Inhomogeneous beams possessing an exponential mode shape

In this study inhomogeneous beams which posses exponential fundamental mode shapes are constructed. Closed-form solutions are derived for five cases of boundary conditions: (a) a beam with two free edges, (b) a cantilevered beam, (c) a beam that is clamped at both ends, (d) a simply supported beam and (e) a beam that is clamped at one end and simply supported at the other. The solutions can be utilized for axially varying the moment of inertia of the cross-section and/or functionally grading the material's modulus of elasticity in order to construct beams with prescribed vibration modes.