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SolSec 5.4

# SolSec 5.4 - 5 29 Problems and Solutions Section 5.4(5.37...

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5- 29 Problems and Solutions Section 5.4 (5.37 through 5.52) 5.37 A machine, largely made of aluminum, is modeled as a simple mass (of 100 kg) attached to ground through a spring of 2000 N/m. The machine is subjected to a 100-N harmonic force at 20 rad/s. Design an undamped tuned absorber system (i.e., calculate m a and k a ) so that the machine is stationary at steady state. Aluminum, of course, is not completely undamped and has internal damping that gives rise to a damping ratio of about ζ = 0.001. Similarly, the steel spring for the absorber gives rise to internal damping of about ζ a = 0.0015. Calculate how much this spoils the absorber design by determining the magnitude X using equation (5.32). Solution: From Eq. (5.21), the steady-state vibration will be zero when a a m k = 2 ω Choosing µ = 0.2 yields m m m a a a = = ( )( ) = = = ( )( ) = µ ω 0 2 100 20 20 20 8000 2 2 . kg k N/m a With damping of ζ = 0.001 and ζ a = 0.0015, the values of c and c a are c km c k m a a a a = = ( ) ( )( ) = = = ( ) ( )( ) = 2 2 0 001 2000 100 0 894 2 2 0 0015 8000 20 1 2 ζ ζ . . . . kg/s kg/s From Eq. (5.32), X k m F c F j K M jC a a a = ( ) + + ( ) ω ω ω ω 2 0 0 2 det Since M C K = = = 100 0 0 20 2 0944 1 2 1 2 1 2 10 000 8000 8000 8000 . . . . , the denominator is –6.4 × 10 7 -1.104 × 10 6 j , so the value of X is X k m F c F j K M jC a a a = ( ) + ( ) + ( ) ω ω ω ω 2 0 0 2 det

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