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Unformatted text preview: Problems and Solutions for Section 7.4 (7.107.19) 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. 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 7.12 Consider a twodegreeoffreedom 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 11 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 1 ρ+ 2(.0129 10 2(.0129 15 ρ+ 10 2 15 2 ρ = 1 2 1 11 1 1 φ ( τ 29 = 1 2 11 φ ( τ 29 y = 1 2 1 { } 11 1 1 ρ= 1 2 1 1 { }ρ This is the representation of the system in modal coordinates, if proportional...
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This note was uploaded on 11/28/2010 for the course ME 4440 taught by Professor Hill during the Winter '09 term at Detroit Mercy.
 Winter '09
 Hill

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