# Active Torque Control for Speed Ripple Elimination: A Mechanical Perspective

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## Abstract

**:**

## 1. Introduction

#### 1.1. Problem Statement and Literature Review

#### 1.2. Aim and Scope

#### 1.3. Paper Structure

## 2. Modeling and Control

#### 2.1. Mechanical Drivetrain Model

#### 2.1.1. Rigid Drivetrain

#### 2.1.2. Elastic Drivetrain

#### 2.2. Electrical PMSM Model

#### 2.3. Field-Oriented Control

#### 2.4. Resonant Control

## 3. The Resonant Controller: Simulation and Experimental Validation

#### 3.1. The Experimental Drivetrain

#### 3.2. Controller Stability and Design

#### 3.3. Simulation Results

#### 3.4. Experimental Results

## 4. The Inherent Drivetrain Factors Affecting Active Control

#### 4.1. The Inertial Attenuation

#### 4.2. The Elastic Filter

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Station | Inertia of … | ${\mathit{J}}_{\mathit{i}}$$(\mathbf{k}\mathbf{g}{\mathbf{m}\mathbf{m}}^{2}$) | Connection | ${\mathit{k}}_{\mathit{i}\mathit{j}}$$\left(\mathbf{k}\mathbf{N}\mathbf{m}\mathbf{r}\mathbf{a}{\mathbf{d}}^{-1}\right)$ |
---|---|---|---|---|

1 | Load Motor | 3749 | 1–2 | 18.4 |

2 | Coupling | 160 | 2–3 | 5.5 |

3 | Transducer | 280 | 3–4 | 17.1 |

4 | Coupling | 167 | 4–5 | 5.5 |

5 | Coupling | 160 | 5–6 | 11.7 |

6 | Gear | 7216 | 6–7 | 261.9 |

7 | Rotor | 7060 | 7–8 | 27.6 |

8 | Coupling | 160 | 8–9 | 5.5 |

9 | Coupling | 167 | 9–10 | 17.1 |

10 | Transducer | 280 | 10–11 | 5.5 |

11 | Coupling | 160 | 11–12 | 24.6 |

12 | Drive Motor | 632 |

Parameter | $\mathbf{V}\mathbf{a}\mathbf{l}\mathbf{u}\mathbf{e}$ |
---|---|

$B\left(\mathrm{N}\mathrm{m}\mathrm{s}\mathrm{r}\mathrm{a}{\mathrm{d}}^{-1}\right)$ | 0.013 |

${J}_{t}\left(\mathrm{k}\mathrm{g}{\mathrm{m}}^{2}\right)$ | 0.02 |

${\tau}_{m}\left(\mathrm{s}\right)$ | 1.55 |

Station | ${\mathit{J}}_{\mathit{i}}$$(\mathbf{k}\mathbf{g}{\mathbf{m}\mathbf{m}}^{2}$) | Connection | $\mathbf{Stiffness}(\mathbf{k}\mathbf{N}\mathbf{m}/\mathbf{r}\mathbf{a}\mathbf{d})$ | $\mathbf{Damping}(\mathbf{m}\mathbf{N}\mathbf{m}\mathbf{s}/\mathbf{r}\mathbf{a}\mathbf{d})$ |
---|---|---|---|---|

1 | 4160 | 1–2 | 1.78 | 99 |

2 | 15,013 | 2–3 | 2.01 | 55 |

3 | 1019 |

Parameter | $\mathbf{P}{\mathbf{I}}_{\mathit{c},\mathit{d}}$ | $\mathbf{P}{\mathbf{I}}_{\mathit{c},\mathit{q}}$ | $\mathbf{P}{\mathbf{I}}_{\mathit{s}}$ |
---|---|---|---|

Cut-off frequency (Hz) | 200 | 200 | 10 |

Time constant ${\tau}_{I}$ (ms) | 8.48 | 11.21 | 1.55 |

Gain ${K}_{p}$ | 3.36 | 4.44 | 1.27 |

Parameter | $\mathbf{P}{\mathbf{I}}_{\mathit{c},\mathit{d}}$ | $\mathbf{P}{\mathbf{I}}_{\mathit{c},\mathit{q}}$ |
---|---|---|

Cut-off frequency (Hz) | 200 | 200 |

$\mathrm{Time}\mathrm{constant}{\tau}_{I}$(ms) | 15.13 | 39.87 |

$\mathrm{Gain}{K}_{p}$ | 2.89 | 7.62 |

Parameter | Drive Motor | Load Motor |
---|---|---|

Switching frequency (kHz) | 4 | 4 |

Sampling frequency (kHz) | 20 | 20 |

Inverter time constant (ms) | 0.3 | 0.3 |

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**Figure 1.**A 3-DOF reduced mechanical drivetrain model. ${J}_{1}$, ${J}_{2}$, and ${J}_{3}$ represent the load machine, central shaft (representing a lumped set of shaft elements), and drive motor, respectively. The spring ${k}_{ij}$ and damper ${c}_{ij}$ elements represent couplings that interconnect elements i and j. ${T}_{d}$ and ${T}_{l}$ represent the drive and load torques, respectively. ${T}_{ij}$ is the transmitted torque over the couplings. The arrows show in which direction the (positive) torque acts.

**Figure 3.**Block diagram of the field-oriented control strategy with the addition of a resonant controller in the parallel path of the PI-speed controller.

**Figure 4.**The drivetrain test setup is used for the experimental validation of the models and associated simulations.

**Figure 5.**The 12-DOF mechanical drivetrain model with inertia–spring–damper elements. The 12-DOF model is only used for determining the mode shapes, where damping elements are not included by definition.

**Figure 6.**Mode shapes for the elastic drivetrain model. (

**a**) 12-DOF model, with the first and second resonant frequencies at 114 Hz and 235 Hz, respectively; (

**b**) reduced 3-DOF model, with the first and second resonant frequencies at 117 Hz and 232 Hz, respectively.

**Figure 7.**Color plots were obtained through torsional vibration analysis, visualizing the Fourier amplitude spectrum of the measured torque signal for various rotational speeds of the drivetrain. (

**a**) Load-side torque transducer. (

**b**) Drive-side torque transducer.

**Figure 8.**Bode Diagrams of the open loop control system. The reference values of the resonant controller used are ${K}_{res}=10$, ${f}_{0}=5\mathrm{H}\mathrm{z}$ and ${f}_{c}=0.5\mathrm{H}\mathrm{z}$. (

**a**) Sensitivity of ${K}_{res}$, (

**b**) sensitivity of ${f}_{0}$, and (

**c**) sensitivity of ${f}_{c}$.

**Figure 9.**Bode diagrams of the studied system with resonant control parameters: ${K}_{res}=10$, ${f}_{0}=5\mathrm{H}\mathrm{z}$ and ${f}_{c}=0.5\mathrm{H}\mathrm{z}$. (

**a**) Open-loop bode diagram with gain and phase margins; (

**b**) disturbance rejection closed-loop bode diagram.

**Figure 10.**Simulated speed of the drive motor is subject to a load torque of multiple frequencies (3, 5, and 15 Hz). The load torque is 3 Nm, with an additional sinewave ripple of 3 Nm. A quasi-resonant controller is switched on at 5 s. Two enlarged figures are provided: one to show the decreasing ripple for increasing frequencies before the use of the quasi-resonant control (3–4 s) and another to show the (relative) effect of the quasi-resonant control (8.5–9 s).

**Figure 11.**Simulated shaft torques of the 3-DOF model of the drivetrain, where the load torque is 3 Nm with an additional sinewave ripple of 3 Nm. The quasi-resonant controller is switched on at 5 s. The sinewave load torque ripple is (

**a**) 1 Hz and (

**b**) 5 Hz.

**Figure 12.**Experimental results. The reference sinewave load torque has an offset of 3 Nm and an amplitude of 3 Nm, and the frequency is speed-dependent and set to $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$ of the actual speed. The quasi-resonant controller is set to mitigate the speed ripple at 1000 rpm and turned on at 2.5 s. The speed reference goes in steps from 500 to 1000 to 1500 rpm, while the load torque frequencies go from 1.67 to 3.33 to 5 Hz, respectively. (

**a**) The measured speed; (

**b**) the measured shaft torques when the quasi-resonant controller is turned on; (

**c**) the measured shaft torques in a steady state at different speeds.

**Figure 13.**Comparison between simulation and experimental load- and drive-side torques of the quasi-resonant controller at 5 Hz load torque ripple frequency.

**Figure 14.**Bode plot of a rigid drivetrain for decreasing total inertias. The biggest inertia (${J}_{t}=0.02\mathrm{k}\mathrm{g}{\mathrm{m}}^{2})$ is the actual experimental setup inertia, the smallest inertia (${J}_{t}=0.005\mathrm{k}\mathrm{g}{\mathrm{m}}^{2}$) represents the setup inertia without the gear present.

**Figure 15.**Forced response torsional analysis of the 3-DOF model of the drivetrain. The forced torque of 1 Nm is applied by the drive motor. (

**a**) The absolute shaft torques both at the drive and load side and (

**b**) the ratio between drive- and load-side torques.

**Figure 16.**Forced response torsional analysis of the 3-DOF model of the drivetrain. The forced torque of 1 Nm is applied by the load-side motor. (

**a**) The absolute shaft torques both at the drive and load sides and (

**b**) the ratio between load- and drive-side torques.

**Figure 17.**Ratio between load-side and drive-side shaft torque for a forced response with a torque ripple applied on the drive motor. In blue, with the central shaft, as in Figure 15, is the 3-DOF model representing the experimental setup. In black, without a central shaft, for the direct-drive version of the experimental test setup modeled with a 2-DOF system.

**Figure 18.**Simulated speed of the drive motor in the stiff drivetrain, subject to a load torque of multiple frequencies (15, 25, and 50 Hz). The load torque is 3 Nm, with an additional sinewave ripple amplitude of 3 Nm. A quasi-resonant controller is switched on at 5 s. Two enlarged figures are provided: one to show the decreasing ripple for increasing frequencies before the use of the quasi-resonant control (3–3.12 s) and another to show the (relative) effect of the quasi-resonant control (8.8–8.92 s).

Parameter | Unit | Drive Side | Load Side |
---|---|---|---|

Nominal power | kW | 7.9 | 15 |

Nominal shaft torque | Nm | 8.9 | 24 |

Nominal speed | rpm | 8500 | 6000 |

Nominal RMS phase current | A | 15.5 | 27 |

Nominal RMS line voltage | V | 399 | 372 |

Pole pair number | / | 5 | 2 |

Stator phase resistance | $\mathsf{\Omega}$ | 0.315 | 0.152 |

D-axis inductance | mH | 2.67 | 2.30 |

Q-axis inductance | mH | 3.53 | 6.06 |

PM flux linkage | Wb | 0.0514 | 0.158 |

Station | ${\mathit{J}}_{\mathit{i}}$$(\mathbf{k}\mathbf{g}{\mathbf{m}\mathbf{m}}^{2}$) | Connection | ${\mathit{k}}_{\mathit{i}\mathit{j}}$$(\mathbf{k}\mathbf{N}\mathbf{m}/\mathbf{r}\mathbf{a}\mathbf{d})$ | ${\mathit{c}}_{\mathit{i}\mathit{j}}$$(\mathbf{m}\mathbf{N}\mathbf{m}\mathbf{s}/\mathbf{r}\mathbf{a}\mathbf{d})$ |
---|---|---|---|---|

1 | 4160 | 1–2 | 1.78 | 99 |

2 | 15.013 | 2–3 | 2.01 | 55 |

3 | 1019 |

Parameter | $\mathbf{V}\mathbf{a}\mathbf{l}\mathbf{u}\mathbf{e}$ |
---|---|

${K}_{res}$ | 10 |

${f}_{0}$ | Equal to load torque frequency |

${f}_{c}$ | 0.5 Hz |

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## Share and Cite

**MDPI and ACS Style**

Croonen, J.; Deraes, A.L.J.; Beckers, J.; Devesse, W.; Hegazy, O.; Verrelst, B.
Active Torque Control for Speed Ripple Elimination: A Mechanical Perspective. *Machines* **2024**, *12*, 222.
https://doi.org/10.3390/machines12040222

**AMA Style**

Croonen J, Deraes ALJ, Beckers J, Devesse W, Hegazy O, Verrelst B.
Active Torque Control for Speed Ripple Elimination: A Mechanical Perspective. *Machines*. 2024; 12(4):222.
https://doi.org/10.3390/machines12040222

**Chicago/Turabian Style**

Croonen, Julien, Adrien Leopold J Deraes, Jarl Beckers, Wim Devesse, Omar Hegazy, and Björn Verrelst.
2024. "Active Torque Control for Speed Ripple Elimination: A Mechanical Perspective" *Machines* 12, no. 4: 222.
https://doi.org/10.3390/machines12040222