example_IV_1_08

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

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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 ( ) EOMs 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 EOMs 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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example_IV_1_08 - Example IV.1.8 L L L ! 1 ! 2 ! 3 m m m K...

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