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In mechanical vibration and acoustics, mechanical impedance is defined as the ratio of which two quantities?

Difficulty: Easy

Correct Answer: rms force to rms velocity

Explanation:


Introduction / Context:
Mechanical impedance is the dynamic analog of electrical impedance, representing how a mechanical system resists motion when subjected to a harmonic force. It is crucial in vibration isolation, acoustics, and structural dynamics.


Given Data / Assumptions:

  • Force is harmonic and measured as RMS value.
  • Velocity is steady-state sinusoidal, measured as RMS value.
  • Displacement and acceleration are related but not directly in the impedance definition.


Concept / Approach:
Definition: Mechanical impedance Z_m = F_rms / v_rms. It has units of N·s/m, analogous to electrical ohms. It combines effects of mass (inertial reactance), damping (resistive), and stiffness (elastic reactance) on system response.


Step-by-Step Solution:

Let applied force = F0 sin(ωt).Resulting velocity v(t) = V0 sin(ωt + φ).Impedance magnitude Z_m = F0_rms / V0_rms.General form: Z_m = R + j(X_m), where R is damping, X_m = ωM − (1/ωK).


Verification / Alternative check:

Textbooks confirm: mechanical impedance = force / velocity.


Why Other Options Are Wrong:

Force/displacement ratio is stiffness, not impedance.Velocity/displacement ratio is frequency (ω), not impedance.Acceleration/force is inverse of mass, not impedance.


Common Pitfalls:

Mixing stiffness and impedance; forgetting impedance links force to velocity, not displacement.


Final Answer:

rms force to rms velocity
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