Ch2-2(3-4)c - 1 MAE351 Mechanical Vibrations Lecture 7...

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1 1 MAE351 Mechanical Vibrations Lecture 7 (Chap. 2.3-2.4) 2 Section 2.3 Alternative Representations A variety of methods for solving differential A variety of methods for solving differential equations So far, we used the method of undetermined coefficients Now we look at 3 alternatives: a geometric approach a frequency response approach a transform approach These also give us some insight and additional useful tools
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2 3 Geometric Approach Position, velocity, and acceleration phase shifted each by /2 Therefore, write each as a vector Therefore, write each as a vector Compute X in terms of F 0 through vector addition Im C D X c X m 2 C 0 F Re ) cos( t kX A B E kX 0 F B A X m k ) ( 2 X c 4 F 0 2 ( k m 2 ) 2 X 2 ( c ) 2 X 2 Using vector addition on the diagram: X F 0 ( k m 2 ) 2 ( c ) 2 At resonance: C B A X m k ) ( 2 X c 0 F  2 , X F 0 c
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3 5 Frequency response/complex function approach   inpu harmonic part imaginary part real ) ( ) ( ) ( ) sin ( cos t j t j e F t kx t x c t x m j t A t A Ae 0 input harmonic Real part of this complex solution corresponds to the physical solution 6 ) ( Xe t x t j p Choose complex exponential as solution ) ( ) ( ) ( ) ( H F j H j c m k F X e F Xe k cj m t j t j 0 2 0 0 2 1 function response frequency the ) ( ) ( ) ( j c m k j 2 Note: These are all complex functions
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4 7 j e F X 0 Using complex arithmetic: ) ( tan ) ( ) ( t j e F x m k c c m k 0 2 1 2 2 2 ) ( ) ( ) ( p c m k t 2 2 2 Has real part = to previous solution 8 Comments: Label x axis Re(e j t )and y axis Label x- ) and y- axis Im(e j t ) which results in the graphical approach It is the real part of this complex solution that has physical meaning The approach is useful in more The approach is useful in more complicated problems
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5 9 The Laplace Transform approach: See appendix B and section 3.4 for details Transforms the time variable into an
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Ch2-2(3-4)c - 1 MAE351 Mechanical Vibrations Lecture 7...

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