tailieunhanh - An interesting material that appears to be fit to possibly all future mechanical vibration textbooks

A material is suggested for future mechanical vibration textbooks. Both mathematically and conceptually it is simpler than most of the material that is already included in the existing textbooks. It pertains to the inverse vibration problem for inhomogeneous beam, . the beam with the modulus of elasticity that varies along the axial coordinate. Specifically, the solution of the following problem is presented: Find a distribution of the modulus of elasticity of an inhomogeneous beam such that the beam would possess the preselected simple, polynomial vibration mode shape. | Vietnam Journal of Mechanics, VAST, Vol. 30, No. 4 (2008), pp. 253 – 258 Special Issue of the 30th Anniversary AN INTERESTING MATERIAL THAT APPEARS TO BE FIT TO POSSIBLY ALL FUTURE MECHANICAL VIBRATION TEXTBOOKS Isaac Elishakoff Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991 Abstract. A material is suggested for future mechanical vibration textbooks. Both mathematically and conceptually it is simpler than most of the material that is already included in the existing textbooks. It pertains to the inverse vibration problem for inhomogeneous beam, . the beam with the modulus of elasticity that varies along the axial coordinate. Specifically, the solution of the following problem is presented: Find a distribution of the modulus of elasticity of an inhomogeneous beam such that the beam would possess the preselected simple, polynomial vibration mode shape. 1. INTRODUCTION The equation of the vibration of the uniform and homogeneous beam ∂ 2w ∂ 4w + ρ =0 (1) ∂x4 ∂t2 was first derived by Jacob Bernoulli and Leonhard Euler in 1730s. In Eq. (1) w(x, t) is the transverse displacement, x= axial coordinate, t= time, E = modulus of elasticity, I = moment of inertia, ρ = mass density, A = cross-sectional area. Since then the solution of Eq. (1) for uniform beams for various boundary conditions became a classic, and is rightfully included perhaps in all structural vibration textbooks (see ., Timoshenko et al. 1974; Rao, 1995, Meirovitch, 2001). We will recapitulate that solution briefly to define the motivation of this study. First of all we look for harmonic vibrations in time, getting EI w(x, t) = Y (x)eiωt, (2) where Y (x) is the vibration mode, ω = natural frequency. Both are saught in conjunction with the solution of Eq. (1) satisfying the appropriate boundary conditions. Substitution of Eq. (2) into Eq. (1) leads to the ordinary differential equation for the vibration mode Y (x) : d4 Y EI 4 − ρAω 2 Y = .