Force-torque vector

Here is a generalized (6N dimensional) force-torque vector, U -u (6N dimensional) is the particle translational-angular velocity relative to the bulk fluid flow evaluated at the particle centre, (3x3 matrix) is the traceless symmetric rate of the strain tensor (supposed to be constant in space). The resistance matrices Rfu (6N x6N) and Rfe (6N x 3 x 3) which depend only on the instantaneous relative particle configurations (position and orientation) relate the force-torque exerted by the suspending fluid on the particles to their motion relative to the fluid and to the imposed shear flow, respectively. Note that in ER (MR) fluids torques can be neglected. 

Equation (7.20) is to be regarded as applying throughout all of space, including the particle interiors. Boundary conditions imposed at the sphere surfaces implicitly furnish the relations determining the F s, these being related to the forces, torques, etc., exerted on the fluid by the particles. For instance, the vector F, is related to the external force acting on each particle via the expression... 

Torque-based impedance controller, x is the robot actual pose in the task space computed from the actual joint configuration q with the forward kinematics (FK) block J is the robot Jacobian is the desired pose in the task space x is the equilibrium pose of the environment is the net stiffness of the sensor and of the envirotunent f j and are the external enviroiunent forces expressed in the task space and in the joint space, respectively fj is the desired force vector is the desired torque vector computed from the force equilibrium r is the torque input vector of the inner torque control loop and is the commanded motor torque vector. The command force f is defined as f = Z(x - x), where Z is the impedance matrix. When the environmental forces are available (dotted lines), the measurements are used to decouple the dynamic of the system. 

Ha 0 Hb angle of 116°. While the products and Ha.-0-Hb are separating, the repulsive forces causing the separation tend to operate along the line of centers between Hg and Hq. Since the 0-Hq line approximately defines the scattering direction, it is clear that the torque vector generated by these repulsive forces must be perpendicular to the helicity axis, and thus correspond to m = 0 final states. 

Where T is the input torque vector Q, Q, and 0 are the joint displacement velocity and acceleration of the manipulator, respectively. Mass matrix, M(0) and the gravitational force vector, H(0,0), and the friction matrix, F(0,0) are complex functions of 0 and 0. Forward and backward recursive equations are applied to propagate kinematics information from the inertial frame to end-effector frame and to propagate the forces and moments exerted on each link from the end-effector frame to the manipulator base frame. It is assumed that each link of a manipulator as a rigid body. 

Hence dridt = o and p = mv, the first term is then zero since both vectors are coUinear. Taking into account that dpidt = F, the second term can be written as [rF], This vector is called the momentum of the force (torque) F with respect to pole O. 

Two vectors commonly represented in terms of cross products are the angular momentum of a particle about some point, equal to the cross product of the momentum vector of the particle and the radius vector from the origin to the particle and torque, equal to the cross product of the force vector and the vector representing the lever arm. 

OO distance and the covalent OH bond length, denoted r, are assumed to be fixed. We also assume that a dipole moment p is rigidly connected with a molecule, so its turn on the angle (3 is accompanied by the same turn of the dipole-moment vector from the position p to p (see Fig. 56b). For simplicity we consider rotation of a molecule OBB in a plane. Then the equation of motion under the torque due to stipulated by this force is given by... 

A dipole is acted on not only by a force, but also by a torque, in an external field, and this torque is proportional to the field strength rather than to its rate of change with position. The x component of this torque, regarded as a vector, is seen to be... 

Problem 7-9. Motion of a Force- and Torque-Free Axisymmetric Particle in a General Linear Flow. We consider a force- and torque-free axisymmetric particle whose geometry can be characterized by a single vector d immersed in a general linear flow, which takes the form far from the particle y°°(r) = U00 + r A fl00 + r E00, where U°°, il, and Ex are constants. Note that E00 is the symmetric rate-of-strain tensor and il is the vorticity vector, both defined in terms of the undisturbed flow. The Reynolds number for the particle motion is small so that the creeping-motion approximation can be applied. 

A moment in mechanics is generally defined as Uj = Qd, where Uj is the jth moment, about a specified line or plane a of a vector or scalar quantity Q (e.g., force, weight, mass, area), d is the distance from Q to the reference line or plane, and j is a number indicating the power to which d is raised. [For example, the first moment of a force or weight about an axis is defined as the product of the force and the distance of the fine of action of the force jfrom the axis. It is commonly known as the torque. The second moment of the force about the same axis (i.e., i = 2) is the moment of inertia.] If Q has elements Qi, each located a distance di from the same reference, the moment is given by the sum of the individual moments of the elements ... 

Torque (T) - For a force f that produces a torsional motion, T = r X F, where r is a vector from some reference point to the point of application of the force. 

The hydrodynamic force F and torque about O, Tq, exerted by the fluid on the particle (exclusive of buoyant forces and torques) are linear vector functions of the velocity and spin of the particle. In particular (B22)... 

Equations (109)-(110) show that the force and torque on the particle are linear vector functions of the translational slip velocity Uo — Uo, the angular slip velocity - tOy, and the shear rate S. 

Equations (167) and (168) express the fact that the force and torque on each particle are linear vector functionals of the slip velocities of all the particles. It should be clearly noted that these operators are calculable solely from a knowledge of the intrinsic translational and rotational solutions of Stokes equations resulting from motion of the yth particle when all the other particles, as well as the fluid at inflnity, are at rest. In the case where u = 0, Eqs. (167) and (168) reduce to (148) and (149), respectively. As yet, there exists no system for which these operators are known. 

The most likely manifestation of this screw-sense lies in the existence of a torque exerted by the fluid on the particles comprising the porous medium and vice versa. The vector —S idVgives the force exerted by the surroundings on the contents of a small volume element dV. Hence, the external moment of the forces acting on a volume V of the medium, about some arbi-tray origin, is... 

The 3-D ground reaction force vector, the vertical ground reaction torque and the point of application of the ground reaction force vector (i.e., center of pressure) are measured with force platforms embedded in the walkway. Force plates with typical measurement surface dimensions of 0.5 x 0.5 m are comprised of several strain gauges or piezoelectric sensor arrays rigidly mounted together. 

Deciding whether to measure force (a translational quantity) or torque (a rotational quantity) is an important issue in testing strength. Even when the functional units of interest produce rotational motion, force measurement at some point along the moment arm is common. This is due to the evolution from manual muscle tests to the use of objective measurements where a force sensor replaces the human examiner sense of force resisted or generated. If d is the distance from the point of rotation to the point of force measurement and the force vector is tangent to the arc of motion, then... 

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