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23_trussbwa_04

# 23_trussbwa_04 - Truss Structures Natural Frequency...

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Frequency Manipulation via SDP Brian W. Anthony Robert M. Freund Truss Structures - Natural 1

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8 Background : Mechanical Systems

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Simple Mechanical System m 1 u 1 k 1 9
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System wants to vibrate when forced. 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 Frequency (Hz or Cycles/Sec) Amplitude 1 DOF System (k/m) .5 Natural Frequencies – How a Mechanical Examples • Ball on a string • Beam(s) • For a simple mechanical systems it is relatively easy to systematically effect a change in the natural frequency. 11
Beam Vibration Narrow Beam How would we change the frequency of vibration? Beam Vibration Narrow Beam Narrow Beam - Natural Frequency = 5.6 Hz Narrow and Short Beam - Natural Frequency = 11.7 Hz Wide Beam - Natural Frequency = 7.8 Hz 12

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Another Mechanical System F(t) m 2 m 3 m 1 u 1 k 1 k 2 k 3 u 2 u 3 Write Newton’s Law for Each Mass F = ma Another Mechanical System F(t) m 2 m 3 m 1 u 1 k 1 k 2 k 3 u 2 u 3 m 1 (d 2 u 1 /dt 2 ) + k 1 u 1 + k 2 (u 1 2 ) = 0 m 2 (d 2 u 2 /dt 2 ) + k 2 (u 2 1 ) + k 3 (u 2 3 ) = 0 m 3 (d 2 u 3 /dt 2 ) + k 3 (u 3 2 ) = F(t) M (d 2 U /dt 2 ) + KU = F (t) The Dynamics Model (The Equations of motion). In Matrix Form -u -u -u -u 13
Equations of Motion M K are the natural frequencies of vibration (squared).

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23_trussbwa_04 - Truss Structures Natural Frequency...

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