---
title: "Parametric Resonance Playground Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/parametric-resonance/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/parametric-resonance/"
last_modified: "2026-10-22"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

**Background** A 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 matters** Section 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 misconception** If a multiplier is unstable, every starting state has to grow.

**Better mental model** A 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 misconception** Changing the starting phase moves the stability tongue.

**Better mental model** The 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 misconception** When the recording stops, the motion has physically saturated.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

An unstable linear system amplifies a perturbation but cannot create motion from an exactly zero initial state.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)