JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 315 (2008) 58–64 Forced response of a viscoelastically damped rod using the superposition of modal contribution functions Fernando Corte´ s à , Marı´a Jesu´s Elejabarrieta Department of Mechanical Engineering, Mondragon Unibertsitatea, Loramendi 4, 20500, Mondragon, Spain Received 3 July 2007; received in revised form 10 January 2008; accepted 14 January 2008 Handling Editor: C. Morfey Available online 04 March 2008 Abstract In this communication the axial vibration problem of a uniform elastic rod with a viscoelastic end damper is studied. The analysis is carried out in the frequency domain, the properties of the damper being characterised by a complex stiffness, and the viscoelastic damping being represented by an exponential model. First, an analytical solution for frequency response functions is obtained using a direct method. Next the computation of the system response is proposed, by means of the modal contribution functions (MCF) superposition method. This method allows evaluating the individual participation of the eigenmodes in the total response (even if the system is not self-adjoint and thus the classical modal superposition cannot be applied), providing important information for practical engineering applications that is lost otherwise. Finally, a numerical example is presented, in which the response provided by both, direct and MCF methods, is compared aimed at validating the latter. r 2008 Elsevier Ltd. All rights reserved. 1. Introduction In machinery, transmission elements such as ball screws transmit dynamic axial forces [1], and consequently, axial vibrations are induced. These vibrations can be mitigated using dampers, which dissipate mechanical energy into heat. If dampers are characterised by viscous damping, the dissipative forces are proportional to the actual velocity, and the mechanical response may be analysed by means of classical methods for damped linear system vibrations (see e.g. Refs. [2,3]). However, if viscoelastic damping is considered, the response of the system depends on the complete history of the load, the relationship between forces and velocity being nonlinear (see the exhaustive work of Adhikari [4] for details on viscoelastic damping). In this sense, Golla and Hughes [5] proposed a time domain formulation for the analysis of structural systems with viscoelastic damping, and Adhikari [6] extends classical modal analysis to treat lumped-parameter nonviscously damped linear dynamic systems. For the analysis of structural systems in the frequency domain, it should be taken into account that viscoelasticity implies frequency-dependent damping properties. Indeed, Corte´ s and Elejabarrieta developed ARTICLE IN PRESS www.elsevier.com/locate/jsvi 0022-460X/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2008.01.027 à Corresponding author. Tel.: +34 9437 94700; fax: +34 9437 91536. E-mail addresses: fcortes@eps.mondragon.edu, mjelejabarrieta@eps.mondragon.edu (F. Corte´ s).