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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:
2 Μαΐ 2017 · Then, for critically damped (equal roots) the solution is of the form: \$\small I_L=(A+Bt)e^{-t}\$, where \$\small A\$ and \$\small B\$ are constants. The voltage across the inductor (and R and C) is \$v=\small L\large \frac{dI_L}{dt}\$, giving: \$v=\small L(B-A-Bt)e^{-t}\$ Which has the characteristic shape of Graph 1
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.
The response is referred to as a critically damped response, and the system is called a critically damped system, with a double pole: x1 = x2 = −b 2a x 1 = x 2 = − b 2 a. if Δ <0 Δ <0, there are two complex, conjugate roots, and the response is a sinusoid with an exponential envelope.
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.
Critically damped systems are dynamic systems that return to equilibrium as quickly as possible without oscillating. This damping condition is characterized by a specific amount of damping that is just enough to prevent oscillations while still allowing the system to settle down efficiently.
The response of a single-degree-of-freedom system to a step input depends critically on the damping in the system. There is a critical damping value for the system. Above critical damping value, the response is an exponential decay.
