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SolSec 2.5

# SolSec 2.5 - 2 36 Problems and Solutions Section 2.5(2.43...

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2- 36 Problems and Solutions Section 2.5 (2.43 through 2.50) 2.43 A lathe can be modeled as an electric motor mounted on a steel table. The table plus the motor have a mass of 50 kg. The rotating parts of the lathe have a mass of 5 kg at a distance 0.1 m from the center. The damping ratio of the system is measured to be ζ = 0.06 (viscous damping) and its natural frequency is 7.5 Hz. Calculate the amplitude of the steady-state displacement of the motor, assuming r ω = 30 Hz. Soltuion: Given: m = 50 kg, 5 = o m , e = 0.1m, 06 . 0 = ζ , ω n = 7 5 . Hz Let ω r =30 Hz So, r r n = = ω ω 4 From Equation 2.51, 2 2 2 2 2 2 2 2 )] 4 )( 06 . 0 ( 2 [ ) 4 1 ( 4 50 ) 1 . 0 )( 5 ( ) 2 ( ) 1 ( + = + = r r r m e m X o ζ X = 0.011m X = 1.1 cm 2.44 The system of Figure 2.15 produces a forced oscillation of varying frequency. As the frequency is changed, it is noted that at resonance, the amplitude of the displacement is 10 mm. As the frequency is increased several decades past resonance the amplitude of the displacement remains fixed at 1 mm. Estimate the damping ratio for the system. Solution: Equation 2.75 is 2 2 2 2 ) 2 ( ) 1 ( r r r m e m X o ζ + = At resonance, X = 10 mm = ζ 2 1 m e m o ζ 2 1 10 = e m m o When r is very large, 1 = e m Xm o and X = 1 mm, so 1 = e m m o

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SolSec 2.5 - 2 36 Problems and Solutions Section 2.5(2.43...

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