The torsional dynamic model of double-helical gear pair considering time-varying meshing stiffness, constant backlash, dynamic backlash, static transmission error, and external dynamic excitation was established. The frequency response characteristics of the system under constant and dynamic backlashes were solved by the incremental harmonic balance method, and the results were further verified by the numerical integration method. At the same time, the influence of time-varying meshing stiffness, damping, static transmission error, and external load excitation on the amplitude frequency characteristics of the system was analyzed. The results show that there is not only main harmonic response but also superharmonic response in the system. The time-varying meshing stiffness and static transmission error can stimulate the amplitude frequency response of the system, while the damping can restrain the amplitude frequency response of the system. Changing the external load excitation has little effect on the amplitude frequency response state change of the system. Compared with the constant backlash, increasing the dynamic backlash amplitude can further control the nonlinear vibration of the gear system.

Double-helical gear is widely used in vehicle, ship, and other heavy machinery transmission systems because of its strong bearing capacity, compact structure, and axial force cancellation. Its nonlinear vibration characteristics will directly affect the stability and reliability of the transmission system. Comparin and Singh [

To sum up, the research on the nonlinear characteristics of the gear transmission system at home and abroad mostly focused on the spur gear and spur planetary gear system. In the reports on the nonlinear dynamics of double-helical gear, there are few reports on the nonlinear frequency response characteristics of double-helical gear. Based on the above research, a torsional dynamic model of the double-helical gear pair system is established in this study. The time-varying meshing stiffness, static backlash, dynamic backlash, static transmission error, external dynamic excitation, and other factors are considered in the model. The influence of parameter changes on the nonlinear dynamic characteristics of the double-helical gear pair system is studied by using the incremental harmonic balance method. The research results can provide reference for vibration reduction analysis and structure optimization of double-helical gear.

As shown in Figure

Torsional nonlinear dynamic model of double-helical gear pair.

According to Newton’s second law, the torsional nonlinear dynamic equation of double-helical gear pair can be written as

In the above formula, _{ij}(_{pL}, _{pR}, _{gL}, and _{gR} at the center of each helical gear, _{ij}(_{ij}(_{j}(_{mj}, _{mj}(_{j}(_{j}(_{it}_{it}(_{pL}(_{pR}(_{gL}(_{gR}(

Each moment can be expressed as follows:_{pL}, _{pR}, _{gL}, and _{gR} are the external nominal moments, and _{pLa}(_{pRa}(_{gLa}(_{gRa}(

The four degrees of freedom _{pL}, _{pR}, _{gL}, and _{gR} in the nonlinear torsional dynamic equation of double-helical gear transmission are not independent of each other. The relative displacement method is used to eliminate the rigid body displacement of the system to solve the differential equation. The absolute torsional angular displacement _{pL}, _{pR}, _{gL}, and _{gR} on the original mass nodes is replaced by the relative meshing displacement _{L}, _{R} along the meshing line direction of the left and right helical gears and the relative torsional line displacement _{P}, _{G} between the coaxial adjacent helical gears.

The relative meshing displacement of the left and right helical gear pairs along the direction of the meshing line is expressed as follows:

The expression of relative torsional line displacement of adjacent coaxial helical gears is as follows:

The actual meshing displacement of double-helical gear along the meshing line is expressed as a piecewise function as follows:

Meshing impact state of double-helical gear pair.

In the existing research, the backlash function is expressed as piecewise linear. Generally, the backlash of gear teeth is obtained by static measurement, and the change of backlash in the process of gear impact is not considered, so the simulation results cannot truly reflect the transmission characteristics of the gear system. In order to reveal the influence of the actual tooth surface characteristics on the vibration and noise of the gear transmission system, the constant backlash model and dynamic backlash model will be considered simultaneously in this study.

The constant backlash model takes the backlash as a constant according to the traditional modeling method. The expression is as follows:

The dynamic backlash model assumes that the backlash is in the form of harmonics. The expression is as follows:_{0} is the equivalent backlash, _{l} is the amplitude of dynamic backlash representing the deformation of the supporting bearing and axle. By substituting equations (_{c} is the equivalent mass of the left and right helical gear pairs, and _{a} is the fluctuation of external load excitation, which are, respectively, expressed as follows:_{l} is the fluctuation coefficient. The time-varying meshing stiffness function and static transmission error function are expressed in the form of Fourier series [_{mean} is the average stiffness, _{l} is the fluctuation coefficient reflecting the change of stiffness, and _{l} is the fluctuation coefficient reflecting the change of static transmission error. When the excitation of static transmission error is maximum, the time-varying meshing stiffness should be the minimum, that is, the two have inverse phase relationship. The time-varying meshing stiffness curve is shown in Figure

Time-varying mesh stiffness curve.

The meshing damping _{mj} and torsional damping _{it} are calculated by the following formula:

According to the knowledge of material mechanics, the calculation formula of torsional stiffness of intermediate shaft section is as follows:

Equation (_{c} is the nominal displacement scale, Δ_{i}, Δ_{j} is the normalized vibration displacement, _{n} is the natural frequency,

The stiffness term in the above formula is as follows:

The static transmission error term in the excitation vector is as follows:

The dynamic excitation of external torque is as follows:

For the piecewise backlash function, if _{c} = _{0} is taken in the dimensionless process, the two kinds of backlash models are transformed into

Equation (

By letting a new time scale

The Newton–Raphson algorithm is used to solve the piecewise linear difference system represented by equation (

By substituting equations (

Order

By using the Galerkin process [

Equation (

According to equation (

The design parameters of double-helical gear pair are given in Table

Basic parameters of the double-helical gear pair.

Parameter | Pinion | Gear |
---|---|---|

Number of teeth | 25 | 75 |

Helix angle (°) | 24.43 | |

Normal pressure angle (°) | 20 | |

Normal module (mm) | 6 | |

Tooth width (mm) | 55 × 2 | |

Moment of inertia (kg·m^{2}) | 0.012 | 0.7279 |

Mass (kg) | 7.11 | 64.75 |

Input torque (Nm) | 1400 | |

Meshing stiffness (N/m) | 1.8456 × 10^{9} | |

Length of intermediate shaft section (mm) | 58 | |

Outer diameter of intermediate shaft section (mm) | 92 | 260 |

Inner diameter of intermediate shaft section (mm) | 55 | 70 |

In order to fully study the nonlinear characteristics of double-helical gear under the coupling effect of time-varying meshing stiffness and backlash, the amplitude frequency characteristic curves of the system under four different models are obtained, and the results are shown in Figure _{1} = 1.3534, _{1} = 0.16, _{2} = 0.07, _{3} = 0.03, _{1} = 0.03, _{2} = 0.01, _{3} = 0.01, _{1} = 0.06, _{2} = 0.03, _{3} = 0.01, _{1} = 0.05, _{2} = 0.02, and _{3} = 0.01.

Amplitude frequency characteristic curve of the system under different models. (a) Constant backlash model under combined internal and external excitation, (b) constant backlash model under external excitation, (c) constant backlash model under internal excitation, and (d) dynamic backlash model under combined internal and external excitation.

In Figure _{Lmax} and minimum Δ_{Lmin} nonlinear vibration displacement of the double-helical gear system along the meshing line, and the abscissa represents the dimensionalized frequency Ω. Four different models are, respectively, the constant backlash model under combined internal and external dynamic excitations as shown in Figure

Figure

The multivalued properties and jump phenomena are corresponding with no impact motion and single-side impact motion. Figure

Steady-state response of the gear pair system with Ω = 0.83. (a) No impact and (b) single-side impact.

The basic system parameters are _{1} = 1.3534, _{1} = 0.16, _{2} = 0.07, _{3} = 0.03, _{1} = 0.03, _{2} = 0.01, _{3} = 0.01, _{1} = 0.06, _{2} = 0.03, _{3} = 0.01, and

Phase diagram and time history diagram of the system at different frequencies. (a) Ω = 1, (b) Ω = ½, and (c) Ω = 1/3.

Figure

The basic system parameters are _{1} = 1.3534, _{1} = 0.16, _{2} = 0.07, _{3} = 0.03, _{1} = 0.03, _{2} = 0.01, _{3} = 0.01, _{1} = 0.06, _{2} = 0.03, _{3} = 0.01, _{1} = 0.05, _{2} _{3}

Influence of backlash on vibration response of the gear system. (a) Constant backlash models and (b) dynamic backlash models.

From the comparison between the linear time-varying model (zero backlash) and the nonlinear time-varying model (with backlash) in Figure

The basic system parameters are _{1} = 1.3534, _{1} = 0.16, _{2} = 0.07, _{3} = 0.03, _{1} = 0.03, _{2} = 0.01, _{3} = 0.01, _{1} = 0.06, _{2} = 0.03, and _{3} = 0.01. The influence of different constant backlash and dynamic backlash amplitude on the amplitude frequency characteristics of the system is shown in Figure

Amplitude frequency characteristic curve of the system under two kinds of backlash models. (a) Constant backlash models and (b) dynamic backlash models.

For constant backlash, five models of constant backlash are considered: (1) _{1} = 0, _{2} _{3} _{1} = 0.05, _{2} _{3} _{1} = 0.10, _{2} _{3} _{1} = 0.15, _{2} _{3}

It can be seen from Figure

The basic system parameters are _{1} = 1.3534, _{1} = 0.03, _{2} = 0.01, _{3} = 0.01, _{1} = 0.06, _{2} = 0.03, _{3} = 0.01, _{1} = 0.05, _{2} _{3}

Influence of time-varying meshing stiffness on amplitude frequency characteristic curve of the system. (a) Constant backlash models and (b) dynamic backlash models.

In this study, three kinds of time-varying meshing stiffness fluctuation coefficients are considered: (1) _{1} = 0.16, _{2} = 0.07, _{3} = 0.03; (2) _{1} = 0.13 _{2} = 0.04, _{3} = 0.01; and (3) _{1} = 0, _{2} = 0, _{3} = 0. It can be seen from Figures

In addition, compared with the constant backlash model, the dynamic backlash model can control the nonlinear vibration of the system more effectively by reducing the fluctuation coefficient of the meshing stiffness.

The basic system parameters are _{1} = 1.3534, _{1} = 0.16, _{2} = 0.07, _{3} = 0.03, _{1} = 0.03, _{2} = 0.01, _{3} = 0.01, _{1} = 0.06, _{2} = 0.03, _{3} = 0.01, _{1} = 0.05, _{2} _{3}

Influence of damping on amplitude frequency characteristic curve of the system. (a) Constant backlash models and (b) dynamic backlash models.

It can be seen from Figure

The basic system parameters are _{1} = 1.3534, _{1} = 0.16, _{2} = 0.07, _{3} = 0.03, _{1} = 0.06, _{2} = 0.03, _{3} = 0.01, _{1} = 0.05, _{2} _{3}

Influence of static transmission error on amplitude frequency characteristic curve of the system. (a) Constant backlash models and (b) dynamic backlash models.

This study considers three kinds of static transmission error amplitudes: (1) _{1} = 0.1, _{2} = 0.05, _{3} = 0.03; (2) _{1} = 0.03, _{2} = 0.01, _{3} = 0.01; and (3) _{1} = 0, _{2} = 0, _{3} = 0. It can be seen from Figure _{1} = 0, _{2} = 0, and _{3} = 0, and the amplitude frequency curve still has a jump phenomenon and multivalue solution phenomenon, which indicates that the static transmission error in double-helical gear is not the main reason for the nonlinearity of the system. When _{1} = 0.1, _{2} = 0.05, _{3} = 0.03 and _{1} = 0.03, _{2} = 0.01, _{3} = 0.01, the larger the amplitude frequency curve of the system appears, and the larger the value of static transmission error amplitude is, the greater the amplitude frequency characteristic curve amplitude is.

The basic system parameters are _{1} = 0.16, _{2} = 0.07, _{3} = 0.03, _{1} = 0.03, _{2} = 0.01, _{3} = 0.01, _{1} = 0.06, _{2} = 0.03, _{3} = 0.01, _{1} = 0.05, _{2} _{3}

Influence of excitation force amplitude on amplitude frequency characteristic curve of the system. (a) Constant backlash models and (b) dynamic backlash models.

It can be seen from Figure _{1} = 3.3534, the amplitude frequency characteristic curve amplitude is the largest, while there is no multivalue solution interval. When _{1} = 1.3534, the amplitude of amplitude frequency characteristic curve is the smallest, and the range of unstable solution interval is the largest. The transition frequency of curve step basically does not change, and the results of the two backlash models have little difference. With the increase of the excitation amplitude, the amplitude frequency characteristic curve of the system still contains the no impact state and the single-side impact state, and the increase of the excitation amplitude has little effect on the change of the meshing impact state of the double-helical gear.

In this study, the torsional dynamic model of double-helical gear transmission is established. The factors such as time-varying meshing stiffness, static transmission error, constant backlash, dynamic backlash, and external dynamic excitation are considered in the model. The frequency response characteristic curve of the system is better solved by using the incremental harmonic balance method. It is found that the system has multivalue solution and jumping nonlinear characteristics, and the results are in good agreement with the numerical integration results.

The conclusions are as follows:

There are main resonance response and superharmonic response in the double-helical gear system. The generation of the superharmonic response solution is caused by the high-order harmonic component in time-varying meshing stiffness and dynamic excitation.

With the increase of the constant backlash, the vibration displacement amplitude of the gear system increases, and the multivalue solution interval becomes larger; with the increase of the dynamic backlash amplitude, the vibration displacement amplitude of the gear system increases slightly, and the multivalue solution range decreases or even disappears.

The results show that the time-varying meshing stiffness and static transmission error can stimulate the amplitude frequency curve of the system, and the damping can restrain the amplitude frequency curve of the system. Compared with the constant backlash model, the vibration of the double-helical gear system can be further controlled by changing the system parameters in the dynamic backlash model.

The excitation amplitude has little effect on the change of the meshing impact state of the double-helical gear. Under the influence of excitation amplitude, the transition frequency of curve step has no change, and the results of the constant backlash model and dynamic backlash model have little difference.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This research was funded by the National Natural Science Foundation of China (51705390), Innovation Capability Support Program of Shaanxi (2020KJXX-016), Scientific Research Program Funded by Shaanxi Provincial Education Department Program (20JC015), Principal foundation project of Xi’an Technological University (XGPY200201), and the Natural Science Foundation of Shaanxi Province (2021JM-428).