SolSec 5.2 - 5Problems and Solutions Section 5.1 (5.6...

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5- 3 Problems and Solutions Section 5.1 (5.6 through 5.26) i) A 100-kg machine is supported on an isolator of stiffness 700 x 103 N/m. The machine contains a rotating unbalance that causes a vertical disturbance force of 350 N at a revolution of 3000 rpm. The damping ratio of the isolator is ζ = 0.2. Calculate (a) the amplitude of motion caused by the unbalanced force, (b) the transmissibility ratio, and (c) the magnitude of the force transmitted to ground through the isolator. Solution: (a) From Window 5.2, the amplitude at steady state is X Fm nn = −+ = 0 22 2 2 2 2 5 2 350 100 83 67 314 2 2 0 2 83 67 314 2 379 10 / () ( ) / ( . . ) ((.) ( . ) .) . ωω ζ ω (b) From Equation (5.7), the transmissibility ratio is F F r rr T 0 2 2 12 0 1368 = + = ( ) . (c) The magnitude is F F F F T T = == 0 0 0 1368 350 47 9 ) . )( ) . N
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5- 4 5.7 Plot the T.R. of Problem 5.6 for the cases ζ = 0.001, ζ = 0.025, and ζ = 1.1. Solution: T.R.= 12 2 2 2 2 + () + ζ r rr A plot of this is given for ζ = 0.001, ζ = 0.025, and ζ = 1.1. The plot is given here from Mathcad:
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5- 5 5.8 A simplified model of a washing machine is illustrated in Figure P5.8. A bundle of wet clothes forms a mass of 10 kg ( m 0 ) in the machine and causes a rotating unbalance. The rotating mass is 20 kg (including m 0 ) and the diameter of the washer basket (2 e ) is 50 cm. Assume that the spin cycle rotates at 300 rpm. Let k be 1000 N/m and ζ = 0.01. Calculate the force transmitted to the sides of the washing machine. Discuss the assumptions made in your analysis in view of what you might know about washing machines. Top view Frictionless support k /2 k /2 c /2 c /2 m e m b Solution: The transmitted force is given by Fk c Tr =+ 22 2 ω where c k m nn r == = = 2 7 071 300 2 60 ζω π ,. rad/s, = 31.42 rad/s, and X is given by equation 2.75 as X me m r rr = () + 0 2 2 2 2 12 ζ Since r r n 4 443 . , then X = 0.1317 m and F T = ( . ) ( ) [ ( . )( )( . )] ( . ) . 0 1317 1000 2 0 01 20 7 071 31 42 132 2 2 N Two important assumptions have been made: ii) The out-of-balance mass is concentrated at a point and iii) The mass is constant and distributed evenly (keep in mind that water enters and leaves) so that the mass actually changes.
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5- 6 5.9 Referring to Problem 5.8, let the spring constant and damping rate become variable. The quantities m , m 0 , e and ω are all fixed by the previous design of the washing machine. Design the isolation system (i.e., decide on which value of k and c to use) so that the force transmitted to the side of the washing machine (considered as ground) is less than 100N. Solution: The force produced by the unbalance is F r = m 0 a where a is given by the magnitude of Eq. (2.72): Fm x e m rr r == = () = 00 2 2 0 25 10 300 2 60 2467 4 ˙˙ .. π N Since F T < 100 N, T.R. = F F T r 100 2467 4 0 0405 . . If the damping ratio is kept at 0.01, this becomes T.R. . . . + [] + 0 0405 12 0 0 1 0 0 1 2 2 2 2 r Solving for r yields r = 5.079. Since r km r = / , k m r r = 2 2 2 2 20 300 2 60 5 079 765 . N/m and ck m ( ) = 2 2 0 01 765 20 2 47 ζ . . kg/s
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5- 7 5.10 A harmonic force of maximum value of 15 N and frequency of 180 cycles/min acts on a machine of 25 kg mass. Design a support system for the machine (i.e., choose c , k ) so that only 10% of the force applied to the machine is transmitted to the base supporting the machine.
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SolSec 5.2 - 5Problems and Solutions Section 5.1 (5.6...

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