Linear System Dynamics Second Order Lienar Systems Systems and Controls Differen

Question

Linear System Dynamics

Second Order Lienar Systems

Systems and Controls

Differential Equations

Laplace Transform

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The fundamental dynamics of many important process and components such as motors, circuits, spring-mass systems, etc, etc (see examples 2.4 & 2.14 in your text) can be described using a 2nd order DE such as: where X is some process variable of interest, such as voltage or displacement, Y is a forcing function (or "excitation") and uh and are positive lumped constants which depend on the physical parameters of the system. Regardless of what specific physical parameters are combined to generate and , they are commonly referred to as the natural (or resonant) frequency and the damping ratio. While the exact form of this equation depends on the specific physical situation being modeled, the form above, which is written in normalized monic form, is perhaps the most general or standard form We shall see that the intrinsic nature of the time and frequency response of any second order system depends entirely on the parameters ,and . A. Write the DE in the "s" domain using the Laplace transforms with the initial conditions = 0 B. Draw a block diagram of the system using only integrators (ie no differentiators) and constant parameters showing the input Write the transfer function x(s)/Y (s) = N(2)/D(s) for this system Factor the denominator (ie solve quadratic D(s) - 0 for s). Remember that the roots of this equation are the system poles which play a vital role in the nature of the response. The only possibilities for these roots are that they are (1) distinct real roots, (2) repeated real roots and (3) complex conjugate roots. Identify which of these possibilities occur if C. D. E. F. Use the final value theorem to predict the final value of X for a unit step input With uh-1, use Matlab to plot X vs time for a step input of Y-1 for 0.1,-1 and = 10 Overlay the 3 plots using the Matlab "hold" function and limit the plot to 100 sec using the property editor. Label the plot to indicate which plot goes with which damping ratio. (You can label by hand to avoid printing in color)* G. Write expressions for the frequency response (ie magnitude and phase) of the transfer function from C above using u, to denote the frequency of the input (ie Y(wi)) Find an expression (in terms of and function when the frequency of the input -uh. This value is sometimes referred to as the Q" of the system. What is the numeric value of the phase at this poin t for any value of 7? H. and ) for the magnitude and phase of the transfer I. Overlay 3 bode plots comparing the magnitude and phase of the system response for 0.1 = 1 and -10. Use the property editor to plot amplitude (abs) and phase (degrees) vs log frequency from W: 0.01 to 1-100. Label the point on the axis where u-u,. Compare the magnitude and phase values at -u, to the values predicted in H above * Looking at table A.2 and the various examples given in the text for doing a partial fraction expansion in order to use the table to get explicit equations for the time response will give you an appreciation for the effort required and also an appreciation for tools such as Matlab and Simulink... and remember, this you can also get an explicit equation using the Matlab symbolic function ilaplace. t a simple 2nd order system. For relatively simple functions

Explanation / Answer

Take Laplace transform on both side:

A). S^2x(s)+2sw£x(s)+w^2 =w^2y(s)

C). X(s)/y(s) = w^2/(s^2 + 2w£s + w^2)

D) for distinct real root £>1

For repeated real root £ = 1

For complex conjugate £<1

E) for unit step input y(s)= 1/s

Then x(steady) = limit. S-->0 sX(s)

=1

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