It has been proved that natural frequencies are stationary at the natural modes. Also it has been shown that at the first mode it is a minimum and for systems with finite number of degrees of freedom the highest frequency is maximum (reference 1 and 2). Beyond these Statements nothing more can be said about the
behavior of the natural frequencies without defining precisely the manner in which the function w(x) is to be varied.
In this paper two forms of Lagrange interpolation formula is used by which the mode shape may be varied smoothly from one natural mode to the next by varying a single parameter P. Thus it is possible to plot the Rayleigh's quotient, R(P),
over as many natural modes desired. Clarifying this interesting and important subject, the method is applied for flextural vibration of beams with .different end conditions and the results are plotted for both of the interpolation.
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