Natural Frequency Simulator
Interactive spring-mass system vibration visualization
About this Simulator
This simulator visualizes how a spring-mass system oscillates when displaced from equilibrium. Watch the interplay between kinetic and potential energy, observe how damping affects the response, and see real-time calculations of natural frequency, period, and damping characteristics.
Physics & Formulas
Natural Frequency:
$$\omega_n = \sqrt{\frac{k}{m}}, \quad f_n = \frac{\omega_n}{2\pi}$$
Damped Response:
$$x(t) = A e^{-\zeta \omega_n t} \cos(\omega_d t + \phi)$$
Damped Frequency:
$$\omega_d = \omega_n \sqrt{1 - \zeta^2}$$
How to Use
- Adjust Mass to see how heavier objects oscillate slower (lower frequency)
- Increase Spring Stiffness to see faster oscillations (higher frequency)
- Change Damping Ratio: 0 = undamped, 0-1 = underdamped, 1 = critically damped, >1 = overdamped
- Set Initial Displacement to change the amplitude of oscillation
- Toggle Show Energy to visualize kinetic/potential energy exchange
- Enable Phase Portrait to see the velocity-displacement trajectory
- Click Start/Pause to begin or pause the simulation
Frequently Asked Questions
What is natural frequency?
Natural frequency is the frequency at which a system tends to oscillate when disturbed from equilibrium. It depends only on mass and stiffness: ωn = √(k/m). Every mechanical system has one or more natural frequencies.
What happens at critical damping (ζ=1)?
Critical damping is the minimum damping that prevents oscillation. The system returns to equilibrium as fast as possible without overshooting. This is often desired in suspension systems and door closers.
Why does energy remain constant when undamped?
In an undamped system (ζ=0), no energy is dissipated. Kinetic energy converts to potential energy and back, maintaining constant total mechanical energy - a demonstration of conservation of energy.
What is the phase portrait showing?
The phase portrait plots velocity vs displacement. For undamped systems, it's a circle/ellipse. For damped systems, it spirals inward toward equilibrium. The current state is shown as a moving point on this trajectory.
