By Edmund Hlawka

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Q¨8 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ . . . . . . . K38 . K48 . . . . K88 ⎤⎧ q3 ⎪ ⎪ ⎪ ⎥⎪ q ⎪ 4 ⎥⎪ ⎥⎨ . ⎥ ⎥⎪ . ⎥⎪ ⎪ ⎦⎪ . ⎪ ⎪ ⎩ q8 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ + ⎧ Q3 ⎪ ⎪ ⎪ ⎪ Q ⎪ 4 ⎪ ⎨ . = ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ Q8 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ The corresponding eigenvalue problem can be easily solved to obtain the frequency and mode shapes. 13 To find the rotating and non-rotating natural frequencies of the cantilever uniform beam for 1 element with 2 degree of freedom. Assume EI (flexural rigidity) = 100000 Nm2 , m (mass per unit length) = 5 Kg/m and R (radius of beam) = 10 m.

Use Rr and Rr as the two admissible functions. 8138 EI R4 m Here, the first frequency is well predicted. 10 Find Y2 (r) for use in the Rayleigh–Ritz method. 1. Using this as admissible function for Rayleigh–Ritz method, find the first two frequencies of a cantilever beam. Solution: w(r, t) = Y1 (r)q1 t + Y2 (r)q2 (t) where, r 2 r 3 1 r 4 −2 + R R 2 R We now assume that r 3 r 4 r 5 Y2 (r) = a +b +c R R R This function already satisfies the geometric boundary conditions Y2 (0) = 0 and Y2 (0) = 0.

Q8 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ The corresponding eigenvalue problem can be easily solved to obtain the frequency and mode shapes. 13 To find the rotating and non-rotating natural frequencies of the cantilever uniform beam for 1 element with 2 degree of freedom. Assume EI (flexural rigidity) = 100000 Nm2 , m (mass per unit length) = 5 Kg/m and R (radius of beam) = 10 m. 1 ⎤ ⎦ ⎤ ⎦ Eigenvalues and Eigenvectors of the matrix can be determined as, The characteristic equation, Kij − Dmij = 0.

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