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

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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? 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 ϖ . Solution: For each mode: i ai bi i ϖ ϖ ϖ ζ 2 - = where bi ϖ and ai ϖ are the frequencies where the magnitude is 2 1 of the resonant magnitude. All values given in the following table are approximate. Mode i ϖ (Hz) ) ( i H ϖ 2 ) ( i H ϖ ai ϖ (Hz) bi ϖ (Hz) i ζ 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 ϖ = 10 rad/s, 2 ϖ = 15 rad/s, and damping ratios 1 ζ = 2 ζ = 0.01. With modal s = 1 2 1 -1 1 1 , 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 1 0 0 1  ρ+ 2(.01 29 10 0 0 2(.01 29 15 ρ+ 10 2 0 0 15 2 ρ = 1 2 1 1 -1 1 1 0 φ ( τ 29 = 1 2 1 -1 φ ( τ 29 y = 1 2 0 1 { } 1 -1 1 1 ρ= 1 2 1 1 { This is the representation of the system in modal coordinates, if proportional damping is assumed. The transfer function is: Y ( s ) = 1 2 1 1 { } Ρ ( σ 29 where R ( s ) = 1
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