Sinusoidal response

Fig. 6.1 Steady-state input and output sinusoidal response.

By rotating the tool, one can differentiate between the level of gamma rays entering from the top and the lower part of the borehole. A sinusoidal response is recorded which depends on the following ... 

Figure 8.1. Schematic response (solid curve) of a first order function to a sinusoidal input (dashed). The response has a smaller amplitude, a phase lag, and its exponential term decays away quickly to become a pure sinusoidal response. 

If one bears in mind that the Fourier theorem states that a periodic non-sinusoidal response can be described by a series of sinusoidal... 

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 ... 

This equation describes a sinusoidal response at frequency, co, to the electric field component at co. This is the basis for the linear optical response. To calculate the optical properties of the Lorenz oscillator the polarization of the medium is obtained as... 

Introduce a delay time Impedance measurements are taken at the sinusoidal steady state, meaning that the sinusoidal response to the sinusoidal input is unchanging with respect to time. A transient is observed as the system responds to a change from one frequency to another, and this transient is incorporated into the integrated value of the impedance. Pollard and Compte have shown that this transient can introduce as much as a 4 percent error in the impedance response measxued by integration over fixe first cycle. To avoid this undesired error caused by fixe transient, it is better to introduce a delay of one or two cycles between the cheinge of frequency and impedance measurement. 

With nonlinear systems, however, all simplicity disappears. No general methods of solving even the simplest, nonlinear differential equations are known. Frequency response characterization is useless since sinusoidal forcing will not produce sinusoidal response. The only recourse other than arbitrary linearization of the equations is to utilize... 

When interpreting the frequency response, we will adopt the following point of view high frequency means fast changing inputs low frequency (in particular ni = 0) signifies steady state. In the case of MIMO sistems, Gijoi) is a matrix. Its elements, gy(/(5j) represent the sinusoidal response from input j to output i. The overall response to simultaneous input signals with the same frequency in several input channels is equal to the sum of individual responses ... 

With the additional information provided by the sinusoidal response of the second channel, the sense of rotation can be determined, and vectors moving at +1/ Hz can be distinguished (Fig. 3.21). Technically, the FT is then complex, with the x and y components being handled separately as the real and the imaginary inputs to the transform, following which the positive and negative frequencies are correctly determined. In the case of the single channel, the data are used as input to a real FT. 

Figure 10.15 Impulse response for 2 HR filters, showing a sinusoid and damped sinusoid response.

Immittance theory is based upon sinusoidal excitation and sinusoidal response. In relaxation theory (and cell excitation studies), a step waveform excitation is used, and the time constant is then an important concept. If the response of a step excitation is an exponential curve, the time constant is the time to reach 63% of the final, total response. Let us for instance consider a series resistor-capacitor (RC)-connection, excited with a controlled voltage step, and record file current response. The current as a function of time I(t) after the step is I(t) = (V/R)e , file time constant x = RC, and I( oo) = 0. 

Mathematical models of ACEO follow other examples of ICEO, as described in the article on nonlinear electrokinetic phenomena. A major simplification in the case of small voltages is to assume sinusoidal response to sinusoidal AC forcing and solve only for the complex amplitudes of the potential and velocity components at a single frequency co (Fourier mode) [2]. In this regime, the basic scaling of time-averaged ACEO flow is... 

C Sinusoidal Response Contiunoiis Glucose Microdiaiysis Monitoring... 

Figure 6.118 shows the integrated Eq. (3) for the steady state of quasi-isothermal experiments (A). The measurements needed for this analysis are displayed for polystyrene in Figs. 6.14 and 6.15 and for poly(ethylene terephthalate) in Figs. 4.129 and 4.130. The parameters in Fig. 6.118 were arbitrarily chosen to clarify the three different contributions to the approximation shown in the figure. For a solution fitted to the experiments for poly(ethylene terephthalate), see Fig. 4.131. The parameters A and A represent the amplitude contributions due to the change in x and N with temperature, and P and y are phase shifts. The plotted (N - No)/N is proportional to the heat flow (and thus to ACp). The curve (A), however, is not a sinusoidal response.

This term is used to cover a range of techniques in which the mean potential is controlled potentiostatically and swept over a range while a small amplitude, relatively high frequency alternating potential superimposed on the slowly-varying sweep is used to excite a sinusoidal response in the current, which is... 

Sinusoidal Response. ConlirYixxjs Glucose Microdialysis Moriloilng... 

In the dynamic mechanical tests, either a vibrational force or a deformation is applied to the specimen, and then the sinusoidal response of either the deformation or force is measured, respectively. The dynamic mechanical properties are measured as a function of frequency at a constant temperature or as a function of temperature. The temperature dependence of dynamic viscoelasticity is conveniently used by the plastic industry to characterize solid polymers. Recently, various kinds of equipment for measuring dynamic viscoelasticity are commercially available and widely used for scientific and practical purposes. 

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.)... 

---