IDENTIFICATION OF NON-STATIONARY LOADINGS APPLIED TO ELASTICALLY DEFORMED ELEMENT OF CONSTRUCTION

Abstract

In the manuscript the technique of identification as functions of time of R non-stationary loads acting on a element of construction is presented. It is assumed that deformation of the element is elastic; its geometry, a type of mount and properties of a material/materials are given; spatial distribution for loads is known. As initial data there are S variables which describe change in time of some deformation parameter and are accessible for experimental measurements. Change in time of these variables are caused by action of required loads.To solve the problem unknown dependences are approximated by set of step functions of type of Heaviside functions. Factors of approximation q are to be calculated. Using S´R so-called “functions of influence” the set of S registered variables are presented as superposition of these functions in view of factors q. It is necessary to note that “functions of influence”, which determine change in time of the s-th (s=1…S) measured variable at acting exclusively the r-th (r=1…R) load in the form of step unit function, can be defined experimentally or by methods of mathematical modelling, so are considered as known.The developed algorithm of factors q calculation is based on the method of least squares. As a result of simple mathematical transformations the problem is reduced to a system of linear algebraic equations (SLAE) with block structure of a computation matrix. To decrease errors in initial data on solution regularized Tikhonov's algorithm is used. The parameter of regularization is calculated on the basis of the principle of relative discrepancy. Having solved the SLAE at the final stage approached profiles of required functions are restored.The algorithm was tested for identification of the concentrated non-stationary load which act axisymmetrically on a round plate with rigidly jammed contour. Experimental investigation of vibration of this plate can be found in literature. The received results (the identified load) coincides with measured values with comprehensible accuracy.The presented technique can be used to solve a wide class of “boundary inverse problems” in mechanics of solids including identification of influences of other physical nature (kinematic, thermal, etc.).