Optimal vibration quenching for an Euler-Bernoulli beam
type: Item
instance of: scholarly article
title: Optimal vibration quenching for an Euler-Bernoulli beam
zbMATH Open document ID: 0937.74048
author: Yanyan Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-06-07T00:00:00Z
review text: The authors study the problem of the optimal vibration quenching for Euler-Bernoulli beam under tension with general linear homogeneous boundary condition. The problem can be written in the form \(m(x)W_{tt}+ L[W]= u(x,t)\), \(0<x<l\), \(0<t<T\), where \(L[W]= [EI(x)W_{xx}]_{xx}- [P(x)W_x]_x\), \(W(x,t)\) -- the displacement, \(m\) -- the unit length mass, \(P\) -- the axial tension force, \(E\) -- the Young's modulus, \(I\) -- the inertia moment, and \(T\) -- the terminal time. The initial and boundary conditions are \(W(x,0)= f(x)\), \(W_t(x,0)= g(x)\), \(f\in H^1(0,l)\), \(g\in L^2(0,l)\); \(B_j[W(x,t,u)]_{x= 0}= 0\), \(j= 1,2\); \(B_j[W(x,t,u)]_{x=l} =0\), \(j= 3,4\), \(B_j\) are some functions. Using the distributed control method, the authors construct the optimal quencher and the quenching modes in the form \(u(x,t)= \sum^\infty_{n=1} U_n(t)\varphi_n(x)\), where \(\varphi_n(x)\in L^2[0,l]\) are complete set of orthogonal eigenfunctions, \(U_n(t)= \sigma_n\cos\lambda_n t+\mu_n\sin \lambda_n t\), \(\sigma_n,\mu_n\in \mathbb{R}\). The authors show that, under some conditions, there exists \(u(x,t)\) that quenches the solution \(W(x,t)\) for all modes, and that this solution is unique. Moreover, it is proved that there exists a number \(N\) and a control \(u_N(x,t)\) which quench the first \(N-1\) modes of \(W\) at \(t=T\), and that the proposed method can be extended to determine a control to quench all modes of the vibration. These theoretical results are applied to some particular cases, which leads to numerical and graphical results.
reviewed by: Ştefan Zanfir
Mathematics Subject Classification ID: 74P10
Mathematics Subject Classification ID: 74H45
Mathematics Subject Classification ID: 74K10
Mathematics Subject Classification ID: 74H20
Mathematics Subject Classification ID: 74H25
zbMATH DE Number: 1384925
zbMATH Keywords: approximate solutions
zbMATH Keywords: uniqueness
zbMATH Keywords: existence
zbMATH Keywords: optimal vibration quenching
zbMATH Keywords: Euler-Bernoulli beam
zbMATH Keywords: distributed control method
zbMATH Keywords: complete set of orthogonal eigenfunctions
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6564
OpenAlex ID: W2075879792
cites work: Active quenching of a set of predetermined vibratory modes of a beam by a single fixed point actuator
cites work: Q3696938
cites work: Q3208205
DOI: 10.1006/JMAA.1999.6564
description: scientific article; zbMATH DE number 1384925