In this problem we consider a higher order eigenvalue problem. The analysis of

Chapter 11, Problem 24

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In this problem we consider a higher order eigenvalue problem. The analysis of transverse vibrations of a uniform elastic bar is based on the differential equation y(4) y = 0, where y is the transverse displacement and = m2/EI; m is the mass per unit length of the rod, E is Youngs modulus, I is the moment of inertia of the cross section about an axis through the centroid perpendicular to the plane of vibration, and is the frequency of vibration. Thus for a bar whose material and geometrical properties are given, the eigenvalues determine the natural frequencies of vibration. Boundary conditions at each end are usually one of the following types: y = y = 0, clamped end, y = y = 0, simply supported or hinged end, y = y = 0, free end.In this problem we consider a higher order eigenvalue problem. The analysis of transversevibrations of a uniform elastic bar is based on the differential equationy(4) y = 0,where y is the transverse displacement and = m2/EI; m is the mass per unit length ofthe rod, E is Youngs modulus, I is the moment of inertia of the cross section about anaxis through the centroid perpendicular to the plane of vibration, and is the frequencyof vibration. Thus for a bar whose material and geometrical properties are given, theeigenvalues determine the natural frequencies of vibration. Boundary conditions at eachend are usually one of the following types:y = y = 0, clamped end,y = y = 0, simply supported or hinged end,y = y = 0, free end.(a) y(0) = y(0) = 0, y(L) = y(L) = 0(b) y(0) = y(0) = 0, y(L) = y(L) = 0(c) y(0) = y(0) = 0, y(L) = y(L) = 0 (cantilevered bar)