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Converted document Forced String Demo

Forced String Demo

Here we consider the Forced String problem. To remind us of the final solution, it is given here:

(1) ν(x, t) = ^∞⎲⎳_i = 1 ⎡⎣⎛⎝δ_i1cos(ω_it) + (2F₀sin(iπ ⁄ 4))/(mℓω²_i)[1 − cos(ω_it)]⎞⎠sin((iπx)/(ℓ))⎤⎦ + ^∞⎲⎳_i = odd⎡⎣⎛⎝δ_i1cos(ω_it) + (4F₀)/(iπm(ω²_i − 16Ω²))(sin(4Ωt) − (4Ω)/(ω_i))⎞⎠(iπx)/(ℓ)⎤⎦ where ω_i = (iπ)/(ℓ)√((T)/(m))

First, set the two slider bars corresponding to the two forces all the way to the left, then hit play under the Time slider bar (click the slower button below the Time slider bar enough times so that you can watch string at a reasonable speed) to see the motion of the string for only the initial condition. When you change the number of mode shapes, what changes do you observe? Why is this the case?

Next, increase the magnitude of the point force to some non-zero value using the slider bar. Observe the effects of increasing the number of mode shapes.

Now set the point force to 0, and increase the value of the distributed force, and observe the motion of the string.

Finally, set the forces and mode shapes all to some high value, to see how all of the forces and initial conditions interact with one another.