Velocity vector rigid-body rotation

The velocity vector for rigid-body rotation of a solid that spins at constant angular velocity is v = ft x r, where ft is the angular velocity vector and r is the position vector from the axis of rotation. Obtain an expression for the vorticity vector V x v for rigid-body rotation in terms of ft. 

Answer Consider rigid-body rotation of a solid sphere about the z axis of a Cartesian coordinate system and calculate the velocity vector at the fluid-solid interface by invoking the no-slip condition... 

Consider a rigid body rotating about a fixed axis with a constant angular velocity, cj, in radians per second. This rotation can be described by a vector CO with length cj and a direction parallel to the axis of rotation. A point P not on the rotation axis will then have a linear velocity given by... 

The upper arm, simplified as a single rigid body, is shown in Fig. 7.8. The velocity and acceleration for the center of mass of the arm are derived and presented in two coordinate systems. Table 7.2 presents the kinematics in an inertial coordinate system, while Table 7.3 utilizes a body-fixed, moving coordinate system. For this system, not unlike the two-segment system of Fig. 7.4, a moving coordinate system bi, b2, b3 is fixed at point B and is allowed to rotate about the bj axis so that the unit vector b, will always lie on segment BC. 

In this equation, M is an A x A nonlinear mass matrix and C is the A-dimensional vector of velocity-dependent (Coriolis and other) forces. 0, 0, and 0 represent the torsional position, velocities and accelerations, respectively. The ability to calculate the accelerations recursively relies on the chainlike structure of the protein, in which each node of the chain represents a rigid body. These rigid bodies consist of one atom or a cluster of atoms whose relative positions are fixed. To simplify the explanation of the algorithm, an unbranched chain will be considered, although the approach can be easily extended to branched systems. For this simple case, the first rigid body, at one end of the chain, defines the base (fe = 0), while the last rigid body, at the other end of the chain, defines the tip k = N). The rotatable torsion angle between bodies k and k — I is defined as 0,. ... 

The connecting points in the system arc also labeled as the bodies. The vector 02k-l at the connecting points represents the rotational direction of the rigid body in the system. The vector 02/fe on each body represents the rotational direction of the deformed body. Thus the absolute angular velocity of any body in the system can be expressed as ... 

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