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Learning Objectives. Learn to analyze a general second order system and to obtain the general solution. Identify the over-damped, under-damped, and critically damped solutions. Convert complex solution to real solution. Suspended “mass-spring-damper” equivalent system.
Figure 2 shows the response of the series RLC circuit with L=47mH, C=47nF and for three different values of R corresponding to the under damped, critically damped and over damped case. We will construct this circuit in the laboratory and examine its behavior in more detail. (a) Under Damped. R=500Ω (b) Critically Damped. R=2000 Ω (c) Over Damped.
Polynomial A(s) is stable (i.e. all roots of A(s) have negative real parts) if there is no sign change in the first column. The number of sign changes in the first column is equal to the number of roots of A(s) with positive real parts. Examples: A(s) = a0s2 + α1 s + α2.
This document discusses the response of a second-order system, such as the mass-spring-dashpot shown in Fig. 1, to a step function. The modeling of a step response in MATLAB and SIMULINK will also be discussed. Fig. 1. Single-degree-of-freedom mass-spring-dashpot system.
1: Critically Damped Case The critically damped case separates the two sets of transient solutions to T/ (1+T) of one, oscillatory sine waves and two, decaying exponentials. We term this the critically damped case. In the approximation that,f lower = 10-1/2Q f 0( of the L-C resonance =1 / 2 π (LC)1/2,we find the lower frequency pole location ...
Under, Over and Critical Damping OCW 18.03SC.. . Example 3. Show that the system x + 4x + 4x = 0 is critically damped and . graph the solution with initial conditions x(0) = 1, x(0) = 0. Solution. Characteristic equation: s2 +4s + 4 = 0. Characteristic roots: (this factors) −2, −2. Exponential solutions: (only one) e−2t. General solution:
Figure 1.20: Free body diagram for second-order system. Initial condition response For this second-order system, initial conditions on both the position and velocity are required to specify the state. The response of this system to an initial displacement x(0) = x0 and initial velocity v(0) = x˙(0) = v0
