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2C) Using the initial conditions, solve for the unknown coefficients for the distributed force:
From the initial condition given,
(22)
ν(x, 0) = sin((πx)/(ℓ)) = ∞⎲⎳i = oddξiφi
= (Aisin(0) + Bicos(0) + Cisin(0))φi
= Biφi
Applying orthogonality, we find that
Now we look to the initial velocity of 0 initial condition
(24)
ν̇(x, 0) = 0
= ∞⎲⎳i = oddξi̇φi = (Aiωicos(0) − Biωisin(0) + 4CiΩcos(0))φi
0
= (Aiωi + 4CiΩ)φi
= ℓ⌠⌡0(Aiωi + 4CiΩ)sin((iπx)/(ℓ))sin((jπx)/(ℓ)) (orthogonality)
= (Aiωi + 4CiΩ)(ℓ)/(2)
Ai = − (4ΩCi)/(ωi)
Now we need to solve the governing equation
(25) Mi(ξï + ω2iξi) = Ξi
Taking 2 derivatives to get
(26) ξï = 4ΩωiCisin(ωit) − Biω2icos(ωit) − 16Ω2Cisin(4Ωt)
We plug in to the governing equation
(27)
(mℓ)/(2)[
4ΩωiCisin(ωit) − Biω2icos(ωit) − 16Ω2Cisin(4Ωt)
+ ω2i(( − 4ΩCisin(ωit))/(ωi) + Bicos(ωit) + Cisin(4Ωt)] = Ξi
Plugging in for Bi, Ξi and cancelling/simplifying we see that
(28) Ci = (4F0)/(iπm(ω2i − 16Ω2))