Free Vibration Generator Online
Synthesize mechanical vibration frequencies, tactile sub-bass tones, and damped harmonic oscillations. Explore natural resonance, logarithmic decrements, and decay envelopes with live physical visualization.
What is Free Vibration and Damped Oscillation?
Free vibration represents the natural oscillatory motion of a mechanical structure following an initial external displacement or impulse strike when no continuous external driving force is applied. The system vibrates under the opposing interplay of elastic potential energy stored in springs and kinetic energy stored in moving inertia mass.
Real physical structures invariably lose vibrational energy through internal material friction, fluid drag, or viscous dashpot dampers—governed by the damping ratio (\(\zeta\)). This generator models free vibration decay waveforms, computes logarithmic decrements, and outputs low-frequency tactile rumble audio tones.
How to Use the Free Vibration Generator
Explore mechanical dynamics and test tactile frequencies in four easy steps.
Set Natural Frequency
Tune fn from 10 Hz to 200 Hz to simulate heavy bridge vibrations up to high-speed engine mounts.
Adjust Damping Ratio
Slide ζ from underdamped (0.01) to overdamped (> 1.0) to model rubber isolators versus steel springs.
Pluck / Strike Impulse
Click the strike button to trigger a physical impulse with exponential decay audio and mobile haptics.
Analyze Waveform Decay
Inspect the real-time oscillogram to measure amplitude decay per cycle and logarithmic decrement δ.
Key Dynamic Vibration Features
Built for mechanical engineers, structural health researchers, and acoustic designers.
Acoustic & Tactile Engine
Uses the Web Audio API to synthesize low-frequency rumble tones synchronized with mobile device vibration motors.
Mass-Spring-Damper Model
2D physical mechanical animation depicts mass block displacement, spring extension, and viscous fluid dashpot motion.
Decay Envelope Analysis
Calculates exact logarithmic decrement δ, damped frequency fd, and time constant τ in seconds.
Governing Vibration Equations of Motion
The classic second-order differential equation for a single-degree-of-freedom (SDOF) mass-spring-damper system is:
Dividing by mass m yields the standard form expressed with natural frequency ωn = √(k/m) and damping ratio ζ = c / (2 · √(k · m)):
For an underdamped system (ζ < 1), the time-domain displacement is an exponentially decaying sinusoid:
Where the damped natural frequency is ωd = ωn · √(1 - ζ²). The logarithmic decrement δ between successive peaks is:
Structural & Engineering Applications
Civil Earthquake & Bridge Dampers
Skyscrapers (like Taipei 101) and long suspension bridges incorporate tuned mass dampers (TMDs) with calibrated damping ratios. When struck by strong typhoon gusts or seismic ground waves, the damper absorbs energy and quells dangerous resonant vibrations before structural fatigue occurs.
Mobile Haptic Feedback & Game Controllers
Modern smartphone linear resonant actuators (LRAs) produce short burst vibrations at specific natural frequencies (typically 170-200 Hz). By shaping the damping envelope, haptic engines simulate mechanical clicks, keyboard taps, and gaming impacts with precise tactile realism.
Frequently Asked Questions
Free vibration occurs when a mechanical system is displaced from its static equilibrium position and released without any continuing external periodic excitation. The system oscillates naturally at its inherent natural frequency under the influence of internal restoring forces (springs) and dissipative damping forces.
In an underdamped system (damping ratio ζ < 1), the system oscillates with exponentially decaying amplitudes. In a critically damped system (ζ = 1), the system returns to equilibrium as fast as possible without oscillating. In an overdamped system (ζ > 1), sluggish viscous resistance prevents oscillation and delays return to equilibrium.
The logarithmic decrement (δ = ln(x1 / x2)) is the natural logarithm of the ratio of two consecutive peak vibration amplitudes. It provides an experimental method to measure the damping ratio ζ = δ / √(4π² + δ²) in real-world structures.
Yes. On supported mobile devices running Android and modern browsers, the HTML5 Vibration API (navigator.vibrate) commands internal eccentric rotating mass (ERM) or linear resonant actuator (LRA) motors to pulse in synchronization with generated vibration frequencies.