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.

Current holding force $6.72\,\mathrm{N}$

Fitted interval slope $49.7\,\mathrm{N\,m^{-1}}$

Heat per complete cycle $0.27\,\mathrm{J}$

Force-fit RMS $0.0231\,\mathrm{N}$

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.

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 misconceptionA harder spring must lose more energy.

Better mental modelTurn 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 misconceptionThe fitted intercept directly measures the initial load.

Better mental modelA 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

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.
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.
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.