SolSec 6.6. - 6-Problems and Solutions Section 6.6(6.48...

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Unformatted text preview: 6-Problems and Solutions Section 6.6 (6.48 through 6.52)6.48Calculate the natural frequencies of the membrane of Example 6.6.1 for the case that one edge x= 1 is free.Solution:The equation for a square membrane iswtt+ϖψψ=ρτϖττwith boundary condition given by w(0,y) = 0, wx(l,y) = 0, w(x,0) = 0, w(x,l) = 0. Assume separation of variables w= X(x)Y(y)q(t) which yields′′ΞΞ+′′ΨΨ=1χ2&&θθ= -ϖ2ϖηερεχ=ρ/τThen&&q+χ2ϖ2θ= 0is the temporal equation and′′ΞΞ= -ϖ2-′′ΨΨ= -α2yields′′Ξ+α2Ξ= 0′′Ψ+γ2Ψ= 0as the spatial equation where γ2= ϖ2– α2and ϖ2= α2+ γ2. The separated boundary conditions are X(0) = 0, ′Ξ(λ29= 0and Y(0) = Y(l) = 0. These yieldX=Ασινα ξ+Βχοσα ξΒ= 0Αχοσαλ= 0ανλ=(2ν- 129π2αν=(2ν- 129π2λ456-Next Y= Csin γy+ Dcos γywith boundary conditions which yield D= 0 and C sin γl= 0. Thusγμ=μπλand forl = 1 we get an= (2n- 129π2,for γm= mπn,m= 1, 2, 3,…ϖνμ2=αν2+γμ2=(2ν- 1292π24+μ2π2=(2ν- 1292+4μ24π2χ2ϖνμ2=χ2(2ν- 1292+4μ24π2So thatϖνμ=(2ν- 1292+4μ2χπ2are the natural frequencies....
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SolSec 6.6. - 6-Problems and Solutions Section 6.6(6.48...

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