# Optimal Design of Fractional-Order Electrical Network for Vehicle Mechatronic ISD Suspension Using the Structure-Immittance Approach

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## Abstract

**:**

## 1. Introduction

## 2. Equivalent Realization of Fractional Passive Network Elements

_{0}are the upper and lower bounds of the operator. The unified definition of fractional calculus operator [26] is:

_{k}and ω

_{k}are zero point and the pole, respectively. ω

_{h}and ω

_{b}are the upper and lower limits of frequency bands, respectively. In general, the weighting parameters b = 10, d = 9. In this paper, the filter frequency band is (10

^{−3}, 10

^{3}) rad/s. The larger the filter order, the higher the approximation accuracy. In this frequency band, the fifth order Oustaloup filtering effect has met the accuracy requirements, so the selection order is five.

## 3. Model Construction of Vehicle Mechatronic ISD Suspension System

#### 3.1. The Ball-Screw Mechatronic Inerter

#### 3.2. Mechatronic ISD Suspension Structure Layout

_{t}, and c are spring stiffness, tire stiffness, and the damping coefficient, respectively, m

_{s}and m

_{u}are the sprung mass and the unsprung mass, respectively. z

_{s}, z

_{u}and z

_{r}are the vertical displacements of the sprung mass, the unsprung mass, and road roughness, respectively, and Z

_{s}, Z

_{u}and Z

_{r}are their Laplace transforms, respectively. B(s) is the impedance expression of the mechatronic inerter, which is shown as follows [17]:

_{e}is the induced electromotive force coefficient of the rotary motor, k

_{t}is the thrust coefficient of the rotary motor. K

_{m}is the electromechanical parameter conversion coefficient of the ball-screw mechatronic inerter, which is taken as 7056 HN/m in this paper. Z

_{e}(s) is the impedance expression of the external electrical network of the rotary motor. The fractional-order external electrical network of the mechatronic inerter includes resistor(s), fractional-order capacitor(s) and fractional-order inductor(s). In order to simplify the electrical network, the number of resistors, fractional capacitors and fractional inductors is limited to one in the optimal design. Eight structures of the three element arrangement are summarized using the structure-immittance approach, and two general structures are used for general expression, as shown in Figure 3 and Figure 4.

_{1}, L

_{2}, L

_{3}, L

_{4}, L

_{5}, and L

_{6}are fractional-order inductors, R and C are the resistor and the fractional-order capacitor, respectively. α and β are the fractional-order inductance order and the fractional-order capacitance order, respectively. In the Y

_{1}(s) structure, at least three of L

_{2}, 1/L

_{3}, L

_{4}and L

_{6}are zero, and in the Y

_{2}(s) structure, at least three of 1/L

_{1}, 1/L

_{2}, L

_{3}and 1/L

_{5}are zero. For example, in Figure 3, when L

_{3}, L

_{4}and L

_{6}are zero, it is a structure in which a fractional-order inductor is connected in parallel with a fractional-order capacitor, and then connected in series with a resistor. In Figure 4, when 1/L

_{2}, 1/L

_{3}and 1/L

_{5}are zero, it is a fractional-order capacitor in series with a fractional-order inductor, and then in parallel with a resistor.

## 4. Parameter Optimization Design

#### 4.1. Pattern Search Optimization Algorithm

#### 4.2. Optimization Results

_{pas}are the RMS values of the vehicle body acceleration of the suspension to be optimized and the traditional passive suspension, respectively, SWS and SWS

_{pas}are the RMS values of the suspension working space of the suspension to be optimized and the traditional passive suspension, respectively, and DTL and DTL

_{pas}are the RMS values of the dynamic tire load of the suspension to be optimized and the traditional passive suspension, respectively. BA

_{pas}, SWS

_{pas}and DTL

_{pas}are calculated by a mature traditional passive suspension [30], and their performances have reached a high level, which are 1.3096 m·s

^{−2}, 0.0130 m and 900.4704 N, respectively. P represents the set of parameters to be optimized for the suspension system, including inertance b, damping coefficient c, fractional-order inductance coefficient L

_{e}, fractional-order capacitance coefficient C

_{e}, resistance coefficient R

_{e}, fractional-order inductance order α, and fractional-order capacitance order β. Their constraints are as follows:

_{2}, 1/L

_{3}and L

_{6}are zero in Y

_{1}(s) structure. Set the fractional-order inductance order α and the fractional-order capacitance order β to 1 for optimization, and get the integer-order ISD suspension system parameters. The optimization parameters of fractional-order ISD suspension and integral-order ISD suspension are shown in Table 3.

## 5. Simulation Analysis

#### 5.1. The Characteristics of Bode Diagram

^{−2}, 2] Hz, the optimized structure shape is similar to the spring. In the range [2,4] Hz, the structure shape is similar to the damper, and above 4 Hz, the optimized structure is similar to the inerter, which is the difference between the traditional passive suspension system and the optimized structure. The traditional suspension system composed of “spring damper” mechanical components cannot show inertia characteristics, which is the main factor limiting the performance improvement of the traditional suspension structure, and also the reason why the ISD suspension of vehicles containing the inerter has better vibration isolation performance.

#### 5.2. Random Road Input

_{r}(t), w(t) and G

_{q}(n

_{0}) are the vehicle speed, the vertical input displacement, the white noise with the mean value of 0, and the road roughness coefficient, respectively. Class C pavement is selected in this paper, and the pavement roughness coefficient is 2.56 × 10

^{−4}m

^{3}. Figure 8, Figure 9 and Figure 10 and Table 4 show the comparison of the RMS values of the vehicle body acceleration, the suspension working space and the dynamic tire load of the three suspension systems at a speed of 20 m/s.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Mechanical Network Elements | Impedance | Electrical Network Elements | Impedance |
---|---|---|---|

Spring | k/s^{α} | Inductor | 1/Ls^{α} |

Damper | c | Resistor | 1/R |

Inerter | bs^{β} | Capacitor | Cs^{β} |

Parameters | Values |
---|---|

Sprung Mass m_{s}/kg | 320 |

Unsprung Mass m_{u}/kg | 45 |

Spring Stiffness k/N m^{−1} | 22,000 |

Tire Stiffness k_{t}/N m^{−1} | 190,000 |

Fractional-Order ISD Suspension | Integer-Order ISD Suspension | ||
---|---|---|---|

Parameters | Values | Parameters | Values |

Inertance b/kg | 5 | Inertance b/kg | 13 |

Damping coefficient c/N·s·m^{−1} | 1074 | Damping coefficient c/N·s·m^{−1} | 232 |

Fractional-order inductance coefficient L_{e}/H | 1.05 | Inductance coefficient L_{e}/H | 1.34 |

Fractional-order capacitance coefficient C_{e}/F | 0.06 | Capacitance coefficient C_{e}/F | 0.03 |

Resistance coefficient R_{e}/Ω | 320.73 | Resistance coefficient R_{e}/Ω | 5.56 |

Fractional-order inductance order α | 0.28 | - | - |

Fractional-order capacitance order β | 0.81 | - | - |

Performance Index | Traditional Passive Suspension | Integer-Order Isd Suspension | Improvement | Fractional-Order Isd Suspension | Improvement |
---|---|---|---|---|---|

RMS of vehicle body acceleration/(m·s^{−2}) | 1.3096 | 1.3051 | 3.44% | 1.3042 | 4.12% |

RMS of suspension working space/(m) | 0.0130 | 0.0101 | 22.31% | 0.0100 | 23.08% |

RMS of dynamic tire load/(N) | 900.4704 | 875.8558 | 2.73% | 852.6704 | 5.31% |

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**MDPI and ACS Style**

Hua, J.; Shen, Y.; Yang, X.; Zhang, Y.; Liu, Y.
Optimal Design of Fractional-Order Electrical Network for Vehicle Mechatronic ISD Suspension Using the Structure-Immittance Approach. *World Electr. Veh. J.* **2023**, *14*, 12.
https://doi.org/10.3390/wevj14010012

**AMA Style**

Hua J, Shen Y, Yang X, Zhang Y, Liu Y.
Optimal Design of Fractional-Order Electrical Network for Vehicle Mechatronic ISD Suspension Using the Structure-Immittance Approach. *World Electric Vehicle Journal*. 2023; 14(1):12.
https://doi.org/10.3390/wevj14010012

**Chicago/Turabian Style**

Hua, Jie, Yujie Shen, Xiaofeng Yang, Ying Zhang, and Yanling Liu.
2023. "Optimal Design of Fractional-Order Electrical Network for Vehicle Mechatronic ISD Suspension Using the Structure-Immittance Approach" *World Electric Vehicle Journal* 14, no. 1: 12.
https://doi.org/10.3390/wevj14010012