Vibration Analysis Of Beams With Variable Stiffness

Free vibration of beams with variable stiffness is studied in the paper, Mainly study is focused on transversely free vibration of axially exponentially graded Timoshenko beams and rotating tapered Rayleigh beams. The main research contents and results are as follows:1. Briefly outlining the research background and significance of functionally graded beams and tapered beams. The research progress of functionally graded beams and tapered beams is also reviewed and main research contents and originality of this paper are indicated.2. Introducing four classical theoretical models of beams. Their own characteristics and differences are especially emphasized. Under four different theoretical models of beams, this paper elucidates the differences between equilibrium equations of free vibration of functionally graded beams and tapered beams, and compares the similarities and differences between them and uniform homogeneous beams.3. Studing transversely free vibration of axially exponentially graded Timoshenko beam, where the cross-sectional stiffness, rotary inertia and distributed mass are assumed to obey a unified exponential form in the axial direction. Governing differential equations of transversely free vibration are obtained. Furthermore frequency equations of transversely free vibration are obtained for various end supports. For various boundary conditions, exact frequency equations are derived in closed form. These frequency equations can reduce to those for homogeneous Timoshenko beams. Moreover, the frequency equations of exponentially graded Rayleigh, shear, and Euler-Bernoulli beams can be obtained as special cases of the present. Obtained analytical solutions can serve as a comparison for examining the accuracy of the numerical results based on other approximate procedures for analyzing transverse vibration of non-uniform axially graded Timoshenko beams.4. Studing free vibration of rotating tapered cantilever beams with rotary inertia of cross-section. The governing differential equation of vibration of rotating Rayleigh beams is derived using Hamilton’s principle, and the integral equation method is proposed to determine natural frequencies. The effects of the rotary inertia, angular velocity, taper ratio and hub radius are discussed in this part. The paper contains18images,9tables and117references.