Solutions : Active Control of Structures
نویسنده
چکیده
0.1 Challenge 1 If we consider that G(x, t) and W (x, t) are harmonic functions in time, then there exist functions g(x) and w(x) such that G(x, t) = g(x) cos(ωt) and W (x, t) = w(x) cos(ωt). Reeplacing these two expressions in the problem (1), we have that −ω2w(x) cos(ωt)− c2w′′(x) cos(ωt) = g(x) cos(ωt), which implies that −ω2w(x) − c2w′′(x) = g(x). On the other hand, the function W (x, t) satisfies the boundary conditions W (0, t) = W (L, t) ∀t > 0, then, we have that W (0) cos(ωt) = W (L) cos(ωt) = 0 ∀t > 0, and as a consequence w(0) = w(L) = 0.
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