RR_kane_impact

# RR_kane_impact - Kanes Dynamical Equations 0 Contents 6...

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Kane’s Dynamical Equations 0 Contents 6 Analytical Dynamics 1 6.1 Kane’s Dynamical Equations . . . . . . . . . . . . . . . . . . . 1 6.2 Lgrange’s Equations of Motion . . . . . . . . . . . . . . . . . . 7

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Kane’s Dynamical Equations 1 6 Analytical Dynamics 6.1 Kane’s Dynamical Equations A two-link kinematic chain is considered in Fig. 5.7. The bars 1 and 2 are homogenuos and have the lengths L 1 = L 2 = L . The masses of the rigid links are m 1 = m 2 = m and the gravitational acceleration is g . The plane of motion is xy plane with the y -axis vertical, with the pos- itive sense directed downward. The origin of the reference frame is at A . The system has two degrees of freedom. To characterize the instantaneous conﬁguration of the system, two generalized coordinates q 1 ( t ) and q 2 ( t ) are employed. The generalized coordinates q 1 and q 2 denote the radian measure of the angles between the link 1 and 2 and the horizontal x -axis. There are two generalized speeds deﬁned as u 1 = ˙ q 1 and u 2 = ˙ q 2 . (6.1) The mass centers of the links are designated by C 1 ( x C 1 ,y C 1 , 0) and C 2 ( x C 2 ,y C 2 , 0). Kinematics The position vector of the center of the mass C 1 of the link 1 is r C 1 = x C 1 ı + y C 1 , where x C 1 and y C 1 are the coordinates of C 1 x C 1 = L 1 2 cos q 1 , y C 1 = L 1 2 sin q 1 . The velocity vector of C 1 is the derivative with respect to time of the position vector of C 1 v C 1 = ˙ r C 1 = ˙ x C 1 ı + ˙ y C 1 , where ˙ x C 1 = - L 1 2 ˙ q 1 sin q 1 and ˙ y C 1 = L 1 2 ˙ q 1 cos q 1 , or v C 1 = - L 1 2 u 1 sin q 1 ı + L 1 2 u 1 cos q 1 . The acceleration vector of C 1 is the double derivative with respect to time of the position vector of C 1 a C 1 = ¨ r C 1 = ¨ x C 1 ı + ¨ y C 1 ,
Kane’s Dynamical Equations 2 where ¨ x C 1 = - L 1 2 ¨ q 1 sin q 1 - L 1 2 ˙ q 2 1 cos q 1 , ¨ y C 1 = L 1 2 ¨ q 1 cos q 1 - L 1 2 ˙ q 2

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RR_kane_impact - Kanes Dynamical Equations 0 Contents 6...

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