Investigating the Effect of Gear Ratio in the Case of Joint Multi-Objective Optimization of Electric Motor and Gearbox

By inergency On Mar 3, 2024

The main cost function performs the detailed evaluation of the objective functions that were not evaluated during initialization. These are typically quantities that require longer FEM simulations, in this case, the loss energy of the motor. The inputs to this module are the vector v and the parameters associated with the operating points. Since the optimization usually involves several operating points, the internal structure is iterative. The boxes in gray are run for each operating point separately.

Since the inputs to the motor model are rotational speed and excitation current amplitude and the working point parameters are rotational speed and expected torque, a current calculation must first be made. In constant speed transient simulations with current excitation, the torque output is proportional to the amplitude of the excitation, so the amplitude value required for the working point can be easily determined from the results obtained when checking the constraints using the (4) relationship.

$I = \frac{I_{m a x} \cdot M_{x}}{M_{m r} \cdot i},$

(4)

where I is the excepted current amplitude, $I_{m a x}$ is the current amplitude used in the reduced motor model, $M_{x}$ is the expected torque at the working point, $M_{m r}$ is torque calculated by the reduced motor model, and i is the gearbox ratio.

The motor model then receives as input the vector of design variables, the vector of implicit parameters, the current amplitude associated with the operating point, and the rotational speed associated with the operating point.

The power losses (P) calculated here therefore only apply to the current operating point. After multiplication by time weighting, they become loss energy. The sum of the loss energies at the operating points (E—marked in gray) will be the objective function value calculated for the individual (E).

2.4.2. The Gearbox Model

The optimization framework uses a single-speed plastic gearbox model, but this can easily be replaced by another model.

Small gears, which are usually subjected to high loads, cannot be manufactured from engineering plastics, because their strength is significantly lower than that of stainless steel gears. For this reason, they are mostly used in applications where their load capacity is still adequate and where their other advantages may be important. Such properties are, e.g., low weight, corrosion resistance, wear resistance, quiet running, economical production, etc. Therefore, the possibility of their application in small electric vehicles based on these properties was investigated. It is worth noting that plastic gears prefer a steady-state mode around a given working point, which is not typical for a vehicle application, as vehicles often accelerate and decelerate, and in the case of modern electric vehicles they can also recuperate, which also involves the transfer of motion and energy through the gearbox, but as these loads are generally lower than the maximum permissible load, they are also well suited to this application (see [20,21]).

The inputs to this model are the design variables, gear ratio, motor torque, and motor rotational speed.

Plastic gears are usually designed for the permissible power rating (

P_{m a x}

) based on the Lewis formula (see [22]). The formula can be given in the (7) form.

$P_{m a x} = \frac{m \cdot y \cdot b \cdot d \cdot n \cdot f_{1} \cdot f_{2} \cdot σ}{6 \cdot 10^{3}},$

(7)

where m is module, y is teeth factor, b is the load-bearing tooth width, d is pitch circle (provided by the gear’s wheelbase and the gear ratio), n is rotational speed, $f_{1}$ is speed factor, $f_{2}$ is operating factor, and $σ$ is bending stress in the gear tooth.

From relation (7), we can extract the minimum value of two parameters, among the parameters that are important for the optimization of the gearbox; one is the module and the other is the tooth width. Since the effect of the tooth width is small from an energy point of view, it is not worth choosing it as the main parameter. Besides the module, it is the gear ratio that has a major impact on the efficiency of the gearbox, typically such that the lower the gear ratio, the better the efficiency. This implies that drive systems with lower ratios will typically have lower loss energy of the gearbox.

Accurate determination of gear losses is a complex task, requiring in-depth knowledge of structural, mechanical, geometrical, tribological, and thermal properties. Most of the losses are due to friction of the associated gear teeth, but bearing friction can also be significant. For high-speed gearboxes, losses due to oil mixing, seal friction, and air turbulence must also be taken into account. At low speeds and a low load torque, sliding friction losses dominate gearbox power losses, while at higher speeds gear rolling losses and bearing losses become more significant. In addition, it can be concluded that, at high speed operation, the windage loss of the gearbox also increases significantly (see [23]). The total power loss of the gearbox can be described by (8).

$P_{l} = P_{z} + P_{r} + P_{s} + P_{l} + P_{w} + P_{b},$

(8)

where $P_{l}$ is the loss power of the gearbox, $P_{z}$ is the tooth friction loss, $P_{r}$ is the rolling resistance loss, $P_{s}$ is the seal friction loss, $P_{l}$ is the oil friction loss, $P_{w}$ is the windage loss, and $P_{b}$ is the losses from the bearings.

Tooth friction loss is a load loss due to friction in the tooth contact. To calculate this, the coefficient of tooth friction is needed, which is determined empirically or by measurement. Determining the coefficient of tooth friction is not simple, as it is influenced by the properties of the lubricant, the lubrication method, the material and design of the gears, the surface condition, surface roughness, operating conditions, load, speed, and temperature. The power loss due to tooth friction can be described by relation (9) (see [24]).

$P_{z} = P_{i n} \cdot μ_{m} \cdot H_{v},$

(9)

where $P_{i n}$ is the input power, $μ_{m}$ is the constant coefficient of friction, and $H_{v}$ is the gear loss factor.

The rolling loss of the gear is usually negligible among the losses, so it is neglected in the model used.

The calculation of the seal friction loss can be properly calculated by knowing the seal design of the gearbox, but since this is not known at the pre-design stage, an estimator (10) is used in the model to describe this loss shape (see [25]).

$P_{s} = 7.69 \cdot 10^{- 6} \cdot d_{a} \cdot n_{a},$

(10)

where $d_{a}$ is the diameter of the axis and $n_{a}$ is the rotational speed.

Oil friction loss occurs when gears are running in oil. At low speeds, the rotating gears at the bottom of the housing, which are in contact with the oil, are subjected to shearing stresses on the fluid. At higher speeds, the centrifugal force forces the oil against the walls of the housing, changing the depth of gear meshing and hence the lubrication quality, and causing fluid flow losses in the gearbox. Since the gearbox models used in this paper are grease lubricated, this type of loss does not occur in the present application.

The windage loss is typical at high speeds and is completely independent of the transmitted load. The main influences on this form of loss are the geometry of the gears, the viscosity of the lubricant, and the peripheral speed of the gears. The loss of power due to air swirl can be described by relationship (11) (see [26]).

$P_{w} = P_{w 1} + P_{w 2},$

(11)

where $P_{w 1}$ is the windage loss of the driving gear and $P_{w 2}$ is the windage loss of the driven gear. These can be calculated from relationships (12) and (13).

$P_{w 1} = 1.16 \cdot 10^{- 8} \cdot (1 + 4.6 \cdot \frac{b}{d_{1}}) \cdot n_{1}^{2.8} \cdot d_{1}^{4.6} \cdot {(0.028 \cdot μ + 0.019)}^{0.2},$

(12)

$P_{w 2} = 1.16 \cdot 10^{- 8} \cdot (1 + 4.6 \cdot \frac{b}{d_{2}}) \cdot {(n_{2} \cdot i)}^{2.8} \cdot d_{2}^{4.6} \cdot {(0.028 \cdot μ + 0.019)}^{0.2},$

(13)

where b is the width of the gear, n is the rotational speed of the gear, i is the gear ratio, d is the diameter of the pitch circle, and $μ$ is the dynamical viscosity.

Rolling bearings are usually used in the gearboxes of electric vehicles. They are less sensitive to continuous lubrication failure and have good efficiency. A wide range of relationships are used to calculate bearing friction. The calculation method for the model used in this paper, defined by SKF, can be described by relation (14) (see [27]).

$\begin{matrix} P_{b} = ω \cdot 0.001 \cdot (Φ_{i s h} \cdot Φ_{r s} \cdot M_{r r} + M_{s l} + M_{s e a l} + M_{d r a g}), \end{matrix}$

(14)

where $ω$ is the angular velocity of the bearing, $Φ_{i s h}$ is the inlet shear heating reduction factor, $Φ_{r s}$ is the kinematic replenishment/starvation reduction factor, $M_{r r}$ is the rolling friction torque, $M_{s l}$ is the sliding friction torque, $M_{s e a l}$ is the seal friction torque, and $M_{d r a g}$ is the drag losses torque.