What it does
Join two pendulums with a spring, or line up masses on springs, and the motion looks complicated: energy sloshes from one to another and back. Underneath, every such system has a set of normal modes: special patterns in which all the masses swing at the same frequency, in step or in opposition, with fixed proportions. Any motion at all is just a sum of these modes, each swinging at its own frequency. Molecules vibrate this way, crystals carry sound this way, and buildings sway this way.
This tool takes up to 12 masses in a line joined by springs, with both ends tied to walls, one end free, or both ends free, and finds all the normal modes by solving the eigenvalue problem K v = ω² M v. It lists each mode’s angular frequency, frequency and period and its shape, draws the shapes, and animates the chain either in one mode or from any starting displacement. It also shows how the energy is shared between the modes and plots every mass’s displacement against time.
How to use it
- Choose the ends: two walls, one wall, or a floating chain.
- Set the number of masses.
- Type the masses in kg from left to right, or one number for all.
- Type the spring constants in N/m from left to right, or one number for all. The page says how many springs are needed: one more than the masses with two walls, the same number with one wall, one fewer for a floating chain.
- Under Watch it move, choose a single mode, “pull mass 1 aside”, or your own starting displacements. Pause, Play and the speed slider control the animation.
- Download the motion as a CSV file or the whole report.
Worked example
The made-up example has masses of 1, 2 and 1.5 kg between two walls, with springs of 10, 20, 15 and 5 N/m. The page finds three modes at 0.272, 0.672 and 1.01 Hz (ω = 1.708, 4.221 and 6.333 rad/s; periods 3.679, 1.489 and 0.9922 s). In the slowest mode all three masses move the same way, the heavy middle one most; in the fastest, neighbours move against each other. Pull mass 1 aside and let go and the energy is shared 1.35 %, 13.2 % and 85.5 % between modes 1, 2 and 3, which is why the motion looks irregular. For three equal 1 kg masses with four 10 N/m springs the shapes are the textbook (0.707, 1, 0.707), (1, 0, −1) and (0.707, −1, 0.707).
Tips
- Start in a single mode and the shape never changes; that is the defining property of a normal mode.
- Two nearly equal frequencies give beats: energy passes slowly back and forth between masses. Try two equal masses joined by a weak middle spring.
- A floating chain has a zero-frequency mode: the whole chain drifting without stretching any spring.
FAQ
Why are the frequencies not simply √(k/m)? Each mass feels several springs, and its neighbours move too, so the modes mix every mass and spring together.
What about damping? Ideal springs and no friction are assumed. Light damping makes each mode die away slowly at almost the same frequency.
Are the shapes unique? Up to a sign and a scale, yes (unless two frequencies are exactly equal). Here each shape is scaled so its largest entry is 1. The tool is for study and planning, not a substitute for professional judgement.
Where the numbers come from
- Small oscillations, the generalised eigenproblem K v = ω² M v and normal-mode superposition: standard classical mechanics. Eigenvalues by the cyclic Jacobi method (own code).
- Checked against SciPy eigh for the generalised problem, the closed form ω = 2√(k/m) sin(jπ/2(N+1)) for a uniform chain, and the matrix exponential of the equations of motion (SciPy expm).
- The example masses and springs are made up.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.