Acceleration resistance

The acceleration resistance is an inertial force that opposes the acceleration of a mass. It determines the power and energy requirements for acceleration. The acceleration resistance is caused by the physical principle of inertia , according to which every body with a mass remains in its state of motion as long as no external force acts on it.

Translational part

Translational inertia force acting on the vehicle

The force resulting from a translational acceleration is obtained using the approach for inertial forces (according to d'Alembert ):

${\displaystyle F_{T}\!\,=-m\cdot a_{x}}$

With the force of inertia, the translational drag force results as: ${\displaystyle \!\,F_{a\ trans}}$

${\displaystyle \!\,F_{a\ trans}=F_{T}=-m\cdot a_{x}}$

With:

${\displaystyle \!\,m=\quad m_{F}+m_{Zu}}$

${\displaystyle \!\,m_{F}=\quad {\text{Fahrzeugmasse}}}$

${\displaystyle \!\,m_{Zu}=\quad {\text{Zuladung}}}$

${\displaystyle \!\,a_{x}=\quad {\text{Beschleunigung des Fahrzeugs}}}$

Rotational part

Rotational drag force acting on the drive wheel

During the translational acceleration of the vehicle, the rotating parts of the drive train (shafts, wheels, gears in the transmission, etc.) must be accelerated rotationally. For this purpose, a rotational drag force must also be overcome, which results from the mass moment of inertia and the angular acceleration of the respective component. To determine the resulting total force, the mass moments of inertia of the rotating parts are reduced on the drive axle.

Similar to the translational calculation, the following applies:

${\displaystyle M_{T}\!\,=-\Theta _{red}\cdot {\ddot {\varphi }}}$

This results in the rotational drag force:

${\displaystyle F_{a\ rot}\!\,={\frac {-M_{T}}{r_{dyn}}}={\frac {\Theta _{red}\cdot {\ddot {\varphi }}}{r_{dyn}}}}$

With:

${\displaystyle \!\,\Theta _{red}=\quad {\text{Tr}}{\ddot {a}}{\text{gheitsmoment aller rotierenden Teile auf die Antriebswelle reduziert}}}$

${\displaystyle \!\,{\ddot {\varphi }}=\quad {\text{Winkelbeschleunigung des Antriebsrades}}}$

${\displaystyle \!\,r_{dyn}=\quad {\text{dynamischer Radhalbmesser}}}$

From the relationship

${\displaystyle \!\,\varphi ={\frac {x}{r_{dyn}}}}$

results from two differentiations according to time:

${\displaystyle \!\,{\ddot {\varphi }}={\frac {\ddot {x}}{r_{dyn}}}}$

Thus, using : ${\displaystyle \!\,{\ddot {x}}=a_{x}}$

${\displaystyle \!\,F_{a\ rot}={\frac {\Theta _{red}}{r_{dyn}^{2}}}\cdot a_{x}}$

The following moments of inertia must be taken into account for the reduced mass moment of inertia:

Moments of inertia to be considered

Component, vehicle component	Moment of inertia (designation)
engine	${\displaystyle \!\,\Theta _{mot}}$
coupling	${\displaystyle \!\,\Theta _{K}}$
Gearbox with respective gear ratio i (based on the gearbox input shaft)	${\displaystyle \!\,\Theta _{G_{i}}}$
Drive shaft, differential	${\displaystyle \!\,\Theta _{Antr}}$
Wheels (mostly including brake discs and axle shafts)	${\displaystyle \!\,\Theta _{R}}$

With regard to the mass moment of inertia of the wheels, make sure that all wheels of the vehicle are taken into account, regardless of whether the front wheels, the rear wheels or all wheels are driven.

Taking into account the gear ratios in the gearbox (for the respective gear) and the axle ratio (for rear or front wheel drive), the mass moment of inertia reduced on the drive axle results for a gear i with the requirement for dynamic equivalence of the output and substitute system: ${\displaystyle \!\,i_{G_{i}}}$ ${\displaystyle \!\,i_{h(v)}}$

${\displaystyle \!\,\Theta _{red_{i}}=\Theta _{R}+i_{h(v)}^{2}\cdot \Theta _{Antr}+i_{h(v)}^{2}\cdot i_{G_{i}}^{2}\cdot (\Theta _{Mot}+\Theta _{K}+\Theta _{G_{i}})}$

Summary of the acceleration components

The total acceleration resistance results from the addition of the translational and rotational drag forces to:

${\displaystyle \!\,F_{a}=F_{a\ rot}+F_{a\ trans}}$

${\displaystyle \!\,F_{a}=\left({\frac {\Theta _{red_{i}}}{r_{dyn}^{2}}}+m_{F}+m_{Zu}\right)\cdot a_{x}}$

For the sake of simplicity and ease of use, a mass factor is now introduced: ${\displaystyle \!\,e_{i}}$

${\displaystyle \!\,e_{i}={\frac {\Theta _{red_{i}}}{m_{F}\cdot r_{dyn}^{2}}}+1}$,

which only contains vehicle-specific data. This results in the following for the entire acceleration resistance:

${\displaystyle \!\,F_{a}=(e_{i}\cdot m_{F}+m_{Zu})\cdot a_{x}}$

Since the gear ratio is included in the determination of the reduced mass moment of inertia as a square, the mass factor can vary over a wide range. For example, in off-road vehicles or commercial vehicles with extremely high gear ratios, a higher power requirement is required to accelerate the rotating masses than to accelerate the vehicle in a purely translational manner . ${\displaystyle \!\,(e_{i}>2)}$

literature

• Hans-Hermann Braess, Ulrich Seiffert: Vieweg manual automotive technology. 2nd Edition. Friedrich Vieweg & Sohn, Braunschweig / Wiesbaden 2001, ISBN 3-528-13114-4

See also

• Breakaway resistance

Individual evidence

1. ^ Karlheinz H. Bill: Introduction to automotive technology . Lecture notes, FHTW-Berlin