SolSec7.4 - Problems and Solutions for Section 7.4...

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Problems and Solutions for Section 7.4 (7.10-7.19) 7.10 Consider the magnitude plot of Figure P7.10. How many natural frequencies does this system have, and what are their approximate values? 0 10 4 10 3 10 2 10 1 10 0 10 20 30 40 Magnitude 50 (Hz) 0.01 0.1 20 40 60 80 (Hz) 1 Solution: The system looks to have 8 modes with approximate natural frequencies of 2, 4, 10, 15, 22, 29, 36, and 47 Hz.
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7.11 Consider the experimental transfer function plot of Figure P7.11. Use the methods of Example 7.4.1 to determine i ζ and i ω . 20 40 60 80 (Hz) 40 0 40 80 Phase (degrees) Solution: For each mode: i ai bi i 2 = where bi and ai are the frequencies where the magnitude is 2 1 of the resonant magnitude. All values given in the following table are approximate. Mode i (Hz) ) ( i H 2 ) ( i H ai (Hz) bi (Hz) i 1 4.80 0.089 0.063 4.56 5.04 0.049 2 15.20 1.050 0.742 14.76 15.48 0.024 3 30.95 1.800 1.270 30.47 31.19 0.012 4 52.62 2.000 1.414 52.14 52.85 0.007 5 80.00 2.100 1.480 79.05 80.48 0.009
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7.12 Consider a two-degree-of-freedom system with frequencies 1 ω = 10 rad/s, 2 = 15 rad/s, and damping ratios 1 ζ = 2 = 0.01. With modal s = 1 2 11 , calculate the transfer function of this system for an input at 1 x and a response measurement at 2 x . Solution: Since the natural frequencies, damping ratios and mode shapes are given, the system can be expressed in modal coordinates as 10 01 20 0 0 0 2 01 15 10 0 5 1 2 1 0 1 2 1 1 2 2 + + = = ˙˙ (. ) (. ) ˙ () rr r ft y = {} = 1 2 1 2 This is the representation of the system in modal coordinates, if proportional damping is assumed. The transfer function is: Ys Rs = 1 2 where ss Fs .
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This note was uploaded on 02/11/2010 for the course MECHANICAL ms316 taught by Professor Abduljaba during the Spring '09 term at Kalamazoo.

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SolSec7.4 - Problems and Solutions for Section 7.4...

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