Question

Critical damping is a function of

• A. mass and stiffness
• B. mass and damping coefficient
• C. mass and natural frequency
• D. damping coefficient and natural frequency

Hint:

Critical damping ensures fastest return to equilibrium without oscillations.

Answer 1

The Correct Option is A

Step 1: Understanding critical damping.
Critical damping is the minimum amount of damping required for a system to return to its equilibrium position without oscillation in the shortest possible time.
Step 2: Expression for critical damping.
For a single degree of freedom system, the critical damping coefficient \( c_c \) is given by: \[ c_c = 2\sqrt{km} \] where \( m \) is the mass of the system and \( k \) is the stiffness.
Step 3: Identifying dependent parameters.
From the expression, it is clear that critical damping depends only on mass and stiffness of the system.
Step 4: Conclusion.
Hence, critical damping is a function of mass and stiffness.

Answer 2

Critical damping is the boundary value of the damping coefficient that separates oscillatory (underdamped) motion from non-oscillatory (overdamped) motion in a vibrating system, and it is a property derived purely from the system's inertia and springiness, before any actual damping coefficient is even introduced. Checking each option:

1. Mass and stiffness: The critical damping coefficient is defined as \( c_c = 2\sqrt{km} \), a quantity that can be calculated from the mass \( m \) and stiffness \( k \) alone, without any reference to the system's actual (applied) damping coefficient. This matches the definition exactly.
2. Mass and damping coefficient: The actual damping coefficient \( c \) is a separate, independently specified property of the dashpot or damper in the system; it is not needed to compute the critical value \( c_c \), so this pairing does not correctly describe what determines critical damping.
3. Mass and natural frequency: While natural frequency \( \omega_n = \sqrt{k/m} \) is related to mass and stiffness, expressing critical damping in terms of mass and natural frequency alone would still implicitly require stiffness information (since \( \omega_n \) already depends on \( k \)); the direct and standard defining formula uses mass and stiffness themselves, not this derived combination.
4. Damping coefficient and natural frequency: Neither of these are the two quantities used to define \( c_c \); the damping coefficient is what critical damping is compared against (via the damping ratio), not an input to computing critical damping itself.

The formula \( c_c = 2\sqrt{km} \) confirms that critical damping is determined directly by mass and stiffness.

Therefore, the correct answer is mass and stiffness.