Transverse vibrations of arbitrary non-uniform beams.pdfVIP

Appl. Math. Mech. -Engl. Ed., 35(5), 607–620 (2014) Applied Mathematics DOI 10.1007/s10483-014-1816-7 and Mechanics c Shanghai University and Springer-Verlag Berlin Heidelberg 2014 (English Edition) Transverse vibrations of arbitrary non-uniform beams∗ 1 2 Shu-qi GUO () , Shao-pu YANG () (1. Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, P. R. China; 2. School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, P. R. China) Abstract Free and steady state forced transverse vibrations of non-uniform beams are investigated with a proposed method, leading to a series solution. The obtained series is verified to be convergent and linearly independent in a convergence test and by the non-zero value of the corresponding Wronski determinant, respectively. The obtained solution is rigorous, which can be reduced to a classical solution for uniform beams. The proposed method can deal with arbitrary non-uniform Euler-Bernoulli beams in principle, but the methods in terms of special functions or elementary functions can only work in some special cases. Key words transverse vibration, non-uniform beam, series solution, convergence Chinese Library Classification O326 2010 Mathematics Subject Classification 74H45 Nomenclature u, displacement; L, length of beam; E ,