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20 Φεβ 2013 · Damping removes energy from the system and so the amplitude of the oscillations goes to zero over time, regardless of the amount of damping. However, the system can have three qualitatively different behaviors: under-damping, critical damping, and over-damping.
Outline. Damped Oscillations in Terms of Undamped Natural Modes. Space-State Formulation & Analysis of Viscous Damped Systems. Assume that the damping mechanism can be described by a viscous, quadratic, dissipation function in the generalized velocities. 1.
The general real solution is found by taking linear combinations of the two basic solutions, that is: x(t) = c1e−bt/2m cos(ωdt) + c2e−bt/2m sin(ωdt) or. x(t) = e−bt/2m(c1 cos(ωdt) + c2 sin(ωdt)) = Ae−bt/2m cos(ωdt − φ). (3) Let’s analyze this physically. When b = 0 the response is a sinusoid.
The damping may be quite small, but eventually the mass comes to rest. If the damping constant is \(b = \sqrt{4mk}\), the system is said to be critically damped , as in curve (\(b\)). An example of a critically damped system is the shock absorbers in a car.
There are examples for under-damped, critically-damped, and over-damped free vibration systems and an under-damped system subjected to sinusoidal forcing (this latter phase-plane is included here for completeness and will be more fully discussed in the Forced Vibration laboratory). These types of
27 Μαΐ 2024 · \(\Delta=0\) : Critically damped vibration. In this case there is only one value \(\lambda=-c /\left(2 m_{A}\right)\). The value of the damping coefficient that belongs to this situation \(\Delta=c_{c}^{2}-4 m_{A} k=0\) is called the critical damping coefficient \(c_{c}=2 \sqrt{m_{A} k}\).
The solution for a critically-damped system is: \[ x(t) = (A + Bt) e^{-\omega_n t} , \] \[ \text{where} \,\, A = x_0 \,\, \text{and} \,\, B = v_0 + x_0 \omega_n.
