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net7 - Rigid Body Kinetics All formula which are applicable...

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Rigid Body Kinetics All formula which are applicable to a system of particles will also be applicable to Rigid Bodies. The additional constraint placed on the system of particles, which requires the distances between particles be constant, simplifies the problem substantially. Recall the equation of motion for a system of particles is X ~ F = d dt ~ L = d dt N X i =1 m i ~v i ! . (1) But for a rigid body, one additional constraint can be placed on the velocities, i.e., ~v i = ~v G + × ~ r i/G = ~v G + × i (2) where i = ~ r i/G is the position vector of mass m i with respect to the center of gravity G . Combining equations (1) and (2), we can write X ~ F = d dt N X i =1 m i ~v G + N X i =1 m i × i ! . (3) Removing now constants from the summation over index i , we have X ~ F = d dt ~v G N X i =1 m i + × N X i =1 m i i ! . (4) x y z ~r i A ~ F j G ~ R G i j ~v i m i ~r j Define now the total mass, M = N X i =1 m i , and recognize that N X i =1 m i i = ~ 0 by the definition of the center of gravity, the equation of motion for rigid bodies in translation can be written simply as X ~ F = d dt ( M~v g ) = M~a G which is identical to that for a system of particles. – 7 / 1 –
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Moment and Angular Momentum of a Rigid Body Recall for a system of particles, the alternate form of Newton’s 2nd Law, that of moment and angular momentum, can be written about the origin O as X ~ M O = d dt ~ H O , (1) where the total moment about point O for all external applied forces is defined as X ~ M O = X j ~ r j × ~ F j (2) and the total angular momentum about point O for N mass particles is defined as ~ H O = N X i =1 ~ r i × m i ~v i . (3) For a rigid body, the additional constraint on the velocities is such that ~v i = ~v A + × ~ r i/A , (4) in which A is a point on the rigid body where the velocity is known. Since Newton’s Second Law for translations utilizes the velocity and acceleration of the center of gravity G , it is convenient to simplify equation (3) by substituting point G as point A in equation (4), i.e., ~v i = ~v G + × ~ r i/G = ~v G + × i , (5) in which ~ r i/G = ~ r i - ~ R G = i . To obtain ~ r i for equation (3), rewrite the above equation as ~ r i = ~ R G + i (6) and then substitute equations (5) and (6) into equation (3) to yield ~ H O = N X i =1 ( ~ R G + i ) × m i ( ~v G + × i ) = N X i =1 ~ R G × m i ~v G + N X i =1 i × m i ~v G + N X i =1 ~ R G × m i ( × i ) + N X i =1 i × m i ( × i ) = ~ R G × N X i =1 m i ! ~v G + N X i =1 m i i ! × ~v G + ~ R G × × N X i =1 m i i ! + N X i =1 m i ( i × × i ) . – 7 / 2 –
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Using again the definition of the center of gravity, i.e., m i i = ~ 0 , and let M be the total mass m i , we can write the angular momentum about the origin O as ~ H O = ~ R G × M~v G + N X i =1 m i ( i × × i ) . (7) The above expression is written in abstract algebra form, which implies it is applicable for any coordinate system.
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