02/08/19

The idea of testing whole vehicles while at rest was first conceived by locomotive engineers before being adopted by the automotive industry. Last steam locomotives build in the UK were tested on multiple axle unit with large eddy-current dynamometers with rollers connected to each driven axle. The tractive force was measured by a mechanical linkage and spring balance.

Testing road vehicles with dynamometer become very popular in 1970’s when first emission regulations were introduced. Since then the small diameter rollers were replaced by larger diameter to better simulate road and inertial mass (flywheel) was substituted mostly by more sophisticated control systems.

Modern chassis dynamometers and these connected directly to the drivetrain or engine have to simulate movement of a particular vehicle. There are few main forces acting on a vehicle during longitudinal movement and equation to describe them is called road load equation (RLE). Common dynamometers used for instance for benchmarking vehicle performance use full or simplified form of the RLE equation

\[ F_{Load}=a_0+a_1 V+a_2 V^2+m \frac{dV}{dt}+mg \sin{θ_r}\]

To model movement of a vehicle the equation should take into account quadratic, linear and constant terms dependent on the vehicle speed V and if no freewheel is present (as is often the case) the full inertia have to be modeled with electromagnetic torque. The inertia force can be calculated simply from the derivative dV/dt or using known vehicle engine or motor torque (the method will be described later). All fundamental parameters of the equation are as follows

RLE Parameter | Term |
---|---|

Vehicle specific: |
- |

Mass of vehicle | \(m [kg]\) |

Component of rolling resistance | \(a_0 [N]\) |

Speed dependent resistance | \(a_1 V [N]\) |

Aerodynamic resistance | \(a_2 V^2 [N]\) |

External: |
- |

Vehicle speed | \(V [m/s]\) |

Road slope | \(θ_r [rad]\) |

The equations used in model the vehicle form a link between performance on the road and performance during the dynamometer testing. Depending on the purpose of testing the RLE can be reduced to i.e. only inertia and constant term or extended by accounting for cornering forces and tire slip.

An example of equation with parameters from the book *Engine Testing* by A. Martyr and M. Plint of typical four-door saloon of moderate performance, mass 1600 kg and combined aerodynamic drag coefficient \(C_{D,tot}=0.43\) is given to understand the magnitudes involved and to have reliable reference for future:

\[ F=150+3V+0.43V^2+1600 \frac{dV}{dt}+1600⋅9.81⋅\sin{θ_r}\]

The sample parameters data was investigated by plotting the equation derived quantities such as power (as energy demanded \(E=∫Fdx\) and power \(P=\frac{dE}{dt}\) then simply \(P=F⋅V\). The force as function of speed of each RLE component shows that at medium to higher speeds air drag resistance is much greater than other forces. While on lower speeds the other forces have greater significance.

Using the same data the plot of derived power was made. The power required to move at certain speed can be very useful for quick calculation. In example the road load equation calls for ~15 kW power at 100 km/h and up to ~69 kW at 180 km/h. Thus the power demand at higher speeds is proportionally higher at higher speeds. In case of electric machines used as source of propulsion without any gearing this may significantly narrow the speed range.

Similarly the plots with force demand during hill climbing as function of the road slope of different mass vehicles was made for quick overview of the magnitudes. Another figure demonstrates force needed during constant acceleration from 0 to 100 km/h as function of duration.

As can be concluded from the plots the force and thus the power required to accelerate a vehicle increases with its mass. If the vehicle is designed to drive in urban areas minimalizing its mass may significantly decrease requirements of the drive motor. However such light cars may be not as safe as regular ones and unable to drive with higher speeds on highways which makes them less desirable. Despite the vehicle type chosen ability to simulate the driving conditions are very useful to optimize the electric machine to decrease the costs.