5mg  α22 l the influence coefficients are: l 5mg      α11 α21 α12 α31 α13 11l 6mg    α22 α32 α23 6mg  α33 11l Approximate methods In many engineering problems it is required to quickly estimate the (fundamental) natural frequency. Approximate methods like Dunkerley’s method, Rayleigh’s method are used in such cases. (i) Dunkerley’s method Dunkerley’s formula can be determined by frequency equation, M K  0 K M 0     ω2 ω2 1 I K 0   2 1 I αM 0   2 For n DOF systems, VTU e-learning Course ME65 Mechanical Vibrations7 . 1 0 . 0 m 0.0 α α α 12 1n 1 1 .0 0 1 . 0 0m2 . α21α22 α2n  0  2 . . . ω .αnn 0. mn 0 0 . 1 0 αn1αn2 1 . 2  α12m2 α1nmn ω 1 0 m . m 0  α α  2nn ω2 2 . . . . 1 m m m  n α α α n11 n22 nn ω2 Solve the determinant 1n 1 n m m  ... α m    α α (1) ω2 111 22 2 nn n ω2 ...   α11α22m1m2  0 ...   α11α33m1m3 αnnmn
l α22  5mg the influence coefficients are: l α11  α21  α12  α31  α13  5mg 11l α22  α32  α23  6mg 11l α33  6mg Approximate methods In many engineering problems it is required to quickly estimate the (fundamental) natural frequency. Approximate methods like Dunkerley’s method, Rayleigh’s method are used in such cases. (i) Dunkerley’s method Dunkerley’s formula can be determined by frequency equation, ω2 M  K   0 K  ω2 M  0 1 I  K  0 2 1 I  αM  0 2 For n DOF systems, VTU e-learning Course ME65 Mechanical Vibrations7 1 0 . 0 α α . α m 0.0 1 12 1n 1 .0 0 1 . 0 α21α22 . α2n 0m2   0 2 ω . . . 0 0 . 1 αn1αn2 .αnn 0 0. mn 1  α12m2 . α1nmn 2 ω 1 0  α m . α m  0 ω2 2 2nn . . . . 1 α m α m  α m n ω2 n11 n22 nn Solve the determinant 1 n 1n α m  α m  ...  α m  ω2 111 22 2 nn n ω2 (1) α11α22m1m2  α11α33m1m3  ...  αnnmn  ...  0