Pulse-forcing functions

Any type of input-forcing function can be used steps, pulses, or a sequence of positive and negative pulses. Figure 14.9a shows some typical input/output data from a process. The specific example is a heat exchanger in which the manipulated variable is steam flow rate and the output variable is the temperature of the process steam leaving the exchanger. 

Consider the illustration of Fig. 5. A system with closed boundaries and a constant flow rate through its single inlet and outlet connections is disturbed by a forcing function, or pulse, of tracer, CA it), a time-varying outlet concentration of tracer, Ca o Ct), is observed. A tracer is any material which behaves in an identical manner to the process fluid flowing through the system but which displays some property which enables it to be differentiated from that fluid such properties could be colour, electrical conductivity, or radioactivity for instance. 

Green s functions appear as the solutions of seismic field equations (acoustic wave equation or equations of dynamic elasticity theory) in cases where the right-hand side of those equations represents the point pulse source. These solutions are often referred to as fundamental solutions. For example, in the case of the scalar wave equation (13.54), the density of the distribution of point pulse forces is given as a product,... 

Commonly encountered forcing functions (or input variables) in process control are step inputs (positive or negative), pulse functions, impulse functions, and ramp functions (refer to Figure 44). 

Although the transfer function gives a complete dynamic description of process elements, some interpretation is necessary. The response of the element to any kind of forcing function can be determined from the transfer function, but only a very few kinds of forcing are of any importance in control problems. These important forcing functions are (a) step (b) pulse (c) ramp (d) steady state sine wave and (e) random. 

The person can create a forcing function (i.e. source) input to the system by moving their hand in a repetitive fashion. In doing so, a series of pulses will form and travel forwards down the rope towards the wall, and these pulses will create a travelling wave, the frequency of which is dictated by the hand movement. When this forward travelling... 

The configuration of the detection is illustrated in Fig. 7.1, showing two cases. One is the case of buried pulse (force)/sft), and the other is that of surface pulse, fsft). As published by Pekeris (Pekeris 1955), these two forces result in the completely different displacement fields at point x. In Fig. 7.2, examples of Lamb s solutions due to a buried step-function force, where/sO) = hsft), are given. The depth of the source, D, is 6 cm and the horizontal distance, R, is varied as 3 cm, 6 cm and 9 cm. Here P-wave velocity Vp is assumed as 4000 m/s and Poisson s ratio is 0.2. These material properties actually represent those of concrete. Near the epicenter, only P-wave and S-wave are observed as shown in Fig. 7.2 (a). 

For integrating objects, the course of the output function corresponds to the accumulation process and to the operation of integration. The object responds to a generated unit pulse with an output signal, which is equivalent to the unit step function the response of the object to the unit step function is a linearly rising function production of the ramp forcing function stimulates the response of the object according to the relation... 

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. 

Dang Sheng and Ren Shen tonify the Qi and strengthen the body s resistance. Ren Shen has a stronger function than Dang Shen. They can be used separately as assistants to tonify the Qi in the formula. The condition where excess heat consumes the Qi is manifested as tiredness, shortness of breath, constant sweating with constant high fever and a forceful pulse that is empty in the deep position. 

An explanation was offered by van Kranendonk many years after the experimental discovery. Van Kranendonk argued that anticorrelations exist between the dipoles induced in subsequent collisions [404], Fig. 3.4. If one assumed that the induced dipole function is proportional to the intermolecular force - an assumption that is certainly correct for the directions of the isotropic dipole component and the force, and it was then thought, perhaps even for the dipole strength - an interference is to be expected. The force pulses on individual molecules are correlated in... 

The wavepacket /(t), on the other hand, is constructed in a completely different way. In view of (4.4), the initial state multiplied by the transition dipole function is instantaneously promoted to the excited electronic state. It can be regarded as the state created by an infinitely short light pulse. This picture is essentially classical (Franck principle) the electronic excitation induced by the external field does not change the coordinate and the momentum distributions of the parent molecule. As a consequence of the instantaneous excitation process, the wavepacket /(t) contains the stationary wavefunctions for all energies Ef, weighted by the amplitudes t(Ef,n) [see Equations (4.3) and (4.5)]. When the wavepacket attains the excited state, it immediately begins to move under the influence of the intramolecular forces. The time dependence of the excitation of the molecule due to the external perturbation and the evolution of the nuclear wavepacket /(t) on the excited-state PES must not be confused (Rama Krishna and Coalson 1988 Williams and Imre 1988a,b)... 

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