UCLA Civil and Environmental Engineering Prof. Ju Spring 2016 MAE M269A Due: 5/11/2016 HOMEWORK #5 Problem 1 (15 Points) Consider a bar of length l and properties ( E, A, ) under axial vibrations. One end of the bar, say x = 0 , is fixed. At the other end, x = l , the bar is attached to a massless linear elastic spring of constant k . (1) Determine the frequency equation for this bar under these end conditions. (2) For k = EA l , determine the lowest three (non-zero) circular frequencies of this system in term of E l 2 . Civil and Environmental Engineering Prof. Ju MAE M269A HOMEWORK #5 (Continued) Problem 2 (15 Points) Consider a system of two identical bars. The properties of each bar of length L are ( E, A, ) for their extensional stiffness, cross-sectional area and mass density per unit volume. Determine the frequency equation for this system. l Extra Credit (5 Points) Show that this frequency equation can be put into the product form, where one part of it applies to natural vibrations that are symmetric about the mid-plane of the system, and the other is for antisymmetric motions. 2 Civil and Environmental Engineering Prof. Ju MAE M269A HOMEWORK #5 (Continued) Problem 3 (10 Points) The solution for spatial dependence beam vibration is Z ( x ) = C1 sin x + C2 cos x + C3 cosh x + C4 sinh x , where 4 = A 2 EI For a beam with the same boundary conditions on both ends, it is possible to explore free vibrations that are symmetric and antisymmetric about the mid-point of the beam by considering separate frequency equations. Consider a beam of length L with the origin of the coordinate system taken at the mid-point of the beam, so that the beam occupies the region x = L 2. Derive the two frequency equations for the symmetric and antisymmetric natural vibrations. 3