M030 · Cable shape and friction

Capstan & Belt Friction

A fixed drum with tangent pull stations reveals the two-sided holding range. Inspect an admissible static tension distribution, acquire noisy probe readings and distinguish mobilized friction from the material coefficient.

Probe coordinate $0$

Physical probe tension $0\,\mathrm N$

Contact arc length $0\,\mathrm m$

Model boundary $\text{Static}$

Acquired tension stations $0$

Fitted mobilized logarithmic slope $\text{Need 3 stations}$

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

Maximum larger / smaller tension ratio $1$

Minimum holding effort $0\,\mathrm N$

Maximum effort before reverse slip $0\,\mathrm N$

Required signed logarithmic slope $0$

Admissible-profile normal force per arc length $0\,\mathrm{N/m}$

Admissible-profile friction per arc length $0\,\mathrm{N/m}$

Required holding torque on the drum $0\,\mathrm{N\,m}$

Physical static power $0\,\mathrm W$

SHAPE / PROBE / CHECK

How little effort can hold this load?

Tension vectorsFixed scale per trialFixed drum radius

R

0.35\,\mathrm m

Contact wrap angle

\beta

180.00\,{}^\circ

Static friction coefficient

\mu_s

0.30\,

Effort end tension

F

30.00\,\mathrm N

Load end tension

W

60.00\,\mathrm N

Probe fraction along contact

\xi

0.50\,

Bounded relative tension noise

\varepsilon

0.00\,\%

Below the slip threshold, static friction does not uniquely determine the tension distribution. The teal curve is one admissible smooth distribution with constant logarithmic slope. The two dashed envelopes are limits. Outside those limits the red curve is only a requested distribution: physical forces, acquisition and static power are unavailable. Slip speed and kinetic friction are not simulated.

The fitted logarithmic slope measures mobilized friction in this selected profile, not the material static coefficient. Equal opposite cut arrows are diagnostic; radius changes contact force density and holding torque, while the ideal tension-ratio limit is unchanged. The drum is fixed and the rope is massless, flexible and flat in one plane.

Static massless flexible flat rope on a fixed cylindrical drum. Uniform Coulomb coefficient, planar partial wrap, no V-groove amplification. End tensions are prescribed by pull stations. The static distribution is one admissible choice, not a unique solution. Beyond the threshold the model declares slip without inventing post-slip motion.

Start with the essentials

Focus question: Does every static hold mobilize the same friction?
One-sentence intuition: Static friction sets a bound; only impending slip saturates it.

Core mathematical model

The local friction cone

\left|\frac{\mathrm d\ln T}{\mathrm d\alpha}\right|\leq\mu_s

The normal contact force on a differential arc supplies the Coulomb bound. Angle is measured from the effort end toward the load.

A two-sided holding range

W e^{-\mu_s\beta}\leq F\leq W e^{\mu_s\beta}

Too little effort lets the load side slip; too much effort demands reverse slip. Equality means impending motion.

One admissible static profile

T(\alpha)=F e^{k\alpha},\qquad k=\frac{\ln(W/F)}{\beta}

This constant logarithmic slope is a selected admissible field when its magnitude is inside the friction bound. Static contact does not select a unique distribution.

Radius changes force density and torque

n=\frac TR,\qquad |f|=\frac{|k|T}{R},\qquad \tau_{\mathrm{hold}}=R(W-F)

Normal and friction force densities have force per metre units. Positive holding torque opposes the positive rope-driven torque on the fixed drum. Static power is zero.

A fit is not a material calibration

\ln T_i^{\mathrm{obs}}\approx b+\widehat k\alpha_i

Only at independently established impending slip does the magnitude approach the static coefficient. Below threshold, the fit recovers mobilized friction in the selected profile.

Common difficulties

Static friction is an inequality

Typical misconceptionEvery held rope must use all available friction.

Better mental modelThe limit bounds many admissible static distributions.

Slip has two directions

Typical misconceptionMore effort is always safer for a fixed load.

Better mental modelCrossing the upper effort limit demands reverse slip.

Mobilized friction is not the material coefficient

Typical misconceptionA fit below threshold calibrates the material coefficient.

Better mental modelIt only recovers the slope of the selected static field.

Run the experiment

01
Find the holding range
Use static hold and compare effort with both limits.
What to observe: The selected static profile lies between the envelopes.
02
Cross each boundary
Try insufficient effort and reverse-slip demand.
What to observe: Red shows an impossible requested equilibrium; no physical acquisition is offered.
03
Change radius independently
Keep wrap, friction and end forces fixed while changing radius.
What to observe: The holding range stays fixed while torque and force densities change.
04
Fit the mobilized slope
Restore a held trial, add noise and scan the contact.
What to observe: The fitted slope approaches the chosen mobilized value, not the material coefficient.