# Modeling and Optimal Shift Control of a Planetary Two-Speed Transmission

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## Abstract

**:**

## 1. Introduction

## 2. Statement of Planetary Two-Speed Transmission

#### 2.1. Planetary Gear Sets Modeling

## 3. Shifting Actuator

#### 3.1. The BLDC Motor Used for Shifting

#### 3.2. Coupling Model

#### 3.3. Worm Gearing with Screw Surface Model

#### 3.4. Modeling of Multi-Disc Wet Brakes

## 4. Optimal Trajectories in the Gear-Shifting Process

#### 4.1. Problem Statement of Gear-Shifting

#### 4.2. Optimal Trajectory Control for Brakes

#### 4.2.1. The B1 Off-Going Stage

#### 4.2.2. The Gap Stage

#### 4.2.3. The B2 On-Coming Stage

## 5. Simulation Results and Discussions

#### 5.1. Simulation Results Following NEDC Drive Cycle

#### 5.2. Verification for Optimal Control Trajectory

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

i_{0} | final ratio |

i_{2} | 1st gear ratio |

i_{2} | 2nd gear ratio |

η_{1} | efficiency of 1st gear |

η_{2} | efficiency of 2nd gear |

η_{0} | efficiency of reducer |

J_{1} | rotational inertia of input shaft (kg m^{2}) |

J_{2} | rotational inertia of output shaft (kg m^{2}) |

J_{C} | inertial of brakes (kg m^{2}) |

J_{m} | inertia of BLDC motor shaft (kg m^{2}) |

T_{CAX} | maximum friction torque of brake (Nm) |

Cds | friction coefficient of driving shaft |

Kds | stiffness of driving shaft (Nm/°) |

Cs | damp coefficient of driving shaft (Nm/(r/min)) |

M | mass of EV (kg) |

ρ | air density (Kg/m^{3}) |

A | area of windward (m^{2}) |

CD | coefficient of air resistance |

F | coefficient of rolling resistance |

R | radius of wheel (m) |

α | slope (°) |

g | acceleration of gravity (m/s^{2}) |

J_{0} | inertia of electronic actuator (kg m^{2}) |

J_{1} | inertial of powertrain system (kg m^{2}) |

J_{2} | inertial of wheel (kg m^{2}) |

ψ_{fl} | flux of driving motor (Wb) |

R_{s} | stator resistance (Ω) |

L_{s} | inductance of driving motor (H) |

L_{sd} | inductance of d axis (H) |

L_{sq} | inductance of q axis (H) |

I_{max} | maximum effective current (A) |

U_{DC} | direct voltage (V) |

P | pole pairs |

R_{0} | the radius of external diameter (m) |

R_{1} | the radius of inner diameter (m) |

N | the number of plates |

μ_{s} | the coefficient of kinetic friction |

μ_{d} | the maximum coefficient of static friction |

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**Figure 1.**Schematic diagram of the planetary two-speed transmission. 1-driving motor; 2-sun gear in planetary system 1(S1); 3-carrier in planetary system 1 (C1); 4-brake 1(B1); 5-ring gear in planetary system 1 (R1); 6-worm shaft; 7-worm gear; 8-ring gear in planetary system 2 (R2); 9-carrier in planetary system 2 (C2); 10-sun gear in planetary system 2(S2); 11-brake 2(B2).

**Figure 4.**Ideal curve. (

**a**) States of the driving motor; (

**b**) States of the brushless DC (BLDC) motor.

**Figure 8.**Curves of gear number and angular velocity in the new European driving cycle (NEDC) driving cycle.

**Figure 9.**Performances during down-shifting. (

**a**) Optimal rotary angle during down-shifting; (

**b**) Jerk with different controller during down-shifting.

**Figure 11.**Test results of the rotary angle of the BLDC motor. (

**a**) Rotary angle; (

**b**) Error of rotary angle.

Component Name | Module | Number of Teeth | Pitch Diameter/mm | Mass/kg | Inertia/kg.m^{2} |
---|---|---|---|---|---|

P1 Sun gear s1 | 2 | 32 | 64 | / | 3.03 × 10^{−4} |

P1 Planet gear p1 | 2 | 22 | 44 | 0.118 | 4.03 × 10^{−4} |

P1 Ring gear r1 | 2 | 76 | 152 | / | 0.01827 |

P1 Carrier c1 | 2 | / | / | / | 5.538 × 10^{−6} |

P2 Sun gear s2 | 2 | 22 | 44 | / | 1.2 × 10^{−4} |

P2 planet gear p2 | 2 | 27 | 54 | 0.28 | 1.5 × 10^{−4} |

P2 Ring gear r2 | 2 | 76 | 158 | / | 0.01515 |

P2 Carrier c2 | 2 | / | / | / | 7.207 × 10^{−6} |

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**MDPI and ACS Style**

Zhao, X.; Tang, J.
Modeling and Optimal Shift Control of a Planetary Two-Speed Transmission. *World Electr. Veh. J.* **2019**, *10*, 53.
https://doi.org/10.3390/wevj10030053

**AMA Style**

Zhao X, Tang J.
Modeling and Optimal Shift Control of a Planetary Two-Speed Transmission. *World Electric Vehicle Journal*. 2019; 10(3):53.
https://doi.org/10.3390/wevj10030053

**Chicago/Turabian Style**

Zhao, Xinxin, and Jing Tang.
2019. "Modeling and Optimal Shift Control of a Planetary Two-Speed Transmission" *World Electric Vehicle Journal* 10, no. 3: 53.
https://doi.org/10.3390/wevj10030053