806_Dynamics 11ed Manual

806_Dynamics 11ed Manual - Engineering Mechanics - Dynamics...

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Unformatted text preview: Engineering Mechanics - Dynamics Chapter 22 sin ( θ ) ≈ θ ⎞ ⎟θ = 0 2 2 ⎝ kG + d ⎠ θ'' + ⎛ ⎜ τ= gd 2π ωn 2 τ = 2π kG + d 2 gd Problem 22-27 The semicircular disk has weight W. Determine the natural period of vibration if it is displaced a small amount and released. Solve using energy methods. Given: W = 20 lb r = 1 ft g = 32.2 ft 2 s Solution: 2 2 2 ⎛ W ⎞ ⎛ r ⎞ − ⎛ W ⎞ ⎛ 4r ⎞ + ⎛ W ⎞ ⎛ r − 4r ⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ IO = ⎜ ⎟ ⎟ ⎝ g ⎠ ⎝ 2 ⎠ ⎝ g ⎠ ⎝ 3π ⎠ ⎝ g ⎠ ⎝ 3π ⎠ T+V= 1 2 ⎛ 4r ⎞ cos ( θ ) IOθ' − W⎜ r − ⎟ 2 ⎝ 3π ⎠ ⎛ ⎝ IOθ'' + W⎜ r − ωn = 4r ⎞ ⎟θ = 0 3π ⎠ 4r ⎞ W⎛ ⎜r − ⎟ IO ⎝ 3π ⎠ θ'' + W⎛ 4r ⎞ ⎜r − ⎟ θ = 0 IO ⎝ 3π ⎠ ωn = 5.34 rad s 804 τ= 2π ωn τ = 1.18 s ...
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