MIT16_07F09_Lec23

# MIT16_07F09_Lec23 - J Peraire S Widnall 16.07 Dynamics Fall...

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Unformatted text preview: J. Peraire, S. Widnall 16.07 Dynamics Fall 2007 Version 1.0 Lecture L23- 2D Rigid Body Dynamics: Impulse and Momentum In lecture L9, we saw the principle of impulse and momentum applied to particle motion. This principle was of particular importance when the applied forces were functions of time and when interactions between particles occurred over very short times, such as with impact forces. In this lecture, we extend these principles to two dimensional rigid body dynamics. Impulse and Momentum Equations Linear Momentum In lecture L22, we introduced the equations of motion for a two dimensional rigid body. The linear momentum for a system of particles is defined as L = m v G , where m is the total mass of the system, and v G is the velocity of the center of mass measured with respect to an inertial reference frame. Assuming the mass of the system to be constant, we have that the sum of the external applied forces to the system, F , must equal the change in linear momentum, L ˙ = F , or, integrating between times t 1 and t 2 , t 2 L 2 − L 1 = F dt (1) t 1 Note that this is a vector equation and therefore must be satisfied for each component separately. Expression 1 is particularly useful when the precise time variation of the applied forces is unknown, but their total impulse can be calculated. Of course, when the impulse of the applied forces is zero, the momentum is conserved and we have ( v G ) 2 = ( v G ) 1 . Angular Momentum A similar expression to 1 can be derived for the angular momentum if we start from the principle of conser- vation of angular momentum, H ˙ G = M G . Here, H G = I G ω is the angular momentum about the center of mass, and M G is the moment of all externally applied forces about the center of mass. Integrating between times t 1 and t 2 , we have, t 2 ( H G ) 2 − ( H G ) 1 = M G dt . (2) t 1 1 In a similar manner, for rotation about a fixed point O , we can write, t 2 ( H O ) 2 − ( H O ) 1 = M O dt , (3) t 1 where H O = I O ω , the moment of inertia, I O , refers to the fixed point O , and the external moments are with respect to point O ....
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MIT16_07F09_Lec23 - J Peraire S Widnall 16.07 Dynamics Fall...

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