Sinusoidal forcing function

Figure 1, Forcing functions for monomer (fu) and initiator (fi) feeds (a) sinusoidal (b) square-wave (c) reception vessel valve operating sequences which are synchronized with the feed policies (see Figure 2 for the location of the valves...

Two typical experiments are described In the first, sinusoidal forcing functions are used for monomer and initiator feeds to the reactor the second experiment is similar except that square-wave forcing functions are used. These forcing functions are shown schematically in Figure l(a,b). 

The ordinary diflcrentiaJ equations could be solved in the time domain with a sinusoidal forcing function, but it is easier to go into the Laplace domain first and then substitute s = iw. 

If xj/ < 0 the output is said to lag the input, and if xj/ > 0 it is said to lead the input. The relationship between input and output for a sinusoidal forcing function as t -> oo constitutes an important tool in the analysis and design of control systems termed frequency response analysis. Of particular importance are the amplitude ratio (AR) and the phase shift xfr. The AR represents the relationship between the output and the input amplitudes... 

Consider the control loop shown in Fig. 7.44. Suppose the loop to be broken after the measuring element, and that a sinusoidal forcing function M sin cot is applied to the set point R. Suppose also that the open-loop gain (or amplitude ratio) of the system is unity and that the phase shift xj/ is -180°. Then the output JB from the measuring element (i.e. the system open-loop response) will have the form ... 

Sinusoidal Forcing. From the standpoint of control analysis the most useful forcing function is the steady state since wave. If a steady-state, low amplitude sinusoidal variation is imposed on some property of an inlet process stream, the same property of the corresponding outlet stream will also vary sinusoidally and at the same frequency. For most process components the output wave will lag behind the input wave, and the output amplitude will be less. 

The phase-plane representation is a plot of dd/dt vs. 6 as families of curves for a given system with initial conditions dd/dt(0) and 0(0) as parameters. This plot or phase portrait, provides a useful indication of the transient response of a nonlinear system. It cannot be applied to sinusoidal or other continuing forcing functions. Furthermore, the method is limited to second-order systems or systems that can be handled as second-order systems. 

Sinusoid Pure periodic sine and cosine inputs seldom occur in real chemical engineering systems. However, the response of systems to this kind of forcing function (called thefrequency response of the system) is of great practical importance, as we show in our Chinese lessons (Part Three) and in multi-variable processes (Part Four). 

Which is half the periodic time of the sinusoidal forcing function v. 

When driven by a sine wave, a valve with hysteresis produces both phase shift and distortion. The former characteristic classifies it as a dynamic element, while the latter distinguishes it as being nonlinear. Controller output and stem position are plotted vs. time for a sinusoidal forcing function in Fig. 5.5. 

ARn represents the effect of the dynamic model parameters (C, t) on the sinusoidal response that is, ARn is independent of steady-state gain K and the amplitude of the forcing function, A, The maximum value of ARn can be found (if it exists) by differentiating (5-64) with respect to to and setting the derivative to zero. Solving... 

Methods of process analysis with forcing functions other than a step input are possible, and include pulses, ramps, and sinusoids. However, step function analysis is the most common, as it is the easiest to implement. 

The ratio of the amplitude of a system s response to its forcing function s amplitude when the forcing function is a continuous sinusoid a form of dynamic gain. 

Here A is the amplitude, cp the initial phase, and coo the frequency of free vibrations. Thus, in the absence of attenuation free vibrations are sinusoidal functions and this result can be easily predicted since mass is subjected to the action of the elastic force only. In other words, the sum of the kinetic and potential energy of the system remains the same at all times and the mass performs a periodic motion with respect to the origin that is accompanied by periodic expansion and compression of the spring. As follows from Equation (3.105) the period of free vibrations is... 

Using the cone and plate rheometer the angle Q is forced in a sinusoidal manner, leading to linear strain being introduced in the polymer. The shear strain, y, is a sinusoidal function of time t with a shear rate amplitude of % as follows ... 

In other words the function x = C coscot, a sinusoidally varying quantity of frequency co and amplitude C, is a solution to the problem of the forced oscillator. The amplitude of the forced motion, C, is given by Equation 7-16. So, our hypothetical molecule will oscillate with the same frequency as that of the driving force, but its amplitude varies with the difference between the natural frequency of the oscillator and the frequency of the applied force. This amplitude becomes very large as co approaches fo0, a condition we call resonance. (In a real molecule it wouldn t become infinite, because the spring (bond)... 

Since the force across the contact only weakly perturbs the motion of the crystal surface, the displacement, i/(f), is mainly governed by the dynamics of the quartz crystal. u t) is sinusoidal with time, and the force F(f) is a function of the displacement and the direction of motion. F(f) can be calculated by inversion of Eq. 25 as ... 

Fig. 6. Sinusoidal forcing of volume with pressure response. A measure of volume and pressure as f(t)i The values of B, the pressure amplitude, and tj), the phase shift, will be constant for a given value of volume amplitude, A, and for a given value of angular frequency, tOi. Both B and (j> will, in general, vary as cu is changed, as they are a function of the system adsorption characteristics. The angular frequency, o), is varied by changing the time required to complete the one cycle (the period).

---