Abstract
The article presents an approximate method of solving direct and inverse problems described by Bernoulli–Euler inhomogeneous equation of vibrations of a beam. A semianalytical solution is approximated by a linear combination of the Trefftz functions (T-functions, solving functions), which satisfies identically the homogenous equation describing the vibrations of a beam. In the paper, the properties of the solving functions have been investigated, theorems concerning their linear independence have been formulated and proved. A method of obtaining the particular solution of the inhomogeneous equation has been shown. To get this solution, recurrent formulas enabling us to determine the inverse operator for monomials have been derived. The paper discusses two kinds of inverse problems. The first one is a boundary inverse problem, in which the boundary conditions are to be determined, based on known displacements within the area. In the second one, the load on the beam needs to be found (identification of the source). The solving functions can be used as a finite element method base functions. This approach is tested for solving inverse problems. The paper includes examples which illustrate the usefulness of the method.
Highlights
The Trefftz functions method (T-functions method) is used for solving linear partial differential equations
A simple method of solving direct and inverse problems described by an inhomogeneous equation of the beam vibration has been presented
The approximate solution is a linear combination of the functions satisfying identically the proper homogenous equation— they are called the Trefftz functions
Summary
The Trefftz functions method (T-functions method) is used for solving linear partial differential equations. The first paper devoted to the Trefftz functions in which the time is considered as a continuous variable, discussed a one-dimensional (one spatial variable) heat conduction equation (Rosenbloom and Widder 1956). Source identification for an Euler–Bernoulli beam equation was considered for example in Liu (2012) and Hasanov (2009) This very paper is a significant development and supplement of the work (Al-Khatib et al 2008), in which recurrent formulas for the Trefftz functions for a homogenous beam vibration equation were derived. The papers published hitherto show a high effectiveness of the Trefftz functions method for solving inverse problems for the heat conduction equation, wave equation and for thermoelasticity problems. This very paper confirms its usefulness for solving inverse problems for the beam vibration equation
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