Abstract
A widespread internal resonance phenomenon is detected in axially moving functionally graded material (FGM) rectangular plates. The geometrical nonlinearity is taken into account with the consideration of von Kármán nonlinear geometric equations. Using d’Alembert’s principle, governing equation of the transverse motion is derived. The obtained equation is further discretized to ordinary differential equations using the Galerkin technique. The harmonic balance method is adopted to solve the above equations. Additionally, stability analysis of steady-state solutions is presented. Research shows that a one-to-one internal resonance phenomenon widely exists in a large range of constituent volume distribution in moving FGM plates. Moreover, it is found that this internal resonance phenomenon can easily happen even though the FGM plates are under extremely small external excitation or with very large damping.
Highlights
In order to meet the demanding requirements for comprehensive behavior of engineering structures in modern industries, a group of Japanese materials researchers composed a new type of non-uniform composite materials in the mid-1980s, namely, functional Gradient Materials (FGMs)[1]
This paper studies dynamic characteristics of longitudinally moving FGM plates, and attention is focused on the internal resonance behavior
On the base of the d’Alembert principle, we can derive the dynamic equilibrium equation governing the transverse vibration of a moving FGM plate:
Summary
Let u, v and w represent displacements of the plate mid-plane along x-, y- and z- axes from static equilibrium (u = v = w = 0), respectively. For a FGM plate, its effective material properties are written as[40,41]: P(z) = PNi VNi(z) + PS VS(z). The constituent volume fraction is considered to vary smoothly along the z-axis and satisfy power law distribution. In which Aij, Bij and Dij (i, j = 1, 2, 6) denote stiffness coefficients. On the base of the d’Alembert principle, we can derive the dynamic equilibrium equation governing the transverse vibration of a moving FGM plate:
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