Consider transverse vibration of the beam with tip mass shown in Fig. P9.11. As in Problem 9.11, the tip mass is m = pAL/5. Use the Rayleigh Method to estimate the fundamental bending frequency of this cantilever beam with tip mass. Let ψ(x) be the non dimensionalized static deflection curve of a uniform cantilever beam with concentrated tip force as shown in Fig. P13.12. (The “exact” fundamental frequency was determined in Problem 13.10.)

Problem 13.10

Consider transverse vibration. v(x, t) of the uniform cantilever beam (AE = constant, pA = constant) with tip mass, as shown in Fig. P9.11. The mass “particle” at x = L is m = pAL/S. (a) Determine the two boundary conditions at end x = L. (b) Obtain the characteristic equation from which the eigenvalues can be determined. Your answer should contain the parameters λ and L, where X is defined by Eq. 13.13. (Hint: See Eq. 6 of Example 13.3.) (c) Solve the characteristic equation [part (b)] for the lowest eigenvalue, Xl7 and then determine the fundamental transverse-bending frequency, ω1,

Example 13.3

Determine the natural frequencies and natural modes of the uniform cantilever beam shown in Fig. 1.



Problem 9.11

Transverse vibration of the uniform cantilever beam with tip mass, shown in Fig. P9.11, is to be modeled by a 2-DOF system based on the shape functions

The tip mass is m = pAL/5. E l = constant and pA = constant, (a) Determine the equations of motion for this 2-DOF transverse-vibration model, (b) Determine the two natural frequencies (squared)

(c) Determine expressions for the two transverse vibration mode shapes, and write them in the form  in the same manner as given in Eqs. 12 and 15 of Example 9.5 for axial-vibration modes, (d) Use M a t l a b to plot these two transverse-vibration mode shapes, in the same manner as the axial-vibration modes shown in Fig. 2 of Example 9.5.

Example 9.5

A 2-DOF assumed-modes math model for axial vibration of the uniform cantilever bar in Fig. 1 was obtained in Example 8.9. Solve for the natural frequencies and modes of this model, and sketch the modes.

Example 8.9

Use the Assumed-Modes Method with a polynomial approximation of u(x, t) to obtain a 2-DOF model for axial vibration of a uniform cantilever bar subjected to an end force P(t), as shown in Fig. 1.