Rolling circumference

The rolling circumference is the distance that a wheel covers without slipping in one revolution ; For motor vehicles , the speed of 60 km / h is to be taken as a basis according to DIN 70020. A fictitious (not geometrically measurable) dynamic rolling radius is calculated from the rolling circumference .

The rolling circumference changes only slightly as a result of the load and speed. The radial tires customary in passenger cars are characterized by a belt that is only slightly stretchable and unrolls once per revolution. The dynamic rolling radius under load is greater than the distance from the center of the wheel to the roadway (static radius ). In particular, the dynamic rolling radius changes significantly less under load than the static radius. The following approximation can be made for the dynamic rolling radius: ${\ displaystyle r _ {\ mathrm {D}}}$ ${\ displaystyle r _ {\ mathrm {S}}}$

{\ displaystyle r _ {\ mathrm {D}} \ approx {\ frac {2} {3}} r_ {0} + {\ frac {1} {3}} r _ {\ mathrm {S}} \;}

with ... unloaded tire radius

{\ displaystyle r_ {0}}

Motor vehicles

In Germany, the rolling circumference of the wheels of motor vehicles is measured at a speed of 60 km / h. A level roadway is required for correct measurement . Guideline 107-01 of the German Rubber Industry Association (WdK) deals with the speed dependency of the rolling circumference of car tires and in Part 02 the test procedure for the static radius and the rolling circumference. The distance covered by one wheel revolution is much more dependent on the speed of tires of the diagonal type than of tires of the radial type. At 180 km / h, diagonal tires "grow" by approx. 5% compared to 60 km / h, radial tires only by approx. 1%.

The rolling circumference of a motor vehicle tire results from the rim size R and the ratio of the height H, which goes beyond the edge of the rim. However, H is not specified directly in the designation of the tire size, but only the ratio of H and the width B of the tread. A calculation is also very easy with the ratio H / B. The information on the size of the tire is structured in such a way that first the width B and after a slash the ratio of H / B multiplied by 100 is given, followed by an R and the amount of the intended rim diameter in inches. For example, the tire with the size specification 245/45 R 18 has a width of 245 mm, H / W corresponds to 0.45 and the tire is intended for an 18-inch rim. The following then applies to the circumference U: U = π (R + 2 (H / B) / 100 B)

It should be noted that the specification of the rim diameter must still be converted into meters or a unit derived from it. One inch corresponds to 2.54 cm or 25.4 mm, for example.

For tires of size 245/45 R 18 the following circumference results: U = π · (18 · 25.4 mm + 2 · 45/100 · 245 mm) = 2129 mm = 212.9 cm = 2.129 m

Deviations due to profile wear and changes in air pressure

The unloaded radius changes as a result of profile wear. According to the estimation formula above, the change in radius only affects 2/3 of the rolling circumference. According to this calculation, 7 mm of tread wear on an average car tire would only cause a deviation of between 1.3% and 2% instead of a deviation of 2–3%. When the air pressure changes, the static radius changes. The effect on the rolling circumference is significantly less. Since the estimation formula is only a guide, the manufacturer's information applies.

When unrolling, the belt retains its length and is only flexible in its shape. It rolls off once per revolution, similar to an armored chain. In the area of the contact point, the profile particles adhere and are easily deformed, but this has no effect on the length of the belt in the area of the contact area. Changes in air pressure or the tread depth therefore only have a minor effect on the rolling circumference. Nevertheless, the minor changes are evaluated in ESP control units and used for roll warning .

Unloaded tires
Tires while driving

Cycles

For bicycle tachometers, too, the rolling circumference is the decisive parameter for correctly determining the driving speed and the distance covered. In addition, the rolling circumference has a direct influence on the development , i.e. H. the distance covered per crank turn.

Background to tire diameter & rim diameter

The size of the tires in inches is only a rough indication of the outer diameter of the tires. For a long time there were hardly any different tire widths, which meant that the size of the rim could also be deduced from the inch specification of the tire diameter. As a result, expressions such as "28 mm rim" have become commonplace, which means a rim with an outer diameter of 622 mm (= 24.5 inches), as this was mostly used for 28 mm tires - and is still used. With more and more different tire widths for "special" uses such as racing bikes or mountain bikes, the relationship between the tire size and the size of the rim is becoming more and more obsolete (see also bicycle tires # tire and rim sizes ).

For example, rims for 27 ″ road tires have a larger outer diameter of 630 mm than those for common 28 ″ tires with 622 mm. Explanation: Due to the thin racing bike tires, the overall diameter of the wheel with 630 rims remains at around 27 inches, whereas the wheel with 622 rims and thicker normal tires has an outer diameter of roughly 28 inches:

Outside diameter wheel ≈ 630 mm rim + 2 × 28 mm tire = 686 mm ≈ 27 ″
Outside diameter wheel ≈ 622 mm rim + 2 × 44 mm tire = 710 mm ≈ 28 ″

Wheels for racing bikes made of 622 rims and the thin 23 mm tires commonly used for this purpose have an outer diameter of only about 26 and a half inches, although these rims are sometimes still called "28" rims, because many people label them as 28 " Tires are used:

Outside diameter wheel ≈ 622 mm rim + 2 × 23 mm tire = 668 mm ≈ 26.3 ″

Approximate calculation of the rolling circumference

One method of calculating the rolling circumference is via the outer diameter of the wheel - i.e. the tire / casing. There are different names on bicycle tires, mostly the outdated - but still common - inch specifications (e.g. 26 × 1 ³ ⁄ ₈  ) and more recent ISO specifications (e.g. 32-571). Although the first number in the inches already indicates the outer diameter of the tire, this can only be seen as a rough classification. Since tires have different widths (the last number in the inch information), this leads to different outside diameters, which ultimately differ from the 26 ″ indicated, for example (see above).

The ISO information according to ETRTO is better suited to the approximate calculation of the rolling circumference than the customs information . The ETRTO dimension is made up of the tire width in millimeters and the outer diameter of the rim (= inner diameter of the tire). The outer diameter of the wheel results from and the height of the tire building up over it . In order to be able to calculate the rolling circumference from the ETRTO information, one uses the fact that an inflated bicycle tube / tire forms roughly a circle in its cross-section. It can therefore be assumed from the approximation that the tire height roughly corresponds to the tire width . This results in the rolling circumference : ${\ displaystyle d}$ ${\ displaystyle d}$ ${\ displaystyle a}$ ${\ displaystyle l}$

{\ displaystyle l \ approx (2 \ cdot a + d) \ cdot \ pi \,}

A 28-inch wheel of ETRTO size 35-622 has a tire width (≈ tire height) of 35 mm and thus a wheel diameter of . This gives the rolling circumference of . ${\ displaystyle 35 + 622 + 35 = 692 \, \ mathrm {mm}}$ ${\ displaystyle l \ approx \ pi \ cdot 692 \, \ mathrm {mm} = 2174 \, \ mathrm {mm}}$

Measurement of the rolling circumference

There are two common methods of measuring the rolling circumference instead of calculating it. One works by measuring the radius of the impeller (= half the diameter) by measuring the distance between the hub and the ground in the loaded state. This distance is then doubled and multiplied by pi results in the rolling circumference.

The second method measures the rolling circumference directly, which is why it is the most accurate of all the methods listed here. It is sometimes called the “ketchup method”, which involves marking a point on the wheel using ketchup or chalk, then moving the wheel forward one full turn and measuring the distance between the two chalk marks on the floor. Although the ketchup method seems very simple, it gives the best result.

Web links

Calculation of the rolling circumference of car tires
Approximate calculation of the rolling circumference of bicycle tires (based on an approximation that the ETRTO specification of the tire width roughly corresponds to the tire height)
Size specification of bicycle tires (in English, rolling circumference only approximately calculated from the ETRTO information)

Individual evidence

↑ Michael Trzesniowski: chassis . Springer Vieweg, 2017, ISBN 978-3-658-15544-5 , p. 12 . ( limited preview in Google Book search)
↑ http://elib.uni-stuttgart.de/opus/volltexte/2007/3196/pdf/dissertation_rau_magnus_abgabe.pdf
^ C. Rill: Vehicle Dynamics . S. 15. http://www.autogumi.com/FDV_Skript.pdf .

[Trzesniowski-1] Michael Trzesniowski: chassis . Springer Vieweg, 2017, ISBN 978-3-658-15544-5 , p. 12 . ( limited preview in Google Book search)

[Rau-2] ttp://elib.uni-stuttgart.de/opus/volltexte/2007/3196/pdf/dissertation_rau_magnus_abgabe.pdf

[RILL-3] C. Rill: Vehicle Dynamics . S. 15. http://www.autogumi.com/FDV_Skript.pdf .