Velocity function, sinusoidal

The sinusoidal velocity profile and the periodic boundary conditions assure nth-order continuity, such that a particle leaving on one side of the system reenters on the opposite side with the same velocity. Using these expressions, the position of any tracer particle within the system can be calculated forward (or backward) for any time simply by knowing its initial (or final) position. Particle trajectories are a function of a single parameter (T), the period of the flow. The simplicity of the solution [eqs. (3-5) and (3-6)] embodies the beautiful paradox of chaotic flows. 

A and B are constants and they are independent of time. Bearing in mind that = X + iy, it is simple to find functions x t) and y t), which describe a motion of the pendulum on the earth s surface. In accordance with Equation (3.88) a solution is represented as a product of two functions. The first one characterizes a swinging of the pendulum with the angular velocity p, which depends only on the gravitational field and the length /, while the second is also a sinusoidal function and its period is defined by the frequency of the earth s rotation and the latitude of the point, (Foucault s pendulum). In order to understand the behavior of the pendulum at the beginning consider the simplest case when a rotation is absent, co — 0. Then, we have... 

The rheometer most often used to measure viscosity at low shear rates is the cone and plate viscometer. A schematic of a cone and plate rheometer is found in Fig. 3.24. The device is constructed with a moving cone on the top surface and a stationary plate for the lower surface. The polymer sample is positioned between the surfaces. Two types of experiments can be performed the cone can be rotated at a constant angular velocity, or it can be rotated in a sinusoidal function. The motion of the cone creates a stress on the polymer between the cone and the plate. The stress transferred to the plate provides a torque that is measured using a sensor. The torque is used to determine the stress. The constant angle of the cone to the plate provides an experimental regime such that the shear rate is a constant at all radii in the device. That is, the shear rate is independent of the radial position on the cone, and thus the shear stress is also independent of the position on the cone. 

Transfer functions involving polynomials of higher degree than two and decaying exponentials (distance-velocity lags) may be dealt with in the same manner as above, i.e. by the use of partial fractions and inverse transforms if the step response or the transient part of the sinusoidal response is required, or by the substitution method if the frequency response is desired. For example, a typical fourth-order transfer function ... 

Figure 1.168 Inlet mean velocity as a function of time in the control case (dashed line) and in the biased sinusoidal pulsing case (solid line) [26] (by courtesy of RSC).

In EHD impedance studies, the relaxation times for the different transport processes are obtained by variation of the modulation frequency of the flow. The utility of the technique relies on being able to separate out individual transport relaxation times. This is possible because, with EHD, each relaxation time will have a different functional dependence upon the perturbation. The theory and methodology were first developed for sinusoidal modulation of the flow velocity in a tube [20, 22]. The results... 

The motion given by this solution is called uniform harmonic motion. It is a sinusoidal oscillation in time with a fixed frequency of oscillation. Figure 8.1 shows the position and the velocity of the suspended mass as a function of time. The motion is periodic, repeating itself over and over. During one period, the argument of the sine changes by In, so that if r is the period (the length of time required for one cycle of the motion). 

Figure 20. Plot showing sinusoidal maternal velocity functions used to create unsteady-state conditions...

The Mossbauer measurement requires the generation of a precise, controllable relative motion between the source and the absorber. A large variety of drive systems has been developed. The majority of drives work on electromechanical, mechanical, hydrauHc, and piezoelectric principle. The spectrometers can be classified into constant-velodty spectrometers and velocity-sweep spectrometers. The mechanical drives, hke a lead screw or a cam, move with constant velocity. They have advantages for the thermal scan method and because their absolute velocity calibration is straightforward. The velocity-sweep spectrometers are usually of electromechanical nature (like loudspeaker-type transducers) and normally used in conjunction with a multichannel analyzer. The most commonly used M(t) functions are rectangular (constant velocity), triangular (constant acceleration), trapezoidal, and sinusoidal. A typical Mossbauer spectrometer is shown schematically in O Fig. 25.24. 

The source (and in a few cases the absorber) is mounted on the vibrating axis of an electromagnetic transducer (loudspeaker system) [ 16J which is moved according to a voltage waveform applied to the driving coil of the system. The usual velocity functions are of the triangular, sawtooth, or sinusoidal form. In special cases the source may be moved with a constant velocity,... 

This equation shows that if the hand velocity becomes faster or if the wavelength, 1, becomes smaller, the frequency,/, increases. We should consider the response characteristics of FA I, which is known as a tactile receptor related to the roughness sensation. It is known that FA I responds to the velocity of mechanical stimuli [18]. Here, when the finger slides across the surface, as shown in Figure 8.10, a displacement of stimulus, y, at a given time, t, is defined as a sinusoidal function ... 

Yang and Kwok [8] presented the analytical solution of fully developed electrokinetic flow subjected to sinusoidal pressure gradient or sinusoidal external electric field. The combined effect of slip flow and electrokinetics was demonstrated on the velocity profile in confined geometries. The velocity profile was observed to be a function of both slip coefficient and external electric field. They observed that both these effects play important roles for flow inside microchannels. 

The solution for the intake, compression and exhaust strokes is very straightforward. The optimal piston velocity in each of these is constant, with a brief acceleration or deceleration at the maximum allowed rate at the juncture of each stroke with the next. The analysis was done both with no constraint on the maximum acceleration and deceleration, and with finite limits on the acceleration. The power stroke required numerical solution of the optimal control equations, in this case a set of non-linear fourth-order differential equations. Figure 14.3 shows the optimal cycle with limits on the acceleration and deceleration, both in terms of the velocity and position as functions of time. The smoother grey curves show the sinusoidal motion of a conventional engine with a piston linked by a simple connecting rod to the drive shaft that rotates at essentially constant speed. The black curves show the optimized pathway. 

A linear system that is forced sinusoidally will have a sinusoidal response at the same frequency. One way to approach the analysis is then to assume an input of the form exp(tot) = cos( >i) -I- i sin( )t), where f = -1 the response will also have the form exp(tot). The linearized equations for isothermal, low-speed Newtonian spinning, for example. Equations ll.lOa-c, will then take the form of Equations 11.13a-c, with A replaced by ico the functions 4>, f, and m are complex and are in fact the normalized Fourier transforms of A, v, and a, respectively. The boundary conditions, however, are no longer zero, but reflect the forcing if we wish to determine the sensitivity of the output area to disturbances in the velocity at z = 0, for example, we would set = 1 + Or at f = 0. (The input condition is the Fourier transform of an impulse, or a delta function, not a sinusoid, because the transfer function is the ratio of output to input in Fourier space. There is no loss of generality in setting the imaginary part to zero at f = 0.)... 

Oscillatory shear experiments using, for example, cone-and-plate devices constitute the third main group of viscometric techniques. These techniques enable the complex dynamic viscosity rj ) to be measured as a function of the angular velocity (cu). The fundamental equations are presented in section 6.2 (eqs (6.22H6.27)). Another arrangement is two rotating parallel excentric discs by which the melt is subjected to periodic sinusoidal deformation. 

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