Study On The Ill-conditioned Problem Of MIMO Environmental Testing

Vibrational environment testing is performed to inspect the reliability and the safety of products under the real loads that they may encounter during their service lifetime.Compared with single-point test,multi-input and multi-output test is suitable in simulating real-life dynamic environment for products.Therefore,a wide-range researches have taken place in this domain and corresponding methods have been documented in military standard 810 G.With the penetration of studies,it is found by researchers that abnormal spectral lines caused by ill-conditioned frequency response functions are hard to be adjusted properly.In engineering,likewise,hardness with the same type frequently entangles engineers who try to perform a successful test.The dissertation demonstrates my findings on this problem based on the traditional multi-input multi-output environmental testing theories.First of all,the factors which caused abnormal spectral lines are thoroughly investigated.Multi-input multi-output vibration test theories and formulations for force identifications are recalled in order to navigate the problematic procedures in environmental testing.Meanwhile,ill-conditioned issues in environmental testing is compared with that in input estimation,several schemes are suggested for the improvement on excessive spectral lines.Secondly,the connection between control algorithms and abnormal spectral lines is discussed.Several control algorithms are applied in order to eliminate abnormality,the spectral densities after being controlled by these algorithms are shown to compare with each other,especially,the flaw of Smallwood algorithm are described.Moreover,the variations of singular values of frequency response function matrices made by control algorithms are observed to test the efficiency of control algorithm in adjusting singular values.Since the shortcomings of control algorithms compromise the efficiency of control process when frequency response function matrices are ill-conditioned,control process without any control algorithm involved is introduced.To begin with,Tikhonov filter is employed to adjust singular values,meanwhile,the influences of ill-conditioning on variables for auto-power spectral and cross-power spectral control are studied.From simulations and experiments,system noises amplified by ill-conditioned frequency response function matrices have great impact on auto-power spectra and cross-power spectra except for phase.As a relatively independent variable,phase cannot be improved totally by Tikhonov filter.As a result,a phase control algorithm is presented to finish the control.It is worth to mention that the combination of Tikhonov filter and phase control algorithm is able to fix the majority of abnormalities caused by ill-conditioned issues,however,to some sensitive spectra,control algorithms should be applied after singular values adjustment in order to achieve a perfect control effect.Traditional environmental testing formulations are established in frequency domain.When the number of control points is not large,the regularization parameters in the vicinity of ill-conditioned frequency points are sometimes hard to be determined.Under that condition,multiple times of adjustment are required,the process can be tedious.To simplify that procedure,a time domain environmental testing formulations are built and described in the dissertation.Instead of using frequency response function matrices,Markov parameters are obtained as a substitution.When conducting a test,Markov parameters in each order are acquired and arranged in Toeplitz form.In contrast with frequency domain method,in time domain procedure,reference spectra are converted into time domain reference signal first.Then,theoretical input signals can be obtained and exerted on under-test articles.After the real responses of the articles are acquired,response spectra are compared with reference spectra.The time domain procedure has its intrinsic advantage.The ill-conditioned problem of Toeplitz matrix is a typical rank-deficient issue,which can be easily handled using truncated singular value decomposition method.In contrast with frequency domain procedure,in which frequency response function matrices are required to be fixed on every ill-conditioned frequencies,singular problem can be corrected in only one time and all the abnormal spectral lines can be improved at once.Besides,another instable problems are analyzed by studying Toeplitz matrix and Markov parameters.Based on the analysis,a new control points selection method is suggested in this dissertation.