Homework 20 Solution

Homework 20 Solution - Lecture 20 MEEN 357 Homework...

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Lecture 20 MEEN 357 Homework Solution 1 Lecture-20-Homework-Solution.doc RMB 1. For a coupled two degree of freedom vibrating system shown in the following figure the equations of motion are 11 2 2 1 1 1 22 1 1 2 () ( ) ( ) mu cu ku c u u k u u f t u u =− + + + = −− −+ ±± ± ± ± ± ± (20.1) where 1 m and 2 m are the two masses, 1 k and 2 k are spring constants, 1 c and 2 c are damping constants and 1 f t and 2 f t are forcing functions. a. Derive the normal form of this system of second order ordinary differential equations (20.1) . b. Consider the special case of (20.1) where 2 0 m = and put the resulting system in normal form or, if necessary, a more general normal form. Solution Part a): Equations (20.1) are of the form (19.8). Following the pattern of (19.9), we define ( ) 1 1 1 2 3 2 4 2 ut xt du t dt t du t dt ⎡⎤ ⎢⎥ == ⎣⎦ x (20.2) Therefore, 1 u 1 m 1 c 1 k 2 m 2 k 2 c 2 u 1 f t 2 f t
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Lecture 20 MEEN 357 Homework Solution 2 Lecture-20-Homework-Solution.doc RMB () ( ) 1 2 1 1 2 1 2 2 2 2 2 2 du t dt u du t u dt dt u dt du t u dt dt ⎡⎤ ⎢⎥ ⇒= = ⎣⎦ x ± ±± ± (20.3) It follows from (20.1) 1 and (20.1) 2 that 112 2 1 11 1 2 12 1 111 1 1 ckc k f t uu u u u mmm m m = + −+ ± ± ± (20.4) and 222 22 1 2 1 2 ckf t u u u mm m = −− ± ± (20.5) Given (20.4) and (20.5), it follows from (20.2) 2 that 1 1 2 1 1 1 21 1 1 1 2 2 2 2 u u k f t u u u u m m u u u dt u t m −−+ + + == + x ± ± ± ± ± (20.6) Next, reuse the definition (20.2) and rewrite (20.6) as
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Homework 20 Solution - Lecture 20 MEEN 357 Homework...

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