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Beam Bending Example

We now solve the governing equation

(24) M_i[ξ_ï + ω²_iξ_i] = Ξ_i

(25) (mℓ)/(2)[ − A_iω²_isin(ω_it) − B_iω²_icos(ω_it) − C_iΩ²cos(Ωt) + A_iω²_isin(ω_it) + B_iω²_icos(ω_it) + C_iω²_icos(Ωt)]

Cancelling and simplifying, we obtain C_i as

(26) C_i = (2F( − 1)^i + 1)/((ω²_i − Ω²)mℓ)

D) Replace and simplify for ν(x, t)

Replacing all of the coefficients in the equation for ν(x, t)

(27) ν(x, t) = ^∞⎲⎳_i = 1⎡⎣ − (2F( − 1)^i + 1)/((ω²_i − Ω²)mℓ)cos(ω_it) + (2F( − 1)^i + 1)/((ω²_i − Ω²)mℓ)cos(Ωt)⎤⎦φ_i(x)

Which simplifies to our final solution

(28) ν(x, t) = ^∞⎲⎳_i = 1⎡⎣(2F( − 1)^i + 1)/((ω²_i − Ω²)mℓ)(cos(Ωt) − cos(ω_it)⎤⎦φ_i(x)

where ω_i and φ_i(x) are given by equations (14) and (15), respectively.