and [M] is a symmetric matrix with elements given by

Mu = ml[Sij + 4fi(—l)'+j] (i, j = 1,2 n). (2.298)

Assuming £ = £ cxp(/V«r), one can write Eq. (2.296) as an eigenvalue problem of the form

Results for the first modal frequency are shown in Table 2.3 and compared therein with the exact solution. As one can see, the approximate solution agrees with the exact solution to within engineering accuracy with only two terms. By way of contrast, results for the second modal frequency are shown in Table 2.4. These results are not nearly as accurate. Results for the higher modes, not shown, are less accurate still. This is one of the problems with modal approximation methods; fortunately, however, aeroelasticians and structural dynamicists are frequently interested only in the lower-frequency modes. Note that the one-term approximation (i.e., the Rayleigh quotient) is within 1.1% for all values of [i displayed.

Example 13: The Ritz Method Using a Simple Power Series

As an alternative to using the mode shapes of a closely related problem, let us repeat the solution of the above using a simple power series to construct a series of functions </>,-. Since the moment vanishes at the free end where x — i, one can make the second derivative of all terms proportional to I — x. To get a complete series, one can multiply this term by a complete power series 1.x. x2, etc. Thus,

Table 2.4. Approximate Values of u>2\lm t4/(EI) for Clamped-Free Beam with Tip Mass of pmt Using n Clamped-Free Modes of Section 2.3.4, Eq. (2.251)


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