Multiple-scale analysis

About: Multiple-scale analysis is a research topic. Over the lifetime, 1360 publications have been published within this topic receiving 27530 citations.

Nonlinear study of the dynamic behavior of a string-beam coupled system under combined excitations

Abstract: In this paper, the nonlinear dynamic behavior of a string-beam coupled system subjected to external, parametric and tuned excitations is presented. The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system which are described by a set of ordinary differential equations with two degrees of freedom. The case of 1:1 internal resonance between the modes of the beam and string, and the primary and combined resonance for the beam is considered. The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system and obtain approximate solutions up to and including the second-order approximations. All resonance cases are extracted and investigated. Stability of the system is studied using frequency response equations and the phase-plane method. Numerical solutions are carried out and the results are presented graphically and discussed. The effects of the different parameters on both response and stability of the system are investigated. The reported results are compared to the available published work.

Combination Resonances of a Circular Plate With Three-Mode Interaction

Abstract: A nonlinear analysis is presented for combination resonances in the symmetric responses of a clamped circular plate with the internal resonance, ω 3 ω 1 + 2ω 2 . The combination resonances occur when the frequency of the excitation are near a combination of the natural frequencies, that is, when Ω 2ω 1 + ω 2 . By means of the internal resonance condition, the frequency of the excitation is also near another combination of the natural frequencies, that is, Ω ω 1 - ω 2 + ω 3 . The effect of two near combination resonance frequencies on the response of the plate is examined. The method of multiple scales is used to solve the nonlinear nonautonomous system of equations governing the generalized coordinates in Galerkin's procedure. For steady-state responses, we determine the equilibrium points of the autonomous system transformed from the nonautonomous system and examine their stability. It has been found that in some cases resonance responses with nonzero-amplitude modes don't exist, and the amplitudes of the responses decrease with the excitation amplitude. We integrate numerically the nonautonomous system to find the long-term behaviors of the plate and to check the validity of the analytical solution. It is found that there exist multiple stable responses resulting in jumps. In this case the long-term response of the plate depends on the initial condition. In order to visualize total responses depending on the initial conditions, we draw the deflection curves of the plate.

Nonlinear normal modes for damage detection

Abstract: The sensitivity of the nonlinear dynamic response to damage is investigated in simply supported beams by computing the nonlinear normal modes (NNMs) and exploiting the rich information they unfold about the nonlinear system behavior. Damage is introduced as a reduction of the flexural stiffness within a small segment of the beam span. The problem is formulated in a piece-wise fashion and analytically tackled by using the method of multiple scales to compute the NNMs and the backbone curves. The latter describe the frequency versus free oscillation amplitude for each mode and as such they represent the skeleton of the nonlinear frequency response around each beam modal frequency. The bending of the backbones is regulated by the so-called effective nonlinearity coefficients associated with each mode. The comparison between the damaged and undamaged beams shows a sensitivity of the effective nonlinearity coefficients higher than the sensitivity of the linear natural frequencies. Moreover, the nonlinear frequency trends unfold an interesting dependence of the nonlinear free response on the stiffness reduction at the damage site. Such dependence yields useful information about the damage position. An effective identification strategy for the damage position is proposed by computing the damage-induced discontinuities in the second derivative of the NNMs (i.e., first order estimate of the bending curvature), without resorting to a baseline model of the undamaged beam. This goal is achieved by computing a high order derivative of the Nadaraya–Watson kernel estimator, which is evaluated from a finite set of sampled beam deflections associated with each individual NNM. The results show that the second derivatives of higher NNMs are more sensitive to damage-induced discontinuities than the linear modal curvatures. The robustness against noise of the damage identification process is also discussed.