# Identification of Control-Related Signal Path for Semi-Active Vehicle Suspension with Magnetorheological Dampers

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Vibration Model of Vehicle with MR Dampers and the Diagnostic Setup

- i: g—Stationary ground of the diagnostic station; r—road-induced excitation signal or related to the diagnostic station; e—steel plate of the mechanical exciter; $ge$/$re$—interaction of the steel plate with the ground/exciter’s motor; u—vehicle wheel; t—vehicle tire; s—vehicle body; $us$—vertically oriented suspension; $sa$—tilted suspension shock absorber;
- j: f—Front; r—rear;
- b: r—Right; l—left.

#### 2.1. Road-Induced Excitation Generated by Mechanical Exciters

#### 2.2. Mathematical Description of the Vehicle Vibration Model

#### 2.3. Bouc–Wen Model of MR Dampers

#### 2.4. Implementation of the Vehicle Vibration Simulator

## 3. Preprocessing and Analysis of Experimental Data

- x—General usage of quantity x, valid for both simulation and measurement results;
- $\widehat{x}$—Quantity x evaluated based on simulation results;
- $\stackrel{\u02d8}{x}$—Quantity x evaluated on the basis of the measurement results.

#### 3.1. Identification Procedure for Control-Related Signal Paths

- Confirmation that the sinusoidal road-related excitation generated using the diagnostic station is appropriate for the considered identification, the selection of its frequency equal to 12 Hz, and the selection of a wideband low-frequency control current signal with a bandwidth equal to 25 Hz;
- The implementation of experiments for different amplitudes and average values of MR damper control currents;
- The filtering of harmonic components of the road-induced excitation from the analysed force and acceleration measurement signals and the compensation of the influence of suspension springs;
- The simultaneous identification of ${H}_{r}$ and ${H}_{l}$ in the frequency domain based on previously preprocessed signals.

#### 3.2. Measurement and Control System of the Experimental Vehicle

- Current transducers (LTS 6-NP) configured for a measuring range equal to ±2 A, manufactured by LEM, separately measuring electric currents controlling the front suspension MR dampers (MR dampers located in the rear parts of the vehicle suspension were unpowered during experiments—passive mode of operation); these measurements were used for the identification of the vehicle model, not the control-related signal paths;
- Force transducers (U93) with a measuring range equal to ±5 kN, manufactured by HBM, separately measuring forces generated by front shock absorbers;
- LVDT (linear variable differential transformer) sensors with a measuring range equal to 60 mm, manufactured by Peltron, separately measuring suspension deflections of front shock absorbers;
- Three-axis MEMS (microelectromechanical system) low-noise accelerometer (ADXL356) configured for a measuring range equal to ±10 g (gravitational acceleration), manufactured by Analog Devices, attached to and measuring the acceleration of the middle front part of the vehicle body.

#### 3.3. Modulation Effect of Road-Related and Control-Related Excitation Signals

#### 3.4. Filtering of Road-Induced Excitation from Measurement Data

## 4. Identification of Control-Related Signal Paths and Vehicle Vibration Model

#### 4.1. Coherence of Force Excitation and Response Signals

#### 4.2. Experimental Results of Identification of Control-Related Signal Paths

#### 4.3. Identification of Vehicle Vibration Model

#### 4.4. Influence of Vehicle Load on Control-Related Signal Paths

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**An experimental all-terrain vehicle with MR dampers subjected to harmonic excitation generated by mechanical exciters included in the vehicle diagnostic station.

**Figure 2.**Mechanical representation of the vehicle vibration model (in black) including Bouc-Wen models of suspension MR dampers tested on a diagnostics station using two mechanical exciters (in red). Notations of vehicle directions: $fr$—front right, $fl$—front left, $rr$—rear right, $rl$—rear left.

**Figure 3.**Elements of the measurement and control system installed in the experimental vehicle: shock absorber, consisting of an MR damper supplied with control current and steel coil spring, and suspension force and LVDT deflection sensors.

**Figure 4.**Force–velocity characteristics of the implemented Bouc–Wen damper model demonstrated for selected configurations of sinusoidal piston excitation and control current ${\widehat{i}}_{\mathrm{mr}}$: (

**a**) model responses obtained for sinusoidal excitation at 6 Hz, 5 mm, and several control currents; (

**b**) model responses obtained for control current equal to 1.4 A and different configurations of sinusoidal excitation.

**Figure 6.**Sensors and actuators of the measurement and control system available in the experimental vehicle and used during presented experiments.

**Figure 7.**Comparison of time diagrams of force generated by the front left shock absorber ${\stackrel{\u02d8}{F}}_{safl}$ related to constant and random control currents ${\stackrel{\u02d8}{i}}_{\mathrm{mr},fl}$ evaluated based on experimental results and related to the second configuration with an average control current equal to 0.36 A and a standard deviation equal to 0.15 A: (

**a**) the whole experiment, (

**b**) time range of 50–52 s, and (

**c**) time range of 51.6–52 s.

**Figure 8.**Simulated effect of Bouc–Wen MR damper model subjected to piston displacement road-related excitation ${\widehat{z}}_{sa}$ with 1 mm amplitude and 12 Hz frequency modulated by a constant or a wide-band low-pass random control current ${\widehat{i}}_{\mathrm{mr}}$ with bandwidth of 5, 11, or 25 Hz: (

**a**) PSD frequency characteristics of different cases of control current ${\widehat{i}}_{\mathrm{mr}}$; (

**b**) PSD frequency characteristics of corresponding force responses of Bouc–Wen model ${\widehat{F}}_{\mathrm{mr}}$.

**Figure 9.**Frequency characteristics of PSDs (calculated based on FFTs, which were used for filtering of road-induced harmonic components) evaluated for experimental results of the second configuration of average control current equal to 0.36 A and standard deviation equal to 0.15 A: (

**a**) PSD of front vehicle body acceleration ${\stackrel{\u02d8}{a}}_{sf}$; (

**b**) PSD of right MR damper force ${\stackrel{\u02d8}{F}}_{\mathrm{mr}fr}$. Frequency ranges of filtering are marked in red.

**Figure 10.**Comparison of frequency characteristics of coherence defined from the inputs ${\stackrel{\u02d8}{F}}_{\mathrm{mr},fr}$ and ${\stackrel{\u02d8}{F}}_{\mathrm{mr},fl}$ to the output ${\stackrel{\u02d8}{a}}_{sf}$, evaluated for three configurations of average values and standard deviations of control current ${\stackrel{\u02d8}{i}}_{\mathrm{mr}}$.

**Figure 11.**Comparison of frequency characteristics of absolute transfer functions of control-related signal paths evaluated based on experimental results ($|{\stackrel{\u02d8}{H}}_{r}|$,$|{\stackrel{\u02d8}{H}}_{l}|$) and related to three configurations of average values and standard deviations of control current ${\stackrel{\u02d8}{i}}_{\mathrm{mr}}$, related to (

**a**) the right suspension part; (

**b**) the left suspension part.

**Figure 12.**Frequency characteristics of absolute transfer functions of control-related signal paths evaluated based on experimental results ($|{\stackrel{\u02d8}{H}}_{r}|$,$|{\stackrel{\u02d8}{H}}_{l}|$) and related to the second configuration with an average control current equal to 0.36 A and a standard deviation equal to 0.15 A, compared to transfer functions evaluated for the simulated responses of the identified vehicle model including mechanical exciters ($|{\widehat{H}}_{r}|$,$|{\widehat{H}}_{l}|$) related to (

**a**) the right suspension part; (

**b**) the left suspension part.

**Figure 13.**Comparison of frequency characteristics of absolute transfer functions of control-related signal paths $|{\widehat{H}}_{r}|$ and $|{\widehat{H}}_{l}|$, evaluated for the simulated responses of the identified vehicle model unloaded or with passengers, related to (

**a**) the right suspension part; (

**b**) the left suspension part.

**Figure 14.**Comparison of frequency characteristics obtained for phase shift of transfer functions dedicated to control-related signal paths $\mathrm{arg}\left({\widehat{H}}_{r}\right)$ and $\mathrm{arg}\left({\widehat{H}}_{l}\right)$ and evaluated for the simulated responses of the identified vehicle model unloaded or with passengers, related to (

**a**) the right suspension part; (

**b**) the left suspension part.

[${\gamma}_{bw,0}$, ${\gamma}_{bw,1}$, ${\gamma}_{bw,2}$, ${\gamma}_{bw,3}$] = [12.490, 440.717, 1153.052, −564.019] |

[${\lambda}_{bw,0}$, ${\lambda}_{bw,1}$, ${\lambda}_{bw,2}$, ${\lambda}_{bw,3}$] = [10.898, −1.076, −4.102, 2.100] |

[${c}_{bw,0}$, ${c}_{bw,1}$, ${c}_{bw,2}$, ${c}_{bw,3}$] = [885.609, 130.047, 2009.233, −1342.062] |

${n}_{bw}$ = 2 ${\u03f5}_{bw}$ = 4.8 |

Mechanical vibration exciters | |||

${A}_{rfb}$ = 0.002 m | ${m}_{efr}$ = 85 kg | ${m}_{efl}$ = 85 kg | |

${k}_{re}$ = 160,000 Nm${}^{-1}$ | ${c}_{re}$ = 50 Nsm${}^{-1}$ | ${k}_{ge}$ = 80,000 Nm${}^{-1}$ | ${c}_{ge}$ = 50 Nsm${}^{-1}$ |

Vehicle wheels and tires | |||

${m}_{ufr}$ = 10 kg | ${m}_{ufl}$ = 10 kg | ${m}_{urr}$ = 15 kg | ${m}_{url}$ = 15 kg |

${k}_{tfr}$ = 75,000 Nm${}^{-1}$ | ${k}_{tfl}$ = 110,000 Nm${}^{-1}$ | ${k}_{trr}$ = 70,000 Nm${}^{-1}$ | ${k}_{trl}$ = 70,000 Nm${}^{-1}$ |

${c}_{tfr}$ = 87 Nsm${}^{-1}$ | ${c}_{tfl}$ = 126 Nsm${}^{-1}$ | ${c}_{trr}$ = 102 Nsm${}^{-1}$ | ${c}_{trl}$ = 102 Nsm${}^{-1}$ |

Vehicle body and suspension damping | |||

${m}_{s}$ = 300 kg | ${I}_{sp}$ = 75 kgm${}^{2}$ | ${I}_{sr}$ = 33 kgm${}^{2}$ | |

${l}_{f}$ = 0.543 m | ${l}_{r}$ = 0.607 m | w = 0.135 m | |

${c}_{usfr}$ = 300 Nsm${}^{-1}$ | ${c}_{usfl}$ = 300 Nsm${}^{-1}$ | ${c}_{usrr}$ = 300 Nsm${}^{-1}$ | ${c}_{usrl}$ = 300 Nsm${}^{-1}$ |

Shock absorbers and suspension design | |||

${k}_{safr}$ = 22,208 Nm${}^{-1}$ | ${k}_{safl}$ = 22,208 Nm${}^{-1}$ | ${k}_{sarr}$ = 34,857 Nm${}^{-1}$ | ${k}_{sarl}$ = 34,857 Nm${}^{-1}$ |

${g}_{{z}_{saf}}$ = 0.515 | ${g}_{{z}_{sar}}$ = 0.428 | ${g}_{{F}_{saf}}$ = 0.549 | ${g}_{{F}_{sar}}$ = 0.501 |

${\alpha}_{saf}$ = 67${}^{\circ}$ | ${\alpha}_{sar}$ = 60${}^{\circ}$ | ||

Harmonic oscillator included in vehicle model response ${\widehat{a}}_{sf}$ | |||

${\omega}_{\mathrm{res}}$ = 2$\pi \phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}$5.5 Hz | ${\zeta}_{\mathrm{res}}$ = 0.04 | ${g}_{\mathrm{res}}$ = 8.42 × 10${}^{-4}$ kg${}^{-1}$ |

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**MDPI and ACS Style**

Krauze, P.
Identification of Control-Related Signal Path for Semi-Active Vehicle Suspension with Magnetorheological Dampers. *Sensors* **2023**, *23*, 5770.
https://doi.org/10.3390/s23125770

**AMA Style**

Krauze P.
Identification of Control-Related Signal Path for Semi-Active Vehicle Suspension with Magnetorheological Dampers. *Sensors*. 2023; 23(12):5770.
https://doi.org/10.3390/s23125770

**Chicago/Turabian Style**

Krauze, Piotr.
2023. "Identification of Control-Related Signal Path for Semi-Active Vehicle Suspension with Magnetorheological Dampers" *Sensors* 23, no. 12: 5770.
https://doi.org/10.3390/s23125770