Time periodic solutions to the beam equation with weak damping

Wang, Yu‐Zhu; Li, Yanshuo

doi:10.1063/1.5046821

Cited by 6 publications

(3 citation statements)

References 28 publications

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“…It is worth mentioning that Kosovali ć4 has investigated quasi-periodic traveling waves for Equation (1.1) with 𝜅 = 1 on a 2-dimensional standard torus and established the existence of small amplitude quasi-periodic traveling wave solutions with two frequencies, which are continuations of rotating wave solutions. Some recent results on damped beam equations can be found in previous studies [7][8][9][10] and the references therein.…”

Section: Backgrounds and Main Ideasmentioning

confidence: 95%

Quasi‐periodic traveling waves for a class of damped beams on rectangular tori

Chen

Gao

Nieto

2021

Math Methods in App Sciences

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This article concerns a class of beam equations with damping on rectangular tori. When the generators satisfy certain relationship, by excluding some values of two model parameters, we prove that such models admit small amplitude quasi-periodic traveling wave solutions with two frequencies, which are continuations of two rotating wave solutions with one frequency. The result holds not only for a rational torus but also for an irrational one. The proof is mainly based on a Lyapunov-Schmidt reduction together with the implicit function theorem.

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Section: Backgrounds and Main Ideasmentioning

confidence: 95%

Quasi‐periodic traveling waves for a class of damped beams on rectangular tori

Chen

Gao

Nieto

2021

Math Methods in App Sciences

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“…under Dirichlet boundary conditions. Furthermore, we refer the readers to [22] introduced by Wang and Li for the following beam equation with damping and periodic external forcing…”

Section: Introductionmentioning

confidence: 99%

Periodic Solutions for a Damped Rayleigh Beam Model with Time Delay

Li¹

2020

CMR

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Vibrations of a beam can be described as an Euler-Bernoulli beam, or as a Rayleigh beam or as a Timoshenko beam. In this paper, we establish the existence of periodic solutions in time for a damped Rayleigh beam model with time delay, which is treated as a bifurcation parameter. The main proof is based on a Lyapunov-Schmidt reduction together with the classical implicit function theorem. Moreover, we give a sufficient condition for a direction of bifurcation.

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“…For the space dimension d > 1, recently, Wang and Li [36] studied the beam equation with weak damping…”

Section: Introductionmentioning

confidence: 99%

Periodic solutions of a semilinear Euler-Bernoulli beam equation with variable coefficients

Wei,

2020

Preprint

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This paper is devoted to the study of periodic solutions for a semilinear Euler-Bernoulli beam equation with variable coefficients. Such mathematical model may be described the infinitesimal, free, undamped in-plane bending vibrations of a thin straight elastic beam. When the frequency ω = 2π T is rational, some properties of the beam operator with variable coefficients are investigated. We obtain the existence of periodic solutions when the nonlinear term is monotone and bounded.

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Time periodic solutions to the beam equation with weak damping

Cited by 6 publications

References 28 publications

Quasi‐periodic traveling waves for a class of damped beams on rectangular tori

Quasi‐periodic traveling waves for a class of damped beams on rectangular tori

Periodic Solutions for a Damped Rayleigh Beam Model with Time Delay

Periodic solutions of a semilinear Euler-Bernoulli beam equation with variable coefficients

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