Random Vibration Transmission Of One-dimensional Structure Based On Multifractal Method

With the advancement of science and technology,random vibration problems have been paid more and more attention by scholars at home and abroad.Usually these problems are a threat to the stability of the structure.However,it is difficult to estimate these random vibration problems using traditional vibration theory With the deepening of research,there is an urgent need to explore mathematical analysis methods to solve complex random vibration problems,and to reveal the statistical characteristics of the response of random systems,which has important practical significance and practical engineering for improving the reliability and stability of actual construction machinery background.As for vibration systems with a random variable in the boundary conditions,prestress of elastic constraint is introduced into the modeling of beam-like structures to deal with arbitrary constraint conditions as well as random boundary conditions when embedding the local Monte Carlo simulation(MCS),based on Saint-Venant's principle in Levinsion beam model.Within the framework of the Carrera Unified Formulation,one dimensional(ID)higher-order models are presented based on Taylor-type expansion of the generalized displacements.After prestress is equivalently considered as the substitution of elastic constraint in LBM modeling,not only associated natural boundaries conditions equations but also the differential governing equations can be derived by means of the principal of the virtual displacements.Free vibration study of 1D structures is carried out in conjunction with Dynamic Stiffness Method(DSM),whereas for random boundary conditions the local MCS is embedded in DSM.The numerical simulation showed that the proposed process gives high degree of accuracy in the calculation of natural frequencies for 1D beams with arbitrary boundary conditions as soon as the structural model was established.Moreover,the proposed model is particularly useful when analyzing 1D structures with random boundary conditions since it matches encouragingly well with the corresponding MCS results.Considering multivariate random boundary conditions,a novel approach is proposed to describe stochastic structural vibration of the systems with random boundary conditions,to make a statistical analysis on eigenvectors of the system's characteristic matrix and to the dynamic compliance in stochastic system.In order to investigate the random vibration of elastic structures under random boundary conditions,the random matrix theory was introduced to establish the Gaussian matrix model of random elastic structure systems called column-constrained Gaussian ensemble(CCGE).After making an exact diagonalization on CCGE ensemble,a multifractal theory was employed to statistically analyze on random mode shapes and to yield obtain spectrum of fractal dimensions and probability density functions(PDF)of random mode shapes.It follows that the spectrum of fractal dimensions was obtained,which differs from N describes numerically the nonergodic states of random processes,and the asymptotic analysis was then carried out for an infinite degree of freedom in structural system,thus implies the statistical properties of mode shapes of the continuous system.In addition,vibration analysis on the beam structure was presented as an illustration to discuss the random dynamic compliance of the vibration system and to further obtain the PDF of them.Subsequently,narrow-band spectrum analysis on our model,showed that the error between proposed method and Monte Carlo simulation is in an acceptable range.The proposed method is expected to establish the statistical characteristics of random response and vibration transmission of random vibration system...