M088 · One spring, two kinds of motion

Swinging Spring & Internal Resonance

Can a bounce lend its energy to a swing? Yes, when the bounce frequency is close to twice the swing frequency. Then a tiny sideways nudge grows: energy flows from stretching into swinging, then back again, over and over. The total energy stays nearly constant through the exchange.

Stretch the spring, give the bob a small nudge and let go. Tune the ratio of the two small-amplitude frequencies and watch energy pass back and forth between stretching and swinging. Add a sideways launch to see the bob move in three dimensions.

Model time $\text{Ready}$

Stretch energy $\text{Ready}$

Swing energy $\text{Ready}$

Total energy above equilibrium $\text{Ready}$

Relative energy error $\text{Ready}$

Vertical angular momentum $\text{Ready}$

Energy trades between stretching and swinging

A small nudge changes the motion.

Small-amplitude frequency ratio

\omega_r/\omega_s

2.00\,

Initial stretch from loaded equilibrium

r_0-\ell_e

0.18\,

Initial tilt in degrees

\theta_0

3.00\,

Initial sideways velocity

v_{z,0}

0.00\,

Use half timestep

b_h

0.00\,

The units set the loaded equilibrium length, the bob mass and gravity to one. Time is measured in the square root of equilibrium length divided by gravity, and energy in mass times gravity times equilibrium length. Up is positive. The bob is released in the plane of the horizontal and vertical axes, and any sideways velocity is along the third axis.

The frequency control changes the stiffness and the unloaded length together, so the loaded equilibrium length stays fixed. The ratio shown is a small-amplitude reference, not a measured nonlinear frequency. The spring is massless, pushes and pulls only along its axis and obeys Hooke's law in tension and compression; the coil shape and bob size are schematic. Bending, coil collisions, air drag and external driving are not modeled.

Velocity Verlet in Cartesian coordinates uses a time step of 0.005 or 0.0025 and saves a state every 0.04 up to time 120. If the radius drops below 0.12, the record ends at the previous saved state. The error readout is scaled by the initial energy above equilibrium, and the reported maximum covers only the saved samples. The stretch and swing energies split the energy exactly by coordinate; they are not separately conserved normal modes.

In this ideal symmetric model, a perfectly vertical start stays vertical, so a small sideways seed is needed to start the swinging. The spatial preset has real out-of-plane motion. The Lab does not diagnose Lyapunov exponents, chaotic windows, mode locking or a universal stepwise precession. The floor is a visual reference, not a collision surface.

A point mass hangs on an ideal spring that pulls only along its own length, and it can move in three dimensions. The spring has no mass, does not buckle and has no damping. The motion is simulated, not measured.

Start with the essentials

Focus question: Why can a mostly vertical release grow into a large swing?
One-sentence intuition: When the small-amplitude bounce frequency is close to twice the swing frequency, a tiny sideways seed can draw energy out of the bounce.

Core mathematical model

Equation of motion in Cartesian coordinates

\ddot{\mathbf r}=-k(r-\ell_0)\frac{\mathbf r}{r}-\hat{\mathbf y}

Units are chosen so that the mass and gravity both equal one. The vertical coordinate points up, and the spring force always points along the line to the pivot.

Small-amplitude bounce and swing frequencies

\omega_r=\sqrt{k},\quad\omega_s=1,\quad\ell_0=1-\frac1k

The equilibrium length with the mass hanging is fixed at one. These frequencies come from the linearized equations; they are not measured from the nonlinear motion.

Energy of the bounce

E_r=\frac{\dot r^2}{2}+\frac{k(r-1)^2}{2}

The spring potential is written about the loaded equilibrium, so it already includes the part of gravity that acts along the spring.

Energy of the swing

E_s=\frac{|\dot{\mathbf r}|^2-\dot r^2}{2}+r+y,\quad E=E_r+E_s

This split of the total energy is exact and never negative. The swing part holds the sideways kinetic energy and the gravity term that depends on the spring's direction. Neither part is conserved on its own.

Angular momentum about the vertical axis

L_y=z\dot x-x\dot z=\mathrm{constant}

It stays constant because the spring force is central and gravity exerts no torque about the vertical axis.

Common difficulties

A swing needs a seed

Typical misconceptionIf I release it exactly vertically, it will start swinging sooner or later.

Better mental modelA motion that starts exactly vertical stays exactly vertical, because nothing in it breaks the symmetry. Use the tilt or sideways control to give it a small sideways start.

Internal resonance needs no drive

Typical misconceptionWhen the swing grows, energy is being added to the system.

Better mental modelThe energy comes from the bounce. This model is free and conservative, with no motor and nothing that varies the stiffness in time.

Run the experiment

01
Release near resonance
Choose Resonant exchange and press Play. Scrub through the whole record.
What to observe: The orange stretch bar shrinks while the blue swing bar grows. Then the energy flows back.
02
Detune and compare
Choose Detune the spring. It starts with the same stretch and tilt.
What to observe: The sideways motion grows much less. The loaded equilibrium length stays the same.
03
Remove the seed
Choose Perfectly vertical.
What to observe: The bob bounces straight up and down in exact simple harmonic motion, and the swing energy stays at zero.
04
Open the third direction
Choose Spatial launch and compare the front and top views.
What to observe: The trail leaves the starting plane, and the angular momentum about the vertical axis stays constant.
05
Check the numerical accuracy
Turn on Use half timestep and compare the largest recorded energy error.
What to observe: The second-order integrator gives a smaller energy error, and the physical starting state does not change.

Core mathematical model

Equation of motion in Cartesian coordinates

Small-amplitude bounce and swing frequencies

Energy of the bounce

Energy of the swing

Angular momentum about the vertical axis

Common difficulties

A swing needs a seed

Internal resonance needs no drive

Run the experiment

Release near resonance

Detune and compare

Remove the seed

Open the third direction

Check the numerical accuracy