# MIMO LS-SVR-Based Multi-Point Vibration Response Prediction in the Frequency Domain
###### Abstract  
To predict the multi-point vibration response in the frequency domain when the uncorrelated multi-source loads are unknown, a data-driven and multi-input multi-output least squares support vector regression (MIMO LS-SVR)-based method in the frequency domain is proposed. Firstly, the relationship between the measured multi-point vibration response and unmeasured multi-point vibration response is formulated using the transfer function in the frequency domain. Secondly, the data-driven multiple regression analysis problem of multi-point vibration response prediction in the frequency domain is described formally, and its mathematical model is established. With the measured multi-point vibration response as the input and the unmeasured multi-point vibration response as the output, the vibration response history data are assembled as a MIMO training dataset at each frequency. Thirdly, using the MIMO LS-SVR algorithm and MIMO history training dataset, the multi-point vibration response prediction model is built at each frequency point. By comparing the transmissibility matrix method, multiple linear regression model-based method, and MIMO neural network method, the application scope of the proposed method and its advantages are analyzed. The experimental results for acoustic and vibration experiment on a cylindrical shell verified that the MIMO LS-SVR-based method predicts the multi-point vibration response effectively when the loads are unknown, and has higher precision than the transfer function method, multiple linear regression method, MIMO neural network method, and transmissibility matrix method. m 20202  
Because they solve the ordinary differential or partial differential vibration equation, equation deduction and theoretical analytical solution methods [6; 7] are only adapted to simple continuum structures (e.g., beam, plate and shell) and are powerless for complex structures. In the experiments and measure methods [8; 9; 10], the multiple loads and transfer functions together determine the multi-point vibration response of the mechanical structure. As part of the structural characteristics, transfer functions between multiple loads excitation and multi-point vibration response are obtained in the transfer function experiment. For example, the response function of the system in the frequency domain can be acquired by simultaneously measuring the input spectrum function of the system's multiple loads excitation and the output spectrum of the multi-point vibration response [11]. When transfer function or frequency response function between loads excitation and vibration response is unknown, the vibration response of the linear time-invariant (LTI) structure can also be acquired by finite element methods (FEM). By discretizing the vibration structure and considering the appropriate boundary conditions and connection conditions, it is easy to determine the multi-point vibration response of complex structures under complex loads excitation [7]. If the model is reasonable, more satisfactory vibration response prediction results can be obtained. With recent further research on multiple regression analysis prediction models, the development of data-driven vibration response prediction methods is at the highest level of activity [12; 13; 14; 15]. [PERSON] [16] used least squares methods for vibration response prediction. However, if the system has suffered ill-posed problems, the prediction performance is often poor. [PERSON] [17] used an artificial neural network to predict the response of an offshore floating structure. However, because of the poor performance of the neural network in small-sample experiments and because many super-parameters need to be set, it is not easy to obtain a good prediction model. Support vector regression (SVR) has high prediction accuracy on small training samples and is not easy to overfit. By replacing nonequality constraints with equality constraints in SVR optimization, least squares-SVR (LS-SVR) reduces the high time complexity of the SVR algorithm.