Article Estimation of geometrical parameters of machine elements based on statistical methods to avoid resonance Siddappa M Khot and Nitesh P Yelve Abstract Resonance is a non-desirable situation, because amplitudes of high magnitude may cause malfunctioning of the machine element. The machine element can be redesigned or a vibration control technique can be implemented to take care of such situations. But these are post-design remedial solutions to the problem of resonance. If such situations are tackled in the initial stage of design itself by appropriately selecting the geometrical parameters so as to keep the natural frequency away from the operating frequency, it will be of great advantage as the time, cost and efforts involved in the redesign process, and expensive methods of controlling vibrations can be minimized, if not completely evaded. If it is tried analytically by considering the basic equations of natural frequencies which contain many parameters, a solution may not be obtained as the number of unknowns is more than that of the equations. Hence to give due consideration to resonance in the initial design stage and also to overcome the problem of analytical approach, the use of statistical methods is suggested here. This paper deals with two avenues of using statistical methods for estimating geometrical parameters of machine elements so as to preclude resonance. First, prediction of significant parameters, which contributes more in the total variation of natural frequency by using the concept of Design of Experiments and Analysis of Variance techniques; and second, by performing regression analysis, formulation of a mathematical function in terms of significant geometric parameters, which is capable of giving appropriate combination of parameters in a given range so as to rule out resonance. To demonstrate both these analyses a cantilever beam is considered. To extend the use of a regression model to a practical case and to check whether it gives appropriate results, it is applied to a compressor reed valve which is structurally equivalent to the cantilever beam. Keywords Cantilever beam, design of experiments, regression model, reed valve, resonance Received: 12 December 2010; accepted: 15 December 2010 1. Introduction In the majority of design processes the dynamic testing of the machine elements is done after the strength design and development of prototypes. However during the dynamic testing, if resonance occurs in the purview of the operating frequency, the geometrical parameters of the machine elements need to be changed, which alto- gether decides its natural frequency. This leads to total redesign work and in turn wastage of time. Vibration control strategies (Lim, 2003) can be implemented to evade resonance, which is an intricate and costlier affair. Thus if a designer has a mathematical function to estimate the geometrical parameters of the machine element for a desired natural frequency, he or she can make changes during the initial stage of the design pro- cess itself. If basic equations of natural frequencies are tried, trivial solutions may not be obtained because of the redundancy of the parameters in the equation. To resolve this problem wherein multiple parameters affect the natural frequency, use of two statistical methods is Department of Mechanical Engineering, Fr. C. Rodrigues Institute of Technology, India Corresponding author: Nitesh P Yelve, Department of Mechanical Engineering, Fr. C. Rodrigues Institute of Technology, Vashi, Navi Mumbai, Maharashtra State, India Email: niteshpy@yahoo.co.in Journal of Vibration and Control 18(2) 209–213 ! The Author(s) 2011 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546311403793 jvc.sagepub.com