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Consider axial free vibration of the uniform bar (AE = constant, pA = constant) in Fig. P9.8. The stiffness of the linear spring at x = L is k = AEf2L; ignore its mass, (a) Determine the boundary condition at end x = L. (b) Obtain the characteristic equation from which eigenvalues can be determined. Your answer should contain the parameters λ and L, where λ is defined by Eq. 13.6. (c) Determine an expression for the fundamental axial frequency of the bar (to, rad/s), (d) Determine an expression for the fundamental axial mode shape, φ_{i}(x) and use M a t l a b , or another computer program, to plot the fundamental mode. Scale the mode so that its maximum value is 1.0.
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