The displacement Y (x, t) of a vibrating cantilevered beam of length L are gover

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The displacement Y (x, t) of a vibrating cantilevered beam of length L are governed by the equation m delat2y/deltat2+ei delta4y/deltax4=0, where m is its mass per unit length, E is the modulus of rigidity, I is the moment of inertia of the beam cross-section, and x is the distance along the beam from the clamped end. The free-oscillations boundary conditions are Y(O) = 0, Yx(O) = 0, YXX(L) = 0, YXXX(L) = 0.use separation of variables Y(x,t) = F(x)G(t) to solve for the free oscillations of the beam (use trigonometric and hyperbolic functions rather than exponentials). Introducing beta4 = m omega2/EI, where omega is angular frequency (you may assume omega2 > 0), show that the eigenmodes correspond to sech betaL + cos betaL = 0,and hence find the eigenfrequencies u>n and eigenfunctions Fn(x). Using whichever computer software you wish, numerically determine the first three betan and omegan and plot their Fomega.

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