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: The damping ratio of a vibrating system is defined as \( \zeta = \dfrac{c}{c_c} \), where critical damping occurs at \( \zeta = 1 \), the boundary between oscillatory and non-oscillatory response.
Step 2: The equation of motion for a damped system is \( m\ddot{x} + c\dot{x} + kx = 0 \); the character of its solution (oscillatory or not) depends on the discriminant of its characteristic equation, \( c^2 - 4km \).
Step 3: Setting this discriminant to zero, the exact borderline (critical) case, and solving for \( c \):
\[ c_c^2 = 4km \implies c_c = 2\sqrt{km} \] which depends only on mass \( m \) and stiffness \( k \).

Answer 2

Dimensional analysis offers another quick way to see which quantities determine critical damping. The damping coefficient has units of force per unit velocity, so the combination that produces these units from other system properties must involve mass and stiffness together. Checking each option:

1. Mass and stiffness: Combining mass and stiffness as \( \sqrt{km} \) gives exactly the units of a damping coefficient (force per unit velocity), consistent with \( c_c = 2\sqrt{km} \).
2. Mass and damping coefficient: This pairing is circular, since the damping coefficient is precisely the quantity critical damping is meant to define a threshold value for; it cannot be an input used to derive that same threshold.
3. Mass and natural frequency: While a combination of mass and natural frequency can also match the required units, matching units alone does not identify the standard, universally used defining formula, which is expressed directly in terms of mass and stiffness.
4. Damping coefficient and natural frequency: This combination again involves the damping coefficient itself, which is not needed to compute the critical damping value; critical damping must be derivable independent of any actual applied damping.

The dimensional match and the standard defining formula both confirm critical damping depends on mass and stiffness.

Therefore, the correct answer is mass and stiffness.