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Unformatted text preview: 5 3Problems and Solutions Section 5.1 (5.6 through 5.26) 5.6A 100kg machine is supported on an isolator of stiffness 700 103N/m. The machine causes a vertical disturbance force of 350 N at a revolution of 3000 rpm. The damping ratio of the isolator is = 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 steadystate is X=Fo/m!n2"!2( )2+2#!n!( )2$%&()1/2Since !n=km= 83.67 rad/s and !=30002"60#$%&(= 314.2 rad/s, (b) From equation (5.7), the transmissibility ratio is FTF=1+2!r( )21"r2( )2+2!r( )2Since r=!!n= 3.755, this becomes FTF=0.1368(c) The magnitude is FT=FTF!"#$%&F=0.1368( )350( )=47.9( )5 45.7Plot the T.R. of Problem 5.6 for the cases = 0.001, = 0.025, and = 1.1. Solution: T.R.=1+2!r( )21"r2( )2+2!r( )2A plot of this is given for = 0.001, = 0.025, and = 1.1. The plot is given here from Mathcad: 5 55.8A simplified model of a washing machine is illustrated in Figure P5.8. A bundle of wet clothes forms a mass of 10 kg (mb) in the machine and causes a rotating unbalance. The rotating mass is 20 kg (including mb) and the diameter of the washer basket (2e) is 50 cm. Assume that the spin cycle rotates at 300 rpm. Let kbe 1000 N/m and = 0.01. Calculate the force transmitted to the sides of the washing machine. Discuss the assumptions made in your analysis in view of what you might know about washing machines. Solution:The transmitted force is given by FT=k2+c2!r2where c=2!"n, "n=km=7.071 rad/s, "r=3002#60=31.42 rad/s,and Xis given by equation (2.84) as X=memr21!r2( )2+2"r( )2Sincer=!r!n=4.443, then X= 0.1317 m and FT=(0.1317) (1000)2+[2(0.01)(20)(7.071)]2(31.42)2=132.2 NTwo important assumptions have been made: i)The outofbalance mass is concentrated at a point and ii)The mass is constant and distributed evenly (keep in mind that water enters and leaves) so that the mass actually changes. 5 65.9Referring to Problem 5.8, let the spring constant and damping rate become variable. The quantities m, mb, eand are all fixed by the previous design of the washing machine. Design the isolation system (i.e., decide on which value of k and cto use) so that the force transmitted to the side of the washing machine (considered as ground) is less than 100N. Solution: The force produced by the unbalance is Fr= mbawhere ais given by the magnitude of equation (2.81): Fr=m!!xr=em!r2=0.25( )10( )3002"60#$%&()*+,.2=2467.4 NSince FT< 100 N, T.R.=FTFr=1002467.4=0.0405If the damping ratio is kept at 0.01, this becomes T.R.=0.0405=1+2 0.01( )r!"#$21%r2( )2+2 0.01( )r!...
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