretical predictions and simulation results illustrate the effectiveness of the analyses.

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Section: Primary Frequency Responsementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Vibration Identification of Folded-MEMS Comb Drive Resonators

Han

Jin

et al. 2018

Micromachines

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“…Repka et al 6 applied the Timoshenko beam model in the analysis of the flexoelectric effect for a cantilever beam under large deformations, and considered the geometric nonlinearity with von Kármán strains. Meanwhile, some methods, such as a homotopy analysis method 7 , a rational elliptic balance method 8 , an enriched multiple scales method 9 , and an improved homotopy analysis method 10 , 11 , etc., have been gradually developed to solve nonlinear differential equations.…”

Section: Introductionmentioning

confidence: 99%

Chaotic vibration control of a composite cantilever beam

Liu,

Sun

2023

Sci Rep

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In this research, an adaptive control strategy adapted from fuzzy sliding mode control is established and applied in chaotic vibration control of a multiple-dimension nonlinear dynamic system of a laminated composite cantilever beam. The third order shearing effect on the vibration of the beam is considered in the nonlinear dynamic model establishment, and the Hamilton principle as well as the Galerkin method is employed. It is discovered that a multi-dimensional nonlinear dynamic system of the cantilever beam needs to be considered for accurate vibration estimation. Therefore, the control strategy appropriate for the chaotic vibration control of a multiple-dimension system of the laminated composite beam is necessary, and then proves to be effective in chaotic vibration control in numerical simulation.

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“…Repka [6] et al applied the Timoshenko beam model to the analysis of the flexoelectric effect for a cantilever beam under large deformations, and considered the geometric nonlinearity with von Kármán strains. Meanwhile, some methods, such as homotopy analysis method [7], rational elliptic balance method [8], enriched multiple scales method [9], improved homotopy analysis method [10][11], etc, have been gradually developed to solve nonlinear differential equations.…”

Section: Introductionmentioning

confidence: 99%

Chaotic Vibration Control of a Composite Cantilever Beam

Liu

Sun

2023

Preprint

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In this research, an adaptive control strategy adapted from Fuzzy Sliding Mode Control is established and applied in chaotic vibration control of a multiple-dimension nonlinear dynamic system of a laminated composite cantilever beam. The 3rd order shearing effect on the vibration of the beam is considered in the nonlinear dynamic model establishment, and the Hamilton principle as well as the Galerkin method is employed. It is discovered that a multi-dimensional nonlinear dynamic system of the cantilever beam needs to be considered for accurate vibration estimation. Therefore, the control strategy appropriate for the chaotic vibration control of a multiple-dimension system of the laminated composite beam is necessary, and then proves to be effective in chaotic vibration control in numerical simulation.

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An enriched multiple scales method for harmonically forced nonlinear systems

Cited by 16 publications

References 40 publications

Vibration Identification of Folded-MEMS Comb Drive Resonators

Vibration Identification of Folded-MEMS Comb Drive Resonators

Chaotic vibration control of a composite cantilever beam

Chaotic Vibration Control of a Composite Cantilever Beam

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