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

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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 x t x t nT Ae t Ae t nT n n t d t nt d d ( ) + ( ) = + ( ) + + ( ) + ( ) ζω ζω ω φ ω ω φ (1) Since n T n t n T t d d d d ω π ω ω φ ω φ = ( ) + + ( ) = + ( ) 2 , sin sin Hence, Eq. (1) becomes ln sin sin ln Ae t Ae e t nt e n T n n n n t d t nt nt d d nt n + ( ) + ( ) + + ( ) = ( ) = ζω ζω ζω ζω ω φ ω ω φ ζω Since ln , x t x t T T n ( ) + ( ) = ζω δ Then ln x t x t nT n ( ) + ( ) = δ Therefore, δ = 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: T I I o max ˙ ˙ = + 1 2 1 2 2 2 θ θ
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SolSec1.6 - Problems and Solutions Section 1.6(1.66 through...

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