lecture810

lecture810 - ME 563 Mechanical Vibrations Lecture #8...

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Unformatted text preview: ME 563 Mechanical Vibrations Lecture #8 Linearizing Equations of Motion (Direct Method) Sprung Pendulum 1 r g One option for linearizing is to derive the equations of motion FIRST and then: - Identify equilibria - Transform coordinates - Discard higher-order terms Identify Equilbria 2 Makes sense a)Pendulum can be straight down or straight up. b)Spring compresses accordingly. Transform Coordinates 3 r eq = r u + Mg / K , eq = M r d r d + r eq ( ) d 2 ( ) + K r d + r eq r u ( ) Mg cos d + eq ( ) + C r d = M r d r d + r eq ( ) d 2 ( ) + K r d + r eq r u ( ) Mg cos d cos eq sin d sin eq ( ) + C r d = M r d r d + r eq ( ) d 2 ( ) + C r d + K r d + Mg K Mg cos d = M 2 r d + r eq ( ) r d d + r d + r eq ( ) 2 d ( ) + Mg r d + r eq ( ) sin d + eq ( ) = M 2 r d + r eq ( ) r d d + r d + r eq ( ) 2 d ( ) + Mg r d + r eq ( )...
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lecture810 - ME 563 Mechanical Vibrations Lecture #8...

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