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SolSec 2.6 - 2-42 Problems and Solutions Section 2.6(2.51...

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2-42 Problems and Solutions Section 2.6 (2.51 through 2.54) 2.51 Calculate damping and stiffness coefficients for the accelerometer of Figure 2.15 with moving mass of 0.04 kg such that the accelerometer is able to measure vibration between 0 and 50 Hz within 5%. ( Hint : For an accelerometer it is desirable for Y Z b 2 / ω = constant.) Solution: Use equation (2.71): Given: m = .04 kg with error < 5% .2 f = 50 Hz f = 250 Hz ω = 2 f π = 1570.8 rad/s Thus, k = 2 ω m = 98,696 N/m When r = .2, .95 < 2 2 2 ) 2 ( ) 1 ( 1 r r ζ + <1.05 (± 5% error) This becomes .8317+.1444 2 ζ <1<1.016+.1764 2 ζ Therefore, km c 2 7 . = = ζ ) 04 )(. 98696 ( ) 7 (. 2 = c c = 87.956 Ns/m 2.52 The damping constant for a particular accelerometer of the type illustrated in Figure 2.15 is 50 N s/m. It is desired to design the accelerometer (i.e., choose m and k ) for a maximum error of 3% over the frequency range 0 to 75 Hz. Solution: Given .2 f = 75 Hz f = 375 Hz ω n = 2 f π = 2356.2 rad/s. Using equation (2.84) when r = .2: .97 < 2 2 2 ) 2 ( ) 1 ( 1 r r ζ + <1.03 (± 3% error) This becomes .8671 + .1505 2 ζ <1<.9777+.1697 2 ζ Therefore, 0.3622 < ζ <0.9395 Choose ) 2 . 2356 ( 2 50 2 7 . m m c = = =
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