SolSec1.6 - Problems and Solutions Section 1.6 (1.66...

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Problems and Solutions Section 1.6 (1.66 through 1.72) 1.66 Show that the logarithmic decrement is equal to δ = 1 0 n x x n ln where x n is the amplitude of vibration after n cycles have elapsed. Solution: ln ln sin sin xt xt nT Ae t Ae t nT n n t d tn t dd () + = + ++ −+ ζω ωφ ωω φ (1) Since nTn tnT t d d ωπ ω = =+ 2 , sin sin Hence, Eq. (1) becomes ln sin sin ln Ae t Ae e t nt en T n nn n t d t n t nt n + = = Since ln , xt T T n + =≡ Then ln n + = Therefore, = 1 n x x n o n ln original amplitude amplitude cycles later
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1.67 Derive the equation (1.68) for the trifalar suspension system. Solution: Using the notation given for Figure 1.29, and the following geometry: r θ r θ φ l r θ l h Write the kinetic and potential energy to obtain the frequency: Kinetic energy: TII o max ˙˙ =+ 1 2 1 2 22 θθ From geometry, θ r x = and ˙ ˙ xr = TI I x r o max ˙ () 1 2 2 2 Potential Energy: Um m g l l o max cos φ
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This note was uploaded on 02/11/2010 for the course MECHANICAL ms316 taught by Professor Abduljaba during the Spring '09 term at Kalamazoo.

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SolSec1.6 - Problems and Solutions Section 1.6 (1.66...

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