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Applied Math: Beam Deflection using Separation of Variables

Beam's dynamic deflection is governed by the following PDE

EI(∂^4w/∂x^4)+(rho)*A* (∂^2w/∂t^2)=f(x,t)

where E = the modulus of elasticity, I = the moment of inertia, p = mass density, A = cross-section area, w(x,t) = deflection, 0 ≤ x ≤ L = axial coordinate, L = length, t = time. The load is given as the following function

f(x,t) = sin(pi*x/L)sin(omega*t)

Use separation of variable and the normal mode approach to find w(x,t) if the boundary conditions are

w = 0 at x = 0 and x = L;

∂^2w/∂x^2=0

And the initial conditions read w=O,

∂w/∂t =0 at t =0