Meccanica DOI 10.1007/s11012-013-9695-x Rotating beams and non-rotating beams with shared eigenpair for pinned-free boundary condition Korak Sarkar · Ranjan Ganguli Received: 8 September 2011 / Accepted: 9 January 2013 © Springer Science+Business Media Dordrecht 2013 Abstract In this paper we look for a rotating beam, with pinned-free boundary conditions, whose eigen- pair (frequency and mode-shape) is same as that of a uniform non-rotating beam for a particular mode. It is seen that for any given mode, there exists a flex- ural stiffness function (FSF) for which the i th mode eigenpair of a rotating beam with uniform mass dis- tribution, is identical to that of a corresponding non- rotating beam with same length and mass distribu- tion. Inserting these derived FSF’s in a finite element code for a rotating pinned-free beam, the frequencies and mode shapes of a non-rotating pinned-free beam are obtained. For the first mode, a physically realis- tic equivalent rotating beam is possible, but for higher modes, the FSF has internal singularities. Strategies for addressing these singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test functions for rotating beam codes and also for targeted destiffening of rotating beams. Keywords Rotating beams · Design · Free vibration · Destiffening · Test functions 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 1 Introduction Rotating elastic beams are generally used for model- ing a variety of important mechanical structures like helicopter blades, propellers, turbine blades, satellite booms etc. These structures suffer from resonance if their natural frequencies coincide with integer multi- ples of the rotating speed. The vibration analysis of beams have been the subjects of considerable research for many years [1–5]. So the accurate determination of the natural frequencies and mode shapes of these rotat- ing elastic beams becomes a very important problem [6, 7]. The long and slender beams are modeled using the Euler-Bernoulli beam theory, which is our main in- terest of study in this paper. The dynamics of a uniform rotating Euler-Bernoulli beam is governed by a 4th order differential equation which does not yield sim- ple closed form solutions such as those obtained for non-rotating uniform Euler-Bernoulli beams. So gen- erally we depend on some approximate methods like Galerkin method [8, 9] or Rayleigh Ritz Method [10, 11], or numerical methods like Finite Element Method (FEM) [12–18], for the accurate determination of the frequencies and mode shapes of rotating beams. Storti and Aboelnaga [19] used hypergeometric functions to analytically solve the 4th order govern- ing differential equation of rotating beams, for a wide class of variable cross-section beams. Giurgiutiu and Stafford [20] calculated the frequencies and mode- shapes of a rotor blade using the semi-analytic method of series solution of differential equations. Another