M068 · A spring with changing stiffness

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.

A three-dimensional cart is tied to a fixed anchor by a spring whose stiffness varies, and a bar shows the current stiffness. Set the depth, frequency and phase of the modulation, release the cart and follow the pump work, which can be positive or negative. Run two independent starting states for one period to measure the Floquet multipliers, then build a stability map and select any cell in it.

Acquired time $0\,\mathrm{s}$

Primary displacement $0\,\mathrm{m}$

Primary velocity $0\,\mathrm{m\,s^{-1}}$

Mechanical energy $0\,\mathrm{J}$

Integrated signed actuator work $0\,\mathrm{J}$

Independent viscous dissipation $0\,\mathrm{J}$

Independent energy balance defect $0\,\mathrm{J}$

Position change with half step · first four cycles $0\,\mathrm{m}$

Actual integration step $0\,\mathrm{ms}$

Model domain / record status $\text{Ready}$

Primary spring force $0\,N$

Primary acceleration $0\,\mathrm{m/s^2}$

Stiffness pump power $0\,W$

Total viscous power $0\,W$

Current stiffness $0$

Largest Floquet multiplier magnitude $0$

Asymptotic linear growth rate $0$

Numerical linear stability $0$

Monodromy determinant defect $0$

Multiplier change with half step $0$

Computed map points $0$

Map acquisition status $0$

Set up a vibration and record its response

Can changing stiffness add energy without an applied drive force?

Physical force vectorsEvery force on both masses is drawn to one common scale at each instant.Velocity vectorsBlue arrows; rightward is positive. Velocity has its own scale, separate from force.Recent acquired pathsShows only the last two natural periods; future positions are never drawn.Moving mass

m

1.00\,\mathrm{kg}

Mean stiffness

k_0

16.00\,\mathrm{N\,m^{-1}}

Damping ratio

\zeta

0.03\,

Stiffness modulation depth

h

0.28\,

Modulation to natural rate

\Omega/\omega_0

2.00\,

Modulation phase at release

\phi

0.00\,{}^{\circ}

Release displacement

x_0

0.04\,\mathrm{m}

Initial velocity

v_0

0.00\,\mathrm{m\,s^{-1}}

Recording modulation cycles

N

48.00\,

Maximum integration step

\Delta t_{\max}

10.00\,\mathrm{ms}

Inspect acquired record

t/t_{\mathrm{end}}

0.00\,\%

Drag to orbit the apparatus. Set the release in the Motion & response view, or use the sliders from the keyboard. Switching to Energy & power keeps the acquisition cursor and any stability maps that still apply. Each fine step advances at most two milliseconds.

The model has linear viscous damping and a positive stiffness that varies sinusoidally. There is no added drive force and no nonlinear saturation. The Floquet multipliers describe the linear periodic system and do not depend on the chosen release; a growing multiplier still cannot make a release from exactly zero move. Changing the modulation phase can change how much the motion grows in a finite time, but not the multiplier magnitudes. Recording stops before the displacement passes two metres; the mathematical instability is still there. The green and red map cells are numerical one-period calculations on a finite grid, so the boundary between them is not certified. The principal-tongue threshold for small depth is only an asymptotic reference. The purple bar shows the current stiffness; it does not move the anchor or shake the base. The coil shape, the size of the mass, the dashpot hardware and the chosen rest length are schematic. Each record is drawn at one fixed scale that covers the whole solution. Ideal horizontal rails balance gravity. The pump power at each instant comes from the rate of change of stiffness, not from an added force times velocity.

Physics tutorial

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.

Start with the essentials

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
Acquire the principal-tongue release and the detuned release.
What to observe: Where the Floquet growth rate is positive, the motion grows over the long run.
02
Compare pumping with damping
Look at the spring and dashpot arrows in the three camera views. Open Energy & power, increase damping, then fine-step through moments of positive and negative pump power.
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
Generate the map, select a cell, then halve the time step.
What to observe: The one-period calculation for the selected cell comes with a determinant check and a half-step refinement check.