Euler parameters

In terms of the Euler parameters, the rotated vector a is given by... 

Calculation of the actuator elongation to place the moving platform in the desired pose from the Euler parameters, the anchorage positions and the encoder actual readings. 

It should be noted that the transformation matrix becomes unbounded for / 7t/2. This is the reason for taking other parameterizations of the rotation matrix if /3 tends towards tt/2. Such a reparameterization introduces discontinuities which can be avoided when using a redundant set of rotation coordinates. One typically uses quaternions often also called Euler parameters. These are four coordinates instead of the three angles and one additional normalizing equation, see Ex. 5.1.10. This normalizing equation describes a property of the motion, a so-called solution invariant. Differential equations with invariants will be discussed in Sec. 5.3. 

Example 5.1.10 As pointed out in the introduction (see Sec. 1.3.11) a three parametric description of rotations may lead to singularities in the equations of motion. Therefore f a description in terms of four parameters 1, 2,93,94 so-called quaternions or Euler parameters, is used. Quaternions have the property... 

Spherical the points and coincide. Using the Euler Parameters p s( 0, 1, 2 3) (which must satisfy the constraint 0 + + 2 + 3 = 1) define... 

The Weber number becomes important at conditions of high relative velocity between the injected Hquid and surrounding gas. Other dimensionless parameters, such as the Ohnesorge ((We /Re), Euler (AP/Pj y i)y and Taylor (Re/ We) numbers, have also been used to correlate spray characteristics. These parameters, however, are not used as often as the Reynolds and Weber numbers. 

These differential equations are readily solved, as shown by Luyben (op. cit.), by simple Euler numerical integration, starling from an initial steady state, as determined, e.g., by the McCabe-Thiele method, followed by some prescribed disturbance such as a step change in feed composition. Typical results for the initial steady-state conditions, fixed conditions, controller and hydraulic parameters, and disturbance given in Table 13-32 are listed in Table 13-33. 

The difference in pressure between vapor and liquid within the evaporation region depends mainly on the Euler and Weber numbers, as well as on the thermal parameter The effect of the Reynolds and Froude numbers on the pressure difference of both phases is negligible. 

At large Euler numbers when AP < 1, the vapor essure may be calculated by the Clausius-Clapeyron equation. In this case Ps and ft in Eq. (9.38) correspond to the saturation parameters. 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

Three more parameters are implicitly included which are the Euler angles that describe the orientation of the principal axes system. 

If D is taken as a traceless tensor, Tr(/)) = Da = 0, there remain only two independent components for D (neglecting the three Euler angles for orientation in a general coordinate system). Usually, these are the parameters D and E, for the axial and rhombic contribution to the ZFS ... 

Figure 10.1 Schematic diagram of the sequential solution of model and sensitivity equations. The order is shown for a three parameter problem. Steps l, 5 and 9 involve iterative solution that requires a matrix inversion at each iteration of the fully implicit Euler s method. All other steps (i.e., the integration of the sensitivity equations) involve only one matrix multiplication each.

Figure 14. The phase diagram of the gradient copolymer melt with the distribution functions g(x) = l — tanh(ciit(x —fo)) shown in the insert of this figure for ci = 3,/o = 0.5 (solid line), and/o — 0.3 (dashed line), x gives the position of ith monomer from the end of the chain in the units of the linear chain length. % is the Flory-Huggins interaction parameter, N is a polymerization index, and/ is the composition (/ = J0 g(x) dx). The Euler characteristic of the isotropic phase (I) is zero, and that of the hexagonal phase (H) is zero. For the bcc phase (B), XEuier = 4 per unit cell for the double gyroid phase (G), XEuier = -16 per unit cell and for the lamellar phases (LAM), XEuier = 0.

In the most general case of a completely anisotropic diffusion tensor, six parameters have to be determined for the rotational diffusion tensor three principal values and three Euler angles. This determination requires an optimization search in a six-dimensional space, which could be a significantly more CPU-demanding procedure than that for an axially symmetric tensor. Possible efficient approaches to this problem suggested recently include a simulated annealing procedure [54] and a two-step procedure [55]. 

Table 5.5 gives values of parameters and steady state conditions. The variables with overscores or bars over them are steadystate values. Note that the time basis used in this problem is hours. Table 5.6 gives a FORTRAN program that simulates this system using Euler integration. The right-hand sides of the... 

From equation (4.26), it is seen that the important dimensionless parameters driving momentum transport are the Reynolds number, the Froude number, the Euler number, and the length and velocity ratios in the flow field. The dimensionless variables all vary between zero and a value close to one, so they are not significant in determining which terms in the governing equation are important. 

Once all of the conditions were determined and parameters chosen, the equations were solved by an implicit Euler method. The program was written with a self adjusting step size and analytic Jacobian to reduce error and run time. 

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