The paper calculates the nonlinear dynamic differential equation model based on the stiffness of the gear teeth and gives the calculation method of the spring stiffness of the transmission system. Choose the Lyapunov energy function and derive the adaptive law that can make the system asymptotically stable globally. At the same time, we discussed the influence of the phase combination of the coupling shaft's torsional stiffness and the gears' meshing stiffness in the multi-stage gear transmission system on the system dynamics. The example calculation shows that the asymptotic method has higher solution accuracy and higher calculation efficiency. This algorithm is a highly versatile analytical solution method.

#### Keywords

- Progressive method
- Gear system
- Nonlinear dynamics differential equation
- Gear stiffness

#### MSC 2010

- 34A34

As the core part of the electro-hydraulic load simulator, the electro-hydraulic loading system is used to generate the desired loading force accurately and apply it to the load object. The system is widely used in aerospace, navigation, and military industries. Therefore, under the premise of ensuring the system's stability, there are also high requirements for its bandwidth and accuracy. The most commonly used in electro-hydraulic servo loading systems is PID control. It has the advantages of reliable work and easy implementation, but it is only suitable for linear systems with simple structure and constant parameters [1]. The loading system is a complex nonlinear system. Factors such as load, pressure, and external interference are constantly changing during the working process, and the system parameters are time-varying within a relatively large range. The current research focus is to ensure the stability, high response, and high precision of the system under this condition.

Model reference adaptive control (MRAC) is a better control method for time-varying parameter systems. It is mainly based on the output error of the controlled object and the reference model to continuously adjust the controller's parameters to suppress the adverse effects of the controlled object's parameter changes and external interference. In recent years, the MRAC method has been widely used in loading systems [2]. The nonlinear dynamic differential equation describing the dynamic characteristics of the gear system is very complicated, and it is impossible to obtain its exact solution directly. Therefore, only high-precision numerical solutions can be obtained for complex high-order nonlinear dynamic equations. However, numerical solutions cannot replace approximate analytical solutions. The perturbation method is the most important tool for studying nonlinear complex differential equations [3]. At present, the perturbation methods mainly include the direct expansion method, Lindstedt-Poincare method, re-normalization method, multi-scale method, parameter variation method, and progressive method. Certain results have been achieved in the research of strongly nonlinear vibration systems. Some scholars have used the phase plane method combined with digital simulation to study the dynamic stability of a single degree of freedom gear transmission system. Some scholars have proposed a vibration equation with a single degree of freedom, constant stiffness, clearance, and parameter excitation of transmission errors. They found the transition frequency, sub-harmonic resonance, and chaos after solving the problem with the harmonic balance method. However, all these methods are approximate solutions proposed for a particular vibration system and are not universal [4]. The author uses the progressive method to solve the nonlinear dynamic differential equations of the gear system. The example calculation shows that the asymptotic method has higher solution accuracy and higher calculation efficiency. This algorithm is a highly versatile analytical solution method.

We assume that the transmission shaft and the support shaft are rigid in the gear system with backlash, and the dynamic model of a pair of spur gear pairs is shown in Figure 1.

We assume that the damping coefficient is _{e}_{p}_{g}

_{p}_{g}^{2}. _{p}_{g}_{p}_{g}_{e}_{d}

We can use equation (3) to transform the equations described by equations (1) and (2) into a degree of freedom differential equation, namely
_{e}_{aT}_{ah}

_{aT}_{eT}_{eh}

Make

Equation (8) can be transformed into a dimensionless equation

The differential equation of gear system dynamics described by equation (9) is very complicated, and it is impossible to find its exact solution directly. Therefore, only approximate solutions can be obtained [6]. The author introduces the asymptotic method for solving differential equations with a single degree of freedom, also known as the three-series method or the KBM method. This method expresses the solution of the equation and the amplitude and phase angle as a power series function of a small parameter

_{k}_{M}_{k}_{k}_{c}x_{M}_{c}_{1}(_{2}(_{e}_{e}

When _{1}, _{2}, _{1}, _{2} is found, the sum of

The applicability of the asymptotic method does not depend on the convergence of the above two equations, but on the asymptoticity of the equation solution when _{1}, _{2} does not contain the first harmonic, _{1}, _{2} should satisfy the following formula

At this time, we can develop the nonlinear force as a Fourier series and substitute it into the equation (10) to eliminate the first harmonic term to obtain the expression of _{1}, _{2}. The expression for the first approximate solution of the equation is _{1}(

According to the above method, solutions with different accuracy requirements can be obtained. The accuracy of the equation solution is determined by the order

We use ^{−5} ^{−6}. We simulate the step signal with an input amplitude of 6kN when R is 10, 50, 500, and 1000 respectively [10]. The results are shown in Figure 2 and Figure 3.

It can be seen from Figure 2 and Figure 3 that the system with adaptive control can follow the reference model well. The following error e eventually converges to zero and remains stable. From the error indicators shown in Table 1, it can be seen that as the adaptive coefficient R increases, the convergence speed is faster, and the maximum error value is also smaller. But compared with R=500, the maximum error and the decrease in convergence time when R=1000 are already very small [11]. When R>1000, the follow-up effect of the system will not be greatly improved since the maximum error and convergence time at R=1000 can meet the purpose of accurately and quickly following the reference model, the adaptive coefficient R is taken as 1000 in the subsequent simulation analysis.

Error indicators for different R values.

R | Maximum error/kN | Error convergence time/ms |
---|---|---|

10 | 2.62 | 6 |

50 | 1.08 | 4.4 |

500 | 0.34 | 2.4 |

1000 | 0.28 | 2.1 |

The hydraulic stiffness Kh corresponding to the piston displacement x at 50, 75, and 99 mm is 56.3, 75.0, and 1420 MN/m, respectively. At the same time, we consider three kinds of load stiffness K and select K and Kh values as shown in Table 2 for simulation based on Kh=56.3 MN/m and K=51.0 MN/m.

Simulation parameters (unit: MN/M).

Serial number | K | Kh |
---|---|---|

1 | 5.1 | 56.3 |

2 | 25.5 | 56.3 |

3 | 51 | 56.3 |

4 | 51 | 75 |

5 | 51 | 1420 |

We take R=1000, and the simulation result of the stop signal with input amplitude of 6kN before and after the adaptive control is added shown in Fig. 4. The dynamic performance index of the system is shown in Table 3.

Dynamic performance indicators of the system before and after adaptive control.

Original system | System after adding adaptive control | |||||
---|---|---|---|---|---|---|

Serial number | tr/ms | ts/ms | ess/% | tr/ms | ts/ms | ess/% |

1 | 54.6 | 85.8 | 8.33 | 4.8 | 14.6 | 0.03 |

2 | 10.7 | 43.6 | 8.33 | 4.8 | 14.6 | 0.03 |

3 | 8 | 38.7 | 8.33 | 4.8 | 14.6 | 0.03 |

4 | 8.7 | 56.3 | 8.33 | 4.8 | 14.6 | 0.03 |

5 | 10.4 | 59.5 | 8.33 | 4.8 | 14.6 | 0.03 |

It can be seen from Table 3 that the dynamic response of the original system changes greatly when K and Kh change. When Kh is constant, the smaller K is, the larger tr and ts are. When K is constant, the greater the Kh, the greater the tr and ts. The dynamic performance of the original system is relatively best when K=51.0 MN/m and Kh=56.3 MN/m. The steady-state error of the original system under the five sets of parameters is 8.33%, which indicates that the system accuracy is very poor. After adding adaptive control, the response of the system under 5 sets of parameters shows excellent consistency. Compared with the situation when the dynamic performance of the original system is at its best, it is greatly reduced, and the steady-state error is very small.

Compared with the existing methods for solving nonlinear differential equations, the asymptotic method has higher solving accuracy and higher efficiency. This algorithm is an effective method for solving complex nonlinear differential equations.

#### Dynamic performance indicators of the system before and after adaptive control.

Original system | System after adding adaptive control | |||||
---|---|---|---|---|---|---|

Serial number | tr/ms | ts/ms | ess/% | tr/ms | ts/ms | ess/% |

1 | 54.6 | 85.8 | 8.33 | 4.8 | 14.6 | 0.03 |

2 | 10.7 | 43.6 | 8.33 | 4.8 | 14.6 | 0.03 |

3 | 8 | 38.7 | 8.33 | 4.8 | 14.6 | 0.03 |

4 | 8.7 | 56.3 | 8.33 | 4.8 | 14.6 | 0.03 |

5 | 10.4 | 59.5 | 8.33 | 4.8 | 14.6 | 0.03 |

#### Simulation parameters (unit: MN/M).

Serial number | K | Kh |
---|---|---|

1 | 5.1 | 56.3 |

2 | 25.5 | 56.3 |

3 | 51 | 56.3 |

4 | 51 | 75 |

5 | 51 | 1420 |

#### Error indicators for different R values.

R | Maximum error/kN | Error convergence time/ms |
---|---|---|

10 | 2.62 | 6 |

50 | 1.08 | 4.4 |

500 | 0.34 | 2.4 |

1000 | 0.28 | 2.1 |

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