Definition

When the damping ratio \( \xi > 1 \), the system is classified as over-damped. In this case, the system returns to equilibrium without oscillating.

Displacement Function

Using Eq. (4.13), the displacement function is expressed as:

\[ x(t) = A_1 e^{\left( -\xi + \sqrt{\xi^2 - 1} \right) \omega_n t} + A_2 e^{\left( -\xi - \sqrt{\xi^2 - 1} \right) \omega_n t} \]

This solution contains two exponentially decaying terms. Both components decrease with time due to the negative exponent, resulting in an aperiodic or non-oscillatory motion.

Initial Conditions

The constants \( A_1 \) and \( A_2 \) are determined using initial conditions:

• Displacement at \( t = 0 \): \( x(0) \)
• Velocity at \( t = 0 \): \( \dot{x}(0) \)

These conditions allow for a unique solution tailored to the system’s initial state.

System Behavior

Once disturbed, an over-damped system takes a long time to return to its equilibrium position. Although it does not oscillate, the return is slow and exponential.