Body-fixed frame of reference

Applications of the theory described in Section III.A.2 to malonaldehyde with use of the high level ab initio quantum chemical methods are reported below [94,95]. The first necessary step is to define 21 internal coordinates of this nine-atom molecule. The nine atoms are numerated as shown in Fig. 12 and the Cartesian coordinates x, in the body-fixed frame of reference (BF) i where n= 1,2,... 9 numerates the atoms are introduced. This BF frame is defined by the two conditions. First, the origin is put at the center of mass of the molecule. 

The first step is to define 21 internal coordinates of the nine-atomic malonaldehyde molecule. We assume the numbering of atoms as shown in Figure 6.2 and introduce the Cartesian coordinates x, in the body-fixed frame of reference (BF) as... 

As was explained in Section 6.2, the body-fixed frame of reference is specified by imposing six conditions on the 15 body-fixed Cartesian coordinates r = Xi i n = 1,2,..., 5). The first three conditions fix the origin to the center of mass. The other three conditions specify the orientation of the body-fixed axes... 

The Kirkendall effect is a well-known phenomenon resulting from the difference in intrinsic diffusivities of chemical constituents of substitution solid solutions (non-reciprocal diffusion). Many textbooks provide a detailed and quantitative treatment of this important phenomenon (Philibert, 1991) schematized in Fig. 2.2 for a homogeneous diffusion couple of constituents A and B forming a continuous substitutional solid solution. In Fig. 2.2a, the initial position of the contact surface is marked by fixed inert markers that define the origin, also named the Matano plane (M), of the reference frame centred on the mass or the number of moles this reference frame is conunonly used to define interdiffusion processes and the unique interdiffusion coefficient that permits the characterization of the transport of A and B. Moving inert markers determine the actual position of the initial contact surface (Fig. 2.2b) and therefore visualize the drift of lattice planes within the diffusion zone. They mark the origin, also named the Kirkendall plane, of the lattice-fixed frame of reference that permits the definition of the different intrinsic diffusion coefficients of the A and B constituents. Relative to the Matano plane, this drift of lattice planes is equivalent to a translation of the diffusion couple without apparent deformation and without the action of any externally applied stress or, in other words, to a rigid body translation of the phase lattice. 

Here, we consider Stokes problem of uniform, streaming motion in the positive z direction, past a stationary solid sphere. The problem corresponds to the schematic representation shown in Fig. 7-11 when the body is spherical. This problem may also be viewed as that of a solid spherical particle that is translating in the negative z direction through an unbounded stationary fluid under the action of some external force. From a frame of reference whose origin is fixed at the center of the sphere, the latter problem is clearly identical with the problem pictured in Fig. 7-11. Because we have already derived the form for the stream-function under the assumption of a uniform flow at infinity, we adopt the latter frame of reference. The problem then reduces to applying boundary conditions at the surface of the sphere to determine the constants C and Dn in the general equation (7-149). The boundary conditions on the surface of a solid sphere are the kinematic condition and the no-slip condition,... 

The absolute velocity and acceleration of G, are expressed in terms of the body-fixed coordinate system, b ba, b3, which is fixed to the pendulum and rotates about the b3 axis as before. Although it is equivalent to expressing within the inertial frame of reference, I, j, k, the body-fixed coordinate system, b bj, bj, uses fewer terms. The velocity and acceleration for G, are respectively as follows ... 

The Born-Oppenheimer diagonal correction is given in Eq. (2a). In that equation, the gradients refer to space fixed frame (SFF) coordinates. For diatomic molecules, considerable savings result from a transformation to body fixed frame (BFF) coordinates. This transformation is accomplished in two steps. The SFF coordinates are transformed to center of mass fixed frame (CMFF) coordinates and then the CMFF coordinates are transformed to BFF coordinates. The details of the transformation are beyond the scope of this review. Here we sketch the ideas involved. A detailed treatment, based on the pioneering work of Kronig, can be found in Ref. 7. In particular, first the rigorously removable center of mass of the nuclei and... 

Kinematics deals with time and distances. To measure time one needs to have a particular instant as a reference. Similarly, the motion of a body remains ambiguous unless it is relative to something else, e.g. to two objects, or points, with a fixed distance separating them. In effect, such a collection of points can be thought of as an observer who is watching the motion, or as a frame of reference . Since such a frame is constructed out of points with fixed separations, it can only undergo rigid body motions, i.e. translations and rotations. Thus, let A (f). c(0 and o be points referred to an old frame and be the transform of X(t) into a... 

Figure 1 shows the undeformed and deformed (shadowed) states of a flexible body k. r is the position vector of an arbitary point M, while a is the position vector of the mass center. Both r and a are defined on the deformable body. X is the vector from the mass center (Gd) to an arbitrary point M on the deformed body and Y is the vector from the mass center (Go) to an arbitrary point Mq on the undeformed body. The following formulations use an inertial reference frame (Ro) and a body fixed frame (R[Pg.63]

Eulerian [82] and the Lagrangian methods [83-85]. The Eulerian approach uses a coordinate system fixed in the frame of reference of the laboratory, and it takes account of the velocity of the body relative to that frame as the volume of the body changes. The Lagrangian method uses a coordinate system fixed in the gel, such that a fixed volume of solid phase is contained in any volume element. The solution is obtained in terms of the material coordinate, m, where... 

Finally, we demonstrate a strong effect of out-of-plane vibrational motions on the tunneling splitting [183]. In order to do this, first we have to define 15 internal coordinates to describe in-plane motions and construct the corresponding Hamiltonian. Following the same method as that explained in Section 6.3.1, we introduce Cartesian coordinates in the body-fixed (BF) frame of reference. 

This work introduced the concept of a vibronic R-matrix, defined on a hypersurface in the joint coordinate space of electrons and intemuclear coordinates. In considering the vibronic problem, it is assumed that a matrix representation of the Schrodinger equation for N+1 electrons has been partitioned to produce an equivalent set of multichannel one-electron equations coupled by a matrix array of nonlocal optical potential operators [270], In the body-fixed reference frame, partial wave functions in the separate channels have the form p(q xN)YL(0, radial channel orbital function i/(q r) and antisymmetrized in the electronic coordinates. Here 0 is a fixed-nuclei A-electron target state or pseudostate and Y] is a spherical harmonic function. Both and i r are parametric functions of the intemuclear coordinate q. It is assumed that the target states 0 for each value of q diagonalize the A-electron Hamiltonian matrix and are orthonormal. 

Each choice of the body-axes specifies a reference orientation, in which the body frame coincides with the space-fixed frame. Let us choose one of the body axes, say u 3, to be in the direction of... 

One should realize that these calculations are based on an expression for Vr which corresponds to potential flow past a stationary nonde-formable bubble, as seen by an observer in a stationary reference frame. However, this analysis rigorously requires the radial velocity profile for potential flow in the Uquid phase as a nondeformable bubble rises through an incompressible liquid that is stationary far from the bubble. When submerged objects are in motion, it is important to use liquid-phase velocity components that are referenced to the motion of the interface for boundary layer mass transfer analysis. This is accomplished best by solving the flow problem in a body-fixed reference frame which translates and, if necessary, rotates with the bubble such that the center of the bubble and the origin of the coordinate system are coincident. Now the problem is equivalent to one where an ideal fluid impinges on a stationary nondeformable gas bubble of radius R. As illustrated above, results for the latter problem have been employed to estimate the maximum error associated with the neglect of curvature in the radial term of the equation of continuity. 

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