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- -nme^^tejMWjW' /=i j=i one obtains the total kinetic energy as

With the use of the mode shapes of Eq. (2.251), one finds that </>, (£) = 2( —1)'+1 and so the kinetic energy simplifies to ml " "

/=i j=l where the Kronecker symbol S,j — 1 for i — j and S,j — 0 for i ^ j. For free vibration, there are no additional forces. Thus, Lagrange's equations can be now written in matrix form as

where [iT] is a diagonal matrix with the diagonal elements given by

Table 2.3. Approximate Values of co\y/ml4/(EI) for Clamped-Free Beam with Tip Mass of pint Using n Clamped-Free Modes of Section 2.3.4, Eq. (2.251)

Table 2.3. Approximate Values of co\y/ml4/(EI) for Clamped-Free Beam with Tip Mass of pint Using n Clamped-Free Modes of Section 2.3.4, Eq. (2.251)

n

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