Abstract:
The knowledge of the dynamic load acting on the structure is always requiredand important in many practical engineering problems, such as structuralstrength analysis, health monitoring and fault diagnosis, and vibrationisolation. However, it is difficult to directly measure the dynamic load ona structure in some situations, such as the wind load on the tall building,the exciting force from road on the vehicle, etc. Meanwhile, the dynamicresponse measurement is correspondingly easy and accurate on a structure.Therefore, it is necessary to develop some inverse analysis techniques forload identification based on the measured dynamic responses.With the linearity and time-invariant suppositions, the loads are firstlyexpressed as a series of kernels of impulse functions or step functions intime domain and the total response of the system can be obtained using theproduct of the convolution integral of the kernel response and the loads.Through the discretization of convolution integral, the forward model forload identification is established. In fact, the inverse analysis for theload identification is to solve a deconvolution problem, but thedeconvolution is an ill-conditioned problem in which the noisy responses andhigh condition numbers of the kernel matrix will induce the amplified errorsin the identified load. Therefore, it is difficult to obtain a stable andaccurate solution for such inverse problems. To deal with ill-condition ofload reconstruction from the noisy responses, zero-phase digital filter,several regularization methods and optimized strategy for stable loadidentification are discussed. Through general filter, the noisy responsesignal will be smooth. But, it has a phase delaying compared with theoriginal signal, and the errors will also be amplified in the identifiedload. The zero-phase digital filter, whose phase error is zero in the curveof phase-frequency characteristic, is realized through reversing the timeserials of the signal. Moreover, a new extension algorithm is applied toimprove the performance of the filter. Comparing with the common differencefilter, this zero phase digital filter can not only avoid phase delaying,but also improve the wave aberration of the start and end section. After theinvestigation the ill-posedness arising from the inverse problem of loadreconstruction, Tikhonov regularization, truncated singular valuedecomposition and total least squares method are adopted to provideefficient and numerically stable solution of the desired unknown load, andthe $L$-curve method is proposed to determine the optimal regularizationparameter. In order to avoid the inverse operation of the matrix, manyoptimized methods can be available and here the conjugate gradient method isadopted. In the numerical example, the reconstruction of dynamic loads fromtwo sources with the noisy responses in the hood structure is investigated.The result indicates that the presented computational inverse technique iseffective and stable for the load identification with the noisy response intime domain.
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