Reducer modeling for electric vehicles considering driving condition

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[Power transmission of electric vehicle]

The power of an electric vehicle starts from a motor. The power is transmitted from the motor to the reducer and is converted to the appropriate torque and rotational speed to be transmitted to the wheel. Conversely, the driving resistance generated from the wheels is transmitted to the motor. <Figure 1>

Motor 그림1.jpg

<Figure 1>

Driving resistance is the resistance that occurs when driving a vehicle. If the load is increased due to the objects loaded in the vehicle, or if you are driving on uneven ground or on a steep road, the driving resistance will increase. Conversely, if the vehicle's load is small, or if you are driving on a well-paved road or downhill, the driving resistance will decrease. Even if the vehicle is running at the same speed, if the driving resistance is large, a larger output is required for the motor. If the driving resistance is small, the desired speed can be achieved with a relatively small output.

[Reducer dynamics model]

그림 2.jpg

<Figure 2>

The constraints for creating a dynamic model of the reducer shown in <Figure 2> are simple. Rotating joints are used for shafts, and the contact between gears can be defined. The reducer boundary condition (motor input/output) can be entered simply by using speed or torque. For example, the reducer input is the power transmitted from the motor, which is defined as the rotational speed or torque. The output of the reducer can be defined by the rotational speed of wheel or the resistance torque. Therefore, in order to obtain the desired result through simulation, it is necessary to define appropriate boundary conditions. The question is how to calculate the speed or torque to be used as a boundary condition. It would be ideal if there was a measured value through the test, but if not, it is necessary to calculate and use the input and output values to be used as the boundary condition of the reducer.

[Calculation of boundary condition]

Here is how to calculate the boundary condition using a simple problem considering situation.

<Conditions>

Weight of empty vehicle: 1530Kg
Motor max power: 100Kw
Motor max torque: 295Nm
Gear ratio (Motor to Wheel): 7.68
Outer diameter of wheel: 627mm

<Question>

What is the boundary condition for the input and output of the reducer when a vehicle is traveling at a constant speed of 50kph on an asphalt road with a 30% slope?

<Answer>

a. Input boundary condition (rotation speed of motor)

If the rotation speed of motor is used as input condition of the reducer, the motor’s rotation speed can be calculated from the wheel speed and gear ratio.

If the speed of the vehicle is 50kph and the outer diameter of the wheel is 627mm, the rotation speed of the wheel can be calculated as below. (Eq. 1)

The motor’s rotation speed can be calculated using the wheel rotation speed and the gear ratio. (Eq. 2)

b. Output boundary condition (driving resistance torque)

When using resistance torque as the output condition of the reducer, it can be calculated as follows.

In order to calculate the driving resistance torque, the first thing you need to know is what the driving resistance is. It is possible to calculate ‘Driving Resistance Power’ using the driving resistance and the speed of vehicle and calculate the resistance torque applied to the wheel using the driving resistance power and the rotational speed of the wheel.

(Driving resistance)

There are four main driving resistances. Rolling resistance is caused by contact between the tire and the ground and grade resistance is caused by slope. Acceleration resistance is caused by vehicle’s acceleration and air resistance is caused by air. In this problem, it is assumed that there is no air resistance and since it is a constant-speed driving, the acceleration resistance is not considered.

1. Rolling resistance

Rolling resistance (Rr) is defined as the multiplication of vehicle weight (W) and rolling resistance coefficient (μ). (Eq. 4)

The rolling resistance coefficient is defined as shown in <Table 1> depending on the condition of the road surface.

<Table 1> Rolling resistance coefficient depending on various road surface conditions

The rolling resistance (Rr) is calculated by considering the above conditions. (Eq. 5)

2. Grade resistance

그림3.jpg

<Figure 3>

In <Figure 3>, if θ is not that large, it can be considered as sin⁡ θ≅tan⁡θ, and the slope tan⁡θ of the road is expressed as a percentage (%) of the ratio of height (m) to horizontal distance L (m). In this case, the horizontal distance is based on L=100(m). Therefore, the grade resistance (Rs) can be calculated as Eq. 6.

Therefore, the grade resistance at a 30% inclination angle is calculated as Eq. 7.

3. Total driving resistance

The total driving resistance (R) is the sum of the four main driving resistances described above. Since the air resistance and acceleration resistance are not considered in this issue, the total driving resistance can be the sum of rolling resistance and grade resistance. (Eq. 9)

(Driving resistance power)

The driving resistance power (NR, hp) can be calculated using the vehicle's constant speed (V, km/h), driving resistance (R, kg), and mechanical efficiency (ηt). (Eq. 10)

Assuming that the mechanical efficiency is η_t=1, and using the previously calculated driving resistance result (Eq. 9), the driving resistance power is as follows. (Eq. 11)

To convert the unit to Watt (W), use the calculation of Eq. 12. (1hp = 0.75kW)

(Driving resistance torque)

The relationship between power and torque can be expressed as Eq. 13.

Torque can be calculated as below. (Eq. 14)

The rotation speed of the wheel can be calculated by using the vehicle's driving speed and wheel size. (Eq. 15)

Therefore, the calculation of resistance torque is as follows by substituting Eq. 12 and Eq. 15 into Eq. 14. (Eq. 16)

[Reducer modeling using RecurDyn]

그림 4.png

<Figure 4> Reducer model using RecurDyn

The model in <Figure 4> is created using RecurDyn's DriveTrain toolkit. Composed of gears, shafts and bearings, the DriveTrain toolkit provides an easy interface to create each element. In the case of shaft, it can be analyzed considering the axial deformation since it is a flexible body composed of beam elements. In the case of bearing, you can select and use the desired bearings from various bearing libraries, and customized bearings can also be created by entering shape information. Gears can also be easily generated using shape information, and contact optimized for gears allows quick and accurate calculations.

Set the boundary condition previously calculated to the created model as shown in <Table 2>.

<Table 2> Calculation result

3249.02 RPM, which is the result of the calculation in Eq. 2 and the rotation speed of the motor, is used as the input of the reducer. The calculation result of Eq. 16, 1486.72 Nm, is used as the resistance torque of the reducer’s output.

Since the motor output required for driving condition is the same as the driving resistance power, the input and output powers are the same when assuming there is no output loss. Therefore, the torque acting on the motor can also be calculated using Eq. 14. However, when entering the boundary conditions of input and output, you can select one of these two values; speed or torque. When inputting the speed, the torque result is obtained by reaction of the movement. When inputting the torque, the speed can be obtained as a result.

[Simulation result]

In this example, the motor input is set to speed and the output is set to resistive torque. The torque acting on the motor can be checked in the following image of ‘Gear pair1, Drive gear Torque’ and the results are similar to those of the previous calculation in Table 2. In addition, the output speed can be checked in the image of ‘Gear pair2, Driven gear Speed’, and similarly, the calculation results are close to those in Table 2.

This simulation model can check the precise transmission errors in micrometers by contact calculation using Meta Model, which is one of the functions of the DriveTrain toolkit. Since a Meta Model calculates the stiffness of a gear in advance and then calculates the contact using it, fast and accurate results can be obtained. Even if an iterative analysis such as changing boundary conditions is required on the same system, the pre-computed Meta Model can be reused for quick results.

gear pair1.jpg