# Signal Identification of Gear Vibration in Engine-Gearbox Systems Based on Auto-Regression and Optimized Resonance-Based Signal Sparse Decomposition

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## Abstract

**:**

## 1. Introduction

## 2. Methodology: Auto-Regression Model-Based Optimized Resonance-Based Signal Sparse Decomposition (AR-ORSSD)

#### 2.1. Pre-Whitening with the AR Model

#### 2.2. Optimized Resonance-Based Signal Sparse Decomposition Based on Meshing Frequency Amplitude Ratio

#### 2.2.1. Resonance-Based Signal Sparse Decomposition

#### 2.2.2. Parameter Selection Problem

#### 2.2.3. Parameter Optimization Based on Meshing Frequency Energy Ratio

#### 2.3. The Proposed AR-ORSSD Algorithm

- (1)
- Remove the normal gear meshing vibration using the AR model;
- (2)
- Determine the ranges of ${Q}_{1}$, ${Q}_{2}$, ${Q}_{1}\in \left[4,\text{}12\right]$, ${Q}_{2}\in \left[1,\text{}3\right]$, in steps of 0.5;
- (3)
- Perform the RSSD operation;
- (4)
- Calculate the MF–ER value for each combination of $\left[{Q}_{1},\text{}{Q}_{2}\right]$;
- (5)
- Obtain the optimal $Q$-factors when MF–ER achieves the maximum;
- (6)
- Implement the RSSD with the optimal $Q$-factors;
- (7)
- Identify the gear vibration signal with the optimized RSSD method.

## 3. Simulated Signal Analysis

## 4. Experimental Verification

## 5. Conclusions

- (1)
- The main idea of this paper is that the gear meshing impact has better frequency aggregation than the engine ignition impact. Therefore, the RSSD algorithm is introduced.
- (2)
- The biggest innovation of this paper is that we define the MF–ER index and introduce it into the RSSD algorithm to adaptively choose the optimal Q-factors, which can improve the accuracy of the separation results.
- (3)
- Due to the interferences of the normal gear meshing vibration, the use of the RSSD algorithm alone cannot achieve perfect results. Therefore, the AR model is used as a pre-processing step to eliminate the normal gear meshing vibration.
- (4)
- Both simulated signals and experimental signals acquired from the engine-gearbox system in a forklift validate the effectiveness of the proposed algorithm.
- (5)
- Both simulated signals and experimental signals validate the necessity of adopting the AR model.
- (6)
- Through comparison with the GA-based RSSD method, it is indicated that the AR-ORSSD algorithm achieves superior performance in identifying gear vibration signals especially when under strong interferences.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${x}_{t},\text{}{x}_{t-i}$ | Data points at time $t$, $t-i$ respectively, V |

$n$ | Model order |

$p$ | The optimal model order |

${a}_{i}$ | $i$th coefficient of the AR model |

${\epsilon}_{t}$ | Residual error at time $t$, V |

$Q$ | Quality factor |

${f}_{c}$ | Center frequency, Hz |

$BW$ | Bandwidth, Hz |

$\alpha $ | High-pass scale |

$\beta $ | Low-pass scale |

$r$ | Redundancy |

$L$ | Decomposition layer |

${f}_{s}$ | Sampling frequency, Hz |

${H}_{1}\left(\omega \right),\text{}{H}_{0}\left(\omega \right)$ | High-pass and low-pass filters |

$\omega $ | Angle, rad |

$\theta \left(\bullet \right)$ | Function, $\theta \left(\omega \right)\text{}=\text{}0.5\left(1+cos\omega \right)\sqrt{2-cos\omega},\text{}\left|\omega \right|\le \pi $ |

$x\left(t\right)$ | Vibration signal, V |

${x}_{1}\left(t\right),\text{}{x}_{2}\left(t\right)$ | High and low resonance components, V |

${S}_{1},\text{}{S}_{2}$ | The overcomplete dictionaries for ${x}_{1}\left(t\right),\text{}{x}_{2}\left(t\right)$ |

${W}_{1},\text{}{W}_{2}$ | The wavelet coefficients of ${x}_{1}\left(t\right)$, ${x}_{2}\left(t\right)$ |

${\lambda}_{1},\text{}{\lambda}_{2}$ | The regularization parameters of ${x}_{1}\left(t\right)$, ${x}_{2}\left(t\right)$ |

${W}_{1}^{\ast},\text{}{W}_{2}^{\ast}$ | The wavelet coefficients of ${x}_{1}\left(t\right)$ and ${x}_{2}\left(t\right)$ when cost function achieves the minimum |

${\widehat{x}}_{1},\text{}{\widehat{x}}_{2}$ | The optimal high and low resonance components, V |

${Q}_{1},\text{}{Q}_{2}$ | Quality factors of ${x}_{1}\left(t\right)$ and ${x}_{2}\left(t\right)$ |

${L}_{1},\text{}{L}_{2}$ | Decomposition layers of ${x}_{1}\left(t\right)$ and ${x}_{2}\left(t\right)$ |

${r}_{1},\text{}{r}_{2}$ | Redundancies of ${x}_{1}\left(t\right)$ and ${x}_{2}\left(t\right)$ |

$N$ | The data length |

$\left[\bullet \right]$ | The rounding operation |

MF–ER | Meshing frequency energy ratio |

$\overline{{H}_{1}}\left(t\right)$ | The Hilbert transform of ${x}_{1}\left(t\right)$, V |

${z}_{1}\left(t\right)$ | The analytical signal of ${x}_{1}\left(t\right)$, V |

$e\left(t\right)$ | The envelop waveform of ${z}_{1}\left(t\right)$, V |

$E\left(f\right)$ | The envelop spectrum of ${x}_{1}\left(t\right)$, V |

${f}_{m}$ | Gear meshing frequency, Hz |

$K$ | The number of meshing frequency harmonics |

${f}_{r,p},\text{}{f}_{r,g}$ | The rotating frequencies of the pinion and gear, Hz |

${f}_{mr}$ | The resonance frequency excited by gear meshing impact, Hz |

${A}_{g},\text{}{B}_{g}$ | The magnitudes of the amplitude and phase modulations, V |

${A}_{m}$ | The amplitude of the impulses due to meshing impacts, V |

${\beta}_{m}$ | Damping characteristic frequency, Hz |

${f}_{en}$ | Ignition frequency, Hz |

${t}_{k,\text{}m}$ | The time of occurrence of the kth impulse, s |

${A}_{e}$ | The amplitude of the ignition impulses of the engine, V |

${\beta}_{e}$ | The structural damping characteristic frequency of the ignition impact, Hz |

${f}_{er}$ | The resonance frequency induced by the engine ignition impact, Hz |

${t}_{n,\text{}e}$ | The time of occurrence of the mth impulse, s |

${N}^{\prime}$ | The cylinder number |

$D$ | The engine stroke constant |

$v$ | Engine rotating speed, rpm |

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**Figure 3.**The time domain impulses and frequency spectrums of: (

**a**,

**b**) the gear meshing impact; (

**c**,

**d**) the engine ignition impact.

**Figure 5.**The framework of the proposed auto-regression model-based optimized resonance-based signal sparse decomposition (AR-ORSSD) algorithm.

**Figure 6.**Simulated signals: (

**a**) gear vibration signal, (

**b**) engine ignition signal, (

**c**) compound signal of the gear and engine.

**Figure 7.**The meshing frequency energy ratio (MF–ER) values with different $Q$-factors of the simulated signal. (

**a**) The result without pre-whitening using AR model; (

**b**) The result of the proposed AR-ORSSD algorithm. (The arrows in the picture indicate the $Q$-factors with the maximum MF–ER).

**Figure 8.**The decomposition results of RSSD using parameters obtained from Figure 7a. (

**a**) The high-resonance component; (

**b**) the low-resonance component.

**Figure 10.**The decomposition results of the RSSD using the proposed AR-ORSSD algorithm. (

**a**–

**c**) The high resonance component, its frequency spectrum, and envelope spectrum; (

**d**–

**f**) the low resonance component, its frequency spectrum, and envelope spectrum (the arrows in blue indicate the characteristic frequencies of the gear while the arrows in red indicate the characteristic frequencies of the engine).

**Figure 11.**The decomposition results of the RSSD optimized by genetic algorithm. (

**a**–

**c**) The high-resonance component, its frequency spectrum, and envelope spectrum; (

**d**–

**f**) the low-resonance component, its frequency spectrum, and envelope spectrum (the arrows in blue indicate the characteristic frequencies of the gear while the arrows in red indicate the characteristic frequencies of the engine).

**Figure 12.**The schematic diagrams of (

**a**) the forklift for the experiment, (

**b**) the transmission system.

**Figure 13.**Vibration signals acquired from the engine and gearbox. (

**a**,

**b**) The time domain signal and corresponding frequency spectrum of the engine; (

**c**,

**d**) the time domain signal and corresponding frequency spectrum of the gearbox (the arrows in blue indicate the characteristic frequencies of the gear while the arrows in red indicate the characteristic frequencies of the engine).

**Figure 14.**The MF–ER values with different $Q$-factors of the gear vibration signal. (

**a**) The result without pre-whitening using AR model; (

**b**) the result of the proposed AR-ORSSD algorithm (the arrows in the picture indicate the $Q$-factors with the maximum MF–ER).

**Figure 15.**The decomposition results of the RSSD using parameters obtained from Figure 14a. (

**a**,

**b**) The high-resonance component and its frequency spectrum; (

**c**,

**d**) the low-resonance component and its frequency spectrum (the arrows in blue indicate the characteristic frequencies of the gear while the arrows in red indicate the characteristic frequencies of the engine).

**Figure 16.**The decomposition results of the RSSD using the proposed AR-ORSSD algorithm. (

**a**–

**c**) The high-resonance component, its frequency spectrum and enlarged high-resonance component; (

**d**–

**f**) the low-resonance component, its frequency spectrum, and enlarged high-resonance component (the arrows in blue indicate the characteristic frequencies of the gear while the arrows in red indicate the characteristic frequencies of the engine).

**Figure 17.**The decomposition results of the RSSD optimized by GA. (

**a**–

**c**) The high-resonance component, its frequency spectrum, and enlarged high-resonance component; (

**d**–

**f**) the low-esonance component, its frequency spectrum, and enlarged high-resonance component (the arrows in blue indicate the characteristic frequencies of the gear while the arrows in red indicate the characteristic frequencies of the engine).

Parameters | Value | Parameters | Value |
---|---|---|---|

${A}_{g}$ | 0.5 | ${\beta}_{m}$ | 1000 |

${B}_{g}$ | 0.2 | ${\beta}_{e}$ | 600 |

${A}_{m}$ | 1 | ${f}_{en}$ | 20 |

${f}_{r,p}$ | 5 | ${f}_{mr}$ | 3000 |

${f}_{r,g}$ | 2 | ${f}_{er}$ | 6000 |

${f}_{m}$ | 65 | ${t}_{k,\text{}m}$ | 0.015 |

${A}_{e}$ | 1 | ${t}_{n,\text{}e}$ | 0.05 |

${\mathit{Q}}_{1}$ | ${\mathit{Q}}_{2}$ | ${\mathit{r}}_{1}$ | ${\mathit{r}}_{2}$ |
---|---|---|---|

4.65 | 1 | 5.05 | 6.44 |

${\mathit{Q}}_{1}$ | ${\mathit{Q}}_{2}$ | ${\mathit{r}}_{1}$ | ${\mathit{r}}_{2}$ |
---|---|---|---|

9.21 | 1.74 | 9.83 | 3.88 |

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## Share and Cite

**MDPI and ACS Style**

Huang, Y.; Tong, S.; Tong, Z.; Cong, F.
Signal Identification of Gear Vibration in Engine-Gearbox Systems Based on Auto-Regression and Optimized Resonance-Based Signal Sparse Decomposition. *Sensors* **2021**, *21*, 1868.
https://doi.org/10.3390/s21051868

**AMA Style**

Huang Y, Tong S, Tong Z, Cong F.
Signal Identification of Gear Vibration in Engine-Gearbox Systems Based on Auto-Regression and Optimized Resonance-Based Signal Sparse Decomposition. *Sensors*. 2021; 21(5):1868.
https://doi.org/10.3390/s21051868

**Chicago/Turabian Style**

Huang, Yuanyuan, Shuiguang Tong, Zheming Tong, and Feiyun Cong.
2021. "Signal Identification of Gear Vibration in Engine-Gearbox Systems Based on Auto-Regression and Optimized Resonance-Based Signal Sparse Decomposition" *Sensors* 21, no. 5: 1868.
https://doi.org/10.3390/s21051868