Consider the axial vibration of a bar (of length I, cross-section area A, Young's Modulus E and density p) that has a lumped mass m = apAl attached at its mid-

span and is fixed at x = 0 and free at x = 1 as shown in Figure Q1. Note that a is an arbitrary constant. u(x,t) bar Figure Q1 (a) For the case where a = 8 using a two element FE model, calculate the first two natural frequencies and sketch the corresponding modeshapes. m (8 marks) (b) Calculate the natural frequency of the structure as a function of a using Rayleigh's method with the guessed modeshape modeshape = {2 for (7 marks) (c) The natural frequency estimate for the first mode given by the FE model and that predicted by Rayleigh's methods are close, comment on why this is and how you expect the similarity between the two predictions will vary with the value of a giving your reasoning. Based on these observations suggest how the system might be approximately modelled using a one mass, one spring structure when a is large. M=PAL 2 1 12], 12¹ (5 marks) The mass and stiffness matrices for an element of bar of length L subject to axial vibration are: 2 𐐀u T= T = √²/2PA (3) ²₂ K = 0