# Influence of Pitch Error on Vibration, Bifurcation, and Chaos Characteristics of the Helical Gear Pair System

Authors: Shijie Pan; Youtang Li; Juane Wang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Aiming at the problem of pitch error of helical gear pair in engineering practice, the influence of pitch error on vibration, bifurcation, and chaos characteristics of the helical gear pair system is mainly studied. Due to the periodic time-varying nature of pitch error, a method of simulating the pitch error as a sine function is proposed to calculate pitch error. A nonlinear dynamic model of bending-torsion-shaft coupling of the helical gear pair system is established considering the effect of pitch error. The influence of pitch error on the vibration, bifurcation, and chaos characteristics of the system is analyzed by the Runge–Kutta numerical integration method. The research results show that the introduction of pitch error has the most significant impact on the torsional vibration of the system. With the increase in pitch error, the system exhibits rich bifurcation and chaos characteristics in the torsional direction. Moreover, it is also found that the vibration response in the torsional orientation of the system increases or decreases to the same degree when the system is in a periodic motion state, and the pitch error varies by the same extent. Therefore, the impact of pitch error on the dynamic performance of the helical gear pair system should be considered in engineering practice.


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## Body

## 1. Introduction

Helical gear pairs are widely used in various machines and mechanical equipment. They are one of the most crucial motion and power transmission devices, whose mechanical properties have an important influence on the whole machine’s vibration, noise, and reliability. Due to the constraints of the actual engineering conditions, there are inevitable meshing errors in the operation of gear pairs. As the most critical component of gear meshing error, pitch error significantly impacts the dynamic characteristics of gear transmission systems. Therefore, it is vital to study the effect of pitch error on the vibration, bifurcation, and chaos characteristics of gear transmission systems.In recent years, many scholars have conducted a lot of research on the nonlinear dynamics of gear systems. Yang et al. [1] established a nonlinear dynamics model of the helical gear system considering tooth wear and analyzed the effect of tooth wear on the bifurcation and chaos characteristics of the system. References [2, 3] discussed in detail the bifurcation and chaos and other nonlinear dynamic characteristics of the multibody mechanical systems with revolute clearance joint and imperfect joint. Wang et al. [4] developed a finite element model of a helical gear rotor system considering the effect of time-varying meshing stiffness and investigated the effect of different coupling forms and helix angles on the vibration characteristics of the system. In literatures [5–10], the effects of tooth surface friction, meshing misalignment, meshing impact, tooth side clearance, and geometric eccentricity on the vibration characteristics of helical gear systems were investigated. Han and Qi [11] studied the influence of tooth spalling and local fracture on the meshing stiffness and vibration characteristics of the helical gear pair system. Huang et al. [12] studied the effects of excitation frequency, backlash, meshing damping ratio, and bearing radial clearance on bifurcation and chaos characteristics of the high contact ratio gear system. Zhang et al. [13] investigated the impact of bearing clearance on the vibration characteristics of the spur gear transmission system. Xu et al. [14] established a mathematical model of spur gear tooth surface including tooth profile deviation and analyzed the influence of different tooth profile deviations on transmission error and dynamic characteristics of the system. Arian and Taghvaei [15] developed a nonlinear dynamics model of a spur gear transmission system with an idler and analyzed the effect of the idler on the chaotic characteristics of the system and the control of system chaos. Saghafi and Farshidianfar [16] proposed a control system for the elimination of chaotic behaviors in spur gear systems and verified the effectiveness of the proposed control system in eliminating homoclinic bifurcation and chaos in nonlinear gear systems. Hua and Chen [17] developed a finite element dynamics model of the bevel gear transmission system considering bearing elasticity and analyzed the effect of bearing elasticity on the vibration characteristics of the system. Yin et al. [18] researched the influence of oil film among meshing teeth on vibration characteristics of the herringbone gear transmission system. Li et al. [19] conducted a global dynamics analysis of a mechanical model with three nonsmooth factors, namely, elastic impact, rigid impact, and dry friction, and comprehensively discussed the effects of different nonsmooth factors on the system motions distribution and transition. Makarenkov and Lamb [20] discussed in detail the research directions in the dynamics of nonsmooth systems and the related bifurcation theories such as border-collision bifurcation and grazing bifurcation. References [21, 22] studied in detail the bifurcation and chaos and other nonlinear dynamic characteristics of the nonsmooth dynamic systems based on the OGY state feedback control law with the help of bifurcation diagrams such as border-collision bifurcation and torus bifurcation.Moreover, in the process of nonlinear dynamic analysis of gear systems, the gear pitch error problem has always been a research hotspot. Umezawa et al. [23, 24] built a pure torsional nonlinear dynamics model of the spur gear system and analyzed the effects of pressure angle and pitch error on the vibration characteristics of the system. Liu et al. [25] built a nonlinear dynamics model of a spur gear pair system considering pitch deviation under multistate meshing conditions and investigated the effects of operating parameters and system parameters on the nonlinear dynamic characteristics of the system under multistate meshing conditions. Handschuh et al. [26] studied the influence of pitch error on tooth root stress of the spur gear pair system. In literatures [27–29], the impact of pitch error on the dynamic transmission error and vibration characteristics of the spur gear pair system was investigated by means of numerical simulations and experiments. Franulovic et al. [30] studied the influence of pitch error on load distribution of contact teeth of the high contact ratio spur gear transmission system. References [31–33] founded the nonlinear dynamic model of the two-stage spur gear transmission system and researched the impact of backlash and pitch error on the vibration and chaos characteristics of the system. Chen and Tang [34] studied the effect of pitch error on the vibration characteristics of the herringbone gear system. References [35–37] investigated the influence of pitch error on meshing stiffness and vibration noise of the cylindrical helical gear transmission system. Guo and Fang [38] proposed a new method to calculate the meshing impact based on the measured pitch error and analyzed the effect of the meshing impact on the vibration and chaos characteristics of the helical gear transmission system. The above literature mainly studies the effect of pitch error on the dynamic characteristics of spur gear systems. However, there are not many articles that consider the impact of pitch error on the dynamic characteristics of helical gear systems and analyze the vibration, bifurcation, and chaos characteristics of helical gear systems.Therefore, this paper comprehensively considers the pitch error, time-varying meshing stiffness, meshing damping, and tooth side clearance and establishes the bending-torsion-shaft coupling nonlinear dynamic model of the helical gear pair system. Through the numerical analysis of the dynamic model, combined with the time-domain diagrams, spectrum maps, phase diagrams, Poincaré cross-section graphs, bifurcation diagram, and largest Lyapunov exponent (LLE) chart of the system response, the influence of pitch error on the vibration, bifurcation, and chaos characteristics of the helical gear pair system is studied in detail. The research results can provide theoretical support for analyzing vibration, bifurcation, and chaos characteristics of the helical gear pair system with pitch errors.The structure of this paper is organized as follows: the nonlinear dynamic model and equations of motion of the system are established in Section2, where the pitch error, time-varying meshing stiffness, meshing damping, and tooth side clearance of the system are given. In Section 3, the equations of motion of the system are solved by the numerical integral method, and then the effect of pitch error on the vibration, bifurcation, and chaos characteristics of the system is investigated. Ultimately, some brief conclusions are given in Section 4.


## 2. The Nonlinear Dynamic Model of Helical Gear Pair considering Pitch Error

Comprehensively considering the pitch error, time-varying meshing stiffness, meshing damping, tooth side clearance, and other nonlinear factors, a six degree of freedom bending-torsion-shaft coupling nonlinear dynamic model of the helical gear pair system with pitch error is established using the lumped mass method [39] as shown in Figure 1. Ignoring the effect of friction between gear teeth, only the torsional vibration of the driving and driven gear around the z-axis, the axial vibration of the driving and driven gear along the z-axis, and the transverse bending vibration of the driving and driven gear along the y-axis are considered. In this model, mp and mg represent the mass of the driving and driven gears, respectively; Ip and Ig represent the moment of inertia; rbp and rbg represent the radius of the base circle; kpz and kgz represent the equivalent bearing support stiffness of the driving and driven gears in the axial direction (z-direction), respectively; kpy and kgy represent the equivalent bearing support stiffness in the tangential direction (y-direction); cpz and cgz represent the equivalent damping in the axial direction; cpy and cgy represent the equivalent damping in the tangential direction; and Tp, Tg, k(t), e(t), cm, 2bn, and βb represent the driving torque, load torque, time-varying meshing stiffness, pitch error, meshing damping, tooth side clearance, and base circle helix angle of the helical gear pair, respectively.Figure 1
Bending-torsion-shaft coupling nonlinear dynamic model.


### 2.1. Equations of Motion of the System

According to Newton’s law of motion, the nonlinear dynamic equations of the system with pitch error can be obtained as follows:(1)mpy¨p+cpyy˙p+kpyyp=Fy,mgy¨g+cgyy˙g+kgyyg=−Fy,mpz¨p+cpzz˙p+kpzzp=−Fz,mgz¨g+cgzz˙gp+kgzzg=Fz,Ipθ¨p=Tp−Fyrbp,Igθ¨g=−Tg+Fyrbg.In equation (1), yp and yg represent the translational displacement of the driving and driven gears along the y-direction, respectively; zp and zg represent the translational displacement of the driving and driven gears along the z-direction, respectively; θp and θg represent the angular displacement of the driving and driven gears around the z-axis, respectively; and Fy and Fz represent the dynamic meshing force of the system along the y-direction and z-direction, respectively.It is known that the pitch error, time-varying meshing stiffness, and meshing damping of the helical gear pair system along the meshing line direction arek(t), e(t), and cm, respectively, which are decomposed into the corresponding y-direction and z-direction pitch error, time-varying meshing stiffness, and meshing damping as follows:(2)eyt=etcosβb,kyt=ktcosβb,cmy=cmcosβb,(3)ezt=etsinβb,kzt=ktsinβb,cmz=cmsinβb.According to equations (2) and (3), the relationship between the pitch error and the dynamic meshing force of the system in y and z directions is established as follows:(4)Fy=cosβbktfyyp+θprbp−yg+θgrbg−eyt+cmy˙p+θ˙prbp−y˙g+θ˙grbg−e˙yt,(5)Fz=sinβbktfzzp−tanβbθprbp+yp−zg+tanβbyg−θgrbg−ezt+cmz˙p−tanβbθ˙prbp+y˙p−z˙g+tanβby˙g−θ˙grbg−e˙zt.In equations (4) and (5), fy and fz represent the tooth side clearance functions of the system along the y-direction and z-direction, respectively.To facilitate the calculation and analysis, equation (1) should be dimensionless. Introducing the displacement nominal scale b′, make δ1 = yp/b′, δ2=yg/b′, δ3 = zp/b′, δ4=zg/b′, and δ5=θprbp−θgrbg−et/b′. Suppose the tangential relative displacement is y1 and the relative axial displacement is y2 of the helical gear pair system. Define the dimensionless time as τ and the dimensionless meshing frequency as ω. y1, y2, τ, and ω are, respectively, as follows:(6)y1=δ1−δ2+δ5−eytb′,(7)y2=δ3−δ4−tanδ1−δ2+δ5−eztb′,(8)τ=ωht,(9)ω=ωnωh,(10)ωn=2πf,(11)ωh=kmme,(12)me=IpIgIprg2+Igrp2.In equations (8)–(12), ωh, f, km, and me are the natural frequency, meshing frequency, average meshing stiffness, and equivalent mass of the helical gear pair system, respectively.The last two torsional dynamic equations in equation (1) are combined into a relative torsional dynamic equation, and then equation (1) is dimensionless to obtain its corresponding dimensionless nonlinear dynamic equations as follows:(13)δ¨1+2η11δ˙1+k11δ1=cosβk10fyy1+2ε10y˙1,δ¨2+2η21δ˙2+k21δ2=−cosβk20fyy1+2ε20y˙1,δ¨3+2η12δ˙3+k12δ3=−sinβk10fzy2+2ε10y˙2,δ¨4+2η22δ˙4+k22δ4=sinβk20fzy2+2ε20y˙2,δ¨5+cosβk0fyy1+2ε0y˙1=fn+ω2eτ.In equation (13), η11=cpy/2mpωn; k11=kpy/2mpωn2; η21=cgy/2mgωn; k21=kgy/mgωn2; η12=cpz/2mpωn; k12=kpz/2mpωn2; η22=cgz/2mgωn; k22=kgz/mgωn2; k10=km/mpωn2; ε10=cm/2mpωn; k20=km/mgωn2; ε20=cm/2mgωn; k0=km/meωn2; ε0=cm/2meωn; fn=Tp/meb′rbpωh2; and e(τ) is dimensionless pitch error, eτ=eτ/b′.


### 2.2. Pitch Error

Pitch error is the difference between the actual pitch and the nominal pitch on the gear reference circle, as shown in Figure2. Due to the presence of pitch error, the position of the gear teeth meshing will deviate from the theoretical meshing position, which causes the gear instantaneous transmission ratio change, resulting in collisions and impacts between the gear meshing tooth surfaces.Figure 2
Pitch error.In the process of gear meshing transmission, the pitch error changes periodically with the meshing period of the gear. Therefore, this paper uses the method of simulating the pitch error as a sine function to calculate the pitch error. The pitch error is expressed as a sine function as follows:(14)et=e0+ersin2πtf+φ.In equation (14), e0 is the mean of pitch error; er is the fluctuation amplitude of pitch error; f is gear meshing frequency; and φ is the initial phase.The main parameters of the helical gear pair in this paper are listed in Table1. According to the basic parameters of the helical gear pair in Table 1, the relevant error values of the helical gear pair are obtained by checking GB/T 10095.1-2008. In addition, make e0 = 0 and φ = 0 and get the simulation curve of pitch error in a meshing cycle of this helical gear pair as shown in Figure 3.Table 1
Main parameters of helical gear pair.
ParameterDriving gearDriven gearNumber of teethz2138Modulemn (mm)2.166Pressure angleαn (°)20Helix angleβ (°)20–20Contact ratioεγ2.7573Face widthB (mm)15Rotational speedn (rpm)20001105.26Accuracy/level77Material4545Figure 3
Simulation curve of pitch error.The above calculation results are basically consistent with the experimental measurement results of the pitch error in reference [40], thus verifying the correctness of the method used.


### 2.3. Calculation of Time-Varying Meshing Stiffness

The contact ratio of helical gear is generally greater than 2. In the process of gear meshing,N(N ≥ 1) pairs of teeth and N + 1 pairs of teeth are engaged alternately, making the meshing stiffness of the gear also change continuously and periodically, i.e., time-varying meshing stiffness. The meshing process of gear is assumed to be composed of many momentarily varying meshing contact lines, and the elastic deformation of the gear teeth is obtained by calculating the total length of the contact lines. On this basis, the time-varying meshing stiffness of the gear is determined.For the helical gear pair studied in this paper, the contact ratio is 2.7573. According to the relationship of the contact ratio, the formula for calculating the total contact line lengthl(t) in one meshing period of the helical gear pair is deduced as follows:(15)lt=2lmax+lmaxt1t,0≤t<t2−2tz,2lmax+lmaxt1t2−2tz,t2−2tz≤t<t1,2lmax+lmaxt1t1+t2−2tz−t,t1≤t<t1+t2−2tz,2lmax,t1+t2−2tz≤t≤tz.In equation (15), lmax is the maximum contact line length in one meshing cycle and tz is the meshing period of the gear pair; its calculation equations are, respectively, as follows:(16)lmax=Bcosβb,tz=60n1z1.Therefore, the time-varying meshing stiffnessk(t) can be obtained as follows [41]:(17)kt=λ⋅lt,where λ is as follows:(18)λ=kmlm.In equation (18), λ is the conversion factor between meshing stiffness and contact line length; km is the average meshing stiffness; and lm is the average value of the total contact line length in a meshing period.Combining the data in Table1, the average meshing stiffness of this helical gear pair is calculated to be 13.3548 N/(μm∙mm) according to the relevant formulas in GB/T 3480.1-2019. On this basis, the time-varying meshing stiffness of the helical gear pair is obtained. In order to facilitate subsequent calculations, this paper uses a 6-order Fourier series to fit the time-varying meshing stiffness to get the following equation:(19)kt=km+∑n=16ancos2πntzt+cnsin2πntzt.In equation (19), an and cn are Fourier series coefficients, and the specific values of the Fourier series coefficients are shown in Table 2. The corresponding fitted curve is shown in Figure 4.Table 2
Fourier series coefficients of time-varying meshing stiffness.
Coefficient codeCorresponding valueCoefficient codeCorresponding valuea1–1.202c11.203a2–5.79e − 5c2–1.15e − 4a3–0.1339c3–0.1335a4–5.784e − 5c49.478e − 8a5–0.04809c50.04819a6–5.79e − 5c6–3.819e − 5Figure 4
Fourier series fitting curve of time-varying meshing stiffness.


### 2.4. Meshing Damping

The meshing damping is mainly related to the meshing damping ratio, average meshing stiffness, and mass of the gear. The meshing damping of the gear can be calculated by the following equation [1]:(20)cm=2ξkmrbp2rbg2IpIgrbp2Ip+rbg2Ig.In equation (20), ξ is the damping ratio, which usually takes the value range of [0.03, 0.17] and is taken as 0.1 in this paper; km is the average meshing stiffness; rbp and rbg are the base circle radius of the driving and driven gears, respectively; and Ip and Ig are the moments of inertia of the driving and driven gears, respectively.According to the main parameters of the helical gear pair in Table1, the relevant values are substituted into equation (20). By calculation, the value of the meshing damping of this helical gear pair can be obtained as follows:(21)cm=247.331Nm⋅s−1.


### 2.5. Tooth Side Clearance

To facilitate the storage of lubricating oil and prevent the tooth surface from being stuck by thermal expansion, gear in the assembly process along the direction of the meshing line to leave a specific tooth side clearance. The tooth side clearance function can be expressed as follows [12]:(22)fxt=xt−bn,xt>bn,0,xt≤bn,xt+bn,xt<−bn.In equation (22), bn is the half tooth side clearance.The correspondingy-direction and z-direction tooth side clearance functions of the helical gear pair system in this paper can be obtained as follows:(23)fyxt=xt−bncosβb,xt>bncosβb,0,xt≤bncosβb,xt+bncosβb,xt<−bncosβb,fzxt=xt−bnsinβb,xt>bnsinβb,0,xt≤bnsinβb,xt+bnsinβbxt<−bnsinβb.


## 2.1. Equations of Motion of the System

According to Newton’s law of motion, the nonlinear dynamic equations of the system with pitch error can be obtained as follows:(1)mpy¨p+cpyy˙p+kpyyp=Fy,mgy¨g+cgyy˙g+kgyyg=−Fy,mpz¨p+cpzz˙p+kpzzp=−Fz,mgz¨g+cgzz˙gp+kgzzg=Fz,Ipθ¨p=Tp−Fyrbp,Igθ¨g=−Tg+Fyrbg.In equation (1), yp and yg represent the translational displacement of the driving and driven gears along the y-direction, respectively; zp and zg represent the translational displacement of the driving and driven gears along the z-direction, respectively; θp and θg represent the angular displacement of the driving and driven gears around the z-axis, respectively; and Fy and Fz represent the dynamic meshing force of the system along the y-direction and z-direction, respectively.It is known that the pitch error, time-varying meshing stiffness, and meshing damping of the helical gear pair system along the meshing line direction arek(t), e(t), and cm, respectively, which are decomposed into the corresponding y-direction and z-direction pitch error, time-varying meshing stiffness, and meshing damping as follows:(2)eyt=etcosβb,kyt=ktcosβb,cmy=cmcosβb,(3)ezt=etsinβb,kzt=ktsinβb,cmz=cmsinβb.According to equations (2) and (3), the relationship between the pitch error and the dynamic meshing force of the system in y and z directions is established as follows:(4)Fy=cosβbktfyyp+θprbp−yg+θgrbg−eyt+cmy˙p+θ˙prbp−y˙g+θ˙grbg−e˙yt,(5)Fz=sinβbktfzzp−tanβbθprbp+yp−zg+tanβbyg−θgrbg−ezt+cmz˙p−tanβbθ˙prbp+y˙p−z˙g+tanβby˙g−θ˙grbg−e˙zt.In equations (4) and (5), fy and fz represent the tooth side clearance functions of the system along the y-direction and z-direction, respectively.To facilitate the calculation and analysis, equation (1) should be dimensionless. Introducing the displacement nominal scale b′, make δ1 = yp/b′, δ2=yg/b′, δ3 = zp/b′, δ4=zg/b′, and δ5=θprbp−θgrbg−et/b′. Suppose the tangential relative displacement is y1 and the relative axial displacement is y2 of the helical gear pair system. Define the dimensionless time as τ and the dimensionless meshing frequency as ω. y1, y2, τ, and ω are, respectively, as follows:(6)y1=δ1−δ2+δ5−eytb′,(7)y2=δ3−δ4−tanδ1−δ2+δ5−eztb′,(8)τ=ωht,(9)ω=ωnωh,(10)ωn=2πf,(11)ωh=kmme,(12)me=IpIgIprg2+Igrp2.In equations (8)–(12), ωh, f, km, and me are the natural frequency, meshing frequency, average meshing stiffness, and equivalent mass of the helical gear pair system, respectively.The last two torsional dynamic equations in equation (1) are combined into a relative torsional dynamic equation, and then equation (1) is dimensionless to obtain its corresponding dimensionless nonlinear dynamic equations as follows:(13)δ¨1+2η11δ˙1+k11δ1=cosβk10fyy1+2ε10y˙1,δ¨2+2η21δ˙2+k21δ2=−cosβk20fyy1+2ε20y˙1,δ¨3+2η12δ˙3+k12δ3=−sinβk10fzy2+2ε10y˙2,δ¨4+2η22δ˙4+k22δ4=sinβk20fzy2+2ε20y˙2,δ¨5+cosβk0fyy1+2ε0y˙1=fn+ω2eτ.In equation (13), η11=cpy/2mpωn; k11=kpy/2mpωn2; η21=cgy/2mgωn; k21=kgy/mgωn2; η12=cpz/2mpωn; k12=kpz/2mpωn2; η22=cgz/2mgωn; k22=kgz/mgωn2; k10=km/mpωn2; ε10=cm/2mpωn; k20=km/mgωn2; ε20=cm/2mgωn; k0=km/meωn2; ε0=cm/2meωn; fn=Tp/meb′rbpωh2; and e(τ) is dimensionless pitch error, eτ=eτ/b′.


## 2.2. Pitch Error

Pitch error is the difference between the actual pitch and the nominal pitch on the gear reference circle, as shown in Figure2. Due to the presence of pitch error, the position of the gear teeth meshing will deviate from the theoretical meshing position, which causes the gear instantaneous transmission ratio change, resulting in collisions and impacts between the gear meshing tooth surfaces.Figure 2
Pitch error.In the process of gear meshing transmission, the pitch error changes periodically with the meshing period of the gear. Therefore, this paper uses the method of simulating the pitch error as a sine function to calculate the pitch error. The pitch error is expressed as a sine function as follows:(14)et=e0+ersin2πtf+φ.In equation (14), e0 is the mean of pitch error; er is the fluctuation amplitude of pitch error; f is gear meshing frequency; and φ is the initial phase.The main parameters of the helical gear pair in this paper are listed in Table1. According to the basic parameters of the helical gear pair in Table 1, the relevant error values of the helical gear pair are obtained by checking GB/T 10095.1-2008. In addition, make e0 = 0 and φ = 0 and get the simulation curve of pitch error in a meshing cycle of this helical gear pair as shown in Figure 3.Table 1
Main parameters of helical gear pair.
ParameterDriving gearDriven gearNumber of teethz2138Modulemn (mm)2.166Pressure angleαn (°)20Helix angleβ (°)20–20Contact ratioεγ2.7573Face widthB (mm)15Rotational speedn (rpm)20001105.26Accuracy/level77Material4545Figure 3
Simulation curve of pitch error.The above calculation results are basically consistent with the experimental measurement results of the pitch error in reference [40], thus verifying the correctness of the method used.


## 2.3. Calculation of Time-Varying Meshing Stiffness

The contact ratio of helical gear is generally greater than 2. In the process of gear meshing,N(N ≥ 1) pairs of teeth and N + 1 pairs of teeth are engaged alternately, making the meshing stiffness of the gear also change continuously and periodically, i.e., time-varying meshing stiffness. The meshing process of gear is assumed to be composed of many momentarily varying meshing contact lines, and the elastic deformation of the gear teeth is obtained by calculating the total length of the contact lines. On this basis, the time-varying meshing stiffness of the gear is determined.For the helical gear pair studied in this paper, the contact ratio is 2.7573. According to the relationship of the contact ratio, the formula for calculating the total contact line lengthl(t) in one meshing period of the helical gear pair is deduced as follows:(15)lt=2lmax+lmaxt1t,0≤t<t2−2tz,2lmax+lmaxt1t2−2tz,t2−2tz≤t<t1,2lmax+lmaxt1t1+t2−2tz−t,t1≤t<t1+t2−2tz,2lmax,t1+t2−2tz≤t≤tz.In equation (15), lmax is the maximum contact line length in one meshing cycle and tz is the meshing period of the gear pair; its calculation equations are, respectively, as follows:(16)lmax=Bcosβb,tz=60n1z1.Therefore, the time-varying meshing stiffnessk(t) can be obtained as follows [41]:(17)kt=λ⋅lt,where λ is as follows:(18)λ=kmlm.In equation (18), λ is the conversion factor between meshing stiffness and contact line length; km is the average meshing stiffness; and lm is the average value of the total contact line length in a meshing period.Combining the data in Table1, the average meshing stiffness of this helical gear pair is calculated to be 13.3548 N/(μm∙mm) according to the relevant formulas in GB/T 3480.1-2019. On this basis, the time-varying meshing stiffness of the helical gear pair is obtained. In order to facilitate subsequent calculations, this paper uses a 6-order Fourier series to fit the time-varying meshing stiffness to get the following equation:(19)kt=km+∑n=16ancos2πntzt+cnsin2πntzt.In equation (19), an and cn are Fourier series coefficients, and the specific values of the Fourier series coefficients are shown in Table 2. The corresponding fitted curve is shown in Figure 4.Table 2
Fourier series coefficients of time-varying meshing stiffness.
Coefficient codeCorresponding valueCoefficient codeCorresponding valuea1–1.202c11.203a2–5.79e − 5c2–1.15e − 4a3–0.1339c3–0.1335a4–5.784e − 5c49.478e − 8a5–0.04809c50.04819a6–5.79e − 5c6–3.819e − 5Figure 4
Fourier series fitting curve of time-varying meshing stiffness.


## 2.4. Meshing Damping

The meshing damping is mainly related to the meshing damping ratio, average meshing stiffness, and mass of the gear. The meshing damping of the gear can be calculated by the following equation [1]:(20)cm=2ξkmrbp2rbg2IpIgrbp2Ip+rbg2Ig.In equation (20), ξ is the damping ratio, which usually takes the value range of [0.03, 0.17] and is taken as 0.1 in this paper; km is the average meshing stiffness; rbp and rbg are the base circle radius of the driving and driven gears, respectively; and Ip and Ig are the moments of inertia of the driving and driven gears, respectively.According to the main parameters of the helical gear pair in Table1, the relevant values are substituted into equation (20). By calculation, the value of the meshing damping of this helical gear pair can be obtained as follows:(21)cm=247.331Nm⋅s−1.


## 2.5. Tooth Side Clearance

To facilitate the storage of lubricating oil and prevent the tooth surface from being stuck by thermal expansion, gear in the assembly process along the direction of the meshing line to leave a specific tooth side clearance. The tooth side clearance function can be expressed as follows [12]:(22)fxt=xt−bn,xt>bn,0,xt≤bn,xt+bn,xt<−bn.In equation (22), bn is the half tooth side clearance.The correspondingy-direction and z-direction tooth side clearance functions of the helical gear pair system in this paper can be obtained as follows:(23)fyxt=xt−bncosβb,xt>bncosβb,0,xt≤bncosβb,xt+bncosβb,xt<−bncosβb,fzxt=xt−bnsinβb,xt>bnsinβb,0,xt≤bnsinβb,xt+bnsinβbxt<−bnsinβb.


## 3. Numerical Results and Discussion

### 3.1. Analysis of System Vibration Characteristics with and without Pitch Error

According to whether the pitch error is added to the nonlinear dynamic equations of the system, the system equations are solved, and the vibration acceleration time-domain curves of the system in each degree of freedom direction with and without pitch error are obtained as shown in Figure5.Figure 5
The time-domain curves of vibration acceleration of the helical gear pair system with and without pitch error: (a) tangential vibration acceleration of driving gear; (b) tangential vibration acceleration of driven gear; (c) axial vibration acceleration of driving gear; (d) axial vibration acceleration of driven gear; (e) vibration acceleration in the torsional direction.
(a)(b)(c)(d)(e)It can be seen from Figure5 that(1)
Since the system operates at 2000 rpm, its dimensionless meshing frequency is calculated to be 0.1394 and the corresponding dimensionless meshing period is 7.1736. From Figure5, it can be seen that the vibration acceleration response of the system in each direction has apparent periodicity and the corresponding period is very close to the dimensionless meshing period, so the accuracy of the calculation results can be confirmed from this perspective.(2)
The changing trend of the time-domain curves of vibration acceleration in each direction after adding the pitch error is basically the same as the changing trend of the time-domain curves of vibration acceleration in each direction without adding the pitch error, still showing obvious periodicity, but the phase changes to a certain extent. In addition, the amplitudes of vibration acceleration in each direction increase slightly after adding the pitch error.(3)
The amplitudes of vibration acceleration in the torsional direction of the system are much greater than those in the tangential and axial directions. This shows that torsional vibration is the primary vibration form of the system, while tangential and axial vibration can be considered as a coupling vibration generated by torsional vibration as the excitation force.(4)
The variations of vibration acceleration amplitude in each direction of the system after adding pitch error are shown in Table3. It can be seen from Table 3 that the maximum change degree of vibration acceleration amplitude in each direction after pitch error is in the torsional direction, followed by the axial direction, and the minimum change degree is in the tangential direction. This indicates that the pitch error has the most significant influence on the torsional direction of this helical gear system, followed by the axial direction and the least tangential direction.Table 3
Variations of vibration acceleration amplitude in each degree of freedom direction of the helical gear pair system after adding pitch error.
Tangential directionAxial directionTorsional directionDriving gearDriven gearDriving gearDriven gearVariation of vibration acceleration amplitude0.00090.00070.00220.00190.0113To further study the characteristics of the system vibration response with and without pitch error, the fast Fourier transform (FFT) can be done for the time-domain response of the vibration acceleration in each direction to analyze its frequency domain characteristics. The vibration acceleration spectrum maps of the system in the direction of each degree of freedom are shown in Figure6.Figure 6
Vibration acceleration spectrum maps of the helical gear pair system with and without pitch error: (a) tangential vibration acceleration of driving gear; (b) tangential vibration acceleration of driven gear; (c) axial vibration acceleration of driving gear; (d) axial vibration acceleration of driven gear; (e) vibration acceleration in the torsional direction.
(a)(b)(c)(d)(e)It can be seen from Figure6 that(1)
The main frequency components of the vibration acceleration response of the system in each direction with and without pitch error are all the dimensionless meshing frequency (0.1394). Therefore, in order to prevent the resonance of the system, the overlap between the meshing frequency and the system’s natural frequency should be avoided in the system operation process.(2)
From the perspective analysis of the magnitude of the acceleration amplitude change, the conclusion obtained is consistent with the analysis result of the vibration acceleration in the time domain: after adding the pitch error, the amplitudes of the vibration acceleration vary the extent the most in the torsional direction, followed by the axial direction, and the degree of tangential direction variation is minimal.Through the analysis of Figures5 and 6, it can be seen that the vibration in the torsional direction of the system is the dominant vibration, and the degree of influence on the torsional orientation of the system after adding the pitch error is the greatest, followed by the axial direction, and the tangential direction is the least.


### 3.2. Influence of Pitch Error on Bifurcation and Chaos Characteristics of the System

From the analysis in Section3.1, it can be seen that the introduction of pitch error has a much more significant impact on the torsional direction of the helical gear pair system than on the tangential and axial directions. Therefore, this section mainly investigates the effect on the bifurcation and chaos characteristics of the helical gear pair system in the torsional direction when the pitch error varies numerically.When the pitch error takes the values of [0e(τ), 15e(τ)] and other parameters are kept constant, the system equations are solved. The bifurcation diagram of the relative torsional displacement of the system with the variation of pitch error is obtained as shown in Figure 7(a), and the corresponding LLE chart is shown in Figure 7(b).Figure 7
(a) Bifurcation diagram. (b) LLE chart.
(a)(b)As shown in Figure7, with the increase in pitch error, the motion state of the system undergoes six stages of change: one time periodic motion→two times periodic motion→ quasiperiodic motion→chaotic motion→three times periodic motion→chaotic motion. When the pitch error ∈ [0e(τ), 10.07e(τ)], the system is in one time-periodic motion state, and the corresponding LLE is less than 0. When the pitch error ∈ [10.08e(τ), 11.16e(τ)], the system undergoes a doubling bifurcation, bifurcating from one time-periodic motion state to two times periodic motion state, and the corresponding LLE is less than 0. When the pitch error ∈ [11.17e(τ), 11.77e(τ)], the system enters a short quasiperiodic motion state; at this time, the LLE becomes 0. When the pitch error ∈ [11.78e(τ), 12.82e(τ)], the system enters an unstable, chaotic motion state, and the corresponding LLE is greater than 0. After a short period of chaotic motion, when the pitch error ∈ [12.83e(τ), 13.04e(τ)], the system enters a short period of three times periodic motion state again, and the corresponding LLE is less than 0. Finally, the system stays in an unstable and complexly changing state, namely, chaotic motion state, and the corresponding LLE is greater than 0. Moreover, the phenomenon of pitch error hopping occurs at the point of 1.12e(τ), where the LLE is −0.0043. Such sudden transition is named as a discontinuous bifurcation (marked by hopping in Figure 7(a)). This hopping occurs for some bifurcation parameter values for which the nonlinear dynamics can be either periodic or chaotic. In Figure 7(a), the hopping induces a transition from periodic gait to periodic gait.The above is a global analysis of the movement characteristics of the entire helical gear pair system with the change of pitch error. In order to verify whether the system motion state described in Figure7 is correct, the following local analysis is used to further investigate the motion characteristics of the system at some special stages.When the pitch error takes different values, the corresponding time-domain diagrams, phase diagrams, and Poincaré cross-section graphs are obtained as shown in Figures8–13.Figure 8
Pitch error = 5.86e(τ). (a) Time-domain diagram. (b) Phase diagram. (c) Poincaré cross-section graph.
(a)(b)(c)Figure 9
Pitch error = 10.75e(τ). (a) Time-domain diagram. (b) Phase diagram. (c) Poincaré cross-section graph.
(a)(b)(c)Figure 10
Pitch error = 11.64e(τ). (a) Time-domain diagram. (b) Phase diagram. (c) Poincaré cross-section graph.
(a)(b)(c)Figure 11
Pitch error = 12.47e(τ). (a) Time-domain diagram. (b) Phase diagram. (c) Poincaré cross-section graph.
(a)(b)(c)Figure 12
Pitch error = 12.97e(τ). (a) Time-domain diagram. (b) Phase diagram. Poincaré cross-section graph.
(a)(b)(c)Figure 13
Pitch error = 14.68e(τ). (a) Time-domain diagram. (b) Phase diagram. (c) Poincaré cross-section graph.
(a)(b)(c)As shown in Figures8, 9, and 12, when the pitch error takes the values of 5.86e(τ), 10.75e(τ), and 12.97e(τ), the corresponding time-domain diagrams have apparent periodicity, the phase diagrams are a closed curve, and the Poincaré cross-section graphs consist of a finite number of points, so the system is in a periodic motion state at this time. And because the phase diagram corresponding to Figure 8 is a closed curve and the Poincaré cross-section graph is a point, the phase diagram corresponding to Figure 9 is two closed curves, and the Poincaré cross-section graph is two discrete points; the phase diagram corresponding to Figure 12 is three closed curves, and the Poincaré cross-section graph is three discrete points, so the specific motion states of the system at this time are one time, two times, and three times periodic motion states, respectively. As shown in Figure 10, when the pitch error takes the value of 11.64e(τ), its corresponding time-domain diagram has a certain periodicity, the phase diagram is two closed bands of curves, and the Poincaré cross-section graph consists of a finite set of points, and the number of point sets is two, so the system is in a two times quasiperiodic motion state at this time. As shown in Figures 11 and 13, when the pitch error is taken as 12.47e(τ) and 14.68e(τ), the corresponding time-domain diagrams have no apparent periodicity, the phase diagrams are irregular and fill a closed area, and the Poincaré cross-section graphs consist of patches of dense points, so the system is in a chaotic motion state at this time.From the above local analysis, it can be seen that the motion states of the system during the special phase are consistent with the results of the global analysis, thus verifying the correctness of the global analysis.The global and local analysis shows that when the pitch error∈ [0e(τ), 10.07e(τ)], [10.08e(τ), 11.16e(τ)], and [12.83e(τ), 13.04e(τ)], the system is in one time, two times, and three times periodic motion state, respectively, which is the ideal working region of the system. At this time, the system’s movement is more stable, and the vibration noise is small.


### 3.3. Impact of Pitch Error on the Vibration Characteristics of the System

The analysis of Section3.2 shows that the system is in a periodic motion state when the pitch error ∈ [0e(τ), 11.16e(τ)]. Therefore, this section mainly studies the effect on the vibration characteristics of the system in the torsional direction when the pitch error varies by the same degree in the cyclic motion state.The pitch errors ofe(τ), 3e(τ), 5e(τ), 7e(τ), 9e(τ), and 11e(τ) are taken into the system equations, respectively, and the other parameters remain unchanged. The system equations are solved to obtain the time-domain curves of vibration acceleration in the torsional direction of the system corresponding to different pitch errors, as shown in Figure 14.Figure 14
Time-domain curves of vibration acceleration in the torsional direction of the system with different pitch errors.It can be seen from Figure14 that(1)
When the value of pitch error increases, the corresponding amplitude of vibration acceleration also increases accordingly.(2)
When the pitch error changes frome(τ) to 3e(τ), 3e(τ) to 5e(τ), 5e(τ) to 7e(τ), 7e(τ) to 9e(τ), and 9e(τ) to 11e(τ), their corresponding vibration acceleration amplitude changes are 0.2163, 0.0354, 0.0359, 0.0356, and 0.0357, respectively. It can be seen that when the pitch error changes from e(τ) to 3e(τ), the corresponding vibration acceleration amplitude changes significantly more than in the other cases. The main reason is that the pitch error causes the jump phenomenon when the pitch error changes from e(τ) to 3e(τ).(3)
When the pitch error varies by the same degree (2e(τ)) and there is no jump phenomenon in the process of pitch error variation, the system will produce a certain vibration response change in the torsional direction.To further study the characteristics of the system torsional direction vibration response when the pitch error changes, the FFT is done on the time-domain response of the system torsional direction vibration acceleration, and the spectrum map of the system torsional direction vibration acceleration at different pitch errors is obtained as shown in Figure15.Figure 15
Vibration acceleration spectrum map of the system in the torsional direction at different pitch errors.It can be seen from Figure15 that(1)
The main frequency components of the vibration acceleration response of the system in the torsional direction for different pitch errors are all dimensionless meshing frequencies (0.1394).(2)
From the perspective analysis of acceleration response amplitude change, the conclusion obtained is consistent with the result of vibration acceleration in the time-domain analysis: when the pitch error varies by the same degree (2e(τ)) and there is no jump phenomenon in the process of pitch error variation, it will produce a certain impact on the torsional direction vibration response of the system.Through the analysis of Figures14 and 15, it can be seen that when the system is in a periodic motion state, the pitch error changes to the same degree (2e(τ)) and there is no jump phenomenon during the change of pitch error, and the system will produce a specific vibration response change in the torsional direction, i.e., the increased or decreased degree of the vibration response in the torsional orientation of the system caused by the change of the pitch error by the same degree each time is almost the same.


## 3.1. Analysis of System Vibration Characteristics with and without Pitch Error

According to whether the pitch error is added to the nonlinear dynamic equations of the system, the system equations are solved, and the vibration acceleration time-domain curves of the system in each degree of freedom direction with and without pitch error are obtained as shown in Figure5.Figure 5
The time-domain curves of vibration acceleration of the helical gear pair system with and without pitch error: (a) tangential vibration acceleration of driving gear; (b) tangential vibration acceleration of driven gear; (c) axial vibration acceleration of driving gear; (d) axial vibration acceleration of driven gear; (e) vibration acceleration in the torsional direction.
(a)(b)(c)(d)(e)It can be seen from Figure5 that(1)
Since the system operates at 2000 rpm, its dimensionless meshing frequency is calculated to be 0.1394 and the corresponding dimensionless meshing period is 7.1736. From Figure5, it can be seen that the vibration acceleration response of the system in each direction has apparent periodicity and the corresponding period is very close to the dimensionless meshing period, so the accuracy of the calculation results can be confirmed from this perspective.(2)
The changing trend of the time-domain curves of vibration acceleration in each direction after adding the pitch error is basically the same as the changing trend of the time-domain curves of vibration acceleration in each direction without adding the pitch error, still showing obvious periodicity, but the phase changes to a certain extent. In addition, the amplitudes of vibration acceleration in each direction increase slightly after adding the pitch error.(3)
The amplitudes of vibration acceleration in the torsional direction of the system are much greater than those in the tangential and axial directions. This shows that torsional vibration is the primary vibration form of the system, while tangential and axial vibration can be considered as a coupling vibration generated by torsional vibration as the excitation force.(4)
The variations of vibration acceleration amplitude in each direction of the system after adding pitch error are shown in Table3. It can be seen from Table 3 that the maximum change degree of vibration acceleration amplitude in each direction after pitch error is in the torsional direction, followed by the axial direction, and the minimum change degree is in the tangential direction. This indicates that the pitch error has the most significant influence on the torsional direction of this helical gear system, followed by the axial direction and the least tangential direction.Table 3
Variations of vibration acceleration amplitude in each degree of freedom direction of the helical gear pair system after adding pitch error.
Tangential directionAxial directionTorsional directionDriving gearDriven gearDriving gearDriven gearVariation of vibration acceleration amplitude0.00090.00070.00220.00190.0113To further study the characteristics of the system vibration response with and without pitch error, the fast Fourier transform (FFT) can be done for the time-domain response of the vibration acceleration in each direction to analyze its frequency domain characteristics. The vibration acceleration spectrum maps of the system in the direction of each degree of freedom are shown in Figure6.Figure 6
Vibration acceleration spectrum maps of the helical gear pair system with and without pitch error: (a) tangential vibration acceleration of driving gear; (b) tangential vibration acceleration of driven gear; (c) axial vibration acceleration of driving gear; (d) axial vibration acceleration of driven gear; (e) vibration acceleration in the torsional direction.
(a)(b)(c)(d)(e)It can be seen from Figure6 that(1)
The main frequency components of the vibration acceleration response of the system in each direction with and without pitch error are all the dimensionless meshing frequency (0.1394). Therefore, in order to prevent the resonance of the system, the overlap between the meshing frequency and the system’s natural frequency should be avoided in the system operation process.(2)
From the perspective analysis of the magnitude of the acceleration amplitude change, the conclusion obtained is consistent with the analysis result of the vibration acceleration in the time domain: after adding the pitch error, the amplitudes of the vibration acceleration vary the extent the most in the torsional direction, followed by the axial direction, and the degree of tangential direction variation is minimal.Through the analysis of Figures5 and 6, it can be seen that the vibration in the torsional direction of the system is the dominant vibration, and the degree of influence on the torsional orientation of the system after adding the pitch error is the greatest, followed by the axial direction, and the tangential direction is the least.


## 3.2. Influence of Pitch Error on Bifurcation and Chaos Characteristics of the System

From the analysis in Section3.1, it can be seen that the introduction of pitch error has a much more significant impact on the torsional direction of the helical gear pair system than on the tangential and axial directions. Therefore, this section mainly investigates the effect on the bifurcation and chaos characteristics of the helical gear pair system in the torsional direction when the pitch error varies numerically.When the pitch error takes the values of [0e(τ), 15e(τ)] and other parameters are kept constant, the system equations are solved. The bifurcation diagram of the relative torsional displacement of the system with the variation of pitch error is obtained as shown in Figure 7(a), and the corresponding LLE chart is shown in Figure 7(b).Figure 7
(a) Bifurcation diagram. (b) LLE chart.
(a)(b)As shown in Figure7, with the increase in pitch error, the motion state of the system undergoes six stages of change: one time periodic motion→two times periodic motion→ quasiperiodic motion→chaotic motion→three times periodic motion→chaotic motion. When the pitch error ∈ [0e(τ), 10.07e(τ)], the system is in one time-periodic motion state, and the corresponding LLE is less than 0. When the pitch error ∈ [10.08e(τ), 11.16e(τ)], the system undergoes a doubling bifurcation, bifurcating from one time-periodic motion state to two times periodic motion state, and the corresponding LLE is less than 0. When the pitch error ∈ [11.17e(τ), 11.77e(τ)], the system enters a short quasiperiodic motion state; at this time, the LLE becomes 0. When the pitch error ∈ [11.78e(τ), 12.82e(τ)], the system enters an unstable, chaotic motion state, and the corresponding LLE is greater than 0. After a short period of chaotic motion, when the pitch error ∈ [12.83e(τ), 13.04e(τ)], the system enters a short period of three times periodic motion state again, and the corresponding LLE is less than 0. Finally, the system stays in an unstable and complexly changing state, namely, chaotic motion state, and the corresponding LLE is greater than 0. Moreover, the phenomenon of pitch error hopping occurs at the point of 1.12e(τ), where the LLE is −0.0043. Such sudden transition is named as a discontinuous bifurcation (marked by hopping in Figure 7(a)). This hopping occurs for some bifurcation parameter values for which the nonlinear dynamics can be either periodic or chaotic. In Figure 7(a), the hopping induces a transition from periodic gait to periodic gait.The above is a global analysis of the movement characteristics of the entire helical gear pair system with the change of pitch error. In order to verify whether the system motion state described in Figure7 is correct, the following local analysis is used to further investigate the motion characteristics of the system at some special stages.When the pitch error takes different values, the corresponding time-domain diagrams, phase diagrams, and Poincaré cross-section graphs are obtained as shown in Figures8–13.Figure 8
Pitch error = 5.86e(τ). (a) Time-domain diagram. (b) Phase diagram. (c) Poincaré cross-section graph.
(a)(b)(c)Figure 9
Pitch error = 10.75e(τ). (a) Time-domain diagram. (b) Phase diagram. (c) Poincaré cross-section graph.
(a)(b)(c)Figure 10
Pitch error = 11.64e(τ). (a) Time-domain diagram. (b) Phase diagram. (c) Poincaré cross-section graph.
(a)(b)(c)Figure 11
Pitch error = 12.47e(τ). (a) Time-domain diagram. (b) Phase diagram. (c) Poincaré cross-section graph.
(a)(b)(c)Figure 12
Pitch error = 12.97e(τ). (a) Time-domain diagram. (b) Phase diagram. Poincaré cross-section graph.
(a)(b)(c)Figure 13
Pitch error = 14.68e(τ). (a) Time-domain diagram. (b) Phase diagram. (c) Poincaré cross-section graph.
(a)(b)(c)As shown in Figures8, 9, and 12, when the pitch error takes the values of 5.86e(τ), 10.75e(τ), and 12.97e(τ), the corresponding time-domain diagrams have apparent periodicity, the phase diagrams are a closed curve, and the Poincaré cross-section graphs consist of a finite number of points, so the system is in a periodic motion state at this time. And because the phase diagram corresponding to Figure 8 is a closed curve and the Poincaré cross-section graph is a point, the phase diagram corresponding to Figure 9 is two closed curves, and the Poincaré cross-section graph is two discrete points; the phase diagram corresponding to Figure 12 is three closed curves, and the Poincaré cross-section graph is three discrete points, so the specific motion states of the system at this time are one time, two times, and three times periodic motion states, respectively. As shown in Figure 10, when the pitch error takes the value of 11.64e(τ), its corresponding time-domain diagram has a certain periodicity, the phase diagram is two closed bands of curves, and the Poincaré cross-section graph consists of a finite set of points, and the number of point sets is two, so the system is in a two times quasiperiodic motion state at this time. As shown in Figures 11 and 13, when the pitch error is taken as 12.47e(τ) and 14.68e(τ), the corresponding time-domain diagrams have no apparent periodicity, the phase diagrams are irregular and fill a closed area, and the Poincaré cross-section graphs consist of patches of dense points, so the system is in a chaotic motion state at this time.From the above local analysis, it can be seen that the motion states of the system during the special phase are consistent with the results of the global analysis, thus verifying the correctness of the global analysis.The global and local analysis shows that when the pitch error∈ [0e(τ), 10.07e(τ)], [10.08e(τ), 11.16e(τ)], and [12.83e(τ), 13.04e(τ)], the system is in one time, two times, and three times periodic motion state, respectively, which is the ideal working region of the system. At this time, the system’s movement is more stable, and the vibration noise is small.


## 3.3. Impact of Pitch Error on the Vibration Characteristics of the System

The analysis of Section3.2 shows that the system is in a periodic motion state when the pitch error ∈ [0e(τ), 11.16e(τ)]. Therefore, this section mainly studies the effect on the vibration characteristics of the system in the torsional direction when the pitch error varies by the same degree in the cyclic motion state.The pitch errors ofe(τ), 3e(τ), 5e(τ), 7e(τ), 9e(τ), and 11e(τ) are taken into the system equations, respectively, and the other parameters remain unchanged. The system equations are solved to obtain the time-domain curves of vibration acceleration in the torsional direction of the system corresponding to different pitch errors, as shown in Figure 14.Figure 14
Time-domain curves of vibration acceleration in the torsional direction of the system with different pitch errors.It can be seen from Figure14 that(1)
When the value of pitch error increases, the corresponding amplitude of vibration acceleration also increases accordingly.(2)
When the pitch error changes frome(τ) to 3e(τ), 3e(τ) to 5e(τ), 5e(τ) to 7e(τ), 7e(τ) to 9e(τ), and 9e(τ) to 11e(τ), their corresponding vibration acceleration amplitude changes are 0.2163, 0.0354, 0.0359, 0.0356, and 0.0357, respectively. It can be seen that when the pitch error changes from e(τ) to 3e(τ), the corresponding vibration acceleration amplitude changes significantly more than in the other cases. The main reason is that the pitch error causes the jump phenomenon when the pitch error changes from e(τ) to 3e(τ).(3)
When the pitch error varies by the same degree (2e(τ)) and there is no jump phenomenon in the process of pitch error variation, the system will produce a certain vibration response change in the torsional direction.To further study the characteristics of the system torsional direction vibration response when the pitch error changes, the FFT is done on the time-domain response of the system torsional direction vibration acceleration, and the spectrum map of the system torsional direction vibration acceleration at different pitch errors is obtained as shown in Figure15.Figure 15
Vibration acceleration spectrum map of the system in the torsional direction at different pitch errors.It can be seen from Figure15 that(1)
The main frequency components of the vibration acceleration response of the system in the torsional direction for different pitch errors are all dimensionless meshing frequencies (0.1394).(2)
From the perspective analysis of acceleration response amplitude change, the conclusion obtained is consistent with the result of vibration acceleration in the time-domain analysis: when the pitch error varies by the same degree (2e(τ)) and there is no jump phenomenon in the process of pitch error variation, it will produce a certain impact on the torsional direction vibration response of the system.Through the analysis of Figures14 and 15, it can be seen that when the system is in a periodic motion state, the pitch error changes to the same degree (2e(τ)) and there is no jump phenomenon during the change of pitch error, and the system will produce a specific vibration response change in the torsional direction, i.e., the increased or decreased degree of the vibration response in the torsional orientation of the system caused by the change of the pitch error by the same degree each time is almost the same.


## 4. Conclusions and Future Works

In this paper, a bending-torsion-shaft coupling nonlinear dynamic model of the helical gear pair system is established considering the effect of pitch error. The pitch error is calculated by using the method of simulating the pitch error as a sine function. Then, the motion equations of the system are solved by the Runge–Kutta numerical integration method. The effect of pitch error on the vibration, bifurcation, and chaos characteristics of the system is investigated in detail.The major conclusions can be summarized as follows:(1)
The introduction of pitch error has the greatest degree of effect on the vibration acceleration response of the helical gear pair system in the torsional direction, followed by the axial direction and the smallest in the tangential direction.(2)
With the increase in pitch error, the helical gear pair system shows complex motion characteristics in the torsional direction. Its motion state undergoes six stages of changes: one time-periodic motion, two times periodic motion, quasiperiodic motion, chaotic motion, three times periodic motion, and chaotic motion. Ultimately the system stays in an unstable and complexly changing chaotic motion state.(3)
Under the condition that the helical gear pair system is in a periodic motion state when the pitch error varies to the same extent and there is no jump phenomenon in the process of pitch error variation, the system will produce a certain change in vibration response in the torsional direction.Based on the conclusions of the work, the following can be further explored:(1)
We investigated the nonlinear dynamic characteristics of the six degrees of freedom helical gear pair system in the present work. As a future direction, our methodology introduced in the current work can be extended to analyze nonlinear dynamic characteristics of helical gear pair systems with more degrees of freedom.(2)
Based on the optimal control principle, a reasonable control strategy can be proposed, and then the corresponding controller can be designed to realize the smooth operation of the gear system.



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*Source: 6338625-2021-11-11.xml*