Estimation of geometrical parameters of machine elements based on statistical methods to avoid resonance

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.