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An example of a critically damped system is the shock absorbers in a car. It is advantageous to have the oscillations decay as fast as possible. Here, the system does not oscillate, but asymptotically approaches the equilibrium condition as quickly as possible.
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.
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.
Effect of Damping Ratio on System Response . Depending on whether the quantity (ζ 2 −1) is negative, zero, or positive, the system is underdamped, critically damped, or overdamped, respectively. Underdamped: When this quantity is negative (ζ<1), the system is said to be underdamped. This is, by far, the most common case for structural systems.
Free Response of Critically Damped 2nd Order System For a critically damped system, ζ = 1, the roots are real negative and identical, i.e. ss12==−ζωn (15) The solution form X(t) = A e st is no longer valid. For repeated roots, the theory of ODE’s dictates that the family of solutions satisfying the differential equation is () n (12)
How do we model oscillatory phenomena in which air drag causes a decrease in oscillation amplitude? 1.1 Drag and general Damping Forces. To achieve our objective of finding a more accurate model for oscillatory phenomena, we need to first find the correct Newton’s second law equation for such systems.
