Velocity transformer

The kinetic energy stored in the flywheel rotor is proportional to the mass of the rotor and the square of its linear velocity. Transformed into a cylindrical system, the stored kinetic energy, KE, is... 

Figure 11. Our new explanation is that to the traveling observer, her sphere of observation is by the velocity transformed into an ellipsoid of observation. However, to the traveler this ellipsoid appears spherical because A and B are focal points of the ellipsoid and thus ACB — ADB — AEB. Thus, to her the car appears cubic because she observes that all sides of the car are touched by this sphere simultaneously, just as when the car was stationary. To the stationary observer this simultaneity is, however, not true.

The mechanism for ultrasonic emulsification is primarily that of cavitation. A typical sonicator for emulsification consists of a velocity transformer coupled to a transducer, capable of oscillating in a longitudinal mode, where the velocity transformer is immersed in the liquid. Figure 4 illustrates the basic parts of a sonicator with a continuous flow attachment, like the one used in this work. In this case, the flow cell is secured to the velocity transformer by a flange and a Teflon 0-ring. The intensity of cavitation depends on the power delivered to the velocity transformer, which is relayed to the transducer from a variable transformer or some other control device not shown in Fig. 4. 

In the strict kinematic limit in which no forces act, one takes the time derivative of Eq. (3) to conclude that the transformation Eq. (3) also relates the initial and final velocities. Often however there is a strong repulsion between the products and in the sudden limit this force gives rise to an impulsive change of the velocities. It follows that the velocities transform as... 

With the present definitions we can write the velocity transformations of Eqs. (2.14) and (2.26) in vector form,... 

Figure 3.12 Longitudinal velocity transformation behind the leading edge of a duct with the EPR near walls h = 0.3, A = 100. The flow regime Re = 10 (solid lines) is compared with the case where the EPR is absent (dotted).

Figure 5. Behavior of a positive-energy wave packet under a velocity transformation. The right image shows the wave packet after a Lorentz transformation (boost) with the indicated velocity. The relative motion between observer and wave packet causes a Lorentz contraction.

As in ordinary mechanics, the Euler - Lagrange equations demand p = F, which arises from a variational problem. Now we want to transform the position q into another new position. We may express this as the position g is a function of some parameter s, i.e., q = q s). In the same way the velocity transforms when the position is changed, q = q s). When the Lagrangian is invariant for such a transformation, we address this as a kind of symmetry, here the translational homogeneity of space. This means, changing the parameter 5 by a small value, the Lagrangian should remain the same ... 

Recognizing that our BEM solver will return velocities normal to the surface, we employ the velocity transformations... 

On the other hand, according to the law of conservation of matter, the vector sum of components fluxes in the laboratory reference frame (connected with one of the ends of an infinite, on diffusion scale, diffusion couple) must be equal to zero. According to Darken, the alloy provides fluxes balancing at the expense of lattice movement as a single whole at some certain velocity, u, which is measured by inert markers ( frozen in lattice) displacement. Correspondingly in the laboratory reference frame, components fluxes acquire a drift component (similar to Galilean velocity transformation equations) as... 

In the previous section we have shown that the Lorentz transformations also affect time. We will therefore have to abandon our simple one-dimensional model when we turn to velocity transformations. Even velocities perpendicular to the relative motion of the two frames are affected by the time transformation. This is in contrast to the Galilean transformations for velocities, which take the simple form... 

These velocity transformations have a numljer of interesting implications. One of these is that the usual rules for the addition of velocities do not hold. From the Galilean transformations it is easy to verify that... 

AN EFFICIENT IMPLEMENTATION OF THE VELOCITY TRANSFORMATION METHOD FOR REAL-TIME DYNAMICS WITH ILLUSTRATIVE EXAMPLES... 

ABSTRACT. This paper presents an efficient algorithm based on velocity transformations for real-time dynamic simulation of multibody systems. Closed-loop systems are turned into open-loop systems by cutting joints. The closure conditions of the cut joints are imposed by explicit constraint equations. An algorithm for real-time simulation is presented that is well suited for parallel processing. The most computationally demanding tasks are matrix and vector products that may computed in parallel for each body. Four examples are presented that illustrate the performance of the method. 

In this paper, an algorithm for dynamic simulation based on the concept of velocity transformations is presented. This algorithm may be applied to the analysis of open and closed-chain systems. The equations of motion for open chain systems are derived using a direct velocity transformation, called open chain velocity trarvrformation. Closed chain systems are analyzed in two steps. First, they are converted into open chain systems by removing some joints and the open chain velocity transformation is applied then, the closed loop conditions are imposed through a second velocity transformation. The implementation of the proposed algorithm was carried out on a SGI 4D/240 workstation and the results obtained for a series of illustrative examples are presented. 

There are several methods for imposing the loop closure conditions. The Lagrange multipliers method uses a vector of unknown reaction forces that act at the cut joints. The dynamic equilibrium equations together with the constraint equations form a set of DAE s that must be integrated. In order to reduce the size of the final system of equations, Avello et al.(1993) presented an approach based on the penalty formulation developed by Bayo et al. (1988). In this paper, the reduction of the original system of DAE s to a set of ODE s is achieved through a second velocity transformation. The numerical efficiency of the last two approaches in terms of CPU time per function evaluation is very similar, as it is shown by the practical examples solved in section 4. 

Cut joint constraints imposed through a penalty formulation Proposed method (constraints imposed through a second velocity transformation)... 

Avello, A., Jimenez, J, M., Bayo, E. and Garcfa de Jaldn, J., 1993, A Simple and Highly Parallelizable Method for Real-Time Dynamic Simulation Based on Velocity Transformations , Computer Methods for Applied Mechanics and Engineering, Vol. 107, pp. 313-339, (1993). 

Kim, S. S. and Vanderploeg, M. J., 1986, A General and Efficient Method for Dynamic Analysis of Multibody Systems Using Velocity Transformations , Journal of Mechanisms, Transmissions and Automation in Design, vol 108, pp. 176-182. 

The fact that the right-hand side of this equation is independent of the choice of the reference frame immediately proves the invariance of the photon momentum under nonrelativistic velocity transformations. 

But in this case, as noted above, they must coincide in any other nonrelativistic reference frame as well. In connection with these remarks, one should emphasize specially the difference in transformation properties between the wave vector and the radiation frequency for the processes of interaction between a nonrelativistic atom and resonant radiation. Indeed, under a nonrelativistic velocity transformation, the wave vector retains its value. At the same time, the radiation frequency changes according to the first-order Doppler effect. If, for example, ujq is the frequency of radiation emitted by a source that is motionless in the laboratory reference frame, then the radiation frequency in the rest frame of the atom uj will be... 

In the above equations, V is called velocity transformation matrix, q and q... 

A velocity transformation matrix has been presented which conveniently transforms easy to develop Cartesian equations into complex joint space equations for efficient solution of flexible robot dynamics. The method has been applied to an industrial robot to verify the significance of link and joint flexibility on the end-effector motion. 

Kim, S.S. Vanderploeg, M.J. A general and Efficient Method for Dynamic Analysis of Mechanical Systems Using Velocity Transformations. Trans. ASME J. Mech., Transm., and Autom. in Design. 108 (1986) 176-182. 

---