In electrical engineering a simple example would be the output of a 5 volt DC power supply when it is turned on: the transient response is from the time the switch is turned on and the output is a steady 5 volts. At this point the power supply reaches its steady-state response of a constant 5 volts.
The transient response is not necessarily tied to "on/off" events but to any event that affects the equilibrium of the system. If in an RC circuit the resistor or capacitor is replaced with a variable resistor or variable capacitor (or both) then the transient response is the response to a change in the resistor or capacitor.
In a mechanical system a simple example is a mass/spring/damper system. The transient response is the position of the mass x(t) as the system returns to equilibrium after an initial force or a non zero initial condition.
Both mechanical and electrical systems are analogous.
The response can be classified as one of three types of damping that describes the output in relation to the steady-state value.
An underdamped response is one that oscillates within a decaying envelope. The more underdamped the system, the more oscillations and longer it takes to reach steady-state. Here Damping Ratio is always < 1
A critically damped response is the response that reaches the steady-state value the fastest without being underdamped. It is related to critical points in the sense that it straddles the boundary of underdamped and overdamped responses. Here Damping Ratio is always 1 (Unity)
An overdamped response is the response that does not oscillate about the steady-state value but takes longer to reach than the critically damped case. Here Damping Ratio is >1
Time required for system response to rise from:
10% to 90% (Overdamped); 5% to 95%; 0% to 100% (Underdamped)
of the final steady state value of the desired response.
Maximum Overshoot is the maximum peak value of the response curve measured from the desired response of the system.
Time required for response to reach and stay within 2% of final value.
The steady state error of a system is the difference between the input and output of the system in the limit as time goes to infinity, i.e. when the transient response reaches a steady state. With no overshoot the steady state error is eliminated when the steady state velocity of the vehicle reaches the desired velocity.