Low Rank Dynamic Mode Decomposition And Its Application In Mechanical Fault Diagnosis

During the operation of the equipment,the vibration signal contains rich operating health information.Monitoring equipment operating conditions by analyzing vibration signals is important to online monitoring and fault diagnosis of mechanical equipment.However,mechanical equipment usually works under complex working conditions.Due to the harsh working environment,the collected signals usually present strong nonlinear and non-stationary characteristics.Moreover,the collected bearing fault signals are often coupled with noise,making it difficult to identify fault features effectively.Dynamic mode decomposition(DMD)is a novel nonlinear mode decomposition method,which characterizes the original complex system by constructing an approximate dynamic system,and can effectively extract the dynamic characteristics of the signal by analyzing the specific DMD modes.However,due to the nonlinear,non-stationary,strong noise and strong interference characteristics of mechanical signals,the application of DMD algorithm in mechanical equipment fault diagnosis still has certain limitations.Therefore,based on the in-depth study of the DMD theory,this dissertation proposes a variety of improved DMD algorithms for some engineering practice problems.This dissertation aims to optimize the noise robustness and feature extraction ability of DMD,and extend it to a multivariate signal processing method,so that it can obtain more accurate results in the field of mechanical equipment fault diagnosis.The main work of this dissertation includes the following aspects:(1)The theoretical basis of the DMD algorithm is deeply studied,and the effectiveness of the dynamic mode decomposition algorithm in the fault diagnosis of mechanical equipment is verified through simulation experiments and measured data.In the next comparative study with the mainstream fault diagnosis methods,the key problems of DMD in the fault diagnosis of mechanical equipment that need to be improved are pointed out through preset simulation cases under several special conditions.(2)In order to solve the influence of strong noise in mechanical equipment fault signals on DMD,this dissertation proposes the DLRDMD algorithm on the basis of in-depth study of the causes of deviation.The projection depolarization operator is introduced to synchronously eliminate the noise in the bilateral snapshot matrix,avoiding the deviation caused by the noise reduction of single snapshot.The iterative soft threshold algorithm is applied to replace the singular value hard threshold in traditional DMD.The algorithm can adaptively eliminate the noise in the projection operator,so that the projection direction is optimized to retain the fault features as much as possible,and the noise robustness of DMD to fault signals is improved.(3)In order to solve the problem of adaptive extraction of mechanical equipment fault features and separation of periodic interference components,this dissertation proposes the LLRDMD algorithm.In the dynamic system matrix,the low-rank components contain not only fault characteristic components,but also periodic interference components generated by other components.Therefore,this chapter uses the local low-rank approximation framework to optimize the feature extraction ability of DMD,directly extract the low-rank component representing the fault feature in the dynamic system matrix.Thus the proposed method can solve the problem of extracting the fault feature in the global low-rank component.Numerical simulations and experiments show that the proposed method is robust to noise and can adaptively extract fault features under strong periodic interference components.(4)In order to solve the problem that to identify and extract the early weak fault feature amplitude of mechanical equipment,this dissertation proposes the PLRDMD algorithm.Based on the research on the mechanism of bearing fault signal generation,this chapter finds that when the length of the snapshot is exactly the number of points of a fault cycle,the dynamic system matrix is in the lowest rank distribution state under natural conditions.At this time,the distribution of singular value is the most discrete,and the fault characteristic components will be more concentrated.The reduction of fault characteristic energy can be avoided as much as possible when eliminating noise,so that the proposed method has the ability of maintaining amplitude.Both simulation and experiment have verified that the proposed method has the ability of maintaining amplitude,and the ability of extracting early weak fault features is better than traditional DMD and other similar methods while has higher computational efficiency.(5)In order to solve the problem of synchronous extraction of fault features of mechanical equipment multi-channel signals,this dissertation proposes the MLRDMD algorithm that extends DMD to the field of multivariate signal processing.Due to the different signal acquisition paths,the distribution of fault characteristic power between channels is usually uneven,which results in that the common single-channel signal processing methods can not effectively use the fault information of multi-channel signals.In order to solve the above problems,this dissertation introduces tensor operation to replace the matrix operation in traditional DMD,and rewrites the MLRDMD algorithm based on TSVD.Numerical simulation and experiments verify that the low rank distribution of fault features in the dynamic system tensor,and also verify the mode alignment characteristics and the synchronous extraction ability of the proposed method for fault features of multivariate signals.