Elastic impact on finite Timoshenko beam

Abstract

In this paper the analytical solutions of the impact of a particle on Timoshenko beams with four kinds of different boundary conditions are obtained according to Navier's idea, which is further developed. The initial values of the impact forces are exactly determined by the momentum conservation law.. The propagation of the longitudinal and transverse waves along the beam, especially, the effects of boundary conditions on the characteristics of the reflected waves, are investigated in detail. Some results are compared with those by MSC/NASTRAN.Recently the strategy by Navier was realized and developed in Refs. [7,8], and this paper will developed the strategy further, where the impactor and the target are treated as an integrated vibration system, and only the impactor has a distributive velocity. Therefore, the problems of elastic impact between structures are transformed into traditional vibration problems with initial values, and the impact loads are the internal forces on the contact area, not the external forces acting on the integrated system. The analysis method of impact problems based on this strategy might as well be called as Direct Mode Superposition Method (DMSM).Among the methods mentioned above, the direct solution of differential equations is greatly limited because of the difficulty of getting the closed form solutions. IMSM is not suitable for the analysis of locking problems of mechanisms, and the time step size and the number of modes used influence seriously the obtained results. In comparison with IMSM, DMSM is a general method of solving the elastic problems of linear impact between structures and the locking problems of mechanisms. According to DMSM, the software systems such as NASTRAN can be directly applied to the analysis of elastic impact problems.About the applications of Timoshenko theory, much research work [3~5,9~12] has been done, where Anderson [9] gave the responses of finite Timoshenko beam with known loads by mode superposition method; Boley et alfl0] obtained the dynamical responses of semiinfinite Timoshenko beam by Laplace transform, but the loads applied to the beam are assumed to be given as simple pulses; Yamamoto[ 11] carried out a numerical analysis of impact problems with contact deformation of a ball on Timoshenko beam by IMSM; Shueei et al. [ 12] obtained analytical solutions of loaded Timoshenko beam with general elastic boundary conditions, the loads are known and no additional mass is attached to the beam, and the method is not suitable for the analysis of impact problems of structures in which the force boundary condition is defined directly by the acceleration of the impactor and there are initial conditions, and the analysis of impacts between structures such as a rod impacting against a beam and so on, But the complicated impact problems mentioned above can be easily solved by DMSM.

Fundamental solution and its validation by numerical inverse Laplace transformation and FEM for a damped Timoshenko beam subjected to impact and moving loads

“…Correspondingly, the frequency equation and the characteristic vectors are and in which If Ω=λ0Lr2, the solution configurations of equation (26) are composed of two multiple roots 0 and a pair of imaginary roots ±α. The corresponding frequency equation and the characteristic vectors are and in which If Ω>λ0Lr2, the solution configurations of equation (26) are composed of two pairs of imaginary roots ±α and ±β, whose frequency equation and characteristic vectors hold: and in which Equations (29)–(34) show the complete modal solutions of the undamped system, which are also consistent (Yufeng et al., 2002; Su and Ma, 2011; Zhang et al., 2018a) with same or similar boundary conditions. Herein, all three cases are of both mathematical and applicable meanings in regard to the mode analyses.…”

“…(2013), also using a modal superposition method, transformed the governing equations of a Timoshenko beam into a set of ordinary differential equations in state-space manners and solved them via a numerical technique. In Yufeng et al. (2002), analytical solutions of the Timoshenko beam subjected to an impulsive particle with four types of boundary conditions were developed on the basis of a similar mode superposition method.…”