Sauthoff formula

The Sauthoff formula or Sauthoff's resistance formula was derived by Friedrich Sauthoff in 1932 on the basis of extensive test data to describe the running resistance of railroad cars, which includes in particular the rolling and air resistance, as part of his Dr.-Ing. Dissertation. It was used until the 1980s to calculate travel and movement resistance and travel times of railroad cars and trains.

A simplified resistance formula (without curve radius of curvature, slope) used by Sauthoff for the rough calculation:

${\ displaystyle w = c_ {1} \ times G + c_ {3} \ times A \ times v ^ {2}}$

w : Resistance in N / kN corresponds to approx .: N / 100 kg

G : weight in N

G = m 9.81 (with location factor earth)

m : vehicle mass in kg

c ₁ : constant of rolling resistance

c ₃ : Air resistance constant (related to the unit of v )

v : speed in km / h

A : Drag area in m²

tram

c ₁ : 5 grooved track

c ₃ : 0.04

A : 7.8

bus

c ₁ : 15 concrete, 20 plasters, 30 tar

c ₃ : 0.04

A : 7.5

Subway / Metro

c ₁ : 2.5 pawl bearing drive

c ₃ : 0.04-0.2 without-with tunnel

A : 11

Simplified up to 40 km / h with surcharge for curvatures can be expected:

w = 6 G Vignol track

w = 8 G grooved track

The Sauthoff formula for wagon trains (behind the locomotive) is still important today :

${\ displaystyle w = (1 {,} 9 + c_ {b} \ cdot v) +0 {,} 048 \ cdot {\ frac {n + 2 {,} 7} {m}} \ cdot f \ cdot v_ {R} ^ {2}}$ in ${\ displaystyle {\ frac {\ mathrm {N}} {\ mathrm {kN}}}}$

n : number of cars

{\ displaystyle c_ {b}}

: Running resistance coefficient z. B. 0.007, 0.004, 0.0024 for 2, 3, 4-axle cars

f : equivalent cross-sectional area from wind tunnel tests, e.g. B. f = 1.45 m²

m : mass of the wagons behind the locomotive

{\ displaystyle v_ {R}}

: Relative speed

{\ displaystyle v_ {R} = v + 15 \, \ mathrm {km / h}}

The more precise formulas from Sauthoff are very detailed. He has drawn up tables for various cars. The values for the corresponding wagon types can be taken from these and inserted into the formulas. For the locomotives, the vehicle resistance according to beam was used more often.

Erfurt formula , formula by Clark and formula by Strahl ( formula by Röckl ) were / are common formulas for the calculation of the driving resistance forces of rail vehicles.

Nowadays equivalent polynomial series are used in simulation. Sauthoff's results are still used today to determine the coefficients.

${\ displaystyle w = c_ {1} + c_ {2} \ cdot v + c_ {3} \ cdot v ^ {2} + \ dots}$

The following driving resistance forces are important for rail vehicles and wagons:

Rolling resistance wheel-rail (mostly in a straight line)
Air resistance (especially in tunnels; head wind, tail wind)
Consumers such as generators and compressors driven by the rail wheels (wagon)
Breakaway force (increased resistance, which must be overcome until the carriage begins to roll at all; analogy to static friction and sliding friction )
Resistance in the curve (increases rolling resistance)
Resistance on inclines / slopes
Resistance when accelerating / braking

literature

F. Sauthoff, The resistance to movement of the railroad cars with special consideration of the more recent experiments of the Deutsche Reichsbahn, Dr.-Ing.-Dissertation Technische Hochschule Berlin-Charlottenburg 1932, 69 pp.
H. Nordmann, On the possibilities of calculating the braking distances of railway trains, Glasers Annalen 80, 3-20, 1956
N. Boden, To determine the air resistance of rail vehicles, Archiv für Eisenbahntechnik 25, 40-75, 1970.
G. Voß, L. Gackenholz, R. Wiebels, A new formula (Hannoversche formula) for determining the air resistance of track-bound vehicles, ZEV - Glasers Annalen 96, 166-171, 1972.
BR Rockenfelt, travel time and train journey calculations for the Deutsche Bundesbahn, in: Elsners Taschenbuch der Eisenbahntechnik, Tetzlaff Verlag, Darmstadt, p. 255, 1983.