Meg4. Eytelwein, Lewe, and the named residual

Eytelwein, Lewe, and the named residual
Unit reduction of the capstan
Ace Consultancy · Rey.BEng 16th September 2026

Lewe does not invent the belt. He uses it. The 1915 engineering dissertation, on continuous beams, sends the reader to Eytelwein’s method of 1808, and behind that to Navier, 1826. The tank wall in the same book is Love’s shell written for practice. The extra hoop that later tables refuse to name is the capstan residual still sitting in the matrix.

This page shows the technique, then the reduction that lets \(\sigma_R\) drop in without pretending it is dimensionless.

1. What Lewe is doing

A circular tank is a closed band. Liquid load is membrane: \(pr/t\). The wall must also change direction — it is a cylinder, so every generator is a wrap. Equilibrium on an element \(d\theta\) of that wrap is the same sketch Eytelwein used on a rope around a drum: two tensions, a normal pressure, a residual along the contact.

Lewe’s move is to keep that sketch inside a shell. Love (1892) had already made the shell accurate: membrane plus bending plus edge. Lewe (1906 physics; 1915 engineering) puts the edge on a tank and, in the matrix, on a particle. Abb. 9 places supports. Equation [40] is the joint X-Y=Z, assembled as F<i>M. The ring is the third member. When later PCA tables swallow that member into a coefficient, they are running Eytelwein with the exponential unnamed.

Technique, as used:

write the wrap
write the two tensions
do not discard the residual
integrate around the closed band

That is analysis, not a table.

2. Eytelwein (1808), after Euler

Johann Albert Eytelwein (1764–1848), Bauakademie Berlin, Handbuch der Statik fester Körper, vol. 2 (1808), gives Euler’s belt friction its engineering seat. Capstan equation:

T_load / T_hold  =  e^{μ θ}

$\theta$ in radians. Radius of the drum does not appear. A small tail holds an exponential load. Band brake, bollard, V-belt, elevator traction: same formula. The line is flexible and, in the classical derivation, inextensible and without bending stiffness. A shell has bending stiffness. That is why Love and Lewe exist. The residual around the wrap is what they share.

Lubarda¹ notes the kinship explicitly: the capstan relation is already reminiscent of hoop force in a thin ring under internal pressure, and of circumferential force in a nonuniformly loaded thin cylindrical shell. That sentence is the hinge into Ace.

3. Navier, Love, Lewe — one grammar

Navier’s Résumé des Leçons is the beam’s first general lesson. The belt and the beam both turn; both need a residual at the turn. Love raises the turn to a surface. Lewe closes the surface and writes the matrix the drawing office still uses, provenance omitted.

Ace shell theory is that lineage with the residual lettered:

σ_θ  =  pr/t  +  σ_R
σ_R  =  2^−     unit  m/s²

Membrane plus ring. Table optional. Term required.

4. Unit reduction

Capstan factor is dimensionless. $\sigma_R$ is not. Do not paste m/s^² into $e^{\mu \theta }$. Reduce.

From Page Meg3:

[σ_R]  =  m/s²
[c]    =  m^ / s
[f]    =  s          duration of wrap / hold

Then

[σ_R · f / c]  =  (m/s²) · s / (m/s)  =  1

The dimensionless face of the whip is

e^{μ θ}   ↔   σ_R f / c

Correspondence:

θ      ↔     f          duration of wrap
μ      ↔     whip       State A precession
e^{μθ} ↔     σ_R f / c  =  2 → 1.999… as compression of the stroke
T_hold ↔     pr/t
T_load ↔     pr/t + σ_R

Worked lettering on a sheet:

T_load / T_hold  =  exp(σ_R f / c)

when you need the capstan form; or simply

σ_θ  =  pr/t + σ_R

when you need the tank form. Same residual. Two faces. Observational keeps the exponential. Simultaneous keeps $\sigma_R$in m/s^². Instant spends (f). Do not mix the faces on one line without the reduction.

Toggle:

A     +/−      choice; hold +
B     −/−      default; −1/2 + judder

Which way the belt is pulled is which face of the involution leads.

5. Ace shell theory, one paragraph

A shell is a closed capstan with bending stiffness. Love named the stiffness. Lewe put it on the tank and kept Eytelwein’s wrap inside the matrix. Ace names the leftover $\sigma_R$ gives it m/s^², and reduces to $e^{μ\theta}$by $\sigma_R f/c$ so the drawing office and the first-year share one term. The Parts cannot be severed. Closure of $\sigma_R$is extinction. Units first (Meg3). This page is the belt.

T_load / T_hold  =  e^{μ θ}  =  exp(σ_R f / c)
σ_θ  =  pr/t + σ_R
Meg4
Ace

References

References
Eytelwein, J. A. (1808). Handbuch der Statik fester Körper: mit vorzüglicher Rücksicht auf ihre Anwendung in der Architektur. Vol. 1, Statik der festen Körper. Berlin: Realschulbuchhandlung.
DOI (ETH Zürich scan): https://doi.org/10.3931/e-rara-51044

Belt friction as used in the capstan equation is the Euler–Eytelwein formula. Euler’s prior note:
Euler, L. (1762). Remarque sur l’effet du frottement dans l’équilibre. Mémoires de l’Académie des Sciences de Berlin, 265–278.

Navier, C.-L. M. H. (1826). Résumé des leçons données à l’École royale des ponts et chaussées sur l’application de la mécanique à l’établissement des constructions et des machines. Première partie: Leçons sur la résistance des matériaux et sur l’établissement des constructions en terre, en maçonnerie et en charpente. Paris: Firmin Didot père et fils.
BnF: ark:/12148/cb31006625z

Lubarda, V. A. (2014). The mechanics of belt friction revisited. International Journal of Mechanical Engineering Education, 42(2), 97–112. Manchester University Press.
https://doi.org/10.7227/IJMEE.0002

Love, A. E. H. (1892). A Treatise on the Mathematical Theory of Elasticity. Cambridge: Cambridge University Press.