Higher Harmonic Motion

In a simple harmonic model, 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:

Boundary Conditions

The following boundary conditions apply to the beam equation:

U(0,t) = U_x(0,t) = U_xx(L,t) = U_xxx(L,t) = 0

L is the length of the beam.

Eigenfunctions

Using various values of beta, we can determine different functions F_beta, which are called eigenfunctions. Each eigenfunction represents a different mode of the oscillatory motion 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.

Initial Conditions

The following initial conditions allow us to find the particular solution for an oscillating cantilevered beam.

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.

More: Comparison of Model with Data
Next: Conclusion
Up: Introduction
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Jennifer Powell<jpowell@geom.umn.edu>

Fati Liamidi<liamidi@geom.umn.edu>