Driving resistance

The driving resistance describes the sum of the resistance that a land vehicle has to overcome with the help of a driving force in order to drive at a constant or accelerated speed on a horizontal or inclined route.

Components of the driving resistance

The driving resistance is made up of various components:

Air resistance

• The air resistance increases quadratically with the driving speed and depends on the aerodynamic shape of the vehicle (air resistance coefficient) and the air density:
  ${\displaystyle F_{\text{Luft}}=c_{\mathrm {W} }\cdot A\cdot {\frac {\rho _{\text{Luft}}\cdot v_{\mathrm {rel} }^{2}}{2}}}$

${\displaystyle F_{\text{Luft}}\,}$	Drag force in [N]
${\displaystyle \rho _{\text{Luft}}\,}$	Air density in [kg / m³] (at sea level at 20 ° C about 1.2 kg / m³)
${\displaystyle c_{\mathrm {W} }\,}$	Drag coefficient / drag coefficient dependent on the shape of the vehicle (slightly dependent on speed), dimensionless [-]
${\displaystyle A\,}$	Projected frontal area (frontal area in silhouette) in [m²]
${\displaystyle v_{\mathrm {rel} }\,}$	Relative speed of the vehicle in [m / s] ${\displaystyle (v_{\mathrm {Fzg} }+v_{\text{Wind}})}$

Rolling resistance

• The rolling resistance is due to the deformation work of tires and road surface at the contact points. The deformation is caused by the vehicle mass in connection with the elastic properties of the road surface and the tires. The rolling resistance is mass-dependent. It can be calculated using the following formula:
  ${\displaystyle F_{\text{Roll}}=f_{\text{Roll}}\cdot (m_{\mathrm {Fzg} }+m_{\mathrm {Zu} })\cdot g\cdot \cos(\alpha )}$

${\displaystyle F_{\text{Roll}}\,}$	Rolling resistance in [N]
${\displaystyle m_{\mathrm {Fzg} }\,}$	Mass of the vehicle in [kg]
${\displaystyle m_{\mathrm {Zu} }\,}$	Mass of the vehicle's payload in [kg]
${\displaystyle g\,}$	Gravitational acceleration , g = 9.81 m / s²
${\displaystyle f_{\text{Roll}}\,}$	Rolling resistance coefficient (approximately independent of speed), dimensionless [-]
${\displaystyle \alpha \,}$	Pitch angle in rad (i.e. dimensionless) [-]

Incline resistance

• The incline resistance arises when driving on an incline. On a downhill gradient the gradient resistance is negative:
  ${\displaystyle F_{\text{Steig}}=(m_{\mathrm {Fzg} }+m_{\mathrm {Zu} })\cdot g\cdot \sin(\alpha )}$

${\displaystyle F_{\text{Steig}}\,}$	Slope resistance in [N]
${\displaystyle \alpha \qquad }$	Angle of incline in rad (ie dimensionless) [-], to be used as a negative value on downhill slopes
Note: In road traffic it is common to express inclines and declines in%. If you choose the value s for the gradient in%, the relationship and results with the auxiliary variable ${\displaystyle p={\frac {s\,\%}{100\,\%}}\,}$ ${\displaystyle \tan(\alpha )=p\,}$ ${\displaystyle F_{\text{Steig}}=m_{\mathrm {Fzg} }\cdot g\cdot {\frac {p}{\sqrt {1+p^{2}}}}}$

Acceleration resistance

• The acceleration resistance occurs when the vehicle changes speed. A deceleration is to be used as a negative acceleration :
  ${\displaystyle F_{\mathrm {B} }=\left(e_{i}\cdot m_{\mathrm {Fzg} }+m_{\mathrm {Zu} }\right)\cdot a}$

${\displaystyle F_{\mathrm {B} }\,}$	Acceleration resistance in [N]
${\displaystyle e_{i}\,}$	Mass factor (> 1) which takes into account the moments of inertia of the accelerated rotating masses in the drive train (depending on the current gear ratio), dimensionless [-]
${\displaystyle a\,}$	Acceleration of the vehicle in [m / s²]

The driving resistance force is the sum of the mentioned forces:

${\displaystyle F_{\mathrm {FW} }=F_{\text{Luft}}+F_{\text{Roll}}+F_{\text{Steig}}+F_{\mathrm {B} }\,}$

Logically, is the driving force, ie the force required for a vehicle to maintain a constant speed to his (or to speed with the acceleration a), the negative driving resistance: . The sign convention arises from the fact that the frictional forces always counteract the movement and the sign agreement for slope and acceleration were chosen to be the same. ${\displaystyle F_{\mathrm {Antrieb} }=-F_{\mathrm {FW} }}$

Required drive power

The question of the drive power that is required to reach a certain speed and what maximum speed a vehicle can reach is closely linked to the driving resistance.

The drive power results from the drive force multiplied by the speed:

${\displaystyle P_{\text{Antrieb}}=F_{\text{Antrieb}}\cdot v=F_{\text{Luft}}\cdot v+F_{\text{Roll}}\cdot v+F_{\text{Steig}}\cdot v+F_{\mathrm {B} }\cdot v}$

For the calculation of the maximum speed it is assumed that there is no more acceleration and the vehicle is moving on the plane. The maximum speed can thus be determined from the following in v cubic equation :

${\displaystyle v^{3}\cdot \left({\frac {\rho _{\text{Luft}}}{2}}\cdot c_{W}\cdot A\right)+v\cdot \left((m_{\mathrm {Fz} }+m_{\mathrm {Zu} })\cdot g\cdot f_{\text{Roll}}\right)-P_{\text{Antrieb}}=0}$

The following rule of thumb applies to a car:

${\displaystyle v_{\text{max}}=47{\text{km/h}}\cdot {\sqrt[{3}]{\frac {P}{\text{kW}}}}}$

literature

• Hans-Hermann Braess , Ulrich Seiffert : Vieweg manual automotive technology. 2nd Edition. Friedrich Vieweg & Sohn Verlagsgesellschaft, Braunschweig / Wiesbaden 2001, ISBN 3-528-13114-4 (7th edition 2013, ISBN 978-3-658-01690-6 )
• Bernd Heißing, Metin Ersoy, Stefan Gies: Chassis Manual : Basics, Driving Dynamics, Components, Systems, Mechatronics, Perspectives . Springer Vieweg, 2013. Chapter 2.1: Driving resistances and energy requirements ; springer.com (PDF; 7 MB)

See also

• Rolling resistance
• Breakaway resistance
• Frictional resistance
• Acceleration resistance

Web links

• Calculation of air and rolling resistance from simple driving tests. ArsTechnica.de
• Klaus Herzog: Energy requirements of a motor vehicle . (PDF)

Individual evidence

1. ↑ Manfred Mitschke, Henning Wallentowitz: Dynamics of motor vehicles . Ed .: Springer Vieweg, Wiesbaden. Wiesbaden 2014, ISBN 978-3-658-05068-9 , pp. 13 ff .
2. ↑ Table of Contents. (PDF) In: Chassis Manual: Fundamentals, Driving Dynamics, Components, Systems, Mechatronics, Perspectives