Spring-Mass-Damper Simulator
ωn = √(k/m) • ζ = c/2√(km) • Free • Damped • Forced — Simulate • Explore • Practice • Quiz
1 Overview
This free vibration analysis tool simulates a spring-mass-damper system across four vibration types: free undamped, free damped, forced, and forced damped. The interactive canvas shows an animated spring-mass-damper mechanism on the left and a live oscilloscope-style waveform chart on the right, letting you visualise displacement versus time with envelope curves for damped oscillations.
The simulator calculates natural frequency ωn = √(k/m), damping ratio ζ = c/(2√km), period, amplitude, phase angle, and frequency ratio in real time. It covers the complete vibration curriculum from undamped free oscillation to resonance in forced systems, and critical damping behaviour — all essential for mechanical, civil, and aerospace engineering courses.
2 Getting Started
The simulator opens in Simulate mode with Free Undamped vibration active (m = 5 kg, k = 200 N/m). The mass oscillates at its natural frequency, and the waveform chart traces displacement over time. Six readout cards display Natural Frequency, Damping Ratio, Period, Amplitude, Phase Angle, and Frequency Ratio.
Use the Vibration Type pills to switch between the four types. Sliders for mass, spring constant, damping coefficient, forcing frequency, forcing amplitude, and initial displacement are grouped below. Presets like Car Suspension, Bridge Resonance, Tuning Fork, and Earthquake Damper configure real-world scenarios instantly.
3 Simulate Mode
Free Undamped: The ideal oscillator. Adjust mass and spring k to see how ωn changes. The waveform is a pure sine wave with constant amplitude — the system oscillates forever.
Free Damped: Enable the damping slider (c). Watch the amplitude decay exponentially. Adjust ζ from underdamped (ζ < 1) through critically damped (ζ = 1) to overdamped (ζ > 1) and observe the waveform transition from oscillatory to non-oscillatory return.
Forced Vibration: Set a forcing frequency ω and amplitude F0. The steady-state amplitude depends on the frequency ratio r = ω/ωn. Sweep the forcing frequency slider toward ωn to see the amplitude grow — approaching resonance.
Forced Damped: The most complete model. Both damping and forcing are active. The amplitude magnification factor and phase angle are calculated in real time. At resonance (r ≈ 1), damping is the only thing limiting amplitude.
The Bridge Resonance preset dramatically demonstrates what happens when forcing frequency matches natural frequency with low damping — a powerful teaching tool for understanding structural failures.
4 Explore Mode
Explore mode contains concept cards across three categories: Free Vibrations (natural frequency, damping ratio, logarithmic decrement, underdamped/critically damped/overdamped), Forced Vibrations (resonance, magnification factor, phase angle, transmissibility), and Applications (vibration isolation, seismic design, machinery balancing). Each card includes formulas, diagrams, and worked examples.
Key formulas covered: ωn = √(k/m), ζ = c/(2mωn), ωd = ωn√(1−ζ²), and the forced response amplitude X = (F0/k)/√((1−r²)² + (2ζr)²).
5 Practice & Quiz
Practice mode generates unlimited random vibration problems: calculate natural frequency from given m and k, find the damping ratio, determine the damped natural frequency, or compute the steady-state amplitude at a given frequency ratio. Full step-by-step solutions are provided for incorrect answers.
Quiz mode presents 5 randomised questions per session covering all four vibration types. Questions mix conceptual items (e.g., what happens at resonance) with numerical calculations. A score breakdown is displayed at the end.
6 Tips & Best Practices
- Start with Free Undamped to establish a baseline, then add damping to see the decay envelope appear on the waveform chart.
- Sweep the forcing frequency slowly through ωn to watch resonance build and then subside — the peak is dramatic with low damping.
- Use the Bridge Resonance preset to demonstrate the Tacoma Narrows Bridge effect: low damping + forcing near ωn produces dangerously large oscillations.
- Compare underdamped, critically damped, and overdamped by adjusting the damping slider — critically damped (ζ = 1) returns to rest fastest without oscillating.
- Check the frequency ratio readout: When r = 1, the system is at resonance. When r > √2, forced vibration is isolated (transmitted force is less than applied force).
- The simulator runs on mobile devices in landscape mode — useful for quick concept checks between classes.
Understanding Mechanical Vibrations — Free Interactive Spring-Mass-Damper Simulator
Mechanical vibrations are oscillatory motions of bodies and structures that occur in virtually every engineering system. From the suspension of a car to the strings of a guitar, from earthquake-resistant buildings to precision machining, understanding vibrations is critical for mechanical, civil, and aerospace engineers. A spring-mass-damper system is the fundamental model used to study vibrations — it consists of a mass (m) connected to a spring (stiffness k) and a viscous damper (damping coefficient c). This simple yet powerful model captures the essential physics of oscillatory motion, including natural frequency, damping ratio, resonance, and frequency response.
Free vs Forced Vibrations — The Four Key Modes
Free undamped vibration occurs when a system oscillates at its natural frequency ωn = √(k/m) without energy dissipation — an idealised case producing perpetual sinusoidal motion x(t) = A·cos(ωn·t). In reality, all systems have some damping. Free damped vibration introduces the damping ratio ζ = c/(2√(km)), producing three distinct behaviours: underdamped (ζ < 1) with oscillatory decay, critically damped (ζ = 1) with the fastest non-oscillatory return to equilibrium, and overdamped (ζ > 1) with slow exponential return. Forced vibration occurs when an external periodic force drives the system. The steady-state response depends on the frequency ratio r = ω/ωn. Forced damped vibration combines both phenomena, with the amplitude magnification factor X = (F0/k)/√((1−r²)² + (2ζr)²) and phase angle φ = arctan(2ζr/(1−r²)).
Resonance — The Critical Phenomenon
Resonance occurs when the forcing frequency equals or approaches the natural frequency (ω ≈ ωn, r ≈ 1). At resonance, the amplitude of oscillation can grow dramatically — theoretically to infinity in an undamped system. This is why the Tacoma Narrows Bridge collapsed in 1940 and why soldiers break step when crossing bridges. Damping limits the peak amplitude at resonance: the lower the damping ratio, the sharper and higher the resonance peak. Engineers must design systems to either avoid resonance or provide sufficient damping to limit dangerous vibration amplitudes. Vibration isolation, achieved by choosing system parameters so that r > √2, ensures transmitted force is less than the applied force.
How to Use This Simulator
In Simulate mode, select a vibration type (Free Undamped, Free Damped, Forced, or Forced Damped) and adjust mass, spring constant, damping coefficient, forcing frequency, forcing amplitude, and initial displacement. The left side of the canvas shows an animated spring-mass-damper system with a realistic zigzag spring, dashpot, and oscillating mass. The right side displays a live waveform chart (oscilloscope-style) showing displacement vs time with envelope curves for damped oscillations. Readout cards display natural frequency, damping ratio, period, amplitude, phase angle, and frequency ratio in real time. Use presets like Car Suspension, Bridge Resonance, Tuning Fork, and Earthquake Damper to explore realistic configurations. Switch to Explore to study 12 vibration concepts across Free Vibrations, Forced Vibrations, and Applications. Practice generates random calculation problems, and Quiz tests your knowledge with 5 randomised questions.
Key Formulas & Calculations
The natural frequency ωn = √(k/m) determines how fast a system oscillates when released from displacement. The damping ratio ζ = c/(2mωn) classifies the system response. The damped natural frequency ωd = ωn√(1−ζ²) is the actual oscillation frequency of an underdamped system. The logarithmic decrement δ = ln(xn/xn+1) = 2πζ/√(1−ζ²) quantifies the rate of amplitude decay. For forced systems, the transmissibility ratio and magnification factor are essential for vibration isolation design. All these calculations are performed in real time by this simulator.
Who Uses This Simulator?
This simulator serves mechanical engineering students studying vibrations and dynamics, automotive engineers designing suspension systems, civil engineers analysing structural dynamics, aerospace engineers studying flutter and aeroelastic phenomena, physics students learning about oscillatory motion, and instructors teaching mechanical vibrations or dynamics of machinery. It provides hands-on visual understanding without laboratory equipment or commercial software.
Explore Related Simulators
If you found this Vibrations simulator helpful, explore our Simple Harmonic Motion simulator, Hooke’s Law simulator, Spring Design simulator, and Gyroscope simulator for more hands-on practice.