Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under three-to-one internal resonance condition

Mamandi, Ahmad; Kargarnovin, M. H.; Farsi, Salman

doi:10.1007/s11071-012-0520-1

Cited by 15 publications

(7 citation statements)

References 25 publications

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“…By employing the assumptions and transition variables (equation (3)), the dimensionless equation of the EFB model is ( Öz and Pakdemirli, 2006;Mamandi et al, 2012;Sato et al, 2007Sato et al, , 2008…”

Section: Efb Model With Uniform Linear Stiffnessmentioning

confidence: 99%

Two elastic supporting models to simulate the submerged floating tunnel and their equivalence on the free/forced vibrations

Yan

Pan

et al. 2023

Journal of Vibration and Control

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By applying two elastically supporting styles, the elastic supports beam (ESB) and elastic foundation beam (EFB) are proposed to simulate the underwater submerged floating tunnel (SFT) under hinged boundaries. Their equivalence is investigated through the frequencies/modes and forced responses caused by equidistant moving loads. The ESB model is built by regarding each tether as an elastic support and using the novel isolating/assembling technique between beam-segments and global tube, and the order of characteristic matrix for any number of intermediate elastic supports is reduced to 4 × 4. Then, the stiffness of all elastic supports and the uniformity/non-uniformity are well considered in the ESB model. The governing equations of the EFB model with a uniform linear elastic support are also given. The newly discovered results, which are verified by the finite element method (FEM), indicate the merging phenomena appear in the frequency spectrum in the ESB model. The equivalence holds for some uniform cases in the ESB model based on the peak values of responses by using the stiffness relationship and a correction coefficient, whereas the equivalence does not exist for the uneven case of elastic supports.

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Section: Efb Model With Uniform Linear Stiffnessmentioning

confidence: 99%

Two elastic supporting models to simulate the submerged floating tunnel and their equivalence on the free/forced vibrations

Yan

Pan

et al. 2023

Journal of Vibration and Control

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“…where = √ (( / ) 2 + ( / ) 2 ) − ( / ) 2 , and are the acoustic mode numbers, and are the numbers of acoustic modes used in the and directions, and and are coefficients to be determined in the following step. Substituting (23) into (22c) and 22dgives…”

Section: Air Cavitymentioning

confidence: 99%

“…These works are relevant to the nonlinear structural-acoustic problem handled in this paper, but the researchers adopted the solution methods which required heavy computational efforts (e.g., finite element method, numerical integration method, and harmonic balance method). Other solution methods for large amplitude structural vibrations and nonlinear oscillations (e.g., the method of multiple scales, the method of normal forms, the method of Shaw and Pierre, and the method of King and Vakakis [23][24][25][26][27]) also require relatively tedious solution implementation and heavy computational efforts. In [28], the simple and straightforward electroacoustic analogy was adopted for modelling of the sound absorption of a panel absorber.…”

Section: Introductionmentioning

confidence: 99%

Analytic Formulation for the Sound Absorption of a Panel Absorber under the Effects of Microperforation, Air Pumping, Linear Vibration and Nonlinear Vibration

Lee

2014

Abstract and Applied Analysis

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This study includes the first work about the absorption of a panel absorber under the effects of microperforation, air pumping, and linear and nonlinear vibrations. In practice, thin perforated panel absorber is backed by a flexible wall to enhance the acoustic performance within the room. The panel is easily excited to vibrate nonlinearly and the wall can vibrate linearly. However, the assumptions of linear panel vibration and no wall vibration are adopted in many research works. The development of the absorption formula is based on the classical approach and the electroacoustic analogy, in which the impedances of microperforation, air pumping, and linear and nonlinear vibrations are in parallel and connected to that of the air cavity in series. Unlike those finite element, numerical integration, and multiscale solution methods and so forth, the analytic formula to calculate the absorption of a panel absorber does not require heavy computation effort and is suitable for engineering calculation purpose. The theoretical result obtained from the proposed method shows reasonable agreement with that from a previous numerical integration method. It can be concluded that the overall absorption bandwidth of a panel absorber with an appropriate configuration can be optimized and widened by making use of the positive effects of microperforation, air pumping, and panel vibration.

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“…On the other hand, there were four analytical solution techniques, such as the method of multiple scales, the method of Normal Forms, the method of Shaw and Pierre, and the method of King and Vakakis, applied to various nonlinear problems (e.g. King and Vakakis, 1994; Shaw and Pierre, 1994; Ghayesh and Balar, 2008; Nayfeh, 2011; Mamandi et al., 2012). All of these methods are required to solve the large numbers of nonlinear algebraic equations simultaneously for the multi-mode formulations of various structures.…”

Section: Introductionmentioning

confidence: 99%

The multi-level residue harmonic balance solutions of multi-mode nonlinearly vibrating beams on an elastic foundation

Hasan

Lee

Leung

2014

Journal of Vibration and Control

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In this paper, an efficient analytical solution method, namely, multi-level residue harmonic balance, is introduced and developed for the nonlinear vibrations of multi-mode flexible beams on an elastic foundation subject to external harmonic excitation. The main advantage of this solution method is that only one set of nonlinear algebraic equations is generated in the zero level solution procedure while the higher level solutions for any desired accuracy can be obtained by solving a set of linear algebraic equations. In other words, the computation effort to find more accurate nonlinear solutions is much less. In this paper, a multi-mode formulation, which represents the nonlinear beam vibration, is derived and set up. Then, the solution procedures are developed for obtaining the nonlinear multi-mode solution. The results from the multi-level residue harmonic balance method agree well with those from a numerical integration method. The effects of various parameters such as vibration amplitude, foundation modulus coefficient, damping factor and excitation level etc., on the nonlinear behaviors are examined. A convergence study is also performed to verify the solutions. The stability analysis is conducted using the virtue of Floquet theory and steady-state solutions are investigated.

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Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under three-to-one internal resonance condition

Cited by 15 publications

References 25 publications

Two elastic supporting models to simulate the submerged floating tunnel and their equivalence on the free/forced vibrations

Two elastic supporting models to simulate the submerged floating tunnel and their equivalence on the free/forced vibrations

Analytic Formulation for the Sound Absorption of a Panel Absorber under the Effects of Microperforation, Air Pumping, Linear Vibration and Nonlinear Vibration

The multi-level residue harmonic balance solutions of multi-mode nonlinearly vibrating beams on an elastic foundation

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