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SolSec 6.5 - 6 37 Problems and Solutions Section 6.5(6.40...

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6- 37 Problems and Solutions Section 6.5 (6.40 through 6.47) 6.40 Calculate the natural frequencies and mode shapes of a clamped-free beam. Express your solution in terms of E , I , ρ , and l . This is called the cantilevered beam problem. Solution: Clamped-free boundary conditions are w t w t w l t w l t x x xx xxx ( , ) ( , ) ( , ) ( , ) 0 0 0 0 = = = and assume E, I, ρ , l constant. The equation of motion is + = 2 2 4 4 0 w t EI A w x ρ assume separation of variables ) ( ) ( ) , ( t q x t x w φ = to get EI A q q ρ φ φ ω ′′′′ = − = ˙˙ 2 The spatial equation becomes ′′′′ − = φ ρ ω φ A EI 2 0 define 0 that so 4 2 4 = = φ β φ ω ρ β EI A which has the solution: x C x C x C x C β β β β φ cosh sinh cos sin 4 3 2 1 + + + = Applying the boundary conditions: φ φ φ φ ( ) ( ) ( ) ( ) 0 0 0 0 = = ′′ = ′′′ = and l l yields that C 2 + C 4 = 0 C 1 + C 3 = 0
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