where [k] and [m] are the n x n matrices, o2 and cos(ot — f) are scalars, and {C} is the amplitude vector. For nontrivial solution, cos(ot — f) ^ 0, thus solution to Equation 2.389 requires the determinant of [[k] — o2[m]] = 0. The expansion of the determinant yields a polynomial of nth degree as a function of o , the n roots of which are the eigenvalues Oi, O2, ..., on.

If the eigenvalue o for a normal mode is substituted in Equation 2.389, the amplitude vector {C} for that mode can be obtained. {Q}, {C2}, {C3},..., {Cn} are therefore called the eigenvectors, the absolute values of which must be determined through initial boundary conditions. The resulting motion is a sum of n harmonic motions, each governed by the respective natural frequency o, written as n

- Small element dx


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