An inverse coefficient problem of identifying the flexural rigidity in damped Euler-Bernoulli beam from measured boundary rotation

Hasanov A.

PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, vol.380, no.2236, 2022 (SCI-Expanded) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 380 Issue: 2236
  • Publication Date: 2022
  • Doi Number: 10.1098/rsta.2021.0358
  • Journal Name: PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aquatic Science & Fisheries Abstracts (ASFA), Artic & Antarctic Regions, CAB Abstracts, Compendex, EMBASE, INSPEC, MEDLINE, Veterinary Science Database, zbMATH
  • Keywords: inverse coefficient problem, damped Euler-Bernoulli beam, measured boundary rotation, existence of a quasi-solution, Frechet differentiability, gradient formula, CANTILEVERS, EQUATION
  • Kocaeli University Affiliated: Yes

Abstract

We present a new mathematical model and method for identifying the unknown flexural rigidity r(x) in the damped Euler-Bernoulli beam equation rho(x)wtt+mu(x)wt+(r(x)wxx)xx-(Tr(x)wx)x=F(x,t),(x,t) & ISIN;omega T:=(0,l)x(0,T), subject to the simply supported boundary conditions w(0,t)=wxx(0,t)=0, w(l,t)=wxx(l,t)=0, from the available measured boundary rotation theta(t):=wx(0,t). We prove the existence of a quasi-solution and derive an explicit gradient formula for the Frechet derivative of the Tikhonov functional J(r)=||wx(0,.;r)-theta||L2(0,T)2. The results obtained here also form the basis of gradient-based computational methods for solving this class of inverse coefficient problems.This article is part of the theme issue 'Non-smooth variational problems and applications'.