Acta Mech Sin (2011) 27(1):56–62 DOI 10.1007/s10409-011-0401-8 RESEARCH PAPER Elastically restrained Bernoulli–Euler beams applied to rotary machinery modelling Tiago A.N. Silva · Nuno M.M. Maia Received: 24 August 2010 / Revised: 15 September 2010 / Accepted: 15 September 2010 ©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2011 Abstract Facing the lateral vibration problem of a ma- chine rotor as a beam on elastic supports in bending, the authors deal with the free vibration of elastically restrained Bernoulli-Euler beams carrying a finite number of concen- trated elements along their length. Based on Rayleigh’s quo- tient, an iterative strategy is developed to find the approxi- mated torsional stiffness coefficients, which allows the rec- onciliation between the theoretical model results and the ex- perimental ones, obtained through impact tests. The men- tioned algorithm treats the vibration of continuous beams un- der a determined set of boundary and continuity conditions, including different torsional stiffness coefficients and the ef- fect of attached concentrated masses and rotational inertias, not only in the energetic terms of the Rayleigh’s quotient but also on the mode shapes, considering the shape functions de- fined in branches. Several loading cases are examined and examples are given to illustrate the validity of the model and accuracy of the obtained natural frequencies. Keywords Transverse vibration of beams · Elastic supports · Torsional stiffness coefficients T.A.N. Silva ISEL, Polytechnic Institute of Lisbon, Department of Mechanical Engineering, Rua Conselheiro Em´ ıdio Navarro, 1959-007 Lisbon, Portugal N.M.M. Maia ( ) IDMEC-IST, Technical University of Lisbon, Department of Mechanical Engineering, Av. Rovisco Pais, 1049-001 Lisbon, Portugal e-mail: nmaia@dem.ist.utl.pt 1 Introduction The study of beam-like components that present cross sec- tion variations along the length direction and carry concen- trated masses and/or springs is often addressed by means of approximated numerical methods, such as Rayleigh’s quo- tient. The accuracy of such an approach depends on the chosen shape function, according to Rayleigh’s theorem [1]. Figures 1 and 2 show the physical system under considera- tion, test configuration I and II, respectively. The objective of the present work is to develop an accurate model of the system configurations in order to replicate the experimental natural frequencies in lateral bending. While the rotor itself presents no problem and can eas- ily be studied as a Bernoulli–Euler beam with a disc in a certain location, the identification of the adequate torsional and linear stiffness properties associated with the supports remains a challenge. In a similar context, De Rosa et al. [2] supposed the beam is elastically restrained against rotation and translation at both ends, so that it was possible to study all the common boundary conditions. Those authors showed that trigonometric functions work slightly better than the static deflections and highlight the accuracy of Rayleigh’s quotient to the true natural frequencies. References [3– 6] present exact solutions for the frequency equation of a Bernoulli–Euler beam restricting the stiffness coefficients, in order to reproduce some particular cases, and accounting for the rotation inertia of attached discs and their eccentricity. Similar problems are treated in Refs. [7,8], with interme- diate supports. Wu and Chen [9] studied the bending vi- brations of wedge Bernoulli–Euler beams with any number