M088 · One spring, two kinds of motion

Swinging Spring & Internal Resonance

Stretch a spring, nudge its bob and release. Tune the small-amplitude frequency ratio, watch energy pass between stretching and swinging, then add a sideways launch to reveal spatial motion.

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 $\text{Ready}$

STRETCH / SWING / EXCHANGE

A little nudge changes everything.

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\,

Units: loaded equilibrium length, bob mass and gravity are one. Time is measured in the square root of equilibrium length divided by gravity; energy in mass times gravity times equilibrium length. Up is positive vertical; the release starts in the horizontal–vertical plane, with sideways velocity along the third axis.

The frequency control changes stiffness and unloaded length together, keeping the loaded equilibrium length fixed. The displayed ratio is a small-amplitude reference, not a measured nonlinear frequency. The spring is massless, axial and Hookean in tension and compression; coil geometry and bob size are schematic. No bending, coil collision, air drag or external drive.

Cartesian velocity Verlet uses a timestep of 0.005 or 0.0025, with records every 0.04 up to time 120. Radius below 0.12 ends the record at the previous saved state. The error readout uses the initial energy above equilibrium; the reported maximum covers recorded samples. Stretch and swing energies form an exact coordinate partition, not individually conserved normal modes.

A perfectly vertical initial state remains vertical in this ideal symmetric model. A transverse seed is needed to excite swinging. The spatial preset has genuine out-of-plane motion; no Lyapunov, chaos-window, mode-locking or universal stepwise-precession diagnosis is claimed. The floor is a visual reference, not a collision surface.

A three-dimensional, ideal axial spring and point mass. No spring mass, buckling, damping or measured-data claim.

Start with the essentials

Focus question: Why can a mostly vertical release grow into a large swing?
One-sentence intuition: Near the two-to-one small-amplitude frequency ratio, a tiny transverse seed can draw energy from radial motion.

Core mathematical model

One Cartesian force law

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

Mass and gravity are one. The vertical coordinate points upward; the spring force is central about the pivot.

Two small-amplitude clocks

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

Loaded equilibrium length is fixed at one. These are linearized frequencies, not a nonlinear frequency measurement.

Radial coordinate energy

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

The spring potential is completed about loaded equilibrium, including its radial gravitational contribution.

Swing coordinate energy

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

This exact nonnegative partition uses transverse kinetic energy and orientation-dependent gravity. Neither part is separately conserved.

A second conservation check

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

The spring is central and gravity has no torque about the vertical axis.

Common difficulties

A seed matters

Typical misconceptionAn exactly vertical release must eventually start swinging.

Better mental modelThe symmetric vertical subspace is invariant. Use the tilt or sideways control to supply a transverse seed.

Internal is not externally driven

Typical misconceptionThe growth of swinging adds energy to the system.

Better mental modelThe radial motion supplies it. This free conservative model has no motor or periodic stiffness actuator.

Run the experiment

01
Release near resonance
Choose Resonant exchange and press Play. Scrub through the full record.
What to observe: The orange stretch bar shrinks as the blue swing bar grows, then the transfer reverses.
02
Detune and compare
Choose Detune the spring with the same initial stretch and tilt.
What to observe: The transverse motion grows much less. The loaded equilibrium length stays fixed.
03
Remove the seed
Choose Perfectly vertical.
What to observe: The bob follows an exact vertical harmonic limit. The swing energy stays zero.
04
Open the third direction
Choose Spatial launch and compare the front and top views.
What to observe: The trail leaves the initial plane. Vertical angular momentum stays constant.
05
Audit the numerics
Enable half timestep and compare the maximum recorded energy error.
What to observe: Second-order integration reduces the energy error without changing the physical initial state.

Core mathematical model

One Cartesian force law

Two small-amplitude clocks

Radial coordinate energy

Swing coordinate energy

A second conservation check

Common difficulties

A seed matters

Internal is not externally driven

Run the experiment

Release near resonance

Detune and compare

Remove the seed

Open the third direction

Audit the numerics