In mechanical vibration and acoustics, mechanical impedance is defined as the ratio of which two quantities?
Electronics and Communication Engineering
Automatic Control Systems
Difficulty: Easy
Choose an option
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Arms force to rms velocity
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Brms force to rms displacement
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Crms velocity to rms displacement
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Dnone of the above
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Erms acceleration to rms force
Answer
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