Uniqueness in the identification of asynchronous sources and damage in vibrating beams

Abstract

In this paper we analyze the problem of identification of the distribution of asynchronous vibration sources and rigidity perturbations in Euler-Bernoulli beams. Here it is proved that the displacement of a beam, even in an engineered continuous beam resting upon several supports, measured for an arbitrary small interval of time over an arbitrary small open set is enough for the unique determination of vibration sources. Using this result, we prove that perturbations in rigidity can be uniquely determined in a linearized problem.

Identification of an unknown spatial load distribution in a vibrating beam or plate from the final state

“…This contribution focuses on the determination of a unknown spatial load distribution f (x) from the final in time deflection in the nonhomogeneous Euler-Bernoulli beam and Kirchhoff-Love plate equations with arbitrary but separable source term. Other source identification problems for Euler-Bernoulli equations from boundary or final in time observations can be found in [12][13][14][15]20,24,32] and for the Kirchhoff-Love equation in [10]. In [32], using spectral theory, the point source a(x) is uniquely determined in the constant coefficient dynamic Euler-Bernoulli equationü + u = λ(t)a(x) where λ ∈ C 1 ([0, T ]) is given and x ∈ (0, 1).…”

“…The missing information is compensated by the following boundary measurements: respectively u (0, t) or u (0, t) for all t ∈ (0, T ). This source identification problem has been reconsidered in [20] for more general Euler-Bernoulli equation, which includes a constant damping and a constant traction force. An effective combination of the Lie-group adaptive method and the differential quadrature method is proposed in [24] to recover an unknown space and time dependent 3 load in a constant coefficient Euler-Bernoulli beam vibration equation.…”

A simple noniterative method for recovering a space‐dependent load on the Euler‐Bernoulli beam equation

“…Then, we develop a closed‐form expansion coefficients method, which is a noniterative method, to recover the unknown function H ( x ) in where F ( t ) is a given function. The recovery of the spatial load H ( x ) has been studied by Nicaise and Zair, Hasanov, Kawano, and Hasanov and Baysal, assuming that F ( t ) is a given function of t . However, those methods are of iterative types.…”

“…The force identification problems of beams have been studied in many research studies. [3][4][5][6][7][8] The parameter identification issue is a subclass of inverse problems, which was studied as the inverse spectral problems. [9][10][11] Besides, some researchers studied different models of the Euler-Bernoulli nanobeams [12][13][14][15][16][17][18][19][20][21] in a microscopic scale.…”

“…Indeed, we can measure the displacement at a certain time, recover the unknown force, and then find the solution of beam in the whole space‐time domain by solving the Euler‐Bernoulli beam equation under the recovered force. The force identification problems of beams have been studied in many research studies . The parameter identification issue is a subclass of inverse problems, which was studied as the inverse spectral problems .…”