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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 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.
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.
22 Ιαν 2021 · Critically damped systems - Transients in this type of system decay to steady state without any oscillations in the shortest possible time. Underdamped systems - Transients in this type of system oscillates with the amplitude of the oscillation gradually decreasing to zero.
Critically Damped The unit step response of a critically damped system (ζ=1) with zero initial conditions is given by x()t 1 [1 e (1 t)] n t 2 n − n −ω ω = −ω. (5) Over-Damped The response of an over-damped system (ζ>1), again assuming zero initial conditions, is () () ζ+ ζ − + ζ − −ζ+ ζ − +
If = 1, then system is critically damped. If < 1, then the system is underdamped and the step response is oscillation with exponential decay envelope. If = 0, then the system is undamped. The step response is pure oscillation without any decay.
The time domain solution of an overdamped system is a sum of two separate decaying exponentials. The time domain solution of a critically damped system is an interesting sum of a constant and another constant multiplied with time "t", and the sum is further multiplied by a decaying exponential.
