# Simulation of vibro-impact gear model considering the lubricant influence with a new computational algorithm

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## Abstract

In this paper, a vibro-impact gear model incorporating the influences of the lubricant and backlash is formulated. Then, a new computational algorithm validated in comparison with the “stiff” solvers, by defining a transition area and adopting the double-changed time step, is proposed to identify the influences of the lubricant on the dynamic system. The results obtained in this paper indicate that the proposed numerical algorithm not only guarantees the precision of solutions, but also reduces the calculation speed of the whole system. The lubricant can potentially reduce the vibrations in the gear system, and the boundaries for double-sided impacts, single-sided impact and no impact are mainly dependent on the fluctuating driving torque and the stiffness of the lubricant. These results could provide a good source of information on the utilization of vibro-impact modeling and simulation for the study of spur gears dynamic performance, and quantification of the factors such as gear backlash, input power or torque fluctuations, lubrication, rattle, etc. In addition, the proposed numerical method could be used as a basic program of vibro-impact in Matlab environment.

## Keywords

Gear system Computational algorithm Nonlinear vibration Gear backlash## Abbreviations

- \(x\)
Dynamic transmission errors

- \(\dot{x}\)
Relative speed

- \(C\)
Viscous damping

- \(L\)
Total backlash

- \(t\)
Time in seconds

- \(t_{f}\)
Time of one excited period cycle

- \(\Delta t\)
Time interval

- \(F_{n} \left( t \right)\)
Nonlinear elastic contact force

- \(I_{p,g}\)
Rotational inertia of the pinion and gear

- \(I_{eq}\)
Equivalent mass

- \(\dot{\theta }_{p,g}\)
Rotational velocity of the pinion and gear

- \(\ddot{\theta }_{p,g}\)
Rotational acceleration of the pinion and gear

- \(R_{p,g}\)
Pitch radius of the pinion and gear

- \(T_{p,g}\)
Driving and driven torque

- \(T_{pm}\)
Mean part of the drag torque

- \(T_{pp}^{k}\)
Amplitude of vibratory part of the \(k{\text{th}}\) harmonic

- \(\varphi_{p}^{i}\)
Initial phase of \(i{\text{th}}\) harmonic

- \(\zeta_{1,2}\)
Critical viscous damping ratio of lubricant and solid

- \(K_{1,2}\)
Stiffness of lubricant and solid

- \(N\)
Initial resolution of the numerical solution

- \(M\)
Number of the period

- \(\omega\)
Fundamental frequency

- ∈
Profile contact ratio of the gears

- \(\varepsilon\)
Small value defining transition area

- \(\nabla t_{ - , + }^{max}\)
Maximum time step for lubricant and solid contact

- \(\theta_{p,g}\)
Rotational displacements of the pinion and gear

## 1 Introduction

Multi-mesh gear systems are generally used in all kinds of industrial machinery and the tooth impacts generated by input and output torque fluctuations are often caused by the presence of gear backlash. Such impacts could result in intense vibration problem and heavy dynamic loads that could affect the product quality and reliability of the gear drive.

Dynamic modeling of gear vibration can be used to promote our understanding of the vibration generation mechanisms in gear transmissions as well as the dynamic behaviors in the presence of various types of gear tooth damage [1, 2, 3, 4]. Therefore, a lot of research has been undertaken to study the dynamic modeling of the gear pairs. The first model about gear system can be attributed to the paper [5]. And a comprehensive review of the mathematical models used in gear dynamics can be found in the paper [6]. Basically, the studied model could be classified into two groups: the first one is the dry impact model. Singh et al. developed a five-speed manual transmission of a front wheel drive automobile model to understand, quantify and control the vibro-impacts problem inducted by gear backlash [7]. Kahraman and Singh constructed a spur gear model with backlash to examine the nonlinear frequency response characteristics for both external and internal excitations [8, 9, 10]. Sheng et al. proposed a vibro-impact dynamic model of a spur gear train by incorporating the constant meshing stiffness, the linear time-invariant viscous damping values and the gear backlash non-linearity [11]. Liu et al. presented an analytical model of gear system to demonstrate that the sliding friction could not only introduce vibration but also decrease the dynamic transmission errors (DTE) under proper operation condition [12]. Latter, Jiang and Liu developed an analytical helical gear pair model by considering the mass eccentric and mesh stiffness to show the sliding friction play a certain role in inhibiting the amplitude of the frequency [13, 14].

Another one is lubricant impact model, in which the lubricant cannot be negligible because it influences drag torque applied to unloaded gears. Since there is always a formed film of lubricant in the gear backlash under low load condition, thus there is no direct “impacts” with metal-to-metal. However, there are strong or smooth fluctuations of the film thickness instead, which can change the force transmitted between the teeth flanks [15]. Therefore, the gear-impacting surface can be treated as lubricated connections rather than the usually reported solid-to-solid collision. Most researchers mainly described the lubricant effect as the damping: Brancati et al. used the squeeze model to study the gear rattle by assuming that the damping force is proportional to the oil viscosity [16, 17]. Theodossiades et al. introduced a new approach for understanding the interactions between the transmission gears during engine idle conditions by taking into account the effect of driving-side lubrication [18, 19, 20]. Latter, Guilbault et al. calculated the nonlinear damping at different mesh frequencies and torque amplitudes to illustrate the squeeze film damping had main contribution to the global mesh damping [4]. Recently, Liu et al. extended the developments of Theodossiades to characterize the damping and stiffness by considering the double-sided lubricant film, they verified that the effects of the lubricant film on the coast side is significant under low loading condition [21, 22] and the hydrodynamic flank friction has almost no influence on the gear system [23].

In the above studies, the nonlinear dynamic modeling of the gear pairs has achieved successful results and the effects of the lubricant damping on the gear system is general well understood. However, some developments are still needed to ensure a rapid and precise representation of the gear mesh model with a new computational algorithm (NCA). This mainly includes two respects: For one hand, many factors such as gear backlash [8], fluctuation of the driving/driven torque or speed [24] and the lubricant [25, 26] synchronously exert a complicated influence on the gear rattle. Therefore, development of a more accurate mathematical model considered more factors to describe the dynamic behaviors has become an important issue. However, to the authors’ knowledge, constructing a vibro-impact model considered the influence of the lubricant is novel. For another hand, the high value of the contact solid stiffness and the nonlinear backlash impact will cause the ill-conditioning and numerical “stiff” [27], which could present particular difficulties at the simulation stage. In order to overcome this numerical problem, various solution techniques, including analog/digital simulation [28], numerical integration [29, 30], (multi-term) harmonic balance method [31, 32, 33, 34, 35], multi-scale method [36] and finite element method [3, 37, 38, 39] are used to solve the dynamic model in order to obtain the gear dynamic response. Lately, Martin even used the periodic Green’s function in the form of truncated Fourier series to find the analytical periodic solution in a steady state [40]. However, the above methods require for a long time and special concern. Actually, most of the gear dynamic researchers had recognized this problem implicitly, but a few researchers can avoid this obstacle for solving. Actually, most of the solution algorithms available in the literature can not be immediately applied to examine this problem even used the “stiff” solvers, because the Jacobian matrix of the linearizing gear system is discontinuities and varying around different operation point [41]. Hence, it is vital important to develop a NCA which does not use the Jacobian matrix to avoid this “stubborn” as suggested in the paper [7, 42].

Therefore, the objective of this work is to extend the methodology in reference paper [21] and focus on the above-mentioned issues in order to comprehensively understand the gear rattle dynamics. The rest of this paper is organized as follows: in the next section, a vibro-impact model of spur gear is presented by considering the torsional fluctuations resident on both input pinion and drag torque, and lubricant in the gear backlash is considered as the linear element. Then by defining a transition area between solid contact and lubricant contact, a proposed NCA with the double-changed time step is proposed in Sect. 3. Subsequently, the NCA is validated by comparison to the “stiff” solvers in Matlab to illustrate its advantages, and the behaviors of lubricant at different condition and its influences on the dynamic respond of gear system are also discussed. In the last section, concluding remarks are presented.

## 2 Dynamic model formulation

## 3 Proposed numerical method

In this paper, Eqs. (12) and (13) are valid for all intermediate time values, but the parameters (\(k, c,w\)) are changed according to the state of the gear system.

## 4 Results and discussion

Gear data

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

Inertias (kg m | \(I_{p} =\) 7.5886E−5 and \(I_{g} =\) 4.0668E−4 |

Pitch radius (m) | \(R_{p} =\) 0.034 and \(R_{g} =\) 0.052 |

Number of teeth | \(Z_{p} =\) 17 and \(Z_{g} =\) 26 |

Solid stiffness (N/m) | \(K_{s} = 6.20{\text{E}}8\) and \(K_{d} = 1.03{\text{E}}9\) |

Lubricant stiffness (N/m) | \(K_{1} = 1.9{\text{E}}6\) |

Damping ratio | \(\zeta_{1} =\) 0.1 and \(\zeta_{2} =\) 0.04 |

Backlash (m) | \(L = 1{\text{E}} - 5\) |

Pressure angle (°) | 20 |

Contact ratio | 1.567846 |

Computing time

Case | \(K_{1} = 0\) N/m | \(K_{1} = 1.9{\text{E}}6\) N/m | ||||
---|---|---|---|---|---|---|

NCA | ODE23s | ODE15s | NCA | ODE23s | ODE15s | |

Time(s) | 1.12 | 27.07 | 19.39 | 0.90 | 21.78 | 10.58 |

The whole simulation is programmed in Matlab R2015a (The Math Works, Inc., Natick, Massachusetts, USA). From the perspective of comparability, the methods (NCA, ODE15s and ODE23s) are running with the same max time-step (\(1 \times 10^{ - 5} {\text{s}}\)) on an iMac with 3.4 GHz Intel Core i5. Because the steady state responses of the gear system are the focus of this paper, it is necessary to run the numerical program sufficiently long until the difference of DTE between the last two periods is less than \(1 \times 10^{ - 11} {\text{m}}\).

According to the vibration theory [46, 47], it can be known that the equilibrium position only depended on the mean part of the exciting force, and the vibration characteristics mainly subject to the fluctuation part of the exciting force. Therefore, in order to facilitate the analysis of the gear vibration, only the fluctuation part of the driving and driven torques could be considered. Figure 5 shows the steady state responses of the time histories obtained from NCA and ODE23 s by neglecting the lubricant stiffness. The horizontal axis represents the dimensionless time normalized by one mesh circle \(\left( {t_{f} } \right)\) and the dimensionless DTE is normalized by the half of the backlash (\(L/2\)). After a close examination of steady state responses in the time domain, it can be easily found that both time histories obtained by NCA and ODE23 s solver yield virtually the same characteristic even in the nonlinear stiffness transition regions. However, the computation times are very different as shown in Table 2: The required calculation times are 1.11551, 19.39326 and 27.06675 s for NCA, ODE15s and ODE23s, respectively. For NCA case, it is even about 1/20 of the calculation time required with ODE15s case. Figure 6 clearly illustrates time histories obtained from NCA and ODE23s with considering the lubricant stiffness, and the value of the lubricant is choose according to our previous research [21]. It is worth to note that the results for both match precisely in peak-to-peak amplitudes. The required times for both methods are also shown in Table 2. Simulation results indicate that the proposed NCA is useful for accelerating calculation speed of the whole system with a high accuracy of the solution.

*i*and B

*i*(

*i*= 1,2,3), respectively. And the specific values for the maximum DTE are shown in Table 3. As expected, all the maximum DTE increases with the increased drag torque. The two teeth eventually penetrate into each other with an elastic deformation and meanwhile move with a relatively constant acceleration in the non-impact state. However, one must note that two important issues emerge: Firstly, at the start of engaging, impact occurs several times on the driving side before switching over to the coast side. This is mainly because sufficient energy has lost due to viscous damping. Consequently, the normal impact force becomes smaller, and eventually it is not enough to overcome the applied torque. Therefore, the first impact can not drive the tooth to the next contact side, which will require more impacts. Furthermore, the maximum DTE in the state of impact occurs at the second impact, which mainly depended on the excitation frequency. In the state of impact, the acceleration of DTE is very high and the gears suffer tremendous dynamic loads that result in intensive vibration and noise problems. Secondly, the maximum DTE in the state of impact is bigger than that in the state of non-impact when \(T_{gp} = 0\) Nm and the value for the relative location in Table 3 is a minus, which makes clearer that why the gear vibration and noise mostly occurs with unengaged gear pairs.

Values for maximum dimensionless DTE

\(T_{gp}\) (Nm) | \(K_{1} = 0\) N/m | \(K_{1} = 1.9{\text{E}}6\) N/m | ||||
---|---|---|---|---|---|---|

Impact state | Non-impact state | Relative location | Impact state | Non-impact state | Relative location | |

0 | 1.113 | 1.096 | − 0.017 | 1.061 | 1.094 | 0.033 |

20 | 1.123 | 1.124 | 0.001 | 1.074 | 1.121 | 0.047 |

40 | 1.132 | 1.151 | 0.019 | 1.085 | 1.149 | 0.064 |

As anticipated, the lubricant with a higher viscosity could generate a larger stiffness and damping. Depending on the lubricant parameters and the fluctuated input torque, there are three cases at which different solutions can be obtained: (1) single-sided impact, which is more likely happen when the engaged gears is in low speed or the drag torque with the unloaded gear pairs; (2) double-sided impact, which demonstrated multiple impact characteristics: several impacts firstly would occur on the driving side and then the driven gear is pushed back to the coast side; (3) without impact, where the tooth separation is not observed. In order to quantify this contribution, the effects of the pinion’s torque and its variation with different lubricant stiffness on the dynamical motion must be ascertained.

## 5 Conclusions

In this paper, a vibro-impact gear model considering the influences of the lubricant is developed to strengthen the understanding of gear vibration mechanisms with a transition from high to low and vice versa stiffness of the corresponding ODEs describing the dynamics of the gear system. Due to the numerical simulation of such systems deals with highly ‘stiff’ case, especially for the solid-to-solid regime of contact between the gear teeth, a numerical transition technique utilizing a double-change integration time-step adjustment has been proposed for enhancement of analyzing capability in this study. This method overcomes the limitations of known prior research work, in which the solutions are mainly depended on the iteration of the Jacobian matrix of the linearizing gear system.

Numerical results are presented to illustrate and to quantify the effect of the lubricant, as well as the numerical efficiency and accuracy of the NCA in comparison with standard multistep Matlab solvers such as ODE23s and ODE15s. It is observed that the lubricant played an important role in dynamic system, and the velocity and the acceleration of DTE could be much reduced with the lubricant by comparison with the results obtained from pure vibro-impact model (without lubricant). And two different torque assign ratios are considered against driving torque with different lubricant stiffness in order to systematically illustrate the characteristics of tooth impact.

The NCA used in this paper could provide important suggestions that would help researchers to more easily understand the influences of the lubricant on the impingement mode of the gear system. In the end, this NCA presented in this paper also could be extended to study gear system including the contact ratio, friction and multiple clearances in multi-degree-of-freedom systems, which would be addressed in the future’s article.

## Notes

### Acknowledgement

The authors acknowledge the financial support from National Natural Science Foundation of China (Grant No. 51305378), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 17KJB460016), China Postdoctoral Science Foundation funded project (2016M590643), Jiangsu Provincial Science and Technology Department (BY2015057-25) and the Research Laboratory of Mechanical Vibration (MVRLAB).

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