Time-Domain Analysis

The manner in whieh a dynamie system responds to an input, expressed as a funetion of time, is ealled the time response. The theoretieal evaluation of this response is said to be undertaken in the time domain, and is referred to as time domain analysis. It is possible to eompute the time response of a system if the following is known  

The following steps are adopted using a direct implicit integration procedure 

For each time step, calculate the displacement xt+t Constant part of the effective load vector 

In fact, (5/ /, is calculated using the initial-stress method. (I) Iteration convergence 

When the velocity varies linearly and the acceleration is constant across the time interval, appropriate substitutions are made into Eq. (3.13), giving 

For clear-cut solutions, the size of the time step between adjacent iterations should not be more than a factor of 10 in non-linear cases and should not be reduced by more than a factor of 2 where plasticity exists. 

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This tutorial introduees the reader to time domain analysis using MATLAB. It uses eommands from the Control System Toolbox. A list of the eommands ean be found using... 

In addition, it should be noted that frequency-domain analysis can be used to determine the phase relationships for harmonic vibration components in a typical machine-train spectrum. Frequency-domain normalizes any or all running speeds, where time-domain analysis is limited to true running speed. 

Even as we speak of time-domain analysis, we invariably still work with Laplace transform. Time-domain and Laplace-domain are inseparable in classical control. 

From the last example, we may see why the primary mathematical tools in modem control are based on linear system theories and time domain analysis. Part of the confusion in learning these more advanced techniques is that the umbilical cord to Laplace transform is not entirely severed, and we need to appreciate the link between the two approaches. On the bright side, if we can convert a state space model to transfer function form, we can still make use of classical control techniques. A couple of examples in Chapter 9 will illustrate how classical and state space techniques can work together. 

Dorf, R. C., Time-Domain Analysis and Design of Control Systems, Addison-Wesley, Reading, Massachusetts, 1964. 

Xu, Y., Lee, R.K., and Yariv, A., 2000, Finite-difference time-domain analysis of spontaneous emission in a microdisk cavity, Rev. A 61 033808. 

The checkmark in the square El indicates that the sweep is enabled. Note that both the Temperature (Sweep) and the Time Domain analysis are enabled. Fill in the Temperdture (Sweep) parameters as shown ... 

The first line of any SPICE netlist is the title line. It is used for documentation purposes only. The next few lines usually tell SPICE which analysis will be performed and what the bounds of that analysis will be. For example, we may be requesting a time domain analysis of a circuit (called a transient analysis). The information as to how long the waveform is and what increments and what section of it are of interest is defined in this section of the code. SPICE netlists generally have one function, command, or element per line (Fig. 2.1). Also defined upfront are global constants, subcircuits (models) used repeatedly in the main circuit, and instructions on which nodes are of interest in the final solution, though this structure is not mandatory. 

The time step Tstep = lOu determines each point in time starting from zero that the transient solver will calculate a solution. A safe estimation of the time step is an order of magnitude less than the period of a switching waveform. For example, the time step for a 100 kHz oscillator (period = 10 /xs) should be approximately 1 /xs. Tmax, the maximum time step, can be left out (at default) or specified to increase (decrease TMAX) or decrease (increase TMAX) simulation accuracy. This allows the simulator to take larger steps when the voltage levels in the circuit experience little change. A transient time domain analysis can prove to be the most difficult to get to converge. 

Time Domain Analysis. Perhaps the simplest and most traditional use of a DSP is filtering. DSPs are designed to implement both Finite Impulse Response (FIR) and Infinite Impulse Response (HR) filters as fast as possible by implementing (a) a single cycle multiply accumulate instruction (b) circular addressing for filter coefficients. These two requirements can be found in all modem DSP architectures. 

In the next chapter we take a quantitative look at the dynamics of these CSTR systems using primarily rigorous nonlinear dynamic simulations (time-domain analysis). However, some of the powerful linear Laplace and frequency-domain techniques will be used to gain insight into the dynamics of these systems. 

A common time domain analysis involves computing the standard deviation of the potential (oe) and the standard deviation of the current (of) from the time series data and taking their ratio to compute a noise resistance Rn ... 

The PGSE NMR method relies on the use of two sharp gradient pulses separated by a well-defined time interval and is therefore naturally suited to time-domain analysis of motion. However, it is important to realize that this particular form of two-pulse gradient modulation is not unique. In particular, a number of other time-modulation schemes are possible in which the molecular motion is detected in a different manner. However, as we shall see, whenever modulated gradients are used to encode the spin magnetization for motion rather than position, it is appropriate to refocus any phase shift due to absolute spin position by means of a spin echo. Consequently, we refer to this more general type of experiment as modu-... 

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