Dynamic Analysis Of Axially Moving Functionally Graded Beams

Some common equipment in mechanical engineering and vehicle engineering, such as band saw blades, power transmission belts, aerial tramways, high-rise elevator cables, can be modeled as axially moving beams. Owing to axially velocity, the transverse vibration may occur in these structures. The transverse vibration may debase work quality or on the contrary. So, it is of great significance to investigate the vibration and stability of axially moving beams for design and optimization in related engineering component.The axially moving structures composed of traditional homogeneous materials have been widely studied, but the study on the non-homogeneous composite structures is very limited. Functional graded material is a new type of gradient changes in non-uniform composite material,whose mechanical properties vary gradually with respect to a desired spatial coordinate. The use of FGM leads to the elimination of stress concentration and also improves the strength and toughness of the structure. So, a systematic study of the dynamic behavior of axially moving functionally graded beams will have important scientific significance and broad application prospects.In this dissertation, the dynamic behaviors and stability of the axially moving functionally graded beams are investigated. If the speed is constant, the nature frequencies, critical speed and the stability of the beam are studied. The parametric resonance of the beam is investigated when the speed changes over time. The specific details are as follows:(1) Based on the Euler beam theory and Timoshenko beam theory, respectively, the transverse vibration of an axially moving, initially tensioned beam made of functionally graded materials is investigated. The material properties are assumed to vary continuously through the thickness of the beam according to a power law distribution. The Hamilton’s principle is employed to derive the governing equation and the complex mode approach is performed to obtain the transverse vibration modes and natural frequencies, respectively. And then the critical speed of the system is derived. The effects of some parameters including axially moving speed, the power-law exponent and the initial stress on the dynamic responses are examined and revealed in detail. The results show that an increase in the power-law exponent or the axial speed results in a lower natural frequency and then the flexural stiffness of the beam is reduced, while the axial initial stress tends to increase the natural frequency. The power-law exponent tends to decrease the natural frequency, while the axial initial stress increases it.(2) The differential equations are discredited by differential quadrature method, and then the system complex characteristic equation is derived. The complex frequencies of transverse vibration of the axially moving beam made of functionally gradient materials are analyzed with the change of the axial movement speed and different gradient index, and the influences of the axial movement speed and gradient index on the transverse vibration characteristics and instability forms of the axially moving beam made of functionally gradient materials are discussed. The results show that the first three imaginary parts(i.e., natural frequencies) of complex frequencies are reduced in magnitude as the axially moving speed increases. When the speed increases to a certain value, the phenomenon of divergence instability and coupled mode flutter appears. With the increase of the power-law exponent, the critical speed and the coupled velocity decreases obviously, and the divergence instability interval becomes increasingly small.(3) When the axial speed is characterized as a simple harmonic variation about the constant mean speed, with the method of multiple scales the instability conditions are presented for axially accelerating functionally graded beams in parametric resonance. The instability is studied when the axial speed variation frequency approaches the sum or difference of any two natural frequencies or two times of a natural frequency of the system, summation or principal parametric resonance may occur. Numerical results present the instability region of the first three order principal parametric resonance and summation parametric resonance between them. Then, the influence of the power-law exponent on the instability region is analyzed.