example_IV_1_08

# example_IV_1_08 - Example IV.1.8 L L L 1 2 3 m m m K K K F...

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Unformatted text preview: Example IV.1.8 L L L ! 1 ! 2 ! 3 m m m K K K F x y 3 2 1 Kinematics v 1 = v + ˙ ! 1 k ( ) " Lcos ! 1 i + L sin ! 1 j ( ) = L ˙ ! 1 # sin ! 1 i + cos ! 1 j ( ) v 2 = v 1 + ˙ ! 2 k ( ) " L cos ! 2 i + Lsin ! 2 j ( ) = L # ˙ ! 1 sin ! 1 # ˙ ! 2 cos ! 2 ( ) i + L ˙ ! 1 sin ! 1 + ˙ ! 2 sin ! 2 ( ) j v 3 = v 2 + ˙ ! 3 k ( ) " L cos ! 3 i + Lsin ! 3 j ( ) = L # ˙ ! 1 sin ! 1 # ˙ ! 2 cos ! 2 # ˙ ! 3 cos ! 3 ( ) i + L ˙ ! 1 sin ! 1 + ˙ ! 2 sin ! 2 + ˙ ! 3 sin ! 3 ( ) j ! r 1 = L ¡ ! ˙ ¢ 1 sin ¢ 1 i + ! ˙ ¢ 1 cos ¢ 1 j ( ) EOM’s T ¡ 1 2 mv 1 2 ¢ 1 2 mv 2 2 ¢ 1 2 mv 3 2 U ¡ 1 2 K £ 1 2 ¢ 1 2 K £ 2 ¡£ 1 £ ¤ 2 ¢ 1 2 K £ 3 ¡£ 2 £ ¤ 2 ¤ W ¡ F j £ ¤ • ¤ r 1 ¡ f Lcos £ 1 £ ¤ ¤£ 1 Using the standard linearization process (about the equilibrium state of ! 1 = ! 2 = ! 3 = ), the EOM’s become: mL 2 3 2 1 2 2 1 1 1 1 ! " # # \$ % & & ˙ ˙ ' 1 ˙ ˙ ' 2 ˙ ˙ ' 3 ( ) * + * ,- * . * + K 2 / 1 / 1 2 / 1 / 1 1 ! " # # \$ % & & ' 1 ' 2 ' 3 ( ) * + * ,- * . * = f L ( ) * + * ,- * . * sin t or M [ ] ˙ ˙ ! + K [ ] ! = f sin " t Natural Frequencies and Modes Although we do NOT need the natural frequencies and modal vectors for finding the particular...
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## This document was uploaded on 12/23/2011.

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example_IV_1_08 - Example IV.1.8 L L L 1 2 3 m m m K K K F...

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