Parametric Resonance Playground

Can changing stiffness add energy without an applied drive force? It can, if the stiffness changes at the right rhythm. Stiffening and softening the spring at about twice its natural frequency does positive work on the motion, so a small wobble grows. If each modulation cycle multiplies the motion by a factor larger than one, it keeps growing. A cart at exact rest stays at rest, because there is no motion to pump.

Change a spring’s stiffness in rhythm with a sliding mass. Watch its amplitude grow or decay, then explore the instability tongues and the effect of pump phase.

Displacement $0$

Velocity $0$

Pump work so far $0$

Floquet growth rate $0$

Physical time $0$

Mechanical energy $0$

Signed input work $0$

Viscous dissipation $0$

Energy balance error $0$

Half-step displacement change · first four cycles $0$

Integration step $0$

Spring force $0$

Damping force $0$

Additive force $0$

Record boundary $0$

Current stiffness $0$

Largest Floquet multiplier magnitude $0$

Monodromy determinant error $0$

Half-step multiplier change $0$

Position is measured to the right from the unmodulated equilibrium in metres; time is in seconds and angular frequencies are in radians per second. The mass is in kilograms and stiffness in newtons per metre. The positive spring stiffness varies sinusoidally with a fixed anchor. Linear viscous damping acts on velocity; there is no additive forcing or nonlinear saturation. Pump power is one half of the stiffness time derivative times displacement squared. Work and heat are integrated independently with the motion. A zero displacement and zero velocity stay exactly zero. Floquet multipliers come from two independent one-period solutions; their magnitudes are independent of release and modulation phase. The phase can change finite-time amplification. The finite numerical stability grid does not certify its boundary; the shallow principal-tongue threshold is only an asymptotic reference. Recording stops before displacement exceeds two metres. The page loads a complete simulated record and starts replay at a representative instant; Replay begins at release. The camera fits the whole record using a fixed length scale. The coil, hardware and chosen rest length are schematic. Horizontal rails balance gravity. Teal arrows show net force and gold arrows show velocity; they use separate fixed scales and their tails are lifted above the carriage for visibility. Dragging prepares a new release, retaining the selected initial velocity and pump phase, and recomputes the record. Sliders provide equivalent keyboard input.

A pump changes the rule of the spring

BackgroundA spring whose stiffness changes in time can feed energy into the motion, even though no separate force pushes the mass. For a given release, how much the motion grows in a finite time depends on the phase of that change.

Why it mattersSection 28.29 of the NIST DLMF defines the one-period (monodromy) matrix of Hill equations and its Floquet eigenvalues. The version here includes damping, adds an exact identity for the determinant and integrates the work separately.

Focus question: How can measuring a single period predict long-term growth in a linear system?
One-sentence intuition: Run two independent starting states through one period to build the one-period map, then check whether its eigenvalues are larger or smaller than one in magnitude.

Core mathematical model

Equation of motion with a modulated spring

m\ddot x+2m\zeta\omega_0\dot x+k_0[1+h\cos(\Omega t+\phi)]x=0,\quad \omega_0=\sqrt{k_0/m}

The modulation depth stays below one, so the stiffness is positive at every instant.

Power delivered by the pump

P_{\mathrm{pump}}=\tfrac12\dot k(t)x^2,\quad E=\tfrac12m\dot x^2+\tfrac12k(t)x^2

The pump can add or remove energy. Its work is integrated separately from the heat lost to viscous damping.

One-period map

\begin{bmatrix}x(T)\\\dot x(T)/\omega_0\end{bmatrix}=\Phi(T)\begin{bmatrix}x(0)\\\dot x(0)/\omega_0\end{bmatrix},\quad T=2\pi/\Omega

Each column of the matrix comes from one of two independent, normalized starting states.

Growth rate and determinant

\gamma=T^{-1}\ln\rho(\Phi),\quad \det\Phi=e^{-2\zeta\omega_0T}

A positive growth rate means the linear system is unstable. The determinant gives a separate numerical check.

Threshold of the main instability at small depth

\Omega\simeq2\omega_0,\quad h_{\mathrm{crit}}\simeq4\zeta

This approximate relation holds for weak modulation and weak damping; it is not how the stability map is computed.

Common difficulties

Zero is still a solution

Typical misconceptionIf a multiplier is unstable, every starting state has to grow.

Better mental modelA mass that starts exactly at zero with no velocity stays there, and some specially aligned starting states shrink at first.

Phase changes the start, not the stability

Typical misconceptionChanging the starting phase moves the stability tongue.

Better mental modelThe phase changes how much of the starting state lies along the growing solution. The eigenvalues of the one-period matrix stay the same when the phase shifts.

Linear growth does not saturate

Typical misconceptionWhen the recording stops, the motion has physically saturated.

Better mental modelThe model has no nonlinear saturation, so the growth continues past the end of the displayed record.

Run the experiment

01
Find an instability tongue
Compare Growing oscillation with Miss the rhythm; both full simulations load immediately.
What to observe: Where the Floquet growth rate is positive, the motion grows over the long run.
02
Compare pumping with damping
Compare the three camera views, increase damping, then open Potential, work and dissipation. Use Fine step to follow the changes in pump work.
What to observe: The net change in energy matches the separately integrated pump work and damping loss.
03
Test the phase and the zero state
Change the modulation phase, then select the exact zero release.
What to observe: The growth over the record changes, but the multiplier magnitude does not, and the zero state stays still.
04
Build the stability map
Select a cell in the automatically calculated map, then halve the integration step under Release and numerical settings.
What to observe: The one-period calculation for the selected cell comes with a determinant check and a half-step refinement check.