# Non-Invasive Identification of Vehicle Suspension Parameters: A Methodology Based on Synthetic Data Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Framework

- Simulation of the vertical movement of a quarter-vehicle model in different external excitation situations;
- Optimization: a process that allows, after a finite number of iterations, us to determine the characteristic suspension parameters;
- By increasing the versatility of the underlying software, we can implement modifications that allow for half-vehicle analysis (bicycle model). Thus, not only vertical oscillation can be analyzed, but pitching as well. This requires the development of a mathematical model that represents the suspension of the half-vehicle as well as an exhaustive previous study of the different suspension systems and their mathematical representation;
- Theoretical validation of the underlying software in different settings. This way, the inner processes of the parameter identification using the half-vehicle suspension model are known.

## 3. Vertical Behavior of Suspension

#### 3.1. Model with One Degree of Freedom

**y**is the road profile.

#### 3.2. Model with Two Degrees of Freedom

_{1}) and another one for the unsprung mass, which simulates the tire and part of the suspension elements (m

_{2}). Moreover, the rigidity and the damping coefficient of the suspension system appear as, (k

_{1}) and (c

_{1}), respectively. The unsprung mass rigidity (k

_{2}) and its damping (c

_{2}) are considered, which simulates the tire behavior. These are, therefore, the parameters of the model, the mases m

_{1}and m

_{2}, the damping coefficients of the suspension system and the unsprung mass, c

_{1}and c

_{2}, and their corresponding rigidities, k

_{1}and k

_{2}. This model allows for the study of the movements of the suspended mass as a function of the unsprung mass, the tire rigidity, the rigidity of the suspension spring, and the characteristic of the damping element.

#### 3.3. Half-Vehicle Model with Four Degrees of Freedom

_{1}, and m

_{2}, the damping coefficients c

_{1}and c

_{2}, the rigidities k

_{1}, k

_{2}, k

_{3}, and k

_{4}, the moment of inertia I, and the distances from the front and rear ends of the mass m to the pitching axis a

_{1}and a

_{2}.

## 4. Optimization and Parameter Identification Tool

- (1)
- Target parameters that allow for the vertical movement simulation of the quarter- or half-vehicle (Procedure 1 or theoretical) or data coming from the sensorization of the vehicle (Procedure 2 or practical);
- (2)
- User-defined initial parameters from which the identification process will take place.

#### 4.1. Generation of the Target Curve

_{1}, the suspension spring rigidity k

_{1}, the suspension damping coefficient c

_{1}, the unsprung mass m

_{2}, the tire rigidity k

_{2}, and the tire damping coefficient c

_{2}. Also, the initial positions and velocities of the suspended—x

_{01}and v

_{01}—and unsprung masses—x

_{02}and v

_{02}—must be set.

#### 4.2. Parameter Identification

#### 4.3. Results Comparison

## 5. Experimental Recovery of Suspension Parameters with Nelder–Mead Optimization

_{0}, v

_{0}, m, c, k, labelled A, B, C, D, and E, sixteen experiments varying their values according to four levels (values) were carried out. Different values for each of the levels can be found in Table 3. The corresponding order of experiments with its values can be found in Table 4.

_{01}, v

_{01}, x

_{02}, v

_{02}, m

_{1}, c

_{1}, k

_{1}, m

_{2}, c

_{2}, and k

_{2}. Labels L through N remain unused. Twenty-seven experiments were carried out, varying their values according to three levels (values). Different values for each of the levels can be found in Table 5 whereas the corresponding order of experiments with its values can be found in Table 6.

- The first minimizes the sum of squares of the differences between the positions of each of the masses in the system under consideration, where the difference is meant to be obtained between the simulated, theoretical solution, and the Nelder–Mead tested (theoretical procedure) or the one obtained between the sensor curve and the Nelder–Mead tested (practical procedure).
- The second is simply the Frobenius norm of the
**Y**-_{theor}**Y**matrix in the case of the theoretical procedure and the_{test}**Y**-_{sensor}**Y**in the case of the practical procedure._{test}**Y**is the matrix obtained via the simulation through the ode45 MatLab function,_{theor}**Y**is the matrix obtained through this same function in the optimization iterative process, and_{test}**Y**is the one obtained through the embarked sensorics._{sensor}

#### 5.1. One-Degree-of-Freedom Model

#### 5.2. Two-Degrees-of-Freedom Model

## 6. Extension to the Half-Vehicle Model—Optimization-Based Denoising

#### 6.1. Extension to the Half-Vehicle Model

_{1}, a

_{2}, k

_{3}, and k

_{4}were not controlled in the frame of the Design of Experiments theory, and they were set to 0.48 m, 0.52 m, 1000 N/m, and 500 N/m, respectively. They were, however, subject to the parameter identification process. Also, in all cases, x

_{m,ini}= 0.05 m, v

_{m}= 0.05 m/s, ${\delta}_{0}=0.05\text{}\mathrm{r}\mathrm{a}\mathrm{d}$, and ${\dot{\delta}}_{0}=0.05\text{}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$.

#### 6.2. Optimization-Based Denoising

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**One-degree-of-freedom system under base harmonic movement. Parameters of the model: Mass m; Damping coefficient c; Tire rigidity k.

**Figure 2.**Two-degrees-of-freedom model (passive suspension). Parameters of the model: Sprung mass m

_{1}; Unsprung mass m

_{2}; Rigidity of the suspension system k

_{1}; Damping coefficient of the suspension system c

_{1}; The tire rigidity k

_{2}; Damping coefficient c

_{2}.

**Figure 17.**Unrestricted optimization. First cost function. Perfect reconstruction is obtained. However, obtained suspension parameters present a −40% error.

**Figure 18.**Unrestricted optimization. Second cost function. Perfect reconstruction is obtained as well. Obtained suspension parameters remain erroneous by a −40% error.

**Figure 19.**Restricted optimization (mass fixed). First cost function. Perfect reconstruction is obtained. Obtained suspension parameters are correct.

**Figure 20.**Restricted optimization (mass fixed). Second cost function. Perfect reconstruction is obtained. Obtained suspension parameters are correct.

**Figure 21.**Unrestricted optimization. Suspended mass. First cost function. Perfect reconstruction is obtained. However, parameters are erroneous by approximately −40%.

**Figure 22.**Unrestricted optimization. Unsprung mass. First cost function. Perfect reconstruction is obtained. However, parameters are erroneous by approximately −40%.

**Figure 23.**Unrestricted optimization. Suspended mass. Second cost function. Perfect reconstruction is obtained. Parameters are erroneous by approximately −46%.

**Figure 24.**Unrestricted optimization. Unsprung mass. Second cost function. Perfect reconstruction is obtained. Parameters are erroneous by approximately −46%.

**Figure 25.**Restricted optimization. Suspended mass. First cost function. Accurate parameter identification.

**Figure 26.**Restricted optimization. Unsprung mass. First cost function. Accurate parameter identification.

**Figure 27.**Restricted optimization. Suspended mass. Second cost function. Accurate parameter identification.

**Figure 28.**Restricted optimization. Unsprung mass. Second cost function. Accurate parameter identification.

**Figure 29.**Unrestricted optimization. Mass m. First cost function. Erroneous parameter identification.

**Figure 30.**Unrestricted optimization. Mass m

_{1}. First cost function. Erroneous parameter identification.

**Figure 31.**Unrestricted optimization. Mass m

_{2}. First cost function. Erroneous parameter identification.

**Figure 32.**Unrestricted optimization. Mass m. Second cost function. Erroneous parameter identification.

**Figure 33.**Unrestricted optimization. Mass m

_{1}. Second cost function. Erroneous parameter identification.

**Figure 34.**Unrestricted optimization. Mass m

_{2}. Second cost function. Erroneous parameter identification.

**Figure 36.**Restricted optimization. Mass m

_{1}. First cost function. Perfect parameter identification.

**Figure 37.**Restricted optimization. Mass m

_{2}. First cost function. Perfect parameter identification.

**Figure 39.**Restricted optimization. Mass m

_{1}. Second cost function. Perfect parameter identification.

**Figure 40.**Restricted optimization. Mass m

_{2}. Second cost function. Perfect parameter identification.

**Figure 41.**Restricted optimization. Mass m. First cost function. Successful parameter identification (percentual relative errors under 8.83%, minimum −0.44%).

**Figure 42.**Restricted optimization. Mass m

_{1}. First cost function. Successful parameter identification (percentual relative errors under 8.83%, minimum −0.44%).

**Figure 43.**Restricted optimization. Mass m

_{2}. First cost function. Successful parameter identification (percentual relative errors under 8.83%, minimum −0.44%).

**Figure 44.**Restricted optimization. Mass m. Second cost function. Successful parameter identification (percentual relative errors under 2.5%, minimum 0.01).

**Figure 45.**Restricted optimization. Mass m

_{1}. Second cost function. Successful parameter identification (percentual relative errors under 2.5%, minimum 0.01).

**Figure 46.**Restricted optimization. Mass m

_{2}. Second cost function. Successful parameter identification (percentual relative errors under 2.5%, minimum 0.01).

Symbol | Description | Value |
---|---|---|

m_{1} | suspended mass | 275 kg |

k_{1} | suspension spring rigidity | 150,000 N/m |

c_{1} | suspension damping coefficient | 1120 Ns/m |

m_{2} | unsprung mass | 27 kg |

k_{2} | tire rigidity | 310,000 N/m |

c_{2} | tire damping coefficient | 3100 Ns/m |

Symbol | Description | Value |
---|---|---|

x_{01} | initial position of the suspended mass | 0.20 m |

v_{01} | initial velocity of the suspended mass | 0.50 m/s |

x_{02} | initial position of the unsprung mass | 0.10 m |

v_{02} | initial velocity of the unsprung mass | 0.05 m/s |

A | B | C | D | E | |
---|---|---|---|---|---|

Value 1 | 0.05 | 0.05 | 250 | 100 | 100 |

Value 2 | 0.11 | 0.11 | 538.61 | 215.44 | 215.44 |

Value 3 | 0.23 | 0.23 | 1160.40 | 464.16 | 464.16 |

Value 4 | 0.5 | 0.5 | 2500 | 1000 | 1000 |

Name | x_{0} | v_{0} | m | c | k |

Unit | m | m/s | kg | Ns/m | N/m |

Type | Numeric | Numeric | Numeric | Numeric | Numeric |

N. Exp. | A | B | C | D | E |
---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 2 | 2 | 2 |

3 | 1 | 3 | 3 | 3 | 3 |

4 | 1 | 4 | 4 | 4 | 4 |

5 | 2 | 1 | 2 | 3 | 4 |

6 | 2 | 2 | 1 | 4 | 3 |

7 | 2 | 3 | 4 | 1 | 2 |

8 | 2 | 4 | 3 | 2 | 1 |

9 | 3 | 1 | 3 | 4 | 2 |

10 | 3 | 2 | 4 | 3 | 1 |

11 | 3 | 3 | 1 | 2 | 4 |

12 | 3 | 4 | 2 | 1 | 3 |

13 | 4 | 1 | 4 | 2 | 3 |

14 | 4 | 2 | 3 | 1 | 4 |

15 | 4 | 3 | 2 | 4 | 1 |

16 | 4 | 4 | 1 | 3 | 2 |

A | B | C | D | E | F | G | H | J | K | L | M | N | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Val1 | 0.05 | 0.05 | 0.01 | 0.01 | 10 | 100 | 100 | 25 | 50 | 50 | 0 | 0 | 0 |

Val2 | 0.16 | 0.16 | 0.03 | 0.03 | 31.62 | 316.23 | 316.23 | 79.06 | 158.11 | 158.11 | 0 | 0 | 0 |

Val3 | 0.5 | 0.5 | 0.1 | 0.1 | 100 | 1000 | 1000 | 250 | 500 | 500 | 0 | 0 | 0 |

Name | x_{01} | v_{01} | x_{02} | v_{02} | m_{1} | c_{1} | k_{1} | m_{2} | c_{2} | k_{2} | alt11 | alt12 | alt13 |

Unit | m | m/s | m | m/s | kg | Ns/m | N/m | kg | Ns/m | N/m | . | . | . |

Type | Num | Num | Num | Num | Num | Num | Num | Num | Num | Num | - | - | - |

Exp | A | B | C | D | E | F | G | H | J | K | L | M | N |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | - | - | - |

2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | - | - | - |

3 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | - | - | - |

4 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | - | - | - |

5 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | - | - | - |

6 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 1 | 1 | 1 | - | - | - |

7 | 1 | 3 | 3 | 3 | 1 | 1 | 1 | 3 | 3 | 3 | - | - | - |

8 | 1 | 3 | 3 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | - | - | - |

9 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | - | - | - |

10 | 2 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | - | - | - |

11 | 2 | 1 | 2 | 3 | 2 | 3 | 1 | 2 | 3 | 1 | - | - | - |

12 | 2 | 1 | 2 | 3 | 3 | 1 | 2 | 3 | 1 | 2 | - | - | - |

13 | 2 | 2 | 3 | 1 | 1 | 2 | 3 | 2 | 3 | 1 | - | - | - |

14 | 2 | 2 | 3 | 1 | 2 | 3 | 1 | 3 | 1 | 2 | - | - | - |

15 | 2 | 2 | 3 | 1 | 3 | 1 | 2 | 1 | 2 | 3 | - | - | - |

16 | 2 | 3 | 1 | 2 | 1 | 2 | 3 | 3 | 1 | 2 | - | - | - |

17 | 2 | 3 | 1 | 2 | 2 | 3 | 1 | 1 | 2 | 3 | - | - | - |

18 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 2 | 3 | 1 | - | - | - |

19 | 3 | 1 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 2 | - | - | - |

20 | 3 | 1 | 3 | 2 | 2 | 1 | 3 | 2 | 1 | 3 | - | - | - |

21 | 3 | 1 | 3 | 2 | 3 | 2 | 1 | 3 | 2 | 1 | - | - | - |

22 | 3 | 2 | 1 | 3 | 1 | 3 | 2 | 2 | 1 | 3 | - | - | - |

23 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 3 | 2 | 1 | - | - | - |

24 | 3 | 2 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 2 | - | - | - |

25 | 3 | 3 | 2 | 1 | 1 | 3 | 2 | 3 | 2 | 1 | - | - | - |

26 | 3 | 3 | 2 | 1 | 2 | 1 | 3 | 1 | 3 | 2 | - | - | - |

27 | 3 | 3 | 2 | 1 | 3 | 2 | 1 | 2 | 1 | 3 | - | - | - |

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**MDPI and ACS Style**

de Hoyos Fernández de Córdova, A.; Olazagoitia, J.L.; Gijón-Rivera, C.
Non-Invasive Identification of Vehicle Suspension Parameters: A Methodology Based on Synthetic Data Analysis. *Mathematics* **2024**, *12*, 397.
https://doi.org/10.3390/math12030397

**AMA Style**

de Hoyos Fernández de Córdova A, Olazagoitia JL, Gijón-Rivera C.
Non-Invasive Identification of Vehicle Suspension Parameters: A Methodology Based on Synthetic Data Analysis. *Mathematics*. 2024; 12(3):397.
https://doi.org/10.3390/math12030397

**Chicago/Turabian Style**

de Hoyos Fernández de Córdova, Alfonso, José Luis Olazagoitia, and Carlos Gijón-Rivera.
2024. "Non-Invasive Identification of Vehicle Suspension Parameters: A Methodology Based on Synthetic Data Analysis" *Mathematics* 12, no. 3: 397.
https://doi.org/10.3390/math12030397