Ch4_Lecture_Nov10_2010-1

# Ch4_Lecture_Nov10_2010-1 - Chapter 4 Multiple...

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Chapter 4 Multiple Degree-of-Freedom (DOF) Systems 1

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Multiple Degree of Freedom Systems m k x(t) c base y(t) 1-DOF System Model Engineering System 2-DOF System Model 2 D e gr ee -of-F r dom (DO F): Minimum number of coordinates to specify the configuration of a system
Analyzing Multi-DOF Systems ± DOF (degree of freedom): Minimum number of coordinates to specify the configuration of a system ± Many systems have more than one DOF ± Examples of two DOF systems ² car with sprung and un-sprung mass (both heave) ² elastic pendulum (radial and angular) ² motions of a ship (roll and pitch) ² Robotic hands ² Manufacturing robot arms 3

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Example: Two-Degree-of-Freedom Model (c=0) A two-DOF system used to base much of the analysis and conceptual development of MDOF systems on. 4
Free-Body Diagram of Each Mass x 1 2 k 1 1 2 ( 2 - 1 ) m 1 2 5 2 ( 2 - 1 )

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Free-Body Diagram of Each Mass x 1 2 k 1 1 2 ( 2 - 1 ) m 1 2 6 ± ² ± ² 11 2 2 1 22 2 2 1 () () t ³ ´ ³ ³ ³ 2 ( 2 - 1 )
Deriving Coupled Equations of Motion ± ² ± ² 11 2 2 1 22 2 2 1 () () m x t k ³ ´ ³ ³ ³ 1 2 1 2 2 2 1 () ( ) () () 0 ´´ ³ ³´ 7 Rearranging the terms in these two (homogenous) equations of motion results in:

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Initial Conditions at t =0 ± Two coupled, second-order, ordinary differential equations with constant coefficients ± Needs four constants of integration to solve ± Thus, four initial conditions on positions and velocities 11 0 0 2 2 0 2 2 0 (0) , (0) x 8
Two-DOF System A two-DOF system has: ± Two equations of motion ± Two natural frequencies (more on this later) 9

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Two Homogeneous/ Coupled Equations ± Free vibrations, so two homogeneous equations ± Two equations of motion are coupled: ² Both have both coordinates x 1 and x 2 ² If only one mass moves, the other must follow ± In this system, the coupling is due to the spring k 2 ² Mathematically and physically coupled ² If k 2 = 0, no coupling occurs and two equation can be solved as two independent SDOF systems 10 11 1 2 1 2 2 22 2 1 () ( ) () () 0 () m x t k ±± ² ²±
Solution by Matrix Methods Two Equations Can be Written in the Form of a Single Matrix Equation 11 1 2 1 2 2 22 2 1 () ( ) () () 0 () m x t k ±± ² ²± 2 1 12 2 2 1 2 2 () 0 () ( 0( ) + ( ) ( ) ( ) 0 ± ² Review Appendix C to Refresh Your Matrices and Vectors Background.

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Solution in Matrix Form 111 222 () x t , ªº «» ¬¼ x x x 11 1 2 2 1 22 2 2 2 0( ) ( ) 0 ) ( ) 0 m k ±² ª º ª º ª º ± « » « » « » ² ¬ ¼ ¬ ¼ ¬ ¼ 12 2 1 2 1 2 2 12 2 2 1 2 2 () 0 () ( ) () () 0 ) + ( ) ( ) ( ) 0 ±± ±² ²±
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Ch4_Lecture_Nov10_2010-1 - Chapter 4 Multiple...

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