Force vectors

Since this approach maps all possible interactions to processors, no communication is required during force calculation. Moreover, the row assignments are completed before the first step of the simulation. The computation of the bounds for each processor require O(P ) time, which is very negligible compared to N (for N S> P). The communication required at the end of each step to update the position and velocity vectors is done by reducing force vectors of length N, and therefore scales as 0 N) per node per time step. Thus the overall complexity of this algorithm is. 

In contrast to the flexibiUty method, the stiffness method considers the displacements as unknown quantities in constmcting the overall stiffness matrix (K). The force vector T is first calculated for each load case, then equation 20 is solved for the displacement D. Thermal effects, deadweight, and support displacement loads are converted to an equivalent force vector in T. Internal pipe forces and stresses are then calculated by applying the displacement vector [D] to the individual element stiffness matrices. 

Fig. 4. Contact angle at the oil—soHd—air boundary line in the roU-back process of oily soHd detergency. and are force vectors at the surface and...

The spatial Cauchy stress tensor s is defined at time by f = sn, where t(x, t, n) is a contact force vector acting on an element of area da = n da with unit normal i and magnitude da in the current configuration. The element of area... 

Figure 44.20 represents a typical process-roll configuration. The arrows indicate the force vectors generated by the wire, belt, or product wrap around these rolls. The left roll has a force vector at 45° down to the left the right roll has a mirror image force vector and the bottom roll has a vertical vector. 

The primary (X-axis) radial measurement for the bottom roll should be in the vertical plane with the transducer mounted on top of the bearing cap. The secondary radial (Y-axis) measurement should be in the horizontal plane facing upstream of the belt. Since the belt carried by the roll also imparts a force vector in the direction of travel, this secondary point should be opposite the direction of belt travel. 

Figure 44.20 Typical process-roll configuration and wrap-force vectors...

As Figure 60.28 illustrates, a restriction is created by the offset. This restriction creates a high pressure and another force vector in the direction of rotation. Oil whirl accelerates the wear and failure of the bearing and bearing support structure. 

Thus, integration over an arbitrary volume allows us to find the force caused by any distribution of masses. It is essential that the particle p can be located either outside or inside of a body and at any distance from its surface. Equation (1.3) describes the total force that is a result of a superposition of the elementary forces, vectors, at the same point. Correspondingly, this force can cause a translation of the particle only. It is also instructive to consider the force F generated by the particle and acting on an arbitrary body. Each elementary volume is subjected to the force... 

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. 

E and B are the fundamental force vectors, while P and H are derived vectors associated with the state of matter. J is the vector current density. The Maxwell equations in terms of E and B are... 

The vertical lines indicate the magnitude of the force vector and of the displacement vector, and 9 is the angle between the force vector and the displacement vector, as illustrated in Figure 3.2. 

The sign convention for work W follows namrally from the definition in Equation (3.3). When the external force vector is in the same direction as the displacement vector, the surroundings do work on the system, 0 is equal to 0, cos 0 is equal to 1, and W is positive. When the external force vector is in the opposite direction from that of the displacement vector, the system does work on the surroundings, 9 is equal to tt, cos 0 is equal to —1, and W is negative. Thus, the convention that W represents the work done on the system by its surroundings follows namrally from the definition of work. ... 

Accordingly, the selection rules for Raman and IR spectroscopy are different. In Raman spectroscopy, there must be a change in the molecule s polarizability upon excitation, whereas a change in dipole moment is required for IR. A dipole moment is the magnitude of the electronic force vector between the negative and positive charges or partial charges on a molecule. A permanent dipole moment exists in all polar mol-... 

It will be assumed throughout that H and are symmetric positive-definite tensors. We write the mobility as a contravariant tensor, with raised bead indices, to reflect its function the mobility H may be contracted with a covariant force vector Fv (e.g., the derivative of a potential energy) to produce a resulting contravariant velocity Fy. 

The laminate forces and moments are obtained by integrating the stresses of each lamina through the total laminate thickness to obtain a force vector, N, and a moment vector, M, the components of which can be related to the strains, and and curvatures, x, of the laminae, where the superscript indicates that tlie strains are measured... 

The force acting on any differential segment of a surface can be represented as a vector. The orientation of the surface itself can be defined by an outward-normal unit vector, called n. This force vector, indeed any vector, has direction and magnitude, which can be resolved into components in various ways. Normally the components are taken to align with coordinate directions. The force vector itself, of course, is independent of the particular representation. In fluid flow the force on a surface is caused by the compressive (or expansive) and shearing actions of the fluid as it flows. Thermodynamic pressure also acts to exert force on a surface. By definition, stress is a force per unit area. On any surface where a force acts, a stress vector can also be defined. Like the force the stress vector can be represented by components in various ways. 

The force vector f is also the momentum-flux vector M. Thus... 

These work contributions have a negative sign because, by definition, positive velocities are flowing in the coordinate direction and positive stresses are opposite to the coordinate directions. The work is a scalar quantity the subscripts on W[ simply indicate the face and do not represent vector components as would be the case for the force vector. 

In the examples (2.62), (2.63), the force vector F was parallel to the displacement vector dr [i.e., 6 = 0 in (2.61b)]. In these cases, the vectorial aspects of the work expression (2.60) could be ignored, because the integration path from initial to final r has only a 1-dimensional character. More generally, the integration path may vary in 3-dimensional (nonlinear) manner from initial point A (at rA) to final point B (at rB), and the value of the integral is not uniquely determined until the exact A —> B path is specified. In this case, the work wa b must be evaluated as a line integral (or path integral cf. Sidebar 1.6),... 

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