A spring-mass network connects point masses with elastic springs, each obeying Hooke's law, to model how soft, deformable structures bend, sag and vibrate in three dimensions. The whole system is a set of coupled oscillators that settle into an equilibrium shape while energy sloshes between motion and spring tension. The same maths underpins cloth and rope simulation in games, molecular dynamics, and the structural analysis of bridges and buildings.
F = -k * (L - L0).
F = -k * (L - L0) is Hooke's law. A single mass on a spring
undergoes simple harmonic motion with period
T = 2*pi*sqrt(m/k). Energy splits into kinetic
KE = 1/2 * m * v^2 and elastic potential
PE = 1/2 * k * (L - L0)^2. A network of many such springs has
a spectrum of normal modes — the eigenmodes of the stiffness
matrix — exactly like coupled pendulums or vibrating molecules.
k; higher values make the mesh stiffer and oscillations faster.m of each node; heavier masses oscillate more slowly.The Tacoma Narrows Bridge collapsed in 1940 because its deck behaved like a spring-mass system driven at its natural resonance by a steady wind. Modern skyscrapers and bridges fight this with tuned mass dampers — huge counterweights on springs that absorb destructive vibrations.
Connect masses with elastic springs and the system finds its equilibrium while vibrating through normal modes. This is the mathematical backbone of cloth simulation, molecular dynamics, structural engineering and computer animation.
Each spring exerts a restoring force F = −k·Δx (Hooke's law). The equations of motion form a system of coupled ODEs integrated with Verlet. The eigenmodes of the stiffness matrix determine vibration frequencies — the same mathematics describes chemical bond vibrations and building resonance.
Drag any node to displace it and watch wave propagation. Add nodes with Click, connect them by dragging between two nodes. Adjust stiffness k and damping. Freeze one node to create a fixed boundary condition — observe standing waves.
The Tacoma Narrows Bridge collapsed in 1940 because its deck acted like a spring-mass system driven at its natural resonance frequency by 64 km/h winds. Modern suspension bridges include tuned mass dampers — heavy pendulums that absorb vibration energy.
This simulation models a network of point masses connected by elastic springs, integrated in real time with Verlet integration. Each spring applies a restoring force proportional to how far it is stretched or compressed from its rest length, following Hooke's law F = -k·x, while a damping term bleeds off energy so the system can settle. You can drag any mass to perturb the network, adjust the spring stiffness and damping, and switch between chain, grid and circular topologies.
Spring-mass systems are the foundation of computational physics for soft bodies, cloth, hair and deformable objects in games, films and engineering simulation. The same lattice of masses and springs models how real materials store and dissipate elastic energy, how waves propagate through a medium, and how structures vibrate at their natural resonant frequencies.
What is a spring-mass system?
It is a set of point masses linked by springs. Each spring pulls or pushes its two masses back toward a rest length, and the combined forces produce oscillation, wave propagation and elastic deformation across the whole network.
What is Verlet integration and why is it used?
Verlet integration advances each mass using its current and previous positions rather than storing explicit velocities. It is stable, energy-conserving and cheap, which makes it the standard choice for real-time cloth and soft-body simulation.
What does the stiffness control do?
Stiffness is the spring constant k in Hooke's law. Higher stiffness makes springs resist stretching more strongly, so the network is firmer and oscillates faster; lower stiffness makes it floppy and slow to respond.
What does damping change?
Damping removes kinetic energy on each step, like friction or air resistance. More damping makes motion die out quickly and settle; less damping lets the network bounce and ring for longer.
Hooke's law states that the restoring force of an ideal spring is proportional to its displacement from rest, F = -k·x. The minus sign means the force always points back toward the equilibrium length.
Chain connects masses in a single line, grid links them into a 2D mesh like a sheet of cloth, and circular joins them in a closed loop. Each topology produces distinctly different vibration and wave behaviour.
Yes. If stiffness is very high and damping is low, the explicit time step can no longer resolve the fast oscillation, and the simulation may overshoot and blow up. Reducing stiffness or adding damping restores stability.
Cloth is commonly modelled as a grid of masses joined by structural, shear and bending springs. This simulation uses the same core idea, so a grid topology behaves much like a hanging piece of fabric.
Every spring-mass network has natural modes at which it prefers to vibrate, set by the masses and stiffnesses. Driving or plucking the network near one of these frequencies produces large, sustained oscillations.
Because the springs couple neighbouring masses, moving one stretches the springs attached to it, which immediately exerts force on the connected masses and propagates the disturbance through the network as a wave.