# Investigation of Nonlinear Characteristics of a Gear Transmission System in a Braiding Machine with Multiple Excitation Factors

Authors: Lingling Yao; Zhuo Meng; Jianqiu Bu; Yize Sun
Journal: Shock and Vibration
 (2020)

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## Abstract

In this study, we attempt to analyze the influence of different excitation factors on the dynamic behavior of a gear transmission system in a braiding machine. In order to observe nonlinear characteristics, a mathematical model is established with a six-degrees-of-freedom gear system for consideration of multiple excitation factors. Iterative results are used to study the nonlinear characteristics of the gear system with respect to contact temperature, varying levels of friction, and disturbance of yarn tension using bifurcation diagrams, maximum Lyapunov exponents, phase diagrams, Poincare maps, and the power spectrum. The numerical results show that excitation factors such as temperature and surface friction, among others, have considerable influence on the nonlinear characteristics of the gear system in a braiding machine, and the model is evaluated to show the key regions of sensitivity. The analysis of associated parameters can be helpful in the design and control of braiding machines.


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## Body

## 1. Introduction

Braiding is a traditional technique used in textile production. In recent years, the emergence of new materials and different types of braiding machines has led to an upsurge in research on braiding.Most of the studies of 3-D braided composites focus on the braiding process, structure, parameters, and performance analysis of composite materials. Ma et al. [1] proposed a mathematical model of tension versus yarn displacement, and Guyader et al. [2] analyzed the relationships between the process parameters and the geometry of the braid. Hajrasouliha et al. [3] presented a theoretical model for the prediction of braid angle at any point of a mandrel with constant arbitrary cross section by considering the kinematic parameters of a circular braiding machine. Shen and Branscomb [4] proposed a purely mathematical model to generate the 3D geometry of braided structures, and Wehrkamp-Richter et al. [5] studied the damage and failure characteristics of triaxial braided composites. Swery et al. [6] provided a complete simulation process for predictions on the manufacturing of braided composite parts. However, further research holds the promise of improving the performance of 3-D circular braiding machines. Zhang et al. [7] proposed that the performance of a braiding machine depends on the motion system, and the key component of the motion system is the gear system. The dynamic excitation created by the gear transmission system in a braiding machine is the main source of vibrations, and these nonlinear vibrations reduce braid quality and have become an issue of urgent concern.Gear transmissions are widely used in engineering machinery, ocean engineering, traffic and transportation, metallurgy, and building materials, and they exhibit a long life span, smooth operation, high load capacity, and high reliability. There has been considerable research on gear systems since 1990, and one of the main goals is the development of dynamic models. Dynamic modeling methods include the lumped parameter method, the finite element method, the lumped mass method, the transfer matrix method, and the power bond graph method. In 1990, Kahraman and Singh [8] established a nonlinear dynamic model for a single-stage gear system by considering error and backlash. Later, these same authors [9] established a nonlinear dynamic model of a 3DOF gear system that considers comprehensive transmission error, backlash, time-varying meshing stiffness, and bearing clearance. Vaishya and Singh [10] established a gear dynamic model with time-varying friction, and Luo [11] established a gear dynamic model considering friction, collision, and lubrication condition. He et al. [12] developed a single-stage gear dynamic model considering friction and time-varying meshing stiffness, and Liu and Parker [13] proposed a multiple-stage gear dynamic model considering contact looseness, load fluctuation, and tooth profile modification. Chang-Jian [14] developed a model considering nonlinear oil film force, nonlinear support, and nonlinear meshing force, Li and Kahraman [15] considered transient elastohydrodynamic lubrication, and Huang et al. [16] considered variable lubricating oil damping. Eritenel and Parker [17] established the equivalent stiffness model in 2012, and Cui et al. [18] established a gear-rotor dynamic model considering nonlinear meshing force and nonlinear oil film force, while Chen et al. [19] established a gear dynamic model considering backlash and asymmetric meshing stiffness. Baguet and Jacquenot [20] established a gear-rotor-bearing coupling dynamic model considering nonlinear support and nonlinear meshing stiffness, Li and Kahraman [21] established a friction dynamic model considering lateral torsion and hybrid elastohydrodynamic lubrication, and Wei [22] developed a multiple-degree-of-freedom gear dynamic model for high-speed locomotives by considering bearing clearance, backlash, and time-varying meshing stiffness. Gao et al. [23] considered transmission error, time-varying meshing stiffness, backlash, nonlinear oil film force, and gear meshing force, and Zhang [24] established a gear-rotor dynamic model considering backlash and radial clearance. Xiang-Feng [25] established a single-degree-of-freedom torsion-vibration model considering the temperature of the tooth surface, and Zhang [26] investigated the influence of multiple excitation factors operating on the dynamic characteristics of a gear system, including time-varying friction, transmission error, and backlash. This literature review shows that various excitation factors have been considered in modeling equations for gear systems, and these can realize close correspondence to actual working conditions.It is clear that contact temperature, time-varying friction, and transmission error cannot be ignored when modeling the gear transmission system. To the authors’ knowledge, studies on nonlinear dynamic features of gear transmission systems for 3-D circular braiding machines are scarce. Zhang et al. [7] researched nonlinear dynamic characteristics of gear transmission systems in braiding machines and considered disturbance of yarn tension and transmission error, but not contact temperature or time-varying friction. However, contact temperature and time-varying friction have an important influence on the dynamic behavior of a gear transmission system in a braiding machine. In this paper, we analyze nonlinear dynamic characteristics of a gear transmission system in a braiding machine and consider disturbance of yarn tension, transmission error, time-varying friction, and contact temperature; all of these factors are of potential importance in the development of models to improve braiding quality, and the associated parameters will be helpful in the design and control of braiding machines.


## 2. Dynamic Modeling

### 2.1. Braiding Process

To gain a better understanding of the braiding process, a schematic of a radial braiding machine with an industrial robot is shown in Figure1. The radial braiding machine has 88 horn gears, each of which has four slots. The carriers are installed with the 1F1E arrangement (a gap is set between two adjacent carriers in the same group) and 176 carriers are driven during the braiding process. The radial braiding machine has 1 layer and 176 spindles, as is established with the coordinate system shown in Figure 1; the rotational center of the end of the robot effector is the origin, and the X, Y, and Z axes are as shown. Fij refers to the force that yarn of the ith (i = 1, 2, …, n) spindle on the jth (j = 1, 2, …, m) track exerts on the mandrel, and this can be obtained from the actual situation; α refers to the angle between one yarn and the Z-axis; ω refers to the angular velocity of the spindle; ϕij refers to the angle between one yarn and the horizontal plane as projected in the X-Y plane. Half of the carriers move in a clockwise direction, while the other half move in a counterclockwise direction during the braiding process. As shown in Figure 2, the carriers in the CA group move counterclockwise, and the carriers in the CB group move clockwise. Meanwhile, the traction system drags the robot with the mandrel, which causes the mandrel to move along the braiding center.Figure 1
Diagram of the mandrel dragged by the robot in a three-dimensional braiding machine.Figure 2
Diagram of the braiding process.


### 2.2. Dynamic Modeling of the Gear System

Figures3(a) and 3(b) show that the transmission chain of a radial braiding machine consists of many transmission structures, including the transmission shaft, bearing, horn gear, gear, woodruff key, carrier, and shaft sleeve. Obviously, one crucial structure of the motion system in a radial braiding machine is the gear. Optimizing gear meshing to minimize vibration is an effective way of improving the performance of the braiding machine.Figure 3
(a) Diagram of a closed-loop gear transmission system in a 3-D circular braiding machine gear transmission system. (b) Partial diagram of the gear transmission system.
(a)
(b)To aid the consideration of the nonlinear characteristics of the motion system in a radial braiding machine, a schematic illustration is shown in Figure4 and the dynamic equations of the gear system areas are established in equation (1). Here, mp and mg are the masses of the gears; me is the effective mass; cpx, cpy, cgx, and cgy are the equivalent dampings of bearing; δpx is the random disturbance of cpx; δpy is the random disturbance of cpy; δgx is the random disturbance of cgx; δgy is the random disturbance of cgy; kpx, kpy, kgx, and kgy are the equivalent stiffnesses of bearings; fpx, fpy, fgx, fgy, and fhare the displacement functions of the bearing; Λ is the sign function; Ft represents friction at the tooth surface; Fpx, Fpy, Fgx, and Fgy are the forces transmitted from the bearing; Fe1 and Fe2 are the eccentric forces; φp and φgare the angular displacements of the gear; ϕpτ and ϕgτ are the phase angles of eccentric force; ch is the damping coefficient of gear meshing; δch is the random disturbance of ch; Rp and Rg are the base circle radii of the gear; xp, xg, yp, and yg are the center displacements of gears; eτ is the static transmission error; khτ is the time-varying meshing stiffness coefficient; Jp is the rotational inertia of the initiative gear; Tp is the driving torque of the initiative gear; Jg is the rotational inertia of the passive gear; ρp and ρg are the masses of eccentric arms; ωp and ωg are the angular velocities of gears; Tg is the load torque of the passive gear; δ is the relative torsional displacement of gear pairs; kw is the amplitude of the time-varying stiffness fluctuation caused by temperature variation; δF is the random disturbance of load. In order to describe the meshing position and state accurately, a schematic diagram of the spread angle of active gears is shown in Figure 5.Figure 4
Dynamic model of gear transmission system.Figure 5
Schematic diagram of spread angle of initiative gear.The dynamic equations of a gear transmission system in a radial braiding machine are established according to Newton’s laws as follows:(1)mp⋅x¨p+cpx+δpx⋅x˙p+kpx⋅fpxxp−∑i=1nΛiFit+Fpx+Fe1⋅cosϕpτ=0,mp⋅y¨p+cpy+δpy⋅y˙p+kpy⋅fpyyp+ch+δch⋅Rp⋅θ˙p−Rg⋅θ˙g+y˙p−y˙g−e˙τ+…khτ⋅fhRp⋅θp−Rg⋅θg+yp−yg−eτ+Fpy+Fe1⋅sinϕpτ=0,Jp⋅θ¨p+Rp⋅ch+δch⋅Rp⋅θ˙p−Rg⋅θ˙g+y˙p−y˙g−e˙τ+…Rp⋅khτ⋅Rp⋅θp−Rg⋅θg+yp−yg−eτ=Tp+∑i=1nΛiRpitFit+δF,mg⋅x¨g+cgx+δgx⋅x˙g+kgx⋅fgxxg+∑i=1nΛiFit−Fgx−Fe2⋅cosϕgτ=0,mg⋅y¨g+cgy+δgy⋅y˙g+kgy⋅fgyyg−ch+δch⋅Rp⋅θ˙p−Rg⋅θ˙g+y˙p−y˙g−e˙τ−…khτ⋅fhRp⋅θp−Rg⋅θg+yp−yg−eτ−Fgy−Fe2⋅sinϕgτ=0,Jg⋅θ¨g−Rg⋅ch+δch⋅Rp⋅θ˙p−Rg⋅θ˙g+y˙p−y˙g−e˙τ−…Rg⋅khτ⋅Rp⋅θp−Rg⋅θg+yp−yg−eτ=−Tg−∑i=1nΛiRgitFit+δF,(2)δ=Rp⋅θp−Rg⋅θg+yp−yg−eτ,me=Jp⋅JgJp⋅Rg2+Jg⋅Rg2,fpxu,fpyu,fgxu,fgyu,fhu=u−b,u>b,0,−b≤u≤b,u=yp,yg,xp,xg,δ,u+b,u<−b,Fe1=mp⋅ρp⋅ωp2,Fe2=mg⋅ρg⋅ωg2.After Fourier series expansion, we take the first-order components ofkhτ and eτ, which are simplified as follows:(3)khτ=k0+kvcosωh+δωhτ,(4)eτ=evsinωhτ.Here,k0 is the average meshing stiffness; kv is the magnitude of variation of meshing stiffness; ωh is the driving frequency of gear pairs; δωh is the random disturbance of ωh; and ev is the amplitude of transmission error.The frequency of gear pairs is(5a)ωn=k0me.The frequency of gearp is(5b)ωp=kpymp.The frequency of gearq is(5c)ωg=kgymg.The backlash ratio is(5d)b∗=bbc.The stiffness ratio is(5e)ε=khvkh0,where b is the actual backlash, bc is the standard backlash, b=bc, khv is the amplitude of time-varying meshing stiffness, and kh0 is the average stiffness.


### 2.3. Time-Varying Friction Coefficient and Calculation Model

The friction coefficient for a tooth surface is affected by many factors such as the micromorphology of the tooth surface, lubrication state, and meshing position. The predictive models of friction coefficient such as the Coulomb model and the smoothed Coulomb model are used to predict the friction coefficient of gears.


#### 2.3.1. Coulomb Model

The Coulomb model is the simpler model. The friction coefficient does not change with the position of meshing contact point, and it can only change when the direction of friction is at a node:(6)fcit=Savgλt=Savg⋅signαC−αA−modΩpt,αD−αA.Here,Savg is the prescribed friction coefficient, Ωp satisfies t=αD−αA/Ωp, t is the time when the meshing point of the gear passes through a base pitch after meshing, αA=0.324, αB=0.453, αC=0.413, αD=0.717, and αE=0.845, as shown below.


#### 2.3.2. Smoothed Coulomb Model

Duan’s [27] research shows that the time-varying friction coefficient has a functional relationship with the slip-rolling ratio. The relative slip rate near the node is infinitely close to zero. The Coulomb model needs to be smoothed as follows:(7)fsit=2Savgπarctanκ⋅xit+xit2Savgσπ1+κ2xi2t,i=1,2,where xit=RpαC−αA−modΩpt,αD−αA, the smoothness, κ, is between 20 and 100, and σ is the overlap ratio.For convenience in solving, a dimensionless transformation is made:(8)t=ωnτyp∗=ygbyg∗=ygbα∗=αbkb1x,y∗=kb1x,ybkb2x,y∗=kb2x,ybk0∗=k0bkv∗=kvbωh∗=ωhωnωp∗=ωpωnωg∗=ωgωn.For ease in programming, a variable substitution is made:(9)x1=yp∗x2=y˙p∗x3=xp∗x4=x˙p∗x5=yg∗x6=y˙g∗x7=xg∗x8=x˙g∗x9=δ∗x10=δ˙∗.Substituting equations (2)–(9) into (1) gives(10)x˙1=x2,(11)x˙2=−2⋅γpy+δpy⋅x2−ηpy⋅fpyx1−2⋅γh1+δch1⋅x10−…η11⋅fhx9−Fpy∗−ρp∗⋅ωh∗+δωh∗2z1⋅sinωh∗+δωh∗tz1,(12)x˙3=x4(13)x˙4=−2⋅γpx+δpx⋅x4−ηpx⋅fpxx3+F1∗t−Fpx∗−ρp∗⋅ωh∗+δωh∗2z12⋅cosωh∗+δωh∗tz1,(14)x˙5=x6,(15)x˙6=−2⋅γgy+δgy⋅x6−ηgy⋅fgyx5+2⋅γh2+δch2⋅x10+…η22⋅fhx9+Fgy∗+ρg∗⋅ωh∗+δωh∗2z22⋅sinωh∗+δωh∗tz2,(16)x˙7=x8,(17)x˙8=−2⋅γgx+δgx⋅x8−ηgx⋅fgxx7−F2∗t+Fgx∗+ρg∗⋅ωh∗+δωh∗2z22⋅cosωh∗+δωh∗tz2,(18)x˙9=x10,(19)x˙10=x˙2−x˙6−2⋅γh3+δch3⋅x10−η33⋅fhx9+F0∗+ev∗ωh∗+δωh∗2sinωh∗+δωh∗t+δF∗+F3∗t,after dimensionless processing, and with z1 and z2 as the tooth numbers of the gear.Here,γpy=cpy/2⋅mp⋅ωn, ηpy=kpy/mp⋅ωn, γh1=ch/2⋅mp⋅ωn, η11=1−ε+δε⋅cosωh∗+δωh∗t+kw, Fpy∗=Fpy/mp⋅bc⋅ωn2, γpx=cpx/2⋅mp⋅ωn, ηpx=kpx/mp⋅ωn, F1∗t=∑i=1nΛi⋅Fit/mp⋅bc⋅ωn2, F2∗t=∑i=1nΛi⋅Fit/mg⋅bc⋅ωn2, Fpx∗=Fpx/mp⋅bc⋅ωn2, γgy=cgy/2⋅mg⋅ωn, ηgy=kgy/mg⋅ωn, γh2=ch/2⋅m⋅gωn, γh3=ch/2⋅me⋅ωn, η22=1−ε+δε⋅cosωh∗+δωh∗t+kw, Fgy∗=Fgy/mg⋅bc⋅ωn2, γgx=cgx/2⋅mg⋅ωn, ηgx=kgx/mg⋅ωn, Fgx∗=Fgx/mg⋅bc⋅ωn2, F0∗=Jg⋅Rp⋅Tp⋅+Jp⋅Rg⋅Tg/Jp⋅Jg⋅bc⋅ωn2, η33=1−ε+δε⋅cosωh∗+δεh∗t+kw, and F3∗t=∑i=1nΛi⋅Fit/me⋅bc⋅ωn2.


## 2.1. Braiding Process

To gain a better understanding of the braiding process, a schematic of a radial braiding machine with an industrial robot is shown in Figure1. The radial braiding machine has 88 horn gears, each of which has four slots. The carriers are installed with the 1F1E arrangement (a gap is set between two adjacent carriers in the same group) and 176 carriers are driven during the braiding process. The radial braiding machine has 1 layer and 176 spindles, as is established with the coordinate system shown in Figure 1; the rotational center of the end of the robot effector is the origin, and the X, Y, and Z axes are as shown. Fij refers to the force that yarn of the ith (i = 1, 2, …, n) spindle on the jth (j = 1, 2, …, m) track exerts on the mandrel, and this can be obtained from the actual situation; α refers to the angle between one yarn and the Z-axis; ω refers to the angular velocity of the spindle; ϕij refers to the angle between one yarn and the horizontal plane as projected in the X-Y plane. Half of the carriers move in a clockwise direction, while the other half move in a counterclockwise direction during the braiding process. As shown in Figure 2, the carriers in the CA group move counterclockwise, and the carriers in the CB group move clockwise. Meanwhile, the traction system drags the robot with the mandrel, which causes the mandrel to move along the braiding center.Figure 1
Diagram of the mandrel dragged by the robot in a three-dimensional braiding machine.Figure 2
Diagram of the braiding process.


## 2.2. Dynamic Modeling of the Gear System

Figures3(a) and 3(b) show that the transmission chain of a radial braiding machine consists of many transmission structures, including the transmission shaft, bearing, horn gear, gear, woodruff key, carrier, and shaft sleeve. Obviously, one crucial structure of the motion system in a radial braiding machine is the gear. Optimizing gear meshing to minimize vibration is an effective way of improving the performance of the braiding machine.Figure 3
(a) Diagram of a closed-loop gear transmission system in a 3-D circular braiding machine gear transmission system. (b) Partial diagram of the gear transmission system.
(a)
(b)To aid the consideration of the nonlinear characteristics of the motion system in a radial braiding machine, a schematic illustration is shown in Figure4 and the dynamic equations of the gear system areas are established in equation (1). Here, mp and mg are the masses of the gears; me is the effective mass; cpx, cpy, cgx, and cgy are the equivalent dampings of bearing; δpx is the random disturbance of cpx; δpy is the random disturbance of cpy; δgx is the random disturbance of cgx; δgy is the random disturbance of cgy; kpx, kpy, kgx, and kgy are the equivalent stiffnesses of bearings; fpx, fpy, fgx, fgy, and fhare the displacement functions of the bearing; Λ is the sign function; Ft represents friction at the tooth surface; Fpx, Fpy, Fgx, and Fgy are the forces transmitted from the bearing; Fe1 and Fe2 are the eccentric forces; φp and φgare the angular displacements of the gear; ϕpτ and ϕgτ are the phase angles of eccentric force; ch is the damping coefficient of gear meshing; δch is the random disturbance of ch; Rp and Rg are the base circle radii of the gear; xp, xg, yp, and yg are the center displacements of gears; eτ is the static transmission error; khτ is the time-varying meshing stiffness coefficient; Jp is the rotational inertia of the initiative gear; Tp is the driving torque of the initiative gear; Jg is the rotational inertia of the passive gear; ρp and ρg are the masses of eccentric arms; ωp and ωg are the angular velocities of gears; Tg is the load torque of the passive gear; δ is the relative torsional displacement of gear pairs; kw is the amplitude of the time-varying stiffness fluctuation caused by temperature variation; δF is the random disturbance of load. In order to describe the meshing position and state accurately, a schematic diagram of the spread angle of active gears is shown in Figure 5.Figure 4
Dynamic model of gear transmission system.Figure 5
Schematic diagram of spread angle of initiative gear.The dynamic equations of a gear transmission system in a radial braiding machine are established according to Newton’s laws as follows:(1)mp⋅x¨p+cpx+δpx⋅x˙p+kpx⋅fpxxp−∑i=1nΛiFit+Fpx+Fe1⋅cosϕpτ=0,mp⋅y¨p+cpy+δpy⋅y˙p+kpy⋅fpyyp+ch+δch⋅Rp⋅θ˙p−Rg⋅θ˙g+y˙p−y˙g−e˙τ+…khτ⋅fhRp⋅θp−Rg⋅θg+yp−yg−eτ+Fpy+Fe1⋅sinϕpτ=0,Jp⋅θ¨p+Rp⋅ch+δch⋅Rp⋅θ˙p−Rg⋅θ˙g+y˙p−y˙g−e˙τ+…Rp⋅khτ⋅Rp⋅θp−Rg⋅θg+yp−yg−eτ=Tp+∑i=1nΛiRpitFit+δF,mg⋅x¨g+cgx+δgx⋅x˙g+kgx⋅fgxxg+∑i=1nΛiFit−Fgx−Fe2⋅cosϕgτ=0,mg⋅y¨g+cgy+δgy⋅y˙g+kgy⋅fgyyg−ch+δch⋅Rp⋅θ˙p−Rg⋅θ˙g+y˙p−y˙g−e˙τ−…khτ⋅fhRp⋅θp−Rg⋅θg+yp−yg−eτ−Fgy−Fe2⋅sinϕgτ=0,Jg⋅θ¨g−Rg⋅ch+δch⋅Rp⋅θ˙p−Rg⋅θ˙g+y˙p−y˙g−e˙τ−…Rg⋅khτ⋅Rp⋅θp−Rg⋅θg+yp−yg−eτ=−Tg−∑i=1nΛiRgitFit+δF,(2)δ=Rp⋅θp−Rg⋅θg+yp−yg−eτ,me=Jp⋅JgJp⋅Rg2+Jg⋅Rg2,fpxu,fpyu,fgxu,fgyu,fhu=u−b,u>b,0,−b≤u≤b,u=yp,yg,xp,xg,δ,u+b,u<−b,Fe1=mp⋅ρp⋅ωp2,Fe2=mg⋅ρg⋅ωg2.After Fourier series expansion, we take the first-order components ofkhτ and eτ, which are simplified as follows:(3)khτ=k0+kvcosωh+δωhτ,(4)eτ=evsinωhτ.Here,k0 is the average meshing stiffness; kv is the magnitude of variation of meshing stiffness; ωh is the driving frequency of gear pairs; δωh is the random disturbance of ωh; and ev is the amplitude of transmission error.The frequency of gear pairs is(5a)ωn=k0me.The frequency of gearp is(5b)ωp=kpymp.The frequency of gearq is(5c)ωg=kgymg.The backlash ratio is(5d)b∗=bbc.The stiffness ratio is(5e)ε=khvkh0,where b is the actual backlash, bc is the standard backlash, b=bc, khv is the amplitude of time-varying meshing stiffness, and kh0 is the average stiffness.


## 2.3. Time-Varying Friction Coefficient and Calculation Model

The friction coefficient for a tooth surface is affected by many factors such as the micromorphology of the tooth surface, lubrication state, and meshing position. The predictive models of friction coefficient such as the Coulomb model and the smoothed Coulomb model are used to predict the friction coefficient of gears.


### 2.3.1. Coulomb Model

The Coulomb model is the simpler model. The friction coefficient does not change with the position of meshing contact point, and it can only change when the direction of friction is at a node:(6)fcit=Savgλt=Savg⋅signαC−αA−modΩpt,αD−αA.Here,Savg is the prescribed friction coefficient, Ωp satisfies t=αD−αA/Ωp, t is the time when the meshing point of the gear passes through a base pitch after meshing, αA=0.324, αB=0.453, αC=0.413, αD=0.717, and αE=0.845, as shown below.


### 2.3.2. Smoothed Coulomb Model

Duan’s [27] research shows that the time-varying friction coefficient has a functional relationship with the slip-rolling ratio. The relative slip rate near the node is infinitely close to zero. The Coulomb model needs to be smoothed as follows:(7)fsit=2Savgπarctanκ⋅xit+xit2Savgσπ1+κ2xi2t,i=1,2,where xit=RpαC−αA−modΩpt,αD−αA, the smoothness, κ, is between 20 and 100, and σ is the overlap ratio.For convenience in solving, a dimensionless transformation is made:(8)t=ωnτyp∗=ygbyg∗=ygbα∗=αbkb1x,y∗=kb1x,ybkb2x,y∗=kb2x,ybk0∗=k0bkv∗=kvbωh∗=ωhωnωp∗=ωpωnωg∗=ωgωn.For ease in programming, a variable substitution is made:(9)x1=yp∗x2=y˙p∗x3=xp∗x4=x˙p∗x5=yg∗x6=y˙g∗x7=xg∗x8=x˙g∗x9=δ∗x10=δ˙∗.Substituting equations (2)–(9) into (1) gives(10)x˙1=x2,(11)x˙2=−2⋅γpy+δpy⋅x2−ηpy⋅fpyx1−2⋅γh1+δch1⋅x10−…η11⋅fhx9−Fpy∗−ρp∗⋅ωh∗+δωh∗2z1⋅sinωh∗+δωh∗tz1,(12)x˙3=x4(13)x˙4=−2⋅γpx+δpx⋅x4−ηpx⋅fpxx3+F1∗t−Fpx∗−ρp∗⋅ωh∗+δωh∗2z12⋅cosωh∗+δωh∗tz1,(14)x˙5=x6,(15)x˙6=−2⋅γgy+δgy⋅x6−ηgy⋅fgyx5+2⋅γh2+δch2⋅x10+…η22⋅fhx9+Fgy∗+ρg∗⋅ωh∗+δωh∗2z22⋅sinωh∗+δωh∗tz2,(16)x˙7=x8,(17)x˙8=−2⋅γgx+δgx⋅x8−ηgx⋅fgxx7−F2∗t+Fgx∗+ρg∗⋅ωh∗+δωh∗2z22⋅cosωh∗+δωh∗tz2,(18)x˙9=x10,(19)x˙10=x˙2−x˙6−2⋅γh3+δch3⋅x10−η33⋅fhx9+F0∗+ev∗ωh∗+δωh∗2sinωh∗+δωh∗t+δF∗+F3∗t,after dimensionless processing, and with z1 and z2 as the tooth numbers of the gear.Here,γpy=cpy/2⋅mp⋅ωn, ηpy=kpy/mp⋅ωn, γh1=ch/2⋅mp⋅ωn, η11=1−ε+δε⋅cosωh∗+δωh∗t+kw, Fpy∗=Fpy/mp⋅bc⋅ωn2, γpx=cpx/2⋅mp⋅ωn, ηpx=kpx/mp⋅ωn, F1∗t=∑i=1nΛi⋅Fit/mp⋅bc⋅ωn2, F2∗t=∑i=1nΛi⋅Fit/mg⋅bc⋅ωn2, Fpx∗=Fpx/mp⋅bc⋅ωn2, γgy=cgy/2⋅mg⋅ωn, ηgy=kgy/mg⋅ωn, γh2=ch/2⋅m⋅gωn, γh3=ch/2⋅me⋅ωn, η22=1−ε+δε⋅cosωh∗+δωh∗t+kw, Fgy∗=Fgy/mg⋅bc⋅ωn2, γgx=cgx/2⋅mg⋅ωn, ηgx=kgx/mg⋅ωn, Fgx∗=Fgx/mg⋅bc⋅ωn2, F0∗=Jg⋅Rp⋅Tp⋅+Jp⋅Rg⋅Tg/Jp⋅Jg⋅bc⋅ωn2, η33=1−ε+δε⋅cosωh∗+δεh∗t+kw, and F3∗t=∑i=1nΛi⋅Fit/me⋅bc⋅ωn2.


## 2.3.1. Coulomb Model

The Coulomb model is the simpler model. The friction coefficient does not change with the position of meshing contact point, and it can only change when the direction of friction is at a node:(6)fcit=Savgλt=Savg⋅signαC−αA−modΩpt,αD−αA.Here,Savg is the prescribed friction coefficient, Ωp satisfies t=αD−αA/Ωp, t is the time when the meshing point of the gear passes through a base pitch after meshing, αA=0.324, αB=0.453, αC=0.413, αD=0.717, and αE=0.845, as shown below.


## 2.3.2. Smoothed Coulomb Model

Duan’s [27] research shows that the time-varying friction coefficient has a functional relationship with the slip-rolling ratio. The relative slip rate near the node is infinitely close to zero. The Coulomb model needs to be smoothed as follows:(7)fsit=2Savgπarctanκ⋅xit+xit2Savgσπ1+κ2xi2t,i=1,2,where xit=RpαC−αA−modΩpt,αD−αA, the smoothness, κ, is between 20 and 100, and σ is the overlap ratio.For convenience in solving, a dimensionless transformation is made:(8)t=ωnτyp∗=ygbyg∗=ygbα∗=αbkb1x,y∗=kb1x,ybkb2x,y∗=kb2x,ybk0∗=k0bkv∗=kvbωh∗=ωhωnωp∗=ωpωnωg∗=ωgωn.For ease in programming, a variable substitution is made:(9)x1=yp∗x2=y˙p∗x3=xp∗x4=x˙p∗x5=yg∗x6=y˙g∗x7=xg∗x8=x˙g∗x9=δ∗x10=δ˙∗.Substituting equations (2)–(9) into (1) gives(10)x˙1=x2,(11)x˙2=−2⋅γpy+δpy⋅x2−ηpy⋅fpyx1−2⋅γh1+δch1⋅x10−…η11⋅fhx9−Fpy∗−ρp∗⋅ωh∗+δωh∗2z1⋅sinωh∗+δωh∗tz1,(12)x˙3=x4(13)x˙4=−2⋅γpx+δpx⋅x4−ηpx⋅fpxx3+F1∗t−Fpx∗−ρp∗⋅ωh∗+δωh∗2z12⋅cosωh∗+δωh∗tz1,(14)x˙5=x6,(15)x˙6=−2⋅γgy+δgy⋅x6−ηgy⋅fgyx5+2⋅γh2+δch2⋅x10+…η22⋅fhx9+Fgy∗+ρg∗⋅ωh∗+δωh∗2z22⋅sinωh∗+δωh∗tz2,(16)x˙7=x8,(17)x˙8=−2⋅γgx+δgx⋅x8−ηgx⋅fgxx7−F2∗t+Fgx∗+ρg∗⋅ωh∗+δωh∗2z22⋅cosωh∗+δωh∗tz2,(18)x˙9=x10,(19)x˙10=x˙2−x˙6−2⋅γh3+δch3⋅x10−η33⋅fhx9+F0∗+ev∗ωh∗+δωh∗2sinωh∗+δωh∗t+δF∗+F3∗t,after dimensionless processing, and with z1 and z2 as the tooth numbers of the gear.Here,γpy=cpy/2⋅mp⋅ωn, ηpy=kpy/mp⋅ωn, γh1=ch/2⋅mp⋅ωn, η11=1−ε+δε⋅cosωh∗+δωh∗t+kw, Fpy∗=Fpy/mp⋅bc⋅ωn2, γpx=cpx/2⋅mp⋅ωn, ηpx=kpx/mp⋅ωn, F1∗t=∑i=1nΛi⋅Fit/mp⋅bc⋅ωn2, F2∗t=∑i=1nΛi⋅Fit/mg⋅bc⋅ωn2, Fpx∗=Fpx/mp⋅bc⋅ωn2, γgy=cgy/2⋅mg⋅ωn, ηgy=kgy/mg⋅ωn, γh2=ch/2⋅m⋅gωn, γh3=ch/2⋅me⋅ωn, η22=1−ε+δε⋅cosωh∗+δωh∗t+kw, Fgy∗=Fgy/mg⋅bc⋅ωn2, γgx=cgx/2⋅mg⋅ωn, ηgx=kgx/mg⋅ωn, Fgx∗=Fgx/mg⋅bc⋅ωn2, F0∗=Jg⋅Rp⋅Tp⋅+Jp⋅Rg⋅Tg/Jp⋅Jg⋅bc⋅ωn2, η33=1−ε+δε⋅cosωh∗+δεh∗t+kw, and F3∗t=∑i=1nΛi⋅Fit/me⋅bc⋅ωn2.


## 3. Results and Discussion

Equations (10)–(19) are solved by iterative methods with step size t=0.05, z1=z2=30, mp=mq=me=0.456kg, Jp=Jq=6×10−6kg⋅m2, bc=0.068mm, ev=0.034mm, k0=2.23×108N/m, kv=1.12×107N/m, ev∗=0.05, Fpx∗=0.02, Fpy∗=0.2, Fgy∗=0.2, Fgy∗=−0.2, η11=1.1, η22=1.1, η33=1.1, γpx=0.01, γpy=0.01, γgx=0.01, γgy=0.01, γh1=0.012, γh2=0.012, γh3=0.05, ε=0.18, ρp∗=0.1, ρg∗=0.1, F0∗=0.15, αA=0.324rad, αB=0.453rad, αc=0.413rad, αD=0.717rad, αE=0.845rad, Rp=Rg=0.06m, Savg=0.991μm, and initial conditions are as follows:x10=0,x20=−0.1,x30=0,x40=−0.1,x50=0,x60=−0.1,x70=0,x80=−0.1,x90=0,andx100=−0.1.


### 3.1. Analysis of a System without Random Perturbation

The nonlinear characteristics of a radial braiding machine’s gear transmission system without random perturbation are shown in Figures6(a)–12. The vibrational bifurcation diagram of the system is presented in Figure 6(a), the maximum Lyapunov exponent diagram of the system is shown as Figure 6(b), and Poincare maps and corresponding phase trajectories of the system are shown in Figures 7–12.Figure 6
(a) Vibrational bifurcation diagram without perturbation. (b) Maximum Lyapunov exponent curve without perturbation.
(a)
(b)Figure 7
ω
=
110 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 8
ω
=
170 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 9
ω
=
200 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 10
ω
=
232 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 11
ω
=
243 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 12
ω
=
265 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)The nonlinear vibration characteristics of the system and the relationship betweenω and x10 are shown in Figure 6(a). Observations include the following:(1)
From Figure6(a), the system has one periodic point when ω<163.3. The Poincare maps and corresponding phase trajectories of the system at ω=110 are shown in Figure 7, where it can be observed that the system converged rapidly to one periodic point at ω=110; this suggests that the system was stable.(2)
The system has four periodic points when163.3<ω<186.5 and Poincare maps and corresponding phase trajectories of the system at ω=170 are shown in Figure 8.(3)
With the increase ofω, the number of periodic points changes when 186.5<ω<190.(4)
The system has nine periodic points when190<ω<213, and Poincare maps and corresponding phase trajectories of the system at ω=200 are shown in Figure 9.(5)
From Figure6(a), chaos occurs in the system when 213<ω<240. Poincare maps and corresponding phase trajectories of the system at ω=232 are shown in Figure 10.(6)
When240<ω<264.7, the corresponding phase trajectories of the system comprise a limit cycle, and Poincare maps and corresponding phase trajectories of the system at ω=243 are shown in Figure 11.(7)
Whenω<264.7, the system becomes divergent and uncontrollable, and Poincare maps and corresponding phase trajectories of the system at ω=265 are shown in Figure 12. Meanwhile, the maximum Lyapunov diagram of the system can also reflect the nonlinearity of the system to a certain extent, as is shown in Figure 6(b). The Lyapunov exponent is defined as λk=limx⟶∞1/ilnλki, k=1,2,…,n, where λ1i≥λ2i≥⋯≥λni. In Figure 6(b), the maximum Lyapunov exponent is negative or fluctuates near zero when ω<213, which indicates that the system is stable according to the Lyapunov theorem. The maximum Lyapunov exponent is positive when ω<213, which indicates that the system exhibits chaos and becomes divergent and uncontrollable.The nonlinearity shown in Figure6(b) is consistent with the analysis from Poincare maps and the corresponding phase trajectories of the system.


### 3.2. Systematic Analysis ofkw

The nonlinear characteristics of a gear transmission system as a function ofkw are analyzed in Figures 13(a)–20(b). The vibrational bifurcation diagram of the system with kw=0.248 is shown in Figure 13(a), the maximum Lyapunov diagram of the system with kw=0.248 is shown in Figure 13(b), and Poincare maps and corresponding phase trajectories of the system with kw=0.248 are shown in Figures 14–19. In addition, the vibrational bifurcation diagram of a system with kw=0.32 is shown in Figure 20(a), and the maximum Lyapunov diagram of the system with kw=0.32 is shown in Figure 20(b). From Figures 13(a) and 20(a), it appears that kw exerts a significant influence on the nonlinear characteristics of the gear transmission system. If kw is relatively small, such as kw=0.248, there is a smaller influence on the nonlinear characteristics of the gear transmission system. The vibrational bifurcation diagram of the system, the maximum Lyapunov diagram of the system, and Poincare maps and corresponding phase trajectories of the system with kw=0.248 are shown in Figures 13(a)–19. Comparing Figures 6(a) and 13(a), it is observed that the vibration amplitude x10 when kw=0.248 is larger than the x10 resulting when kw=0, and the system exhibits chaos earlier when kw=0.248 as opposed to when kw=0. When ω=200, the system exhibits chaos with kw=0.248, but when ω=213, the system exhibits chaos with kw=0. When ω<200, the form of the solution with kw=0.248 is similar to that of kw=0. When 200<ω<215, the system exhibits chaos and Poincare maps and corresponding phase trajectories of the system at ω=214 are shown in Figure 17. When 215<ω<240, the corresponding phase trajectories of the system form a limit cycle. Poincare maps and corresponding phase trajectories of the system at ω=234 are shown in Figure 18. When ω=240, the system becomes divergent and uncontrollable. Poincare maps and corresponding phase trajectories of the system at ω=240 are shown in Figure 19. From Figure 13(b), the maximum Lyapunov exponent is negative or fluctuates near zero when ω>200, indicating that the system is stable according to the Lyapunov theorem. The maximum Lyapunov exponent is positive when ω>200, indicating that the system exhibits chaos and becomes divergent and uncontrollable. The nonlinearity shown in Figure 13(b) is consistent with the analysis from Poincare maps and corresponding phase trajectories for the system. If kw is relatively large, such as kw=0.32, there is a substantial influence on the nonlinear characteristics of the gear transmission system. The vibrational bifurcation diagram of the system, the maximum Lyapunov diagram of the system, Poincare maps, and corresponding phase trajectories of the system with kw=0.32 are shown in Figures 20(a) and 20(b). From Figure 20(b), it is clear that the maximum Lyapunov exponent is positive if kw is relatively large, such as kw=0.32, indicating that the system exhibits chaos and becomes divergent and uncontrollable. Here, the Poincare maps and corresponding phase trajectories of the system are omitted.Figure 13
(a) Vibrational bifurcation diagram withkw=0.248. (b) Maximum Lyapunov exponent curve with kw=0.248.
(a)
(b)Figure 14
ω
=
110 and kw=0.248 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 15
ω
=
170 and kw=0.248without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 16
ω
=
192 and kw=0.248 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 17
ω
=
214 and kw=0.248without random perturbation. (a) Phase trajectories, (b) Poincare maps.
(a)
(b)Figure 18
ω
=
234 and kw=0.248 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 19
ω
=
240 and kw=0.248 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 20
(a) Vibrational bifurcation diagram curve withkw=0.32. (b) Maximum Lyapunov exponent with kw=0.32.
(a)
(b)


### 3.3. Systematic Analysis of Friction on the Tooth Surface

The dynamic friction resulting from the Coulomb model is shown in Figure21. Friction on the tooth surface exhibits an obvious periodic variation when 0.05<t<0.0581 and ω<210. Bifurcation diagrams, maximum Lyapunov exponents, phase diagrams, Poincare maps corresponding to Figure 21, and specific analyses are shown in Figures 22(a)–29. When 0.05<t<0.0581, the vibrational bifurcation diagram for the system using the Coulomb model to predict the time-varying friction coefficient is shown in Figure 22(a); Figure 22(a) indicates that the nonlinear characteristics of friction on the tooth surfaces of the gear transmission system are complex. Observations include the following:(1)
When0.05<t<0.0581 and ω<163.3, the system has one periodic point. Poincare maps and corresponding phase trajectories of system at ω=110 are shown in Figure 23; this shows that the system rapidly converges to one periodic point, indicating that the system is stable.(2)
When0.05<t<0.0581 and 163.3<ω<186.5, the system has four periodic points, and Poincare maps and corresponding phase trajectories of system at ω=174 are shown in Figure 24.(3)
With increasingω=110, the number of periodic points changes when 0.05<t<0.0581 and 186.5<ω<190.(4)
When0.05<t<0.0581 and 190<ω<203, the system has nine periodic points, and Poincare maps and corresponding phase trajectories of the system at ω=192 are shown in Figure 25.(5)
From Figure22(a), the system has a finite number of periodic points when 0.05<t<0.0581 and 203<ω<210. Poincare maps and corresponding phase trajectories of the system at ω=206 are shown in Figure 26.(6)
From Figure22(a), the phase trajectories of the system when 0.05<t<0.0581 and 210<ω<230 constitute a limit cycle. Poincare maps and corresponding phase trajectories of the system at ω=220 are shown in Figure 27.(7)
When230<ω<272, the system has 13 periodic points. Poincare maps and corresponding phase trajectories of the system at ω=237 are shown in Figure 28.(8)
Whenω=272, the system becomes divergent and uncontrollable. Poincare maps and corresponding phase trajectories of the system at ω=273 are shown in Figure 29.Figure 21
Dynamic friction with the Coulomb model.Figure 22
(a) Vibrational bifurcation diagram with the Coulomb model whent∈0.050.0581. (b) Maximum Lyapunov exponent curve with the Coulomb model when t∈[0.05 0.0581].
(a)
(b)Figure 23
ω
=
110 and kw=0 with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 24
ω
=
174 and kw=0with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 25
ω
=
192 and kw=0with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 26
ω
=
206 and kw=0with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 27
ω
=
220 and kw=0 with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 28
ω
=
237 and kw=0with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 29
ω
=
273 and kw=0with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)From Figure22(b), the maximum Lyapunov exponent is negative or fluctuates near zero when ω<210, indicating that the system is stable according to the Lyapunov theorem. The maximum Lyapunov exponent is positive when ω>210, which indicates that the system exhibits chaos and becomes divergent and uncontrollable. In general, friction in the Coulomb model increases with an increase in tooth surface roughness because of the change in the nonlinear friction coefficient. Meanwhile, friction in the Coulomb model has nonperiodic and sharp fluctuations when t>0.0581. Bifurcation diagrams, maximum Lyapunov exponents, phase diagrams, Poincare maps corresponding to Figure 21, and specific analyses are shown in Figures 30(a) and 30(b). Whent>0.0581, use of the Coulomb model to predict the time-varying friction coefficient exhibits chaos and becomes divergent and uncontrollable. Figure 30(a) shows the vibrational bifurcation diagram of the system using the Coulomb model to predict the time-varying friction coefficient when 0.0581<t<0.1009, and the maximum Lyapunov diagram of the system is shown in Figure 30(b). From Figure 30(b), the maximum Lyapunov exponent is positive when 0.0581<t<0.1009, which indicates that the system exhibits chaos and becomes divergent and uncontrollable. Here, the Poincare maps and corresponding phase trajectories of the system are omitted.Figure 30
(a) Vibrational bifurcation diagram with the Coulomb model whent∈0.05810.1009. (b) Maximum Lyapunov exponent curve with the Coulomb model when t∈0.05810.1009.
(a)
(b)The dynamic friction obtained with the smoothed Coulomb model is shown in Figure31. Friction on the tooth surface exhibits obvious periodic changes when 0.05<t<0.0597 and ω<201. Bifurcation diagrams, maximum Lyapunov exponents, phase diagrams, Poincare maps corresponding to Figure 31, and specific analyses are shown in Figures 32(a)–38. The vibrational bifurcation diagram of the system using the smoothed Coulomb model to predict the time-varying friction coefficient when 0.05<t<0.0597 is given in Figure 32(a), which shows that the nonlinear characteristics of tooth surface friction in the gear transmission system is complex. Observations include the following:(1)
When0.05<t<0.0597 and ω<163.3, the system has one periodic point. Poincare maps and corresponding phase trajectories of the system at ω=110 are shown in Figure 33. In Figure 33, the system rapidly converges to one periodic point, indicating that the system is stable.(2)
When0.05<t<0.0597 and 163.3<ω<182, the system has four periodic points. Poincare maps and corresponding phase trajectories of the system at ω=174 are shown in Figure 34.(3)
With the increase ofω, the number of periodic points changes when 0.05<t<0.0597 and 182<ω<190.(4)
When190<ω<201, the system has nine periodic points. Poincare maps and corresponding phase trajectories of the system at ω=192 are shown in Figure 35.(5)
When0.05<t<0.0597and 201<ω<229, corresponding phase trajectories of the system constitute a limit cycle. Poincare maps and corresponding phase trajectories of the system at ω=210 are shown in Figure 36.(7)
When229<ω<272, the system has 13 periodic points. Poincare maps and corresponding phase trajectories of the system at ω=230 are shown in Figure 37.(8)
Whenω>272, the system becomes divergent and uncontrollable. Poincare maps and corresponding phase trajectories of the system at ω=273 are shown in Figure 38.Figure 31
Dynamic friction from the smoothed Coulomb model.Figure 32
(a) Vibrational bifurcation diagram with the smoothed Coulomb model whent∈0.050.0597. (b) Maximum Lyapunov exponent curve with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)Figure 33
ω
=
110 and kw=0 with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)Figure 34
ω
=
174 and kw=0 with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)Figure 35
ω
=
192 and kw=0 with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)Figure 36
ω
=
210 and kw=0 with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)Figure 37
ω
=
230 and kw=0 with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)Figure 38
ω
=
273 and kw=0 with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)From Figure32(b), the maximum Lyapunov exponent is negative or fluctuates near zero when ω<201, indicating that the system is stable according to the Lyapunov theorem. The maximum Lyapunov exponent is positive when ω>201, which indicates that the system exhibits chaos and becomes divergent and uncontrollable. In general, friction with the smoothed Coulomb model increases with an increase in tooth surface roughness because of the change of the nonlinear friction coefficient. Meanwhile, friction in the smoothed Coulomb model has nonperiodic and sharp fluctuations when t>0.0597. Bifurcation diagrams, maximum Lyapunov exponents, phase diagrams, Poincare maps corresponding to Figure 31, and specific analyses are shown in Figures 39(a) and 39(b). Figure 39(a) shows the vibrational bifurcation diagram of the system using the smoothed Coulomb model to predict the time-varying friction coefficient when 0.0597<t<0.0921, and the maximum Lyapunov diagram of the system is shown in Figure 39(b). From Figure 39(b), the maximum Lyapunov exponent is positive when 0.0597<t<0.0921, which indicates that the system exhibits chaos and becomes divergent and uncontrollable. Here, the Poincare maps and corresponding phase trajectories of the system are omitted.Figure 39
(a) Vibrational bifurcation diagram with the smoothed Coulomb model whent∈0.05970.0921. (b) Maximum Lyapunov exponent curve with the smoothed Coulomb model when t∈0.05970.0921.
(a)
(b)


### 3.4. Analysis of System with Random Perturbation

The nonlinear characteristics of a gear transmission system with random perturbation are analyzed as shown in Figures40–48. The basic parameters are the same as those presented above, but some random perturbations are added here; these include δpx, δpy, δgx, δgy, δch1, δch2, δch3, δε, δF∗∼N0,0.0012, and δωh∗∼N0,0.000052. The Poincare maps and corresponding phase trajectories of the system with random perturbation are shown in Figures 40–44, in which the nonlinear characteristics of the gear transmission system with small random perturbation are basically consistent with the nonlinear characteristics of the system without random perturbation (as shown in Figures 7–12). The system with small random perturbations converges to some degree, but the system without random perturbations converges to a finite point. The power spectra in Figures 45 and 46 were generated with a sampling frequency of 1000.Figure 40
ω
=
110 with random perturbation.
(a)
(b)Figure 41
ω
=
200 with random perturbation.
(a)
(b)Figure 42
ω
=
232 with random perturbation.
(a)
(b)Figure 43
ω
=
238 with random perturbation.
(a)
(b)Figure 44
ω
=
285 with random Perturbation.
(a)
(b)Figure 45
Power spectrum ofx10with random perturbation, ω=175.Figure 46
Power spectrum ofx10 without perturbation, ω=175.Figure 47
Vibrational bifurcation random diagram withδF∗∼N0,0.012.Figure 48
Vibrational bifurcation diagram withδF∗∼N0,0.122.The trend of the power spectrum ofx10 with and without random perturbation is almost identical, but the fluctuation of the power spectrum of x10 with random perturbation is greater than that without random perturbation.The vibrational bifurcation diagrams of the system withδF∗∼N0,0.012, δF∗∼N0,0.122 are shown in Figures 47 and 48. With the increase of δF∗, the dynamic characteristics of the system change from Figure 47 to 48. When δF∗ increases to the extent that δF∗∼N0,0.122, the bifurcation of the system disappears completely and the system becomes uncontrollable.


## 3.1. Analysis of a System without Random Perturbation

The nonlinear characteristics of a radial braiding machine’s gear transmission system without random perturbation are shown in Figures6(a)–12. The vibrational bifurcation diagram of the system is presented in Figure 6(a), the maximum Lyapunov exponent diagram of the system is shown as Figure 6(b), and Poincare maps and corresponding phase trajectories of the system are shown in Figures 7–12.Figure 6
(a) Vibrational bifurcation diagram without perturbation. (b) Maximum Lyapunov exponent curve without perturbation.
(a)
(b)Figure 7
ω
=
110 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 8
ω
=
170 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 9
ω
=
200 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 10
ω
=
232 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 11
ω
=
243 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 12
ω
=
265 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)The nonlinear vibration characteristics of the system and the relationship betweenω and x10 are shown in Figure 6(a). Observations include the following:(1)
From Figure6(a), the system has one periodic point when ω<163.3. The Poincare maps and corresponding phase trajectories of the system at ω=110 are shown in Figure 7, where it can be observed that the system converged rapidly to one periodic point at ω=110; this suggests that the system was stable.(2)
The system has four periodic points when163.3<ω<186.5 and Poincare maps and corresponding phase trajectories of the system at ω=170 are shown in Figure 8.(3)
With the increase ofω, the number of periodic points changes when 186.5<ω<190.(4)
The system has nine periodic points when190<ω<213, and Poincare maps and corresponding phase trajectories of the system at ω=200 are shown in Figure 9.(5)
From Figure6(a), chaos occurs in the system when 213<ω<240. Poincare maps and corresponding phase trajectories of the system at ω=232 are shown in Figure 10.(6)
When240<ω<264.7, the corresponding phase trajectories of the system comprise a limit cycle, and Poincare maps and corresponding phase trajectories of the system at ω=243 are shown in Figure 11.(7)
Whenω<264.7, the system becomes divergent and uncontrollable, and Poincare maps and corresponding phase trajectories of the system at ω=265 are shown in Figure 12. Meanwhile, the maximum Lyapunov diagram of the system can also reflect the nonlinearity of the system to a certain extent, as is shown in Figure 6(b). The Lyapunov exponent is defined as λk=limx⟶∞1/ilnλki, k=1,2,…,n, where λ1i≥λ2i≥⋯≥λni. In Figure 6(b), the maximum Lyapunov exponent is negative or fluctuates near zero when ω<213, which indicates that the system is stable according to the Lyapunov theorem. The maximum Lyapunov exponent is positive when ω<213, which indicates that the system exhibits chaos and becomes divergent and uncontrollable.The nonlinearity shown in Figure6(b) is consistent with the analysis from Poincare maps and the corresponding phase trajectories of the system.


## 3.2. Systematic Analysis ofkw

The nonlinear characteristics of a gear transmission system as a function ofkw are analyzed in Figures 13(a)–20(b). The vibrational bifurcation diagram of the system with kw=0.248 is shown in Figure 13(a), the maximum Lyapunov diagram of the system with kw=0.248 is shown in Figure 13(b), and Poincare maps and corresponding phase trajectories of the system with kw=0.248 are shown in Figures 14–19. In addition, the vibrational bifurcation diagram of a system with kw=0.32 is shown in Figure 20(a), and the maximum Lyapunov diagram of the system with kw=0.32 is shown in Figure 20(b). From Figures 13(a) and 20(a), it appears that kw exerts a significant influence on the nonlinear characteristics of the gear transmission system. If kw is relatively small, such as kw=0.248, there is a smaller influence on the nonlinear characteristics of the gear transmission system. The vibrational bifurcation diagram of the system, the maximum Lyapunov diagram of the system, and Poincare maps and corresponding phase trajectories of the system with kw=0.248 are shown in Figures 13(a)–19. Comparing Figures 6(a) and 13(a), it is observed that the vibration amplitude x10 when kw=0.248 is larger than the x10 resulting when kw=0, and the system exhibits chaos earlier when kw=0.248 as opposed to when kw=0. When ω=200, the system exhibits chaos with kw=0.248, but when ω=213, the system exhibits chaos with kw=0. When ω<200, the form of the solution with kw=0.248 is similar to that of kw=0. When 200<ω<215, the system exhibits chaos and Poincare maps and corresponding phase trajectories of the system at ω=214 are shown in Figure 17. When 215<ω<240, the corresponding phase trajectories of the system form a limit cycle. Poincare maps and corresponding phase trajectories of the system at ω=234 are shown in Figure 18. When ω=240, the system becomes divergent and uncontrollable. Poincare maps and corresponding phase trajectories of the system at ω=240 are shown in Figure 19. From Figure 13(b), the maximum Lyapunov exponent is negative or fluctuates near zero when ω>200, indicating that the system is stable according to the Lyapunov theorem. The maximum Lyapunov exponent is positive when ω>200, indicating that the system exhibits chaos and becomes divergent and uncontrollable. The nonlinearity shown in Figure 13(b) is consistent with the analysis from Poincare maps and corresponding phase trajectories for the system. If kw is relatively large, such as kw=0.32, there is a substantial influence on the nonlinear characteristics of the gear transmission system. The vibrational bifurcation diagram of the system, the maximum Lyapunov diagram of the system, Poincare maps, and corresponding phase trajectories of the system with kw=0.32 are shown in Figures 20(a) and 20(b). From Figure 20(b), it is clear that the maximum Lyapunov exponent is positive if kw is relatively large, such as kw=0.32, indicating that the system exhibits chaos and becomes divergent and uncontrollable. Here, the Poincare maps and corresponding phase trajectories of the system are omitted.Figure 13
(a) Vibrational bifurcation diagram withkw=0.248. (b) Maximum Lyapunov exponent curve with kw=0.248.
(a)
(b)Figure 14
ω
=
110 and kw=0.248 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 15
ω
=
170 and kw=0.248without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 16
ω
=
192 and kw=0.248 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 17
ω
=
214 and kw=0.248without random perturbation. (a) Phase trajectories, (b) Poincare maps.
(a)
(b)Figure 18
ω
=
234 and kw=0.248 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 19
ω
=
240 and kw=0.248 without random perturbation. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 20
(a) Vibrational bifurcation diagram curve withkw=0.32. (b) Maximum Lyapunov exponent with kw=0.32.
(a)
(b)


## 3.3. Systematic Analysis of Friction on the Tooth Surface

The dynamic friction resulting from the Coulomb model is shown in Figure21. Friction on the tooth surface exhibits an obvious periodic variation when 0.05<t<0.0581 and ω<210. Bifurcation diagrams, maximum Lyapunov exponents, phase diagrams, Poincare maps corresponding to Figure 21, and specific analyses are shown in Figures 22(a)–29. When 0.05<t<0.0581, the vibrational bifurcation diagram for the system using the Coulomb model to predict the time-varying friction coefficient is shown in Figure 22(a); Figure 22(a) indicates that the nonlinear characteristics of friction on the tooth surfaces of the gear transmission system are complex. Observations include the following:(1)
When0.05<t<0.0581 and ω<163.3, the system has one periodic point. Poincare maps and corresponding phase trajectories of system at ω=110 are shown in Figure 23; this shows that the system rapidly converges to one periodic point, indicating that the system is stable.(2)
When0.05<t<0.0581 and 163.3<ω<186.5, the system has four periodic points, and Poincare maps and corresponding phase trajectories of system at ω=174 are shown in Figure 24.(3)
With increasingω=110, the number of periodic points changes when 0.05<t<0.0581 and 186.5<ω<190.(4)
When0.05<t<0.0581 and 190<ω<203, the system has nine periodic points, and Poincare maps and corresponding phase trajectories of the system at ω=192 are shown in Figure 25.(5)
From Figure22(a), the system has a finite number of periodic points when 0.05<t<0.0581 and 203<ω<210. Poincare maps and corresponding phase trajectories of the system at ω=206 are shown in Figure 26.(6)
From Figure22(a), the phase trajectories of the system when 0.05<t<0.0581 and 210<ω<230 constitute a limit cycle. Poincare maps and corresponding phase trajectories of the system at ω=220 are shown in Figure 27.(7)
When230<ω<272, the system has 13 periodic points. Poincare maps and corresponding phase trajectories of the system at ω=237 are shown in Figure 28.(8)
Whenω=272, the system becomes divergent and uncontrollable. Poincare maps and corresponding phase trajectories of the system at ω=273 are shown in Figure 29.Figure 21
Dynamic friction with the Coulomb model.Figure 22
(a) Vibrational bifurcation diagram with the Coulomb model whent∈0.050.0581. (b) Maximum Lyapunov exponent curve with the Coulomb model when t∈[0.05 0.0581].
(a)
(b)Figure 23
ω
=
110 and kw=0 with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 24
ω
=
174 and kw=0with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 25
ω
=
192 and kw=0with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 26
ω
=
206 and kw=0with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 27
ω
=
220 and kw=0 with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 28
ω
=
237 and kw=0with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)Figure 29
ω
=
273 and kw=0with the Coulomb model when t∈0.050.0581. (a) Phase trajectories. (b) Poincare maps.
(a)
(b)From Figure22(b), the maximum Lyapunov exponent is negative or fluctuates near zero when ω<210, indicating that the system is stable according to the Lyapunov theorem. The maximum Lyapunov exponent is positive when ω>210, which indicates that the system exhibits chaos and becomes divergent and uncontrollable. In general, friction in the Coulomb model increases with an increase in tooth surface roughness because of the change in the nonlinear friction coefficient. Meanwhile, friction in the Coulomb model has nonperiodic and sharp fluctuations when t>0.0581. Bifurcation diagrams, maximum Lyapunov exponents, phase diagrams, Poincare maps corresponding to Figure 21, and specific analyses are shown in Figures 30(a) and 30(b). Whent>0.0581, use of the Coulomb model to predict the time-varying friction coefficient exhibits chaos and becomes divergent and uncontrollable. Figure 30(a) shows the vibrational bifurcation diagram of the system using the Coulomb model to predict the time-varying friction coefficient when 0.0581<t<0.1009, and the maximum Lyapunov diagram of the system is shown in Figure 30(b). From Figure 30(b), the maximum Lyapunov exponent is positive when 0.0581<t<0.1009, which indicates that the system exhibits chaos and becomes divergent and uncontrollable. Here, the Poincare maps and corresponding phase trajectories of the system are omitted.Figure 30
(a) Vibrational bifurcation diagram with the Coulomb model whent∈0.05810.1009. (b) Maximum Lyapunov exponent curve with the Coulomb model when t∈0.05810.1009.
(a)
(b)The dynamic friction obtained with the smoothed Coulomb model is shown in Figure31. Friction on the tooth surface exhibits obvious periodic changes when 0.05<t<0.0597 and ω<201. Bifurcation diagrams, maximum Lyapunov exponents, phase diagrams, Poincare maps corresponding to Figure 31, and specific analyses are shown in Figures 32(a)–38. The vibrational bifurcation diagram of the system using the smoothed Coulomb model to predict the time-varying friction coefficient when 0.05<t<0.0597 is given in Figure 32(a), which shows that the nonlinear characteristics of tooth surface friction in the gear transmission system is complex. Observations include the following:(1)
When0.05<t<0.0597 and ω<163.3, the system has one periodic point. Poincare maps and corresponding phase trajectories of the system at ω=110 are shown in Figure 33. In Figure 33, the system rapidly converges to one periodic point, indicating that the system is stable.(2)
When0.05<t<0.0597 and 163.3<ω<182, the system has four periodic points. Poincare maps and corresponding phase trajectories of the system at ω=174 are shown in Figure 34.(3)
With the increase ofω, the number of periodic points changes when 0.05<t<0.0597 and 182<ω<190.(4)
When190<ω<201, the system has nine periodic points. Poincare maps and corresponding phase trajectories of the system at ω=192 are shown in Figure 35.(5)
When0.05<t<0.0597and 201<ω<229, corresponding phase trajectories of the system constitute a limit cycle. Poincare maps and corresponding phase trajectories of the system at ω=210 are shown in Figure 36.(7)
When229<ω<272, the system has 13 periodic points. Poincare maps and corresponding phase trajectories of the system at ω=230 are shown in Figure 37.(8)
Whenω>272, the system becomes divergent and uncontrollable. Poincare maps and corresponding phase trajectories of the system at ω=273 are shown in Figure 38.Figure 31
Dynamic friction from the smoothed Coulomb model.Figure 32
(a) Vibrational bifurcation diagram with the smoothed Coulomb model whent∈0.050.0597. (b) Maximum Lyapunov exponent curve with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)Figure 33
ω
=
110 and kw=0 with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)Figure 34
ω
=
174 and kw=0 with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)Figure 35
ω
=
192 and kw=0 with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)Figure 36
ω
=
210 and kw=0 with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)Figure 37
ω
=
230 and kw=0 with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)Figure 38
ω
=
273 and kw=0 with the smoothed Coulomb model when t∈0.050.0597.
(a)
(b)From Figure32(b), the maximum Lyapunov exponent is negative or fluctuates near zero when ω<201, indicating that the system is stable according to the Lyapunov theorem. The maximum Lyapunov exponent is positive when ω>201, which indicates that the system exhibits chaos and becomes divergent and uncontrollable. In general, friction with the smoothed Coulomb model increases with an increase in tooth surface roughness because of the change of the nonlinear friction coefficient. Meanwhile, friction in the smoothed Coulomb model has nonperiodic and sharp fluctuations when t>0.0597. Bifurcation diagrams, maximum Lyapunov exponents, phase diagrams, Poincare maps corresponding to Figure 31, and specific analyses are shown in Figures 39(a) and 39(b). Figure 39(a) shows the vibrational bifurcation diagram of the system using the smoothed Coulomb model to predict the time-varying friction coefficient when 0.0597<t<0.0921, and the maximum Lyapunov diagram of the system is shown in Figure 39(b). From Figure 39(b), the maximum Lyapunov exponent is positive when 0.0597<t<0.0921, which indicates that the system exhibits chaos and becomes divergent and uncontrollable. Here, the Poincare maps and corresponding phase trajectories of the system are omitted.Figure 39
(a) Vibrational bifurcation diagram with the smoothed Coulomb model whent∈0.05970.0921. (b) Maximum Lyapunov exponent curve with the smoothed Coulomb model when t∈0.05970.0921.
(a)
(b)


## 3.4. Analysis of System with Random Perturbation

The nonlinear characteristics of a gear transmission system with random perturbation are analyzed as shown in Figures40–48. The basic parameters are the same as those presented above, but some random perturbations are added here; these include δpx, δpy, δgx, δgy, δch1, δch2, δch3, δε, δF∗∼N0,0.0012, and δωh∗∼N0,0.000052. The Poincare maps and corresponding phase trajectories of the system with random perturbation are shown in Figures 40–44, in which the nonlinear characteristics of the gear transmission system with small random perturbation are basically consistent with the nonlinear characteristics of the system without random perturbation (as shown in Figures 7–12). The system with small random perturbations converges to some degree, but the system without random perturbations converges to a finite point. The power spectra in Figures 45 and 46 were generated with a sampling frequency of 1000.Figure 40
ω
=
110 with random perturbation.
(a)
(b)Figure 41
ω
=
200 with random perturbation.
(a)
(b)Figure 42
ω
=
232 with random perturbation.
(a)
(b)Figure 43
ω
=
238 with random perturbation.
(a)
(b)Figure 44
ω
=
285 with random Perturbation.
(a)
(b)Figure 45
Power spectrum ofx10with random perturbation, ω=175.Figure 46
Power spectrum ofx10 without perturbation, ω=175.Figure 47
Vibrational bifurcation random diagram withδF∗∼N0,0.012.Figure 48
Vibrational bifurcation diagram withδF∗∼N0,0.122.The trend of the power spectrum ofx10 with and without random perturbation is almost identical, but the fluctuation of the power spectrum of x10 with random perturbation is greater than that without random perturbation.The vibrational bifurcation diagrams of the system withδF∗∼N0,0.012, δF∗∼N0,0.122 are shown in Figures 47 and 48. With the increase of δF∗, the dynamic characteristics of the system change from Figure 47 to 48. When δF∗ increases to the extent that δF∗∼N0,0.122, the bifurcation of the system disappears completely and the system becomes uncontrollable.


## 4. Conclusions

This study investigates the nonlinear characteristics of a gear transmission system in a braiding machine experiencing multiple excitation factors. The well-known radial braiding machine with one layer was used to investigate the nonlinear characteristics of the gear transmission system. The results show that the nonlinear characteristics of the gear transmission system under the influence of multiple excitation factors were in accordance with the principles of practical engineering.In conclusion, this research on the nonlinear characteristics of the gear transmission system in a braiding machine is helpful for engineers engaged in future design of this well-known machine. Our findings include the following:(1)
According to the bifurcation diagram of the system without random perturbation, the system exhibits chaos whenω>213. Therefore, ω must be controlled so ω<213, that is, the speed is 25561.25 r/s. If factors such as yarn tension and safety are taken into account, the safe speed in this braiding machine might be under about 302.5r/s, or 1.815×103 r/min.(2)
kw has a great influence on the nonlinear characteristics of the gear transmission system. With an increase in kw, e.g., with kw≥0.32, the system of the radial braiding machine with one layer always exhibits chaos. Therefore, kw must be controlled so that kw<0.32.(3)
Friction on the tooth surface has a large influence on the nonlinear characteristics of the gear transmission system. The Coulomb model and the smoothed Coulomb model are used to predict the time-varying friction coefficient. In general, friction on the tooth surface increases with increasing tooth surface roughness because of the change in the nonlinear friction coefficient. The system always exhibits chaos whent>0.0581 with the Coulomb model used to predict the time-varying friction coefficient. Meanwhile, the system always exhibits chaos when t>0.0597, and the smoothed Coulomb model is used to predict the time-varying friction coefficient.(4)
The system with small random perturbations converges to some degree, and the system without random perturbation converges to a finite point. However, the dynamic characteristics of the system have not changed. The increase of some certain perturbations does lead to changes in the dynamic characteristics of the system. WhenδF∗ increases to the extent that δF∗∼N0,0.122, the bifurcation of the system disappears completely and the system becomes uncontrollable.In this paper, we analyze nonlinear dynamic characteristics of a gear transmission system in a braiding machine and only consider disturbance of yarn tension, transmission error, time-varying friction, and contact temperature because of the time limit. However, many other factors have an important influence on nonlinear dynamic characteristics of a gear transmission system. Future research on the influence of other excitation factors operating on the gear transmission system in a braiding machine would be of value.



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*Source: 2747204-2020-03-25.xml*