Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams Korak Sarkar • Ranjan Ganguli Received: 16 August 2013 / Accepted: 10 March 2014 / Published online: 25 March 2014 Ó The Author(s) 2014 Abstract In this paper, the governing equations for free vibration of a non-homogeneous rotating Timo- shenko beam, having uniform cross-section, is studied using an inverse problem approach, for both cantilever and pinned-free boundary conditions. The bending displacement and the rotation due to bending are assumed to be simple polynomials which satisfy all four boundary conditions. It is found that for certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, the assumed polynomials serve as simple closed form solutions to the coupled second order governing differential equations with variable coeffi- cients. It is found that there are an infinite number of analytical polynomial functions possible for material mass density, shear modulus and elastic modulus distributions, which share the same frequency and mode shape for a particular mode. The derived results are intended to serve as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of rotating non-homogeneous Tim- oshenko beams. Keywords Rotating beams  Timoshenko beams  Free vibration  Test functions 1 Introduction Rotating elastic beams serve as important mathemat- ical models for a wide range of mechanical structures like helicopter rotor blades, turbine blades, propellers, satellite booms etc. Hence, the determination of the natural frequencies and mode shapes is an important problem from a structural dynamics point of view. Beams with variable properties are mostly used in order to optimize the distribution of strength and weight, and also sometimes to satisfy certain func- tional requirements. The Euler-Bernoulli beam theory has been widely applied for the free vibration study of rotating beams [1–4]. The secondary effects such as shear deformation and rotary inertia have a small effect on lower modes but have considerable effect on higher modes. The Rayleigh beam theory includes the effect of rotary inertia [5, 6], while the Timoshenko beam theory includes the effects of both shear deformation and rotary inertia [7–11]. Hence for accurate prediction of higher modes, the Timoshenko beam model is employed. Gas and steam turbine blades are generally short and rigid and can be modeled as Timoshenko beams. There also exists different more precise models in the literature, e.g. models concerning the transportation effect and taking the Coriolis effect into account [12], but the inclusion K. Sarkar  R. Ganguli (&) Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India e-mail: ganguli@aero.iisc.ernet.in K. Sarkar e-mail: korakpom@gmail.com 123 Meccanica (2014) 49:1469–1477 DOI 10.1007/s11012-014-9927-8