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Unformatted text preview: 5 23Problems and Solutions Section 5.3 (5.27 through 5.36) 5.27A motor is mounted on a platform that is observed to vibrate excessively at an operating speed of 6000 rpm producing a 250N force. Design a vibration absorber (undamped) to add to the platform. Note that in this case the absorber mass will only be allowed to move 2 mm because of geometric and size constraints. Solution: The amplitude of the absorber mass can be found from equation (5.22) and used to solve for ka: Xa=0.002 m=Fka=250kaka=125,000 N/mFrom equation (5.21), !2=kamama=ka!2=125,00060002"60#$%&’()*+,.2=0.317 kg5 245.28Consider an undamped vibration absorber with β= 1 and μ= 0.2. Determine the operating range of frequencies for which Xk/F!0.5.Solution: From equation (5.24), with β= !a!p=1(i.e., !a=!p) and μ=0.2,XkF=1!""a#$%&’(1+0.2 1( )2!""a#$%&’(2)*++,..1!""a#$%&’(2)*++,..!0.2 1( )2=1!""a#$%&’(2""a#$%&’(4!2.2""a#$%&’(2+1For XkF= 0.5, this yields 0.5!!a"#$%&’4(0.1!!a"#$%&’2(0.5=Solving for the physical solution gives !!a"#$%&’=1.051Solving for !!a"#$%&’gives !!a"#$%&’=0.955, 1.8135 25Comparing this to the sketch in Figure 5.15, the values for which XkF!5 are 0.955!a"!"1.051!aand !#1.813!a5 265.29Consider an internal combustion engine that is modeled as a lumped inertia attached to ground through a spring. Assuming that the system has a measured resonance of 100 rad/s, design an absorber so that the amplitude is 0.01 m for a (measured) force input of 102 N. Solution: The amplitude of the absorber mass can be found from equation (5.22) and used to solve for ka: Xa=0.01m=Fka=100kaka=10,000 N/mChoose ω= 2ωn= 200 rad/s. From equation (5.21), ma=ka!2=10,0002002=0.25 kg5.30A small rotating machine weighing 50 lb runs at a constant speed of 6000 rpm. The machine was installed in a building and it was discovered that the system was operating at resonance. Design a retrofit undamped absorber such that the nearest resonance is at least 20% away from the driving frequency....
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 Spring '10
 Tang
 Frequency, RPM

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