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SolSec 6.6 - 6 45 Problems and Solutions Section 6.6(6.48...

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6- 45 Problems and Solutions Section 6.6 (6.48 through 6.52) 6.48 Calculate 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 is w w w tt yy tt + = ρ τ with boundary condition given by w (0, y ) = 0, w x ( l,y ) = 0, w ( x ,0) = 0, w ( x,l ) = 0. Assume separation of variables w = X ( x ) Y ( y ) q ( t ) which yields ′′ + ′′ = = − = X X Y Y c q q c 1 2 2 ˙˙ / ω ρ τ where Then ˙˙ q c q + = 2 2 0 ω is the temporal equation and ′′ = − ′′ = − X X Y Y ω α 2 2 yields ′′ + = ′′ + = X X Y Y α γ 2 2 0 0 as the spatial equation where γ 2 = ω 2 α 2 and ω 2 = α 2 + γ 2 . The separated boundary conditions are X (0) = 0, = X l ( ) 0 and Y (0) = Y ( l ) = 0. These yield X A x B x B A l l n n l n n = + = = = = sin cos cos ( ) ( ) α α α α π α π 0 0 2 1 2 2 1 2
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