# Model of a Quarter Car Suspension with a Damper Containing Magnetorheological Fluid and with Damaged Parts Controlled by Backstepping Method

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## Abstract

**:**

## 1. Introduction

## 2. Practical Use—Physical Model

## 3. Backstepping

## 4. Numerical Experiment of Backstepping Control

- Even vehicle mass distribution for each wheel.
- Force applied to the wheel is in the system symmetry axis.
- The model does not take into account any errors in calculated values (Figure 4 shows the already mentioned spring with nonlinear characteristics).

- ${x}_{{1}_{max}}$—the maximum value of displacement of unsprung mass;
- ${x}_{{2}_{max}}$—the maximum value of displacement of the body;
- ${x}_{basic}$—basic displacement of the piston; a value of 0.1 (m) was used in the investigation.

## 5. Simulation Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Model of a quarter car, (

**a**) diagram of the analyzed system, (

**b**) breaking from the constraints of the force arrangement diagram; m

_{1}—tyre mass (unsprung mass), k

_{1}—tyre spring stiffness, c

_{1}—tyre damping coefficient, x

_{1}—wheel displacement, m

_{2}—body mass, k

_{2}—suspension spring stiffness, c

_{2}—controlled damping coefficient, x

_{2}—body displacement, F—force acting in suspension, S

_{1}—spring force in tyre, R

_{1}—damping force in tyre, S

_{2}—spring force in suspension, R

_{2}—controlled damping force in suspension.

**Figure 6.**Body vertical displacement for a damper damaged by magnetorheological fluid leakage with switching into emergency mode.

**Figure 7.**Body vertical acceleration for a damper damaged by magnetorheological fluid leakage with switching into emergency mode.

**Figure 8.**Body vertical displacement for a step-damaged damper with introduction of an emergency mode.

**Figure 9.**Body vertical acceleration for a step-damaged damper with introduction of an emergency mode.

**Figure 10.**Body vertical displacement for a linearly damaged damper with introduction of an emergency mode.

**Figure 11.**Body vertical acceleration for a linearly damaged damper with introduction of an emergency mode.

Body | $\mathbf{Displacement}\left[\mathit{m}\right]$ | $\mathbf{Acceleration}\left[\frac{\mathit{m}}{{\mathit{s}}^{2}}\right]$ | Stabilization Time [s] |
---|---|---|---|

Leakage from magnetorheological damper | 0.025 | 17.94 | 31 |

Damper linear damage | 0.022 | 10.93 | 36 |

Damper step damage | 0.022 | 10.93 | 40 |

Leakage from magnetorheological damper in emergency mode | 0.037 | 19.44 | 8.75 |

Linear damage of damper in emergency mode | 0.024 | 10.97 | 8.25 |

Step damage of damper in emergency mode | 0.040 | 12.15 | 8.45 |

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**MDPI and ACS Style**

Słomczyński, M.; Radkowski, S.; Makowski, M. Model of a Quarter Car Suspension with a Damper Containing Magnetorheological Fluid and with Damaged Parts Controlled by Backstepping Method. *Energies* **2023**, *16*, 3044.
https://doi.org/10.3390/en16073044

**AMA Style**

Słomczyński M, Radkowski S, Makowski M. Model of a Quarter Car Suspension with a Damper Containing Magnetorheological Fluid and with Damaged Parts Controlled by Backstepping Method. *Energies*. 2023; 16(7):3044.
https://doi.org/10.3390/en16073044

**Chicago/Turabian Style**

Słomczyński, Maciej, Stanisław Radkowski, and Michał Makowski. 2023. "Model of a Quarter Car Suspension with a Damper Containing Magnetorheological Fluid and with Damaged Parts Controlled by Backstepping Method" *Energies* 16, no. 7: 3044.
https://doi.org/10.3390/en16073044