Online-Calculator pulling weight needed for cars

Pullingweights (often also named as pullingforces) of locomotives are measured easily with forcemeters or springbalances. Another way is using a rope-roll having the locomotive on the horizontal, the weight on the vertical side. However, often model-railroaders are somehow frightened of these instruments, be it the complexity, their price or the difficulty of the setup. Following an online-calculator is presented, which can with a few instruments used, to get the pulling weight for a given car. Please note: it is an approximation!

Theory

Material rolling has a moving resistance, consisting of friction between wheel and track as well as within the axlebearing. This moving resistance can be calculated from the total weight of the car and a factor "w_t,rel", named specific moving resistance. This factor has no unit. The moving resistance is the sum of rolling resistance and the friction of the axle-bearings.

F_w = w_t,rel*(F_roll + F_fric)
These " resistances" are forces (this is why the formular-symbols of forces). Now regarding the specific moving resistance: it is the quotient of the "leverarm of the moving friction" (f) and the diameter of the wheel (R):

w_t,rel= f/R*(Q+(Q+m_wheels)+(r/R*m*Q))/(Q+m_wheels)
with m_ges = Q + m_wheels and solving the term in parenthesis will result in:

w_t,rel= f/R*(2 - m_wheels/m_ges + r/R*m - m_wheels/m_ges*r/R*m)
The last term is neglectable, since it is less then 1% of the largest term (2).
To get the "leverarm of the moving friction" one needs the dimensions of the wheel:

At the center of the rotational axis the gravity force Q and perpendicular to it P (=mQ) are effective.
Vectoraddition of both results in R, having an angle of tanr = P/Q to the gravity force
At the radius r of the axle the counterforce R^-1 with the distance sin(arctan(r))*r from the prolonged distance QZ is effective
This distance is the above mentioned "leverarm of the moving friction": f = sin(arctan(m))*r

Dividing through the terrestrical acceleration g (9.81 m/s²) one get the needed pulling weight:

m_pull = w_t,rel * m_ges
In order to calculate the pulling weight needed for a car, we need four values:

the total weight m_ges in g
the total weight of the wheelsets m_wheels in g
the radius R of the wheels R in mm
the readius r of the wheel's axles within the bearing mm (conical axles measured at the top end)

The coefficient m has the following approx. values:

1,00: rubbertires
0,60: oily rubbertires
0,30: plastictires
0,20: oily plastictires
0,17: metal-metal pairing
0,15: metal-metal pairing lubricated
0,13: metal-plastic pairing
0,12: metal-plastic pairing lubricated

Literature

F. Sass, Ch. Bouche, A. Leitner, "Dubbel's Taschenbuch für den Maschinenbau", 12. Auflage 1963, Springer-Verlag ohG, Berlin/Göttingen/Heidelberg

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