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Vibrating String: Example 2

Need to solve the governing equation to solve for C_i + D_i

(18) M_i = (mℓ)/(2) M_i(ξ_ï + ω²_iξ_i) = Ξ_i (mℓ)/(2)(ξ_ï + ω²_iξ_i) = F(2ℓ)/(iπ)cos(Ωt) − F( − 1)^{(i − 1) ⁄ 2}

Solving for ξ_ï

(19) ξ_ï = ω²_i(C_i + D_i)cos(ω_it) − Ω²C_icos(Ωt)

and then plugging everything back into the governing equation

(20) ω²_iC_icos(ω_it) + ω²_iD_icos(ω_it) − Ω²C_icos(Ωt) − ω²_iC_icos(ω_it) − ω²_iD_icos(ω_it) + ω²_iC_icos(Ωt) + ω²_iD_i = (4F)/(iπ)cos(Ωt) − (2)/(ℓm)F( − 1)^{(i − 1) ⁄ 2}

Cancelling and simplifying we get

(21) (ω²_i − Ω²)C_icos(Ωt) + ω²_iD_i = (4F)/(miπ)cos(Ωt) − (2)/(ℓm)F( − 1)^{(i − 1) ⁄ 2)}

Then breaking it up into like parts:

(22) (ω²_i − Ω²)C_icos(Ωt) = (4F)/(miπ)cos(Ωt) C_i = (4F)/(miπ)(1)/((ω²_i − Ω²)

(23) D_i = − (2)/(ℓω²_i)F( − 1)^{(i − 1) ⁄ 2}

Combining we have our final solution

(24) ν(x, t) = (1)/(ω₂)sin(ω₂t)cos⎛⎝(2πx)/(ℓ)⎞⎠ + ^∞⎲⎳_i = oddsin⎛⎝(iπx)/(ℓ)⎞⎠⎡⎣(4F)/(miπ)(1)/((ω²_i − Ω²))(cos(Ωt) − cos(ω_it)) − (2)/(mℓω²_i)F( − 1)^{(i − 1) ⁄ 2}(1 − cos(ω_it))⎤⎦

where ω_i = √((T)/(m))(iπ)/(ℓ)

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