The question asks about the condition of the frequency of damped vibration when critical damping occurs. To understand this, let's first explore what damped vibration and critical damping mean in the context of mechanical systems.

Damped Vibration and Its Frequency

Damped vibration refers to the oscillation of a system where energy is gradually lost due to resistive forces, such as air resistance or internal friction. This energy loss causes the amplitude of the oscillations to decrease over time until the motion eventually stops. The frequency of this damped oscillation is known as the damped natural frequency.

For a single-degree-of-freedom system, the equation of motion for damped free vibration is often represented as:

\[ m\ddot{x} + c\dot{x} + kx = 0 \]

Where:

• \(m\) is the mass
• \(c\) is the damping coefficient
• \(k\) is the stiffness
• \(x\) is the displacement
• \(\dot{x}\) is the velocity
• \(\ddot{x}\) is the acceleration

The natural frequency of the undamped system is given by:

\[ \omega_n = \sqrt{\frac{k}{m}} \]

And the damping ratio (\(\zeta\)) is a dimensionless measure describing how oscillations in a system decay after a disturbance. It is defined as:

\[ \zeta = \frac{c}{c_c} \]

Where \(c_c\) is the critical damping coefficient, given by \(c_c = 2\sqrt{km} = 2m\omega_n\).

The frequency of damped vibration, often denoted as \(\omega_d\), is given by the formula:

\[ \omega_d = \omega_n \sqrt{1 - \zeta^2} \]

This formula is crucial for understanding the behavior of damped systems.

Critical Damping Condition

Critical damping is a specific condition where the damping ratio (\(\zeta\)) is exactly equal to 1 (\(\zeta = 1\)). In a critically damped system, the system returns to its equilibrium position as quickly as possible without oscillating. This is often the desired behavior for systems where overshoot or oscillation is undesirable, such as in door closers or shock absorbers.

Impact on Damped Vibration Frequency

Let's substitute the value of the damping ratio for critical damping (\(\zeta = 1\)) into the formula for the frequency of damped vibration:

\[ \omega_d = \omega_n \sqrt{1 - \zeta^2} \]

Substitute \(\zeta = 1\):

\[ \omega_d = \omega_n \sqrt{1 - (1)^2} \]

\[ \omega_d = \omega_n \sqrt{1 - 1} \]

\[ \omega_d = \omega_n \sqrt{0} \]

\[ \omega_d = 0 \]

This calculation shows that when critical damping occurs, the frequency of damped vibration becomes zero. A frequency of zero means that there is no oscillatory motion; the system simply returns to equilibrium without any cycles of oscillation.

Comparing Damping Types

To further clarify, let's compare critical damping with other types of damping:

From the table, it is clear that only underdamped systems exhibit a non-zero frequency of damped vibration. For critically damped systems, the oscillatory component vanishes, and thus the frequency is zero.

Therefore, when critical damping occurs, the frequency of damped vibration is zero.