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Unformatted text preview: Solutions: Homework Set 3 1. Problem 3.1. Spring elongations: e 1 =- Lθ , e 2 = Lθ- y . Draw free-body diagram and sum moments about 0: summationdisplay M O = (- k 1 Lθ ) L- k 2 ( Lθ- y ) L- k T θ = I O ¨ θ EOM: ¨ θ + bracketleftbigg L 2 ( k 1 + k 2 ) I O + k T I O bracketrightbigg θ = k 2 LA I O cos ωt At resonance, ω = ω r = radicalBig L 2 ( k 1 + k 2 ) I O + k T I O , so ω n = radicalbigg k T I O = radicalbig ω 2 r- L 2 ( k 1 + k 2 ) /I O 2. Problem 3.4. Apply Newton’s second law to the manometer fluid, which is subjected to a force due to gravity, if the fluid height is not the same on both sides of the tube, and a force due to the pressure p ( t ) = p cos ωt . This gives ρAL ¨ x =- ρA (2 x ) g + p A cos ωt which gives the EOM ¨ x + 2 g L x = p A cos ωt . Resonance frequency: ω n = radicalbig 2 g/L . 3. Problem 3.5. Draw FBD of M . Upward vertical force on M from beam equals k eq ( x- y ) where k eq = 3 EI L 3 . Without constant weight term Mg , EOM becomes M ¨ y + k eq y = k eq x = k eq A...
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- Spring '10
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