M024 · Cable shape and friction

Catenary & Suspension Cable

An equal-height cable bench joins self-weight catenaries, deck-dominated bridge shapes and mixed loading. Redesign span and sag, compare an independent parabola, inspect anchor reactions and acquire a tension profile.

Probe coordinate $0$

Physical probe tension $0\,\mathrm N$

Required cable length $0\,\mathrm m$

Model boundary $\text{Static}$

Acquired tension stations $0$

Fitted horizontal tension $\text{Need 3 stations}$

Tension fit RMS residual $\text{Need 3 stations}$

Height above the cable low point $0\,\mathrm m$

Local cable slope $0$

Constant horizontal tension $0\,\mathrm N$

Vertical reaction at each anchor $0\,\mathrm N$

Rope plus deck weight $0\,\mathrm N$

Integrated vertical force residual $0\,\mathrm N$

Maximum same-sag parabolic height difference $0\,\mathrm{mm}$

SHAPE / PROBE / CHECK

Is every hanging cable a parabola?

Tension vectorsFixed scale per trialSupport span

L

12.00\,\mathrm m

Sag / span

d/L

0.18\,

Rope mass per arc length

\lambda

1.50\,\mathrm{kg/m}

Deck load per horizontal metre

q

0.00\,\mathrm{N/m}

Probe fraction along the span

\xi

0.50\,

Bounded relative tension noise

\varepsilon

0.00\,\%

Supports have equal height. Rope weight is uniform per cable arc length; deck weight is uniform per horizontal metre. Span and sag determine a different cable length for each trial: changing these controls is a redesign, not stretching a fixed rope. Visible hangers illustrate a continuous load, not discrete point loads.

Record or scan virtual tension observations. The horizontal-force fit uses geometry supplied by the model; it is not an independent geometry measurement. Gray dashed shape is a separate same-span, same-sag parabolic reference. Editing physical parameters or noise clears the record; moving the probe preserves it.

Flexible static planar cable, equal-height rigid supports and continuous vertical loads. Sag fixes the design cable length. No elastic stretch, wind, bending stiffness, structural strength or vibration. Hanger spacing and rope thickness are schematic; the bridge deck carries the selected effective line load.

Start with the essentials

Focus question: When is the parabola a good model?
One-sentence intuition: The measure used to distribute weight determines the differential equation and cable geometry.

Core mathematical model

Two ways to distribute weight

H y^{\prime\prime}=\lambda g\sqrt{1+(y^\prime)^2}+q

Rope weight follows arc length; deck load follows horizontal distance. Horizontal tension is constant.

Self weight gives a catenary

y=a[\cosh(x/a)-1],\qquad a=\frac H{\lambda g}

The origin is the lowest point. Equal-height anchors lie at opposite half spans and have the selected sag above the origin.

Horizontal load gives a parabola

y=\frac{q x^2}{2H},\qquad H=\frac{qL^2}{8d}

This exact limit assumes zero rope weight. A light rope under a large deck load approaches it.

Audit anchors and the entire cable

T=H\sqrt{1+(y^\prime)^2},\qquad 2V=\lambda g\ell+qL

The two anchor vertical reactions carry both contributions. Model quadrature integrates the cable length independently of the drawing.

Fit observed tension using supplied geometry

T_i^{\mathrm{obs}}\approx\widehat H\sqrt{1+p_i^2}

The geometry slopes are supplied by the model. This estimates horizontal force from virtual noisy tension readings; it is not a reconstruction of measured shape.

Common difficulties

A cable shape depends on the load measure

Typical misconceptionAll hanging cables are exact parabolas.

Better mental modelDistinguish weight per arc length from weight per horizontal metre.

Changing sag is a redesign

Typical misconceptionThe same inextensible cable stretches when the sag control moves.

Better mental modelThe model recomputes the required cable length for each span and sag.

Keep reference and acquisition separate

Typical misconceptionThe dashed parabola was reconstructed from sensor readings.

Better mental modelIt is an independently calculated geometric reference.

Run the experiment

01
Start with self weight
Use the self-weight preset and move the probe from the centre to an anchor.
What to observe: Horizontal tension stays fixed while total tension grows.
02
Add a bridge deck
Choose deck-dominated loading and compare with the dashed reference.
What to observe: A light rope approaches the parabolic shape.
03
Redesign the sag
Drag the orange anchor handle or use its keyboard slider.
What to observe: A shallow cable requires larger horizontal force and a different length.
04
Acquire a tension profile
Add bounded noise, scan 25 stations and export the observations.
What to observe: The fitted horizontal force uses the supplied geometry; the residual measures fit mismatch.