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 θ 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 , θ eq = M ˙ ˙ r d − r d + r eq ( ) ˙ θ d 2 ( ) + K r d + r eq − r u ( ) − Mg cos θ d + θ eq ( ) + C ˙ r d = M ˙ ˙ r d − r d + r eq ( ) ˙ θ d 2 ( ) + K r d + r eq − r u ( ) − Mg cos θ d cos θ eq − sin θ d sin θ eq ( ) + C ˙ r d = M ˙ ˙ r d − r d + r eq ( ) ˙ θ d 2 ( ) + C ˙ r d + K r d + Mg K − Mg cos θ d = M 2 r d + r eq ( ) ˙ r d ˙ θ d + r d + r eq ( ) 2 ˙ ˙ θ d ( ) + Mg r d + r eq ( ) sin θ d + θ eq ( ) = M 2 r d + r eq ( ) ˙ r d ˙ θ d + r d + r eq ( ) 2 ˙ ˙ θ d ( ) + Mg r d + r eq ( )...
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lecture810 - ME 563 Mechanical Vibrations Lecture#8...

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