In the model for linear harmonic motion, we saw the presence of beats in the plot of the error. For this reason, we need to use a model that incorporates higher harmonics. The beam equation is such a model:
The solution to this partial differential equation is obtained by separation of variables and application of the boundary and initial conditions. The general solution can be written in the following form:
where
and
and beta is any solution to the following equation:
U(0,t) = U_x(0,t) = U_xx(L,t) = U_xxx(L,t) = 0
L is the length of the beam.
F_beta0
F_beta1
F_beta2
The eigenfunction corresponding to the first value of beta has the most effect on the overall position of the beam, whereas subsequent eigenfunctions have a diminishing effect on the beam's position. The sum of these eigenfunctions describes the overall position of the whole beam as it oscillates.
U_t(x,0) = 0 and U(x,0) = f(x)
where f(x) is a function describing the position of the beam when t=0.
Next: Conclusion
Up: Introduction
Previous: Linear Harmonic Motion
Jennifer Powell<jpowell@geom.umn.edu> Fati Liamidi<liamidi@geom.umn.edu>