Two different order reduction methods of the deterministic and stochastic systems are discussed in this paper. First, the transient proper orthogonal
decomposition (T-POD) method is introduced based on the high-dimensional nonlinear dynamic system. The optimal order reduction conditions of the T-POD
method are provided by analyzing the rotor-bearing system with pedestal looseness fault at both ends. The efficiency of the T-POD method is verified via comparing with the results of the original system. Second, the polynomial dimensional
decomposition (PDD) method is applied to the 2 DOFs spring system considering
the uncertain stiffness to study the amplitude-frequency response. The numerical
results obtained by the PDD method agree well with the Monte Carlo simulation
(MCS) method. The results of the PDD method can approximate to MCS better
with the increasing of the polynomial order. Meanwhile, the Uniform-Legendre
polynomials can eliminate perturbation of the PDD method to a certain extent
via comparing it with the Gaussian-Hermite polynomials. 相似文献
This paper deals with the experimental analysis of the long-term behaviour of periodically excited linear beams supported by a one-sided spring or an elastic stop. Numerical analysis of the beams showed subharmonic, quasi-periodic and chaotic behaviour. Furthermore, in the beam system with the one-sided spring three different routes leading to chaos were found. Because of the relative simplicity of the beam systems and the variety of calculated nonlinear phenomena, experimental setups are made of the beam systems to verify the numerical results. The experimental results correspond very well with the numerical results as far as the subharmonic behaviour is concerned. Measured chaotic behaviour is proved to be chaotic by calculating Lyapunov exponents of experimental data.
Sommario Il presente lavoro concerne l'analisi sperimentale del comportamento a regime di travi lineari, su supporti elastici nonlineari discontinui, eccitate periodicamente. L'analisi numerica dei sistemi in esame ha evidenziato risposte subarmoniche, quasi-periodiche e caotiche, nonchè l'esistenza, nel caso di trave con una molla laterale, di tre differenti percorsi verso il caos. La relativa semplicità dei sistemi di travi ha consentito di procedere ad una verifica sperimentale dei risultati numerici e della varietà dei fenomeni nonlineari da essi evidenziati. La corrispondenza fra risultati sperimentali e numerici è molto buona nel caso di risposta subarmonica. Il comportamento caotico sperimentale è stato convalidato attraverso il calcolo degli esponenti di Lyapunov a partire dai relativi dati.
Many machine elements in common engineering use exhibit the characteristic of “hysteresis springs”. Plain and rolling element bearings that are widely used in motion guidance of machine tools are typical examples. The study of the non-linear dynamics caused by such elements becomes imperative if we wish to achieve accurate control of such machines.
This paper outlines the properties of rate-independent hysteresis and shows that the calculation of the free response of a single-degree-of-freedom (SDOF) mass-hysteresis-spring system is amenable to an exact solution. The more important issue of forced response is not so, requiring other methods of treatment. We consider the approximate describing function method and compare its results with exact numerical simulations. Agreement is good for small excitation amplitudes, where the system approximates to a linear mass-spring-damper system, and for very large amplitudes, where some sort of mass-line is approached. Intermediate values however, show high sensitivity to amplitude variations, and no regular solution is obtained by either approach. This appears thus to be an inherent property of the system pointing to the need for developing further analysis methods. 相似文献