---
title: "Spring Law Calibration Interactive Lab | Sandbox Physics"
description: "When can one stiffness describe a spring’s whole loading stroke? Only over a range where the force is nearly linear and the loading history is fixed."
canonical_url: "https://sandboxphysics.com/labs/spring-law-calibration/"
language: "en"
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  zh-CN: "https://sandboxphysics.com/zh/labs/spring-law-calibration/"
last_modified: "2026-11-04"
---

M017 · Force, displacement and memory

# Spring Law Calibration

**When can one stiffness describe a spring’s whole loading stroke?** Only over a range where the force is nearly linear and the loading history is fixed. Move the fit interval to reveal changing stiffness. A conservative nonlinear spring retraces its path; a frictional slider makes a loop whose area is dissipated work. Pre-extension shifts the force, while reversal can temporarily change the fitted slope.

Load and unload four spring laws. Fit a selected force–displacement interval, then separate pre-extension, nonlinear stiffness and frictional hysteresis.

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

Model assumptions

The actuator prescribes a quasistatic sinusoidal stroke; inertia is omitted. An ideal bidirectional load cell reads the sum of an elastic core force and one parallel Jenkins element: a 100-newton-per-metre spring in series with an ideal Coulomb slider. Two conditioning cycles precede the recorded cycle. Pre-extension applies to the elastic core; the slider keeps its own history. The linear, bilinear, cubic-hardening and asinh-softening laws are teaching choices, not material calibrations of the drawn coil. Their transition length is 60 millimetres. Encoder positions and times are exact; independent Gaussian noise affects force only. A straight line fits only the selected branch and interval. Its intercept includes preload, slider memory and any curvature; its statistical standard errors omit instrument calibration and model uncertainty. External work equals stored-energy change plus frictional heat. No temperature, creep, buckling, failure or real coil geometry is modeled.

Physics tutorial

## One spring, more than one stiffness

**Background** Pulling a spring farther need not increase the force at a constant rate. Returning along a different force path reveals memory, which curvature alone does not imply.

**Why it matters** NIST force calibration separates applied force, readout and fitting uncertainty. Here the sensor is idealized: use a local fit to ask what the data support, rather than treating one slope as a material constant.

Start with the essentials

- **Focus question**: When does a fitted slope represent spring stiffness, and when does it mix curvature with loading history?
- **One-sentence intuition**: Compare both branches and move the fit window. A straight segment can have a different offset on unloading; a curved but conservative spring can return every joule.

### Core mathematical model

#### Separate pre-extension from the imposed stroke

$$
F=F_e(x_p+x)+F_h,\qquad x=\frac A2\left(1-\cos\frac{2\pi t}{T}\right)
$$

The sensor reads the signed holding force. The elastic core is already stretched before the recorded stroke; slider memory also contributes at the start.

#### Linear, hardening and softening cores

$$
F_e(u)=ku,\qquad ku\left[1+(u/\ell)^2\right],\qquad k\ell\operatorname{asinh}(u/\ell)
$$

These are separate teaching laws. The transition length is sixty millimetres. Hardening increases tangent stiffness; asinh softening decreases it while the force still grows.

#### A second linear stiffness

$$
F_e(u)=ku+2k\,\operatorname{sgn}(u)\max(0,|u|-\ell)
$$

The bilinear core changes from the small-extension stiffness to three times that value. At the kink there is no unique tangent.

#### A spring and slider remember reversal

$$
F_h=k_hz,\quad |F_h|\le Y,\quad dQ=Y\,|ds|,\quad W=\Delta U+Q
$$

The Jenkins branch first responds elastically after reversal, then slips at its threshold. The complete conditioned loop loses energy only when the slider moves.

#### Fit the observations you selected

$$
\widehat F=a+bx,\qquad r_i=F_i-a-bx_i
$$

The intercept and slope come only from noisy observations in one branch and interval. Residual curvature warns that one straight line misses structure.

### Common difficulties

#### Curvature is not hysteresis

**Typical misconception** A harder spring must lose more energy.

**Better mental model** Turn off the slider and compare hardening and softening. Their force paths coincide on return, and the net work over a closed cycle is zero.

#### The intercept is not always preload

**Typical misconception** The fitted intercept directly measures the initial load.

**Better mental model** A nonlinear interval extrapolates away from the measured region, and slider memory shifts each branch. State the fitting window and compare both branches.

### Run the experiment

1. 01

   #### Watch a complete loop

   Use A friction loop and follow the gold carriage through one outward and return stroke.

   **What to observe:** The two force branches separate; accumulated heat stays positive after the carriage returns.

2. 02

   #### Find the useful Hooke interval

   Choose Stiffer as it stretches, then move the interval endpoints in Recording and model details.

   **What to observe:** The fitted slope depends on where you look. The residual tests whether a straight line is adequate.

3. 03

   #### Separate preload from memory

   Compare Load before the stroke with A friction loop, then fit the unloading branch.

   **What to observe:** Pre-extension changes the elastic force offset; frictional memory also creates a reversal segment and loop area.

Physical conclusion

A fit belongs to its stated branch and displacement interval. Hysteresis is a loop with dissipated work, not simply a nonlinear spring.

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)