# Information Dynamic Correlation of Vibration in Nonlinear Systems

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O and SO

_{2}in algebraic models under different MQN and initial states, so that different entanglement behaviors in these two molecules can be characterized. Liu et al. [3] used an algebraic model to study the dynamics entanglement of small molecule vibrations, and gave the analytical expressions of linear entropy, Von Neumann entropy and Lyapunov function of the integrable dimer and the actual small molecule in the initial Fock state and coherent state. Kis et al. [36] uses analytical methods to determine the vibrational state of polyatomic molecules excited by the optically limited pulse, and uses Von Neumann entropy to describe the size and vibration mode of the entanglement. Ecker et al. [37] used holography to numerically study the entanglement entropy and quantum zero energy conditions in strongly coupled far non-equilibrium quantum states. Wang et al. [38] studied quantum entanglement in two-dimensional ion trap systems. The quantum entanglement between ions and phonons is discussed by using the Reduced Entropy, and the quantum entanglement between the two degrees of freedom of the vibrational motion in the x and y directions is discussed by using the quantum Relative Entropy. Yuan et al. [15] studied the quantum entropy, energy and entanglement dynamics of different initial states in an important spectral Hamiltonian of the curved triatomic molecules H

_{2}O, D

_{2}O and H

_{2}S.

- (1)
- By establishing the state density matrix of nonlinear mechanical systems, the state characteristics of nonlinear mechanical systems and Lü’s chaotic systems are described. The full-vector multi-scale Rényi entropy based on homology information fusion is constructed. A method is proposed to quanlifies the degree of chaos, nonlinear characteristics and coupling relationship of the system by using Von Neumann entropy and full vector multi-scale Rényi entropy. Von Neumann entropy and Rényi entropy are successfully applied to the field of mechanical system dynamics.
- (2)
- By using Von Neumann entropy and Rényi entropy, the chaotic degree, nonlinear characteristics and coupling relationship of Lü’s chaotic system and nonlinear mechanical system can be quantified, so as to achieve the purpose of mode identification, system time evolution and fault diagnosis.
- (3)
- In the study, we found some rules between Rényi entropy and its scale parameters.

## 2. Theory

#### 2.1. Von Neumann Entropy and Rényi Entropy

#### 2.2. Compressive Sensing

#### 2.2.1. Theory

- (1)
- Sparse representation of the signal.
- (2)
- The observation matrix is designed to ensure that the dimension is reduced and the loss of signal characteristics is minimized.
- (3)
- By using the minimum L_0 norm optimization algorithm, the approximate sparse coefficient is obtained, and the X is restored from the observed value y.

#### 2.2.2. Analysis of Noise Reduction Performance

^{0.5t}cos(10πt − 1) + 4.78sinc(t) + 5diric(t) + 2.5sin(2πt) + 3.11cos(20πt)

- (1)
- The acceleration signal of the mechanical system is collected by the acceleration sensor: the x-direction signal and the y-direction signal.
- (2)
- The compressed sensing technology is used to reconstruct and reduce the noise of the signal.
- (3)
- The state density matrix of the nonlinear mechanical system is constructed, and the degree of coupling between two degrees of freedom signals is calculated by using Von Neumann entropy and Rényi entropy, respectively.

## 3. Chaotic System Analysis

_{c=}

_{12}> Rényi entropy

_{c=}

_{20}> Rényi entropy

_{c=}

_{28}> Rényi entropy

_{c=}

_{30}. The two kinds of entropy measures can accurately distinguish several different states of Lü system and the coupling degree among x, y, z, which is the three degrees of freedom data of the system.

## 4. Experimental Study

#### 4.1. Accelerated Life Test of Rolling bearings

#### 4.1.1. Introduction to the Experiment

#### 4.1.2. Data Analysis

^{−4}. Compared with the later entropy development trend, it can be said that the Von Neumann entropy before the 230th minute is stable in a relatively high range, and the coupling degree of the two degrees of freedom signals is relatively high, which reflects that the mechanical system is currently in a relatively disordered state, further explaining the health of the mechanical system. After that, the entropy decreases rapidly. In 230–400 min, the Von Neumann entropy decreases from −0.5 × 10

^{−4}to −3.5 × 10

^{−4}, the coupling degree between two degrees of freedom signals changes obviously, and the nonlinear characteristics of the mechanical system are enhanced, which shows the evolution process of the occurrence and development of mechanical system faults.

#### 4.2. Fault Diagnosis Experiment of Planetary Gear Transmission System

#### 4.2.1. Experimental Introduction

#### 4.2.2. Data Analysis

- (1)
- Acceleration signals in different states of the same system are collected and obtained by accelerometer: two signals x
_{1}, y_{1}in normal state, x_{2}and y_{2}in relative fault state. - (2)
- The compressed sensing technology is used to reconstruct and reduce the noise of the signal.
- (3)
- Vector fusion of homologous signals
- (4)
- Construct state density matrix of nonlinear mechanical system and calculate the Rényi entropy between the two systems at different scales.

## 5. Discussion

## 6. Conclusions

- (1)
- The state density matrix we have established can well describe the state features of nonlinear mechanical systems in practical tests. The Von Neumann entropy and Rényi entropy are used as indicators to measure the degree of chaos, nonlinear characteristics and coupling between nonlinear systems. By coupling each fault system (or chaotic system) with the healthy system (or non-chaotic system), we can identify the mode of the chaotic system and the nonlinear mechanical system, dynamics inversion and fault diagnosis analysis.
- (2)
- Von Neumann entropy and Rényi entropy, two kinds of measurement methods, can play an important role in the study of chaotic characteristics of L ü systems. These two measures can provide a standard measure of coupling of nonlinear systems and accurately judge the state of the chaotic system.
- (3)
- When using the full vector multi-scale Rényi entropy to study the coupling relationship between the vibration signals of the planetary gear transmission system in different states, there is a certain rule. With the change of Rényi entropy scale parameter, the positive and negative of entropy will change, which is just the expression of different physical meaning, and does not affect the problem of mode identification, dynamic inversion and fault diagnosis of nonlinear mechanical system; When the Rényi entropy scale parameter $\tau \to 5$ the Rényi entropy of the multi-stage planetary gear transmission system will converge to a fixed value.
- (4)
- According to the characteristics of low signal-to-noise ratio of mechanical system signal, the noise in vibration signal is reduced by using the compression sensing technology. The experimental signal processing results show that the compression sensing technology has good noise reduction ability and noise robustness.
- (5)
- When the running state of the mechanical system is disturbed by noise, through the calculation of Von Neumann entropy and multi-scale Rényi entropy, the dynamics characteristics of the system are the same, that is, Von Neumann entropy and multi-scale Rényi entropy are robust to the change of the running state of the mechanical system.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Phase diagram and three-dimensional diagram of attractor of Lü chaotic system. (

**a**) x–y phase plane strange attractors. (

**b**) x–z phase plane strange attractors. (

**c**) y–z phase plane strange attractors. (

**d**) Three-dimensional view.

**Figure 4.**Von Neumann entropy between x, y, z with the variation of control parameter c of Lü chaotic system.

**Figure 5.**Rényi entropy between x, y, z with the variation of control parameter c of Lü chaotic system.

**Figure 6.**Von Neumann entropy between x, y with the variation of chaotic system control parameter c and x, y at c = 12 (that is, non-chaotic state).

**Figure 7.**Rényi entropy between x, y with the variation of chaotic system control parameter c and x, y at c = 12 (that is, non-chaotic state).

**Figure 8.**Curve of Rényi entropy between x-y with control factor $\tau $ when the control parameter c of Lü chaotic system takes different eigenvalues.

**Figure 9.**Curve of Rényi entropy between x-z with control factor $\tau $ when the control parameter c of Lü chaotic system takes different eigenvalues.

**Figure 10.**PRONOSTIA bearing accelerated Aging Test bench. (1: Acquisition system; 2: AC motor 3: Speed recorder; 4: Retarder; 5: Pneumatic jack; 6: Test bearing; 7: Accelerometers).

**Figure 15.**Fault diagnosis test bench for planetary gear boxes. (1: Controller; 2: AC motor 3: Planetary gearbox; 4: Accelerometers; 5: Gearbox; 6: Magnetic powder brake).

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**MDPI and ACS Style**

Wu, Z.; Yang, G.; Zhang, Q.; Tan, S.; Hou, S.
Information Dynamic Correlation of Vibration in Nonlinear Systems. *Entropy* **2020**, *22*, 56.
https://doi.org/10.3390/e22010056

**AMA Style**

Wu Z, Yang G, Zhang Q, Tan S, Hou S.
Information Dynamic Correlation of Vibration in Nonlinear Systems. *Entropy*. 2020; 22(1):56.
https://doi.org/10.3390/e22010056

**Chicago/Turabian Style**

Wu, Zhe, Guang Yang, Qiang Zhang, Shengyue Tan, and Shuyong Hou.
2020. "Information Dynamic Correlation of Vibration in Nonlinear Systems" *Entropy* 22, no. 1: 56.
https://doi.org/10.3390/e22010056