M068 · Vibration design

Parametric Resonance Playground

A stiffness actuator periodically changes the spring. Adjust modulation depth, frequency and phase, release the cart and audit the signed pump work. Integrate two independent basis releases over one period to measure Floquet multipliers; build and select a stability map.

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 ledger 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 $\text{Ready}$

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 $0$

DESIGN / ACQUIRE / AUDIT

Can changing stiffness add energy without an applied drive force?

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 the mass or use the keyboard ranges. Acquire the full record before comparing late-cycle measurements. Changing the experiment cancels incompatible scans.

Linear viscous damping and sinusoidally modulated positive stiffness. There is no additive drive force or nonlinear saturation. Floquet multipliers describe the linear periodic system, independently of the chosen release; a growing multiplier does not make an exactly zero release move. Changing modulation phase can alter finite-time amplification but not multiplier magnitudes. Recording stops before displacement exceeds two metres; the mathematical instability remains. Green and red map cells are numerical period calculations with a finite grid, not a certified boundary. The small-depth principal-tongue threshold is only an asymptotic reference.

Start with the essentials

Focus question: How can one-period measurements predict long-term linear growth?
One-sentence intuition: Evolve two basis states to reconstruct the period map; compare its eigenvalue magnitudes with unity.

Core mathematical model

Modulated spring law

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}

Depth stays below unity, so the instantaneous stiffness remains positive.

Actuator power

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

Pump work can have either sign. Integrate it independently from viscous heat.

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

Two independent normalized basis releases construct the matrix columns.

Growth and determinant

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

A positive growth rate is a linear instability; the determinant supplies an independent numerical check.

Small-depth principal threshold

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

This is a weak-modulation, weak-damping asymptotic relation, not the map calculation.

Common difficulties

Zero is still a solution

Typical misconceptionAn unstable multiplier means every release grows.

Better mental modelThe exact zero state remains zero; specially aligned states can initially shrink.

Finite phase sensitivity

Typical misconceptionChanging the starting phase changes the stability tongue.

Better mental modelIt changes startup projection; monodromy spectra are invariant under a shift of phase.

Linear growth does not saturate

Typical misconceptionThe domain stop is a physical saturation.

Better mental modelThe model has no nonlinear saturation; its growth continues beyond the displayed record.

Run the experiment

01
Find a tongue
Acquire the principal-tongue and detuned releases.
What to observe: A positive Floquet rate aligns with long-term growth.
02
Balance pumping and damping
Increase damping and inspect work and heat.
What to observe: The net energy change follows the independent ledger.
03
Test phase and zero
Change modulation phase, then select exact zero release.
What to observe: Finite amplification changes while multiplier magnitude persists; the zero state remains motionless.
04
Build the map
Generate the map, select a cell, then halve the time step.
What to observe: The selected period calculation has determinant and refinement diagnostics.