Over damped, Critically damped and Under damped Cases

Second-Order Linear Systems

A second-order linear system can be described by the following differential equation:

\[ m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F(t) \]

where:

• \(m\) is the mass of the system,
• \(c\) is the damping coefficient,
• \(k\) is the spring constant,
• \(x\) is the displacement of the system from its equilibrium position,
• \(F(t)\) is an external force applied to the system,
• \(\frac{dx}{dt}\) and \(\frac{d^2x}{dt^2}\) represent the first and second derivatives of \(x\) with respect to time.

Damping Ratio (\(\zeta\)):

The behavior of a second-order system is often characterized by the damping ratio (\(\zeta\)), which is defined as:

\[ \zeta = \frac{c}{2\sqrt{mk}} \]

Over-Damped (\(\zeta > 1\)):

In an over-damped system, damping is strong. The roots of the characteristic equation are real and distinct. The system returns to equilibrium without oscillation. The response is relatively slow. The damping ratio (\(\zeta\)) is greater than 1.

Critically Damped (\(\zeta = 1\)):

In a critically damped system, damping is at a critical level. The roots of the characteristic equation are real and repeated. The system returns to equilibrium as quickly as possible without oscillating. It represents the fastest possible approach to equilibrium without overshooting. The damping ratio (\(\zeta\)) is equal to 1.

Under-Damped (\(0 < \zeta < 1\)):

In an under-damped system, damping is weak. The roots of the characteristic equation are complex conjugates. The system exhibits oscillatory behavior as it returns to equilibrium. The response includes both exponential decay and oscillation. The damping ratio (\(\zeta\)) is less than 1.

Response:

The response of a second-order system to a step input (sudden change) is often characterized by the following parameters:

• Rise time: The time it takes for the system response to go from 10% to 90% of its final value.
• Settling time: The time it takes for the system response to reach and stay within a specified percentage (commonly 2%) of its final value.
• Overshoot: The maximum percentage by which the system overshoots its final value.

Understanding these concepts helps in analyzing and designing control systems, where the behavior of the system in response to different inputs is crucial. The choice of damping ratio influences the performance and stability of the system.