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me352_hw1_soln

# me352_hw1_soln - ME 352 System Dynamics and Control...

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1 ME 352 System Dynamics and Control Homework #1 – Solutions 1. (a) Force equations: ( ) ( ) 2 1 1 1 2 1 1 3 1 2 2 2 1 2 2 2 3 1 2 2 2 2 ( ) d y dy dy dy f t M B B K y y dt dt dt dt dy dy d y dy B K y y M B dt dt dt dt = + + - + - - + - + + Rearrange the equations as follows: ( ) ( ) ( ) ( ) 2 1 3 1 1 3 2 1 2 2 1 1 1 1 2 2 3 2 3 1 2 1 2 2 2 2 2 B B d y dy B dy K f y y dt M dt M dt M M B B d y B dy dy K y y dt M dt M dt M + = - + - - + + = - + - State variables: x y x dy dt x y x dy dt 1 2 2 2 3 1 4 1 = = = = , , , . State equations: dx dt x dx dt K M x B B M x K M x B M x 1 2 2 2 1 2 3 2 2 2 3 3 2 4 = = - - + + + dx dt x dx dt K M x B M x K M x B B M x M f 3 4 4 1 1 3 1 2 1 3 1 3 1 4 1 1 = = + - - + + Transfer functions: ( ) ( ) ( ) [ ] ( ) [ ] ( ) { } ( ) ( ) [ ] ( ) [ ] ( ) { } 2 2 2 3 1 3 2 1 2 1 3 2 2 3 1 1 2 1 2 2 3 1 3 1 2 2 3 3 2 1 2 1 3 2 2 3 1 1 2 1 2 2 3 1 3 1 2 ( ) ( ) ( ) ( ) M s B B s K Y s F s s M M s B B M B B M s K M M B B B B B B s B B K Y s B s K F s s M M s B B M B B M s K M M B B B B B B s B B K + + + = + + + + + + + + + + + + = + + + + + + + + + + + (b) Force equations: ( ) 2 1 2 1 1 2 2 2 1 2 2 2 1 B B d y dy B dy dy dy K f y dt M dt M dt M dt dt B + = - + + = -

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me352_hw1_soln - ME 352 System Dynamics and Control...

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