Body-fixed reference frame

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. 

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. 

Figure 5.13. Schematic representation of the three-site SPC/E water model. Indicated are the principal axes of the body-fixed reference frame (the x axis being normal to the molecular plane) and the coordinates of constituent atoms. The values of the parameters are zo = —0.0646 A, yu = 0.8165 A, zn = 0.5127 A, 9 = 54.74 °.

The use of a body-fixed reference frame together with total angular momentum conservation was discussed by Miller [641. whilst Brodsky and Levich [13] employed a space-fixed frame. However. the first... 

Every body has a body fixed reference frame attached to its center of mass (CM). Initially the axes of these frames are parallel to the corresponding axis of the inertial frame. The location of the centers of mass is given with respect to the inertial frame, i.e. in absolute coordinates. 

The position of a rigid body in space can be unequivocally defined, with respect to a fixed reference frame, by six parameters conventionally the first three define the position of a point... 

The geometry of polyhedrons is defined in terms of corners, edges, and faces the location of corners is given by a series of vectors from the center of mass, and a unit outward normal vector is associated with each face. The location and orientation in space of each polyhedron are defined by the components of a vector to the center of gravity (with respect to a fixed reference frame) and by the principal axes of inertia of the body. The advantage of this type of shape is that complex flat-faced particles can be very accurately represented (Figure 7.11). 

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. 

The kinematic equations that describe the body motion are shown in this section. The motion is restricted to as to be uniform, i.e., the body is moving with constant speed. In order to write these equations, a reference fixed (camera) frame of coordinates must be used also there is a frame fixed to the body. The body position and orientation are established by the position and orientation of the body frame origin and the orientation of this frame with respect to the fixed reference frame. Since the motion is tridimensional, there are three position coordinates and three angles of orientation. Also, these variables are changing, producing three components of speed and rotation. A total of twelve state variables are needed to describe the body motion. To describe the orientation of the body frame, orientation... 

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. 

An important issue in reaction stereodynamics must be eonsidered. Experiment and vector preparations are usually performed in the laboratory reference frame (space fixed) whereas the important preparation for the reaction is the molecular reference frame (body fixed) which rotates during the collisions at a non-constant angular velocity. This leads to numerous difficulties, which have motivated an important literature reviewed in Ref [18], As far as stereodynamics considerations are... 

The frame (B) was chosen such that the rotational diffusion tensor is diagonal. In general, the polarizability tensor a will not be diagonal in the same body fixed frame that diagonalizes. In the special case when a and are simultaneously diagonalized in the frame (B) that is, when the molecule is a true symmetric top, then aij(B) = 0 for i = j. Referring back to Eq. (7.4.1) we see that in this eventuality azz(B) = a and axx(B) = ccyy(B) = a , and... 

Second law Given O is a fixed point on the inertial reference frame, the rate of change of the angular momentum of the body about O is equal to net moment of forces acting on the body about O. 

Figure 6.12 shows a computational algorithm diat can be used to model the centroid path of a musculotendinous actuator fin a given configuration of the joints in the body. There are four steps in the computational algorithm. Given the relative positions of the bones, the locations of all fixed via points are known and can be expressed in the obstacle reference frame (Fig. 6.12, Step 1). The locations of the remaining via points in the actuator s path, the obstacle via points, can be calculated...

FIGURE 6.14 Two bodies, A and B, shown articulating at a joint. Body A is fixed in an inertial reference frame, and body B moves relative to it. The path of a generic muscle is represented by the origin S on B, the insertion N on A, and three intermediate via points P, Q, and R.QandR are via points arising from contact of the muscle path with body B. Via point P arises from contact of die muscle path with body A. The ISA of B relative to A is defined by the angular velocity vector of B in A [Modified from Pandy (1999). ... 

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 ... 

As a mathematical preliminary, we note the following. To deal with molecules, we are required to transform freely between the laboratory reference frame and the body-fixed frame that rotates with the molecule. The rotation from the lab frame (x,y,z)... 

The rotational inertia, which is defined with respect to the center of mass and a fixed reference to the body, will be constant if it is defined in coordinates of this latter frame. It will consequently be added to the 1-junction representing the absolute angular velocity of the body of the frame of fixed coordinates to the body. 

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... 

As shown below, the motion of an arbitrarily shaped body, such as a structured molecule, can be decomposed in terms of translational motion of the center of mass of the molecule (F = dp/df) and rotation of the molecule about its center of mass (Tg = dLc/df). It is to be noted that in writing the force and torque equations, all vector components are with reference to an inertial or space-fixed (laboratory) frame. 

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