The ramp function

Find the Fourier series expansion of the ramp function (Figure 2.11) defined over the interval [—X,X] by... 

It will be left as an exercise for the reader to show that the cosine terms bn, including the mean value b0, are zero. The Fourier series expansion of the ramp function is therefore... 

In order to make the solution consistent with initial and boundary conditions, we will use for u(r,0) the ramp function defined in Chapter 2. For 0Fourier expansion of this function... 

The ramp function is one that changes continuously with time at a constant rate K. 

It is possible to vary the load or, in hard machines, the displacement, either in ramp mode or with a discontinuous increment (step mode). The ramp function needs to be smooth, as well as linear, and there is evidence (Mayo Nix, 1988) that if the ramp is digitally controlled, the data will vary for the same mean loading rate according to the size of the digitally produced load increments, unless these are very small. 

The generation of such a forcing function is used in steering theory as well as in adiabatic and scanning calorimetry. The ramp function has the following Laplace transform ... 

Equation (3.18) describes the nonisothermal-nonadiabatic system in which changes occur in the shield temperature. The temperature rise is usually linear and described by the ramp function (Eq. 2.49) as in the... 

Regardless of which measurement method is used, in each of them the heat effects generated in the sample and in the calorimeter shield are superimposed. In consequence of the same type of inertial properties (of inertial objects) of the devices mentioned, the course of the output function caused by the programmed rise of temperature of the shield is always the same (see 3.2.5). Let us confine ourselves to considering only the changes in temperature Tc(t) that are caused by linear rise of the shield temperature To. When the initial temperature of the shield To(0)= Tq, the temperature of the vessel Tc(0)= T°, and the ramp function is f(t) = Gat... 

Reading et al. [216, 217] proposed a method in which a DSC is used. In this case, the response of the calorimeter as a Mnear system would be a superposition oftwo input functions 1) the ramp function [Eq. (2.49)] generated in the calorimetric shield and 2) the periodic function generated in the sample. When the periodic function is sinusoidal [Eq. (2.53],... 

Let us consider a system composed of two calorimeters (I and II), characterized by the dynamic properties of inertial objects of the first order, placed in a common shield. The sample is situated in one of them, and the reference substance in the other. The forcing input function is the ramp function generated in the thermostat. Let us assume that this function is at the same time the input function of both calorimeters. The influence of the forcing function on a differential calorimeter is shown... 

When the simulation starts, a ramp-function at the input boundary may be applied. The ramp function prevents the impulse-like behavior at the start and consequently reduces the corresponding unnecessary transient waves, which usually wastes computational time to die out. As a result, the simulation is more stable and soon reaches the steady state. 

The integral of u t) is the ramp function u (t) which has the defining relations... 

From Table 7.2, it can be seen that the discrete and continuous step response is identical. Table 7.3 shows the discrete response x kT) and continuous response x t) to a unit ramp function where Xo t) is calculated from equation (3.39)... 

The advantage of using frequency converters is that the possibility exists to use a ramp function when starting and stopping the motor (soft start). By using this function, it is possible to avoid starting both fans at full speed with closed dampers it also reduces stresses on the fan transmission (belts) at the start. 

Figure 2.31. Response of the most common controller modes for step change and ramp function of the error signal.

When the reactor temperature (Tl) becomes greater than Tmax (=240 F), PERIOD = 2, and the program turns the cooling water on with flow rate Fw. This flow is controlled with a proportional controller using control constant Kc, whose set point (Pset) is varied according to the time ramp function with setting kR and whose output to the valve is Pc. This ramp is horizontal until time period Tihold has passed. Then the setpoint is decreased linearly. The temperature is sensed using a pressure transmitter with output Ptt. 

Study the effects of the parameters of the cooling water ramp function (Tihoid and kg) on the selectivity, SEL. 

We can show that with a step input, the output is a ramp function. When we have an impulse input, the output will not return to the original steady state value, but accumulates whatever we have added. (Both items are exercises in the Review Problems.)... 

The transfer function has the distinct feature that a pole is at the origin. Since a step input in either q in or q would lead to a ramp response in h, there is no steady state gain at all. 

For a facial selective assembly ofthe stereogenic centers and the introduction of the amino functionality, chiral nitrogen-containing reagents, such as benzyl(2-pheny-lethyl)amine (2-19) and trimethylsilyl RAMP derivative 2-24 were applied. Treatment of diacrylates 2-18, 2-21, and 2-23 with 2-19 and 2-24, respectively, gave the protected amino acids 2-20, 2-22, and 2-25 in good yields as single isomers. 

In order to control those initial nonequilibrium effects, a ramp function can be added which reduces the variations from one step to the next of the external force applied in a given bin. The external force applied to the system can be chosen equal to... 

In order to ameliorate the sharply sloping background obtained in an STS spectrum, the data are often presented as di,/dFh vs. Vb, i.e. the data are either numerically differentiated after collection or Vb has a small modulation applied on top of the ramp, and the differential di,/d Vb is measured directly as a function of Vb. The ripples due to the presence of LDOS are now manifest as clear peaks in the differential plot. dt,/dFb vs. Vb curves are often referred to as conductance plots and directly reflect the spatial distribution of the surface electronic states they may be used to identify the energy of a state and its associated width. If V is the bias potential at which the onset of a ripple in the ijV plot occurs, or the onset of the corresponding peak in the dt/dF plot, then the energy of the localised surface state is e0 x F. Some caution must be exercised in interpreting the differential plots, however, since... 

Assume a ramp function for the flow rate (F = F0 + kt) and find the values of F0 and k which give the highest value for the objective function. Remember that F must remain positive. 

Figure 2.11 The periodic ramp function with period 2X.

An example of a rig which tests one property as a function of different conditions is the ramp test used to measure friction of flooring. An operator walks on a ramp the angle of which can be varied. The angle at which he slips is related to the coefficient of friction and can be measured for different footwear, or surface conditions, or treatments. 

LAPUCE-DOUAIM DYNAMICS AND OONTKOL Therefore the Laplace transformation of a ramp function is... 

Let us consider the following protocol the dipole starts in the down state —p at H = —Hq. The field is then ramped from —Ho to +Ho at a constant speed r = H, so H t) = rt (Fig. 13a). The protocol lasts for a time tn x = 2//o/r and the field stops changing when it has reached the value Ho- The free energy difference between the initial and final states is 0 because the free energy is an even function of H. To ensure that the dipole initially points down and that this is an equilibrium state, we take the limit Hq oo but we keep the ramping speed r finite. In this way we generate paths that start at H = oo at t = —oo and end up at H = oo at t = oo. We can now envision all possible paths followed by the dipole. The up configuration is statistically preferred for H > 0,... 

Note that we obtain a very nice ramp function through multiplying H(x) by a straight line of unit slope, and that the same result can be obtained by the self-convolution of H(x) ... 

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