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Vibrating String: Example 2

B) Using A), determine the general solution of ξ_i:

Starting from the governing equation of the form:

(8) M_i(ξ_ï + ω²_iξ_i) = Ξ_i

In order to solve this, we need to determine ξ_i. We must consider the odd and even cases separately. For the even forcing function, we simply have a 0. For the odd forcing function, we have a cos(Ωt) and a constant term, so we will have:

(9) ξ_i = A_isin(ω_it) + B_icos(ω_it) + C_icos(Ωt) + D_i if i = odd ξ_i = A_isin(ω_it) + B_icos(ω_it) if i = even

C) Using the initial conditions, solve for the unknown coefficients:

We have been given 2 initial conditions, so we apply them here and solve:

(10) ν(x, 0) = 0 = ^∞⎲⎳_i = 1ξ_iφ_i dx

(11) ξ_i = A_isin(0) + B_icos(0) + C_icos(0) + D_i = B_i + C_i + D_i if i = odd ξ_i = A_isin(0) + B_icos(0) = B_i if i = even

Applying orthogonality here, we get that

(12) B_i = 0 if i = even B_i = − C_i − D_i if i = odd

(13) (∂ν(x, 0))/(∂t) = sin(2πx)/(ℓ) = ^∞⎲⎳_i = 1ξ_i̇φ_i

Taking one derivative of ξ_i we see that for both even and odd cases

(14) ξ_i̇ = ω_iA_i

Plugging this in and applying orthogonality by multiplying both sides by sin((jπx)/(ℓ)) and integrating from 0 to ℓ with respect to x

(15) ^ℓ⌠⌡₀sin⎛⎝(2πx)/(ℓ)⎞⎠sin⎛⎝(jπx)/(ℓ)⎞⎠ dx = ^∞⎲⎳_i = 1ω_iA_i^ℓ⌠⌡₀sin⎛⎝(iπx)/(ℓ)⎞⎠sin⎛⎝(jπx)/(ℓ)⎞⎠ dx

(16) (ℓ)/(2) = ω₂A₂(ℓ)/(2) ⇒ A₂ = 1 ⁄ ω₂ if i = j = 2 0 = ω_iA_i(ℓ)/(2) ⇒ (ℓ)/(2) ≠ 0 and ω_i ≠ 0 so A_i = 0 if i = j ≠ 2

(17) ξ_i = ⎧⎪⎨⎪⎩ (1)/(ω₂)sin(ω₂t) if i = 2 0 if i = even ≠ 2 − (C_i + D_i)cos(ω_it) + C_icos(Ωt) + D_i if i = odd

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B) Using A), determine the general solution of ξi:

C) Using the initial conditions, solve for the unknown coefficients:

B) Using A), determine the general solution of ξ_i: