For professionality in design and development of a vehicle’s **electrical powertrain** -for example on an electrical two-wheeler, a thorough understanding of some physical relationships is esssential.

Frequently, electric vehicles are mere conversions of models with conventional propulsion. In these cases, principal parameters, for example vehicle mass, drag coefficient and projected frontal area, are well known and already availible. By integration of an electric powertrain and a traction battery, for the converted vehicle new performance characteristics arise, frequently affecting acceleration behaviour, maximum speed and vehicle range.

Modelling and calculation of the converted vehicle’s maximum speed in still air on a plain level is quite easy to perform. The top speed is frequently achieved, whenever propulsive force of the vehicle and driving resistance are balanced.

Driving resistance is made up of two components: First a Rolling resistance, induced by the rolling of the wheels on the road, and second an air drag, induced by friction between the moving vehicle’s surface and the surrounding air atmosphere.

The rolling resistance, F.RO, is calculated as product of a rolling resistance coefficient f, the vehicle’s mass m and gravity factor g as follows:

F.RO = f*m*g

Typical values for rolling resistance coefficients for well-filled air tires are between 0.015 on asphalt and 0.05 on dirt roads.

The air drag F.LU is calculated by use of the drag coefficient of the vehicle cW, the vehicle’s projected frontal area A, the density of the surrounding air rho and the vehicle’s speed v as follows:

F.LU = 0.5*Rho*cW*A*V^2

Typical values for cW can be found between 1.2 (city bike) and 0.8 (racing bicycle). Air density on sea level and at 20 degree celsius might be accounted with 1.2kg/m³. Typical projected frontal areas might be applicable between 0.4m² (racing bicycle) and 0.6m² (city bike).

As mentioned before, top speed is achieved whenever the vehicle’s propulsion force and its driving resistance are in balance.

The required propulsion force for a given top speed v therefore can be calculated as a sum of air drag and rolling resistance:

F_Propulsion=F.LU+F.RO

More commmonly in vehicle modelling is an approach via propulsion power. The vehicle’s propulsion power P_Propulsion is calculated as product of propulsion force F_Propulsion and vehicle speed v.

P_Propulsion = F.Propulsion*v

**An Example:**

Our protagonists are: Max (45 years old, 84kg) and his city bike (6 weeks old, 18kg). They would like to travel a 20 km distance on the dyke between Hoernum and Westerland within an hour. Both of them account for a cW of 1.2, a projected frontal area of 0.6m² and a cumulated weight of 102kg. The rolling resistance coefficient on the dyke road is estimated by 0.05, and there is no wind at all.

Q: How do they have to perform?

The calculation is perfomed using SI units- 20km distance within one hour comes up with an average speed of 5.6m/s

Therefore, our rolling resistance comes up with 102kg * 9,81 kg*m/s² * 0,05 = 50N

Air drag all the same with 0.5*1.2kg/m³*1,2*0.6m²*(5,6m/s)^2= 13,5N

For this, the cumulated driving resistance adds up to 13.5+50=63,5N

Now, the propulsion power can be calculated as a product of driving resistance and vehicle speed:

P_Propulsion = 63,5N*5,6m/s = 355,6 N*m/s = 355,60 Watts.

As Max is only a hobby bicyclist without noteworthy training, he contributes only 150 Watts with muscle power. Remaining 205,60 Watt have to be contributed by his electric bicycle for both of them being in time.

## Crash Course: Vehicle Physics

For professionality in design and development of a vehicle’s

electrical powertrain-for example on an electrical two-wheeler, a thorough understanding of some physical relationships is esssential.Frequently, electric vehicles are mere conversions of models with conventional propulsion. In these cases, principal parameters, for example vehicle mass, drag coefficient and projected frontal area, are well known and already availible. By integration of an electric powertrain and a traction battery, for the converted vehicle new performance characteristics arise, frequently affecting acceleration behaviour, maximum speed and vehicle range.

Modelling and calculation of the converted vehicle’s maximum speed in still air on a plain level is quite easy to perform. The top speed is frequently achieved, whenever propulsive force of the vehicle and driving resistance are balanced.

Driving resistance is made up of two components: First a Rolling resistance, induced by the rolling of the wheels on the road, and second an air drag, induced by friction between the moving vehicle’s surface and the surrounding air atmosphere.

The rolling resistance, F.RO, is calculated as product of a rolling resistance coefficient f, the vehicle’s mass m and gravity factor g as follows:

F.RO = f*m*g

Typical values for rolling resistance coefficients for well-filled air tires are between 0.015 on asphalt and 0.05 on dirt roads.

The air drag F.LU is calculated by use of the drag coefficient of the vehicle cW, the vehicle’s projected frontal area A, the density of the surrounding air rho and the vehicle’s speed v as follows:

F.LU = 0.5*Rho*cW*A*V^2

Typical values for cW can be found between 1.2 (city bike) and 0.8 (racing bicycle). Air density on sea level and at 20 degree celsius might be accounted with 1.2kg/m³. Typical projected frontal areas might be applicable between 0.4m² (racing bicycle) and 0.6m² (city bike).

As mentioned before, top speed is achieved whenever the vehicle’s propulsion force and its driving resistance are in balance.

The required propulsion force for a given top speed v therefore can be calculated as a sum of air drag and rolling resistance:

F_Propulsion=F.LU+F.RO

More commmonly in vehicle modelling is an approach via propulsion power. The vehicle’s propulsion power P_Propulsion is calculated as product of propulsion force F_Propulsion and vehicle speed v.

P_Propulsion = F.Propulsion*v

An Example:Our protagonists are: Max (45 years old, 84kg) and his city bike (6 weeks old, 18kg). They would like to travel a 20 km distance on the dyke between Hoernum and Westerland within an hour. Both of them account for a cW of 1.2, a projected frontal area of 0.6m² and a cumulated weight of 102kg. The rolling resistance coefficient on the dyke road is estimated by 0.05, and there is no wind at all.

Q: How do they have to perform?

The calculation is perfomed using SI units- 20km distance within one hour comes up with an average speed of 5.6m/s

Therefore, our rolling resistance comes up with 102kg * 9,81 kg*m/s² * 0,05 = 50N

Air drag all the same with 0.5*1.2kg/m³*1,2*0.6m²*(5,6m/s)^2= 13,5N

For this, the cumulated driving resistance adds up to 13.5+50=63,5N

Now, the propulsion power can be calculated as a product of driving resistance and vehicle speed:

P_Propulsion = 63,5N*5,6m/s = 355,6 N*m/s = 355,60 Watts.

As Max is only a hobby bicyclist without noteworthy training, he contributes only 150 Watts with muscle power. Remaining 205,60 Watt have to be contributed by his electric bicycle for both of them being in time.