Stochastic response analysis of multi-degree-of-freedom vibro-impact system undergoing Markovian jump
- PDF / 1,671,882 Bytes
- 12 Pages / 547.087 x 737.008 pts Page_size
- 104 Downloads / 171 Views
(0123456789().,-volV) (0123456789().,-volV)
ORIGINAL PAPER
Stochastic response analysis of multi-degree-of-freedom vibro-impact system undergoing Markovian jump Rongchun Hu
. Xudong Gu . Zicheng Deng
Received: 4 January 2020 / Accepted: 14 July 2020 Ó Springer Nature B.V. 2020
Abstract The paper treats stationary response of a stochastically excited multi-degree-of-freedom (multi-DOF) vibro-impact system undergoing Markovian jump. The vibro-impact system with sudden abrupt changes in substructures or external excitations is modeled as a continuous-discrete Markovian jump system, which is essentially different from the traditional vibro-impact model. It is demonstrated that the random jump factors switch between a finite number of modes. This salient feature allows us to identify this type of dynamic behaviors as response of hybrid vibro-impact systems undergoing Markovian jump. Utilizing a two-step approximate technique, we can reduce the considered multi-DOF hybrid system to one-dimensional averaged Itoˆ equation of the form of system’s total energy. The approximate analytical solution of the associated Fokker–Planck–Kolmogorov (FPK) equation of system’s energy is derived to predict the stationary response of original hybrid systems. Keywords Vibro-impact system Markovian jump Stationary response Stochastic averaging
R. Hu (&) X. Gu Z. Deng Department of Engineering Mechanics, MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an 710129, China e-mail: [email protected]
1 Introduction Vibro-impact (VI) system when the components collide with rigid obstacles is considered as a complex and strongly nonlinear system. The research on response of VI systems, especially under time-dependent loadings of the stochastic nature, has become a hot issue due to their fundamental importance from both theoretical and practical viewpoints for several decades [1–3]. The response of the VI system under random excitation should be predicted by seeking statistical moments or probability density functions (PDFs). However, it is quite difficult to obtain the dynamical response analytically due to its non-smooth characteristic. Only some exact PDFs or approximate stationary solutions were explored in very special cases [4–7]. Nayak [8] obtained some stationary solutions of a single-DOF VI system with the Hertz contact law [9]. With a non-smooth variable transformation technique, Zhuravlev [10] converted a VI system into one without barriers. Then Dimentberg [11, 12] and Iourtchenko [13] explored impact energy losses for linear vibration systems with impact and the response PDFs of stochastic linear systems with impact numerically. Rong [14] studied the nonlinear VI system under combined deterministic harmonic and random excitations. And for analyzing the stationary response of VI systems, researchers have proposed some analysis methods, such as the linearization method [15], the energy balance method
123
R. Hu et al.
[16], the perturbation method [17] and t
Data Loading...
