Mechanical Resonance
Mechanical resonance occurs when an oscillating system is driven at its natural frequency, causing the amplitude of oscillation to grow dramatically — theoretically without bound in an undamped system. Every mechanical system has one or more natural frequencies at which it vibrates freely after being disturbed. Resonance is critical in engineering design, as it can cause catastrophic structural failure (Tacoma Narrows Bridge, 1940) or be harnessed usefully in musical instruments, clocks, and sensors.
Key Formula
f₀ = (1/2π)√(k/m), Resonance amplitude Ares = F₀/(2mβω₀)
LaTeX: f_0 = \frac{1}{2\pi}\sqrt{\frac{k}{m}}, \quad A_{res} = \frac{F_0 / m}{2\beta\omega_0}
| Symbol | Meaning | Unit |
|---|---|---|
| f_0 | Natural (resonant) frequency | Hz |
| k | Spring constant | N/m |
| m | Mass | kg |
| F_0 | Amplitude of driving force | N |
| \beta | Damping coefficient | s⁻¹ |
| \omega_0 | Natural angular frequency | rad/s |
Worked Example
Problem
A bridge girder modeled as a spring-mass system has mass m = 5000 kg and spring constant k = 2 × 10⁷ N/m. At what frequency would soldiers marching in step cause resonance? What is the natural frequency in Hz?
Solution
Step 1: Natural angular frequency ω₀ = √(k/m) = √(2×10⁷/5000) = √4000 ≈ 63.25 rad/s. Step 2: Natural frequency f₀ = ω₀/(2π) = 63.25/(2π) ≈ 10.07 Hz. Step 3: Soldiers marching at this frequency (≈600 steps per minute) could induce resonance.
Answer
f₀ ≈ 10.1 Hz (T ≈ 0.099 s). Marching at this cadence risks resonant buildup.
Natural frequencies of common mechanical systems
| System | Typical Natural Frequency | Damping | Resonance Risk | Real Example |
|---|---|---|---|---|
| Suspension bridge | 0.1 – 1 Hz | Low | High | Tacoma Narrows (1940) |
| Guitar string (A) | 440 Hz | Low | Acoustic | Musical instruments |
| Human body torso | 4 – 8 Hz | Moderate | Discomfort | Vehicle vibration |
| Quartz crystal | 32,768 Hz | Very low | Precision timing | Watch oscillator |
| Building (10-floor) | 0.3 – 1 Hz | Moderate | Earthquake damage | Seismic design |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Simple Harmonic Motion
Simple Harmonic Motion (SHM) is a type of periodic oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and always directed toward it. The motion follows a sinusoidal pattern over time, characterized by constant amplitude, frequency, and period in the absence of damping. SHM is the basis for understanding pendulums, springs, sound waves, and alternating electric circuits.
Spring-Mass System
A spring-mass system consists of a mass attached to an ideal spring that obeys Hooke's Law, where the restoring force is proportional to the displacement from equilibrium. When displaced and released, the mass oscillates in simple harmonic motion with a period that depends on the mass and the spring constant, but not on amplitude. This system is the canonical model for oscillatory behavior in physics and engineering.
Pendulum
A pendulum is a mass (called a bob) suspended from a fixed point by a string or rod that oscillates back and forth under the influence of gravity. For small angular displacements (less than about 15°), a simple pendulum exhibits simple harmonic motion, and its period depends only on its length and the local gravitational acceleration, not on mass or amplitude. Pendulums have historically been used in clocks and are fundamental to understanding oscillatory systems.
From Latin "resonantia" (echo), derived from "resonare" (to resound), from "re-" (again) + "sonare" (to sound). The physical concept was formalized in the 19th century by Helmholtz and Lord Rayleigh in "The Theory of Sound" (1877).