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L21 2D Rigid Body Dynamics

L21 2D Rigid Body Dynamics - J Peraire S Widnall 16.07...

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J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L21 - 2D Rigid Body Dynamics Introduction In lecture 11, we derived conservation laws for angular momentum of a system of particles, both about the center of mass, point G , and about a fixed (or at least non-accelerating) point O . We then extended this derivation to the motion of a rigid body in two-dimensional plane motion including both translation and rotation. We obtained statements about the conservation of angular momentum about both a fixed point and about the center of mass. Both are powerful statements. However, each has its own sperate requirements for application. In the case of motion about a fixed point, the point must have zero acceleration . Thus the instantaneous center of rotation, for example the point of contact of a cylinder rolling on a plane, cannot be used as the origin of our coordinates. For motion about the center of mass, no such restriction applies and we may obtain the statement of conservation of angular momentum about the center of mass even if this point is accelerating. Kinematics of Two-Dimensional Rigid Body Motion Even though a rigid body is composed of an infinite number of particles, the motion of these particles is constrained to be such that the body remains a rigid body during the motion. In particular, the only degrees of freedom of a 2D rigid body are translation and rotation . Parallel Axes Consider a 2D rigid body which is rotating with angular velocity ω about point O , and, simultaneously, point O is moving relative to a fixed reference frame x and y with origin O . 1
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In order to determine the motion of a point P in the body, we consider a second set of axes x y , always parallel to xy , with origin at O , and write, r P = r O + r P (1) v P = v O + ( v P ) O (2) a P = a O + ( a P ) O . (3) Here, r P , v P and a P are the position, velocity and acceleration vectors of point P , as observed by O ; r O is the position vector of point O ; and r , ( v P ) O and ( a P ) O are the position, velocity and acceleration P vectors of point P , as observed by O . Relative to point O , all the points in the body describe a circular orbit ( r P = constant), and hence we can easily calculate the velocity, ( v P ) O = r P θ ˙ = r ω , or, in vector form, ( v P ) O = ω × r , P where ω is the angular velocity vector. The acceleration has a circumferential and a radial component, (( a P ) O ) θ = r θ ¨ = r ˙ (( a P ) O ) r = r P θ ˙ 2 P ω 2 . P P ω , = r Noting that ω and ω ˙ are perpendicular to the plane of motion (i.e. ω can change magnitude but not direction), we can write an expression for the acceleration vector as, ( a P ) O = ω ˙ × r P ) . P + ω × ( ω × r Recall here that for any three vectors A , B and C , we have A × ( B × C ) = ( A C ) B ( A B ) C . Therefore · · ω × ( ω × r ) = ( ω r ) ω ω 2 r = ω 2 r . Finally, equations 2 and 3 become, P P P P · v P = v O + ω × r (4) P a P = a O + ω ˙ × r P ) . (5) P + ω × ( ω × r Body Axes An alternative description can be obtained using body axes. Now, let x y be a set of axes which are rigidly attached to the body and have the origin at point O .
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