Presets
Playback
Number of carts
Masses
The cart colours match the graph colours.
Springs and drag
Initial motion
Pull the carts and release, or start from a single normal mode.
Actions
Layers
Model checks
The science
The model
Carts of mass m_i sit between two walls, joined by springs: k_1 from the left wall to cart 1, k_2 between carts 1 and 2, and so on. With x_i the displacement of cart i from equilibrium:
M x'' = -K x - b x', where M is the diagonal mass matrix and K is the symmetric tridiagonal stiffness matrix (for three carts: K_11 = k_1 + k_2, K_12 = -k_2, K_22 = k_2 + k_3, ...).
Normal modes
Looking for solutions x = phi cos(omega t) gives the eigenvalue problem K phi = omega^2 M phi. The eigenvectors phi_j are the normal modes: patterns in which every cart oscillates at the same single frequency omega_j. The solver symmetrises the problem to M^(-1/2) K M^(-1/2) and diagonalises it with the Jacobi method.
Beats and energy transfer
A general motion is a sum of modes. Pulling only cart 1 excites mostly modes 1 and 2 together, and their slightly different frequencies make the amplitude sway between carts with beat period 2 pi / (omega_2 - omega_1). For two equal carts the energy passes completely from one to the other in pi / (omega_2 - omega_1).
Energy bookkeeping
The modal energy is E_j = (1/2) (y_j'^2 + omega_j^2 y_j^2) with y_j = phi_j^T M x (mass-normalised). Without drag these stay constant for each mode, which is the clean statement of what normal modes are. Drag b damps the total energy.
Simplifications
Springs are ideal and massless, the air cushion is frictionless apart from the linear drag b, motion is one-dimensional and small enough for Hooke's law, and the carts never touch each other or the walls. Display displacements are scaled up by a factor 10 so that centimetres are visible.
What to try
- Pull cart 1 with the default springs: the amplitude moves along the track and back. Check the spectrum graph: three peaks at the three eigenfrequencies.
- Press Excite mode 1, 2 and 3: only one peak, constant modal energy, and the cart pattern never changes.
- Two carts with weak coupling: the exchange time becomes long. Compare it with the readout of the beat time and the live check.
- Heavy - light - heavy: the mode shapes are no longer symmetric in size; the light middle cart moves most in the high modes.
- Switch on air drag: the high modes die out first because their energy is exchanged faster for the same drag.