# Dynamic Modeling and Chaotic Analysis of Gear Transmission System in a Braiding Machine with or without Random Perturbation

Authors: Zhang Yujing; Meng Zhuo; Sun Yize
Journal: Shock and Vibration
 (2016)

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## Abstract

This paper is aimed at analyzing the dynamic behavior of the gear transmission system in a braiding machine. In order to observe the nonlinear phenomenon and reveal the time-varying gear meshing mechanism, a mathematical model with five degrees-of-freedom gear system under internal and external random disturbance of gear system is established. With this model, bifurcation diagrams, Poincare maps, phase diagrams, power spectrum, time-process diagrams, and Lyapunov exponents are used to identify the chaotic status. Meanwhile, by these analytical methods, spur gear pair with or without random perturbation are compared. The numerical results suggest that the vibration behavior of the model is consistent with that of Clifford system. The chaotic system associated parameters are picked out, which can be helpful to the design and control of braiding machines.


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## Body

## 1. Introduction

Regardless of the different braiding methods, the performance of the same braiding machine type primarily depends on the motion system.In [1], the motion system of braiding machine is briefly introduced. Figure 1 is a detailed view of a large horizontal braiding machine. The yarn material is wound on bobbins, which are mounted on carriers. Every carrier follows the specific track by the guide chunk as their feet, moving forwards or backwards. Every gear is mounted on a shaft. At the same time, a horn disc with several cuts is fixed on the other side of the shaft. Horn disc pushes carrier base from one horn disc to another. Thus, carriers rotate toward the center, while the shafts rotate toward themselves. Obviously, one crucial thing of the motion system is the gear. Analyzing the gear meshing to keep down the vibration is an effective way of improving the performance of the braiding machine.Figure 1
Motion system of the braiding machine.Nonlinear dynamics of gear system has been studied for many years. Blankenship and Kahraman [2] discussed the single-degree-of-freedom gear system with the super-harmonic spur by taking parametric and clearance into account with the HBM method. Kahraman and Singh [3, 4] investigated the time-varying mesh stiffness and clearance nonlinearities in a geared system. Chang-Jian and Chang [5] studied the dynamic behavior of a single-degree-of-freedom spur gear system, revealing that the system exhibits periodic and chaotic vibration. Raghothama and Narayanan [6] have studied the three degrees-of-freedom nonlinear dynamic gear system by using bifurcation diagrams. Wang et al. [7] identified the parameters of gear system with sliding friction. Wang et al. and Xiaolin [8, 9] researched the three degrees-of-freedom gear system under the external random disturbance. But with using the methods of three degrees-of-freedom gear system and ignoring the influence of internal incentive in the modelling of braiding machines, the results cannot be satisfactory.During braiding, there are many spur sources to gears. The reactive force fluctuates severely. Not only external random disturbances must be considered, the internal disturbances cannot be ignored. On one hand, the yarn tension changes, especially when yarns are getting stuck. On the other hand, carriers change and the push force changes, which aggravates the fluctuation of forces. Moreover, as hundreds of gears participating in transmission, the influence of manufacture error and assembly error ought to be considered.Therefore, a study that focuses on analyzing the effects of both tangential and normal displacement, the eccentric force due to the manufacturing error, and assembly error to gear meshing is urgently needed.


## 2. Dynamic Modeling

To get the dynamic model, a schematic illustration is shown in Figure2. Here, the influence from two directions displacement to the stiffness coefficient of bearings is considered. mg1, mg2 are the mass of gear; me is effective mass; Ig1, Ig2 are the moment of inertia; rg1, rg2 are the base circle radius of gear; yg1, yg2, xg1, and xg2 are the center displacement of gear; Fb1x, Fb1y, Fb2x, and Fb2y are the force transmitted from bearing; ch is the damping coefficient of gear mesh; cb1x, cb1y, cb2x, and cb2y are the damping coefficient of bearing; Fe1, Fe2 are the eccentric force; φg1, φg2 are the angular displacement of gear; ϕg1(τ), ϕg2(τ) are the phase angle of eccentric force; fb1x, fb1y, fb2x, and fb2y are the displacement function of bearing; e(τ) is the static transmission error between gears; kb1x, kb1y, kb2x, and kb2y are the stiffness coefficient of bearing; kh(τ) is the time-varying mesh stiffness coefficient; F0 is the mean force of input torque; Fv is the variable amplitude of input torque; δΔ is the random disturbance of relevant parameter (subscript Δ stands for parameter); ρg1,2 is the mass eccentric arm; ωe1, ωe2 are the angular velocity of gear.Figure 2
Dynamic model of gear system.According to Newton’s laws of motion, the differential equations of the gear system dynamics can be deduced as(1)
m
g
1
y
¨
g
1
+
c
b
1
y
+
δ
c
b
1
y
y
˙
g
1
+
c
h
+
δ
c
h
z
˙
-
y
˙
g
1
+
y
˙
g
2
-
e
˙
τ
+
k
b
1
y
f
b
1
y
y
g
1
+
k
h
τ
f
h
z
-
y
g
1
+
y
g
2
-
e
τ
+
F
b
1
y
+
F
e
1
sin
⁡
ϕ
g
1
τ
=
0
,
m
g
1
x
¨
g
1
+
c
b
1
x
+
δ
c
b
1
x
x
˙
g
1
+
k
b
1
x
f
b
1
x
x
g
1
+
F
b
1
x
+
F
e
1
cos
⁡
ϕ
g
1
τ
=
0
,
m
g
2
y
¨
g
2
+
c
b
2
y
+
δ
c
b
2
y
y
˙
g
2
-
c
h
+
δ
c
h
z
˙
-
y
˙
g
1
+
y
˙
g
2
-
e
˙
τ
+
k
b
2
y
f
b
2
y
y
g
2
-
k
h
τ
f
h
z
-
y
g
1
+
y
g
2
-
e
τ
+
F
b
2
y
+
F
e
2
sin
⁡
ϕ
g
2
τ
=
0
,
m
g
2
x
¨
g
2
+
c
b
2
x
+
δ
c
b
2
x
x
˙
g
2
+
k
b
2
x
f
b
2
x
x
g
2
+
F
b
2
x
+
F
e
2
cos
⁡
ϕ
g
2
τ
=
0
,
m
e
z
¨
+
c
h
+
δ
c
h
z
˙
-
y
˙
g
1
+
y
˙
g
2
-
e
˙
τ
+
k
h
τ
f
h
z
-
y
g
1
+
y
g
2
-
e
τ
-
F
0
-
F
v
τ
-
δ
F
=
0
,where(2)z=rg1φg1-δφg1-rg2φg2-δφg2,me=Ig1Ig2Ig1rg22+Ig2rg12,fb1xu,fb1yu,fb2xu,fb2yu,fhu=u-b,u>b0,-b≤u≤bu+b,u<-bu=yg1,yg2,xg1,xg2,α,Fe1=mg1ρg1ωe12,Fe2=mg2ρg2ωe22.Meanwhile,kh(τ) and e(τ) are time-varying functions. They can be expanded into the following equations by Fourier transform:(3)khτ=k0+∑i=1∞kvcos⁡iωh+δωhτ+ϕhi,eτ=ev∑j=1∞sin⁡jωhτ+ϕhj,where k0 is the mean mesh stiffness; kv is the variable amplitude of mesh stiffness; ev is the amplitude of transmission error; ωh is the driving frequency; ϕhi, ϕhj are the phase angle of each harmonic term.As is known, fundamental frequency influences the model most. Therefore, neglecting the influence of phase angle,kh(τ) and e(τ) can be written as(4)khτ=k0+kvcos⁡ωh+δωhτ,eτ=evsinωhτ.For convenience and to simplify the equations, the following parameters are defined:(5)α=rg1φg1-δφg1-rg2φg2-δφg2-yg1+yg2-eτ,t=ωnτ,ωn=k0me,ω1=kb1ymg1,ω2=kb2ymg2,b∗=bbc,yg1∗=yg1bc,yg2∗=yg2bc,α∗=αbc,kb1x,y∗=kb1x,ybc,kb2x,y∗=kb2x,ybc,k0∗=k0bc,kv∗=kvbc,ωh∗=ωhωn,ω1∗=ω1ωn,ω2∗=ω2ωn,ε=khvkh0,where α is the relative torsional displacement of gear system; ω1, ω2 are equivalent frequency; b is half of total gear backlash; bc is characteristic length of gear backlash; Δ∗ is dimensionless parameter (subscript Δ stands for parameter except for coefficient of damping and stiffness).To analyze the equations, letx1=yg1∗, x2=yg1∗˙, x3=xg1∗, x4=xg1∗˙, x5=yg2∗, x6=yg2∗˙, x7=xg2∗, x8=xg2∗˙, x9=α∗, and x10=α∗˙, assign b=bc, and neglect initial phase angle of ϕg1,ϕg2. By using the defined dimensionless parameters above, the state equations (1) of the system can be turned into (6)x1˙=x2,(7)x2˙=-2γ11+δγ11x2-2γ15+δγ15x10-η11fb1yx1-η15tfhx9-Fb1y∗-ρg1∗ωh∗+δωh∗2z12cos⁡ωh∗+δωh∗t/z1,(8)x3˙=x4,(9)x4˙=-2γ22+δγ22x4-η22fb1xx3-Fb1x∗-ρg1∗ωh∗+δωh∗2z12sin⁡ωh∗+δωh∗t/z1,(10)x5˙=x6,(11)x6˙=-2γ33+δγ33x6+2γ35+δγ35x10-η33fb2x5+η35tfhx9-Fb2y∗-ρg2∗ωh∗+δωh∗2z22cos⁡ωh∗+δωh∗t/z2,(12)x7˙=x8,(13)x8˙=-2γ44+δγ44x8-η44fb2xx7-Fb2x∗-ρg2∗ωh∗+δωh∗2z22sin⁡ωh∗+δωh∗t/z2,(14)x9˙=x10,(15)x10˙=x2˙-x6˙-2γ55+δγ55x10-η55tfhx9+F0∗+ev∗ωh∗+δωh∗2sin⁡ωh∗+δωh∗t+δF∗,where z1, z2 are the number of teeth; γij is the dimensionless damping coefficient; ηij is the dimensionless stiffness coefficient (i,j=1,…,5); Fb1∗=Fb1/mg1bcωn2; Fb2∗=Fb2/mg2bcωn2; γ11=cb1y/2mg1ωn; γ15=ch/2mg1ωn; γ22=cb1x/2mg1ωn; γ33=cb2y/2mg2ωn; γ35=ch/2mg2ωn; γ44=cb2x/2mg2ωn; γ55=ch/2meωn; η11=kb1y/2mg1ωn; η22=kb1x/2mg1ωn; η33=kb2y/2mg2ωn; η44=kb2x/2mg2ωn; η55(t)=4η15(t)=4η35(t)=1 − (ε+δε)cos[(ωh∗+δωh∗)t].


## 3. Analysis of Numerical Simulation Results and Discussion

Based on the mathematical model established above, the nonlinear dynamic equations were solved by difference-iteration method. During the numerical simulation, the time step was assigned a value of 0.05, and, for convenience, letω=ωh∗. In the study, the ω is variable, while other parameters are constant: m1=m2=me=0.456 kg, Ig1=Ig2=6×10-6 kg·m2, z1=z2=30, bc=0.068mm, ev=0.034mm, k0=2.23×108 N·m, and kv=1.12×107 N·m. Here, k0, kv are obtained by reference to [10, 11]. Therefore, the value of system parameters can be set as ev∗=0.05, Fb1y∗=0.2, Fb1x∗=0.2, Fb2y∗=-0.2, Fb2x∗=0.2, F0∗=0.2, η11=1.1, η22=1.1, η33=1.1, η44=1.1, ε=0.18, γ11=0.01, γ22=0.01, γ33=0.01, γ44=0.01, γ15=0.012, γ35=0.012, γ55=0.05, ρg1∗=0.1, and ρg2∗=0.1. The initial state is set as x1(0)=0, x2(0)=-0.1, x3(0)=0, x4(0)=-0.1, x5(0)=0, x6(0)=-0.1, x7(0)=0, x8(0)=-0.1, x9(0)=0, and x10(0)=-0.1.


### 3.1. Analysis of System without Random Perturbation

The bifurcation diagram of system without random perturbation, which is generated withω and x10 in (10)–(15), and the maximum Lyapunov exponent curve graph are shown in Figure 3, while Poincare maps and corresponding phase trajectories are drawn in Figures 4–9.Figure 3
System response without random perturbation.
(a)
Vibrational bifurcation diagram
(b)
Maximum Lyapunov exponent curveFigure 4
ω
=
110 without random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 5
ω
=
175 without random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 6
ω
=
180 without random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 7
ω
=
200 without random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 8
ω
=
240 without random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 9
ω
=
262 without random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsOn one hand, from the bifurcation diagram in Figure3(a), it can be observed that the system presents a nonlinear vibration periodically. (1) When ω<153.5, the system only has one periodic point, which is shown in Figure 3(a). From the phase trajectory of ω=110 in Figure 4, it can be seen that iteration converges on one periodic point quickly, which means the motion of the nonlinear dynamic system is steady. (2) As the parameter ω increases from 153.70 to 174.18, the number of periodic points varies. (3) Nonetheless, when 174.19<ω<179.15, there are 7 periodic points according to Figure 3(a). And a case of ω=175 is shown in Figure 5. (4) During 179.16<ω<180.05, there are 18 periodic points in the diagram in Figure 3(a), which is a consequence of period bifurcations. And Figure 6 is an example of ω=180. (5) In Figure 3(a), while 180.06<ω<220, the system exhibits a chaotic behavior. And Figure 7 is an illustration of ω=200. (6) By observing Figures 3(a) and 8, when 220<ω<261.17, the periodic points form a limit-circle. (7) Until ω is above 261.18 in Figure 3(a), with amplitudes expanding, the vibration and noise increase rapidly. At last, in Figure 9, while ω=262 system becomes divergent and uncontrollable.On the other hand, from the maximum Lyapunov exponent curve graph in Figure3(b), the nonlinear phenomenon also can be deduced. Here, the maximum Lyapunov exponents are calculated by using Jacobi matrix. Meanwhile, Lyapunov exponent is defined as(16)λk=limi→∞1iln⁡λki,k=1,…,nwhereλ1i≥λ2i≥⋯≥λni.Whenω is assigned about 160 to 180, the maximum Lyapunov exponents oscillate around zero. According to the L-S theory, system bifurcates, which is consistent with the bifurcation diagram. When ω keeps increasing, the maximum Lyapunov exponents always stay positive. Chaotic behavior occurs, and after that, system becomes diverging gradually.


### 3.2. Analysis of System with Random Perturbation

While observing system with random perturbation, the following parameters are assigned:(17)δγ11,δγ15,δγ22,δγ33,δγ35,δγ44,δγ55,δε,δF∗~N0,0.0012,δωn∗~N0,0.000052.Meanwhile, values of other parameters are the same as in Section3.1.The Poincare maps and corresponding phase trajectories are drawn from Figures10–15. It can be inferred that, after taking random perturbation into consideration, the system diverges in different degree. In Figures 10 and 11, the system still keeps stable. Yet, while ω=180, the system advanced exhibits a chaotic behavior in Figure 12. By comparison, the system without random perturbation only has 18 periodic points.Figure 10
ω
=
110 with random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 11
ω
=
175 with random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 12
ω
=
180 with random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 13
ω
=
200 with random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 14
ω
=
240 with random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 15
ω
=
262 with random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsTo obtain more detail of the system under random perturbation, power spectrum, time-process diagrams, and so forth are drawn in the following. Here, takeω=175, for instance. In the time-process diagrams in Figure 16, the amplitude without random perturbation is -0.09510.9517, while that with random perturbation is -0.25861.0949. The power spectrum in Figure 17 was depicted by a sampling frequency of 1000. And the Lyapunov exponents in Figure 18 were drawn by iteration time from 100 to 1000. Neither of them changes much. In a nutshell, random perturbation does not change the system performance fundamentally.Figure 16
Time-process diagrams,ω=175.
(a)
Without random perturbation
(b)
With random perturbationFigure 17
Power spectrum,ω=175.
(a)
Without random perturbation
(b)
With random perturbationFigure 18
Lyapunov exponents,ω=175.
(a)
Without random perturbation
(b)
With random perturbationHowever, if the value ofδF∗ fluctuates more violently, the dynamical property of the system will be changed. (1) When δF∗~N(0,0.012), the bifurcation of the system in Figure 19(a) is not clearly seen. (2) When δF∗~N(0,0.12), the bifurcation of the system disappears completely, and the system becomes uncontrollable in Figure 19(b).Figure 19
Bifurcation diagrams with random perturbation.
(a)
δ
F
∗
~
N
(
0,0.0
1
2
)
(b)
δ
F
∗
~
N
(
0,0
.
1
2
)


## 3.1. Analysis of System without Random Perturbation

The bifurcation diagram of system without random perturbation, which is generated withω and x10 in (10)–(15), and the maximum Lyapunov exponent curve graph are shown in Figure 3, while Poincare maps and corresponding phase trajectories are drawn in Figures 4–9.Figure 3
System response without random perturbation.
(a)
Vibrational bifurcation diagram
(b)
Maximum Lyapunov exponent curveFigure 4
ω
=
110 without random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 5
ω
=
175 without random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 6
ω
=
180 without random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 7
ω
=
200 without random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 8
ω
=
240 without random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 9
ω
=
262 without random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsOn one hand, from the bifurcation diagram in Figure3(a), it can be observed that the system presents a nonlinear vibration periodically. (1) When ω<153.5, the system only has one periodic point, which is shown in Figure 3(a). From the phase trajectory of ω=110 in Figure 4, it can be seen that iteration converges on one periodic point quickly, which means the motion of the nonlinear dynamic system is steady. (2) As the parameter ω increases from 153.70 to 174.18, the number of periodic points varies. (3) Nonetheless, when 174.19<ω<179.15, there are 7 periodic points according to Figure 3(a). And a case of ω=175 is shown in Figure 5. (4) During 179.16<ω<180.05, there are 18 periodic points in the diagram in Figure 3(a), which is a consequence of period bifurcations. And Figure 6 is an example of ω=180. (5) In Figure 3(a), while 180.06<ω<220, the system exhibits a chaotic behavior. And Figure 7 is an illustration of ω=200. (6) By observing Figures 3(a) and 8, when 220<ω<261.17, the periodic points form a limit-circle. (7) Until ω is above 261.18 in Figure 3(a), with amplitudes expanding, the vibration and noise increase rapidly. At last, in Figure 9, while ω=262 system becomes divergent and uncontrollable.On the other hand, from the maximum Lyapunov exponent curve graph in Figure3(b), the nonlinear phenomenon also can be deduced. Here, the maximum Lyapunov exponents are calculated by using Jacobi matrix. Meanwhile, Lyapunov exponent is defined as(16)λk=limi→∞1iln⁡λki,k=1,…,nwhereλ1i≥λ2i≥⋯≥λni.Whenω is assigned about 160 to 180, the maximum Lyapunov exponents oscillate around zero. According to the L-S theory, system bifurcates, which is consistent with the bifurcation diagram. When ω keeps increasing, the maximum Lyapunov exponents always stay positive. Chaotic behavior occurs, and after that, system becomes diverging gradually.


## 3.2. Analysis of System with Random Perturbation

While observing system with random perturbation, the following parameters are assigned:(17)δγ11,δγ15,δγ22,δγ33,δγ35,δγ44,δγ55,δε,δF∗~N0,0.0012,δωn∗~N0,0.000052.Meanwhile, values of other parameters are the same as in Section3.1.The Poincare maps and corresponding phase trajectories are drawn from Figures10–15. It can be inferred that, after taking random perturbation into consideration, the system diverges in different degree. In Figures 10 and 11, the system still keeps stable. Yet, while ω=180, the system advanced exhibits a chaotic behavior in Figure 12. By comparison, the system without random perturbation only has 18 periodic points.Figure 10
ω
=
110 with random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 11
ω
=
175 with random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 12
ω
=
180 with random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 13
ω
=
200 with random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 14
ω
=
240 with random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsFigure 15
ω
=
262 with random perturbation.
(a)
Phase trajectories
(b)
Poincare mapsTo obtain more detail of the system under random perturbation, power spectrum, time-process diagrams, and so forth are drawn in the following. Here, takeω=175, for instance. In the time-process diagrams in Figure 16, the amplitude without random perturbation is -0.09510.9517, while that with random perturbation is -0.25861.0949. The power spectrum in Figure 17 was depicted by a sampling frequency of 1000. And the Lyapunov exponents in Figure 18 were drawn by iteration time from 100 to 1000. Neither of them changes much. In a nutshell, random perturbation does not change the system performance fundamentally.Figure 16
Time-process diagrams,ω=175.
(a)
Without random perturbation
(b)
With random perturbationFigure 17
Power spectrum,ω=175.
(a)
Without random perturbation
(b)
With random perturbationFigure 18
Lyapunov exponents,ω=175.
(a)
Without random perturbation
(b)
With random perturbationHowever, if the value ofδF∗ fluctuates more violently, the dynamical property of the system will be changed. (1) When δF∗~N(0,0.012), the bifurcation of the system in Figure 19(a) is not clearly seen. (2) When δF∗~N(0,0.12), the bifurcation of the system disappears completely, and the system becomes uncontrollable in Figure 19(b).Figure 19
Bifurcation diagrams with random perturbation.
(a)
δ
F
∗
~
N
(
0,0.0
1
2
)
(b)
δ
F
∗
~
N
(
0,0
.
1
2
)


## 4. Conclusions

This paper proposed a nonlinear dynamic model appropriate for the gear transmission system in a braiding machine. The influence of time-varying mesh stiffness, eccentric forces, multidisplacement functions of bearings, reactive forces of carriers, and so forth is fully considered in the mathematical model. The differential equations are similar to the Clifford system but contain more nonlinear terms. And the numerical results are accordant to those in [12]. Here, the Clifford system can be written as follows:(18)xk+1=sin⁡ayk-zkcos⁡bxk,yk+1=zksin⁡cyk-cos⁡dxk,zk+1=esin⁡bxk.Based on the differential equations, bifurcation diagrams, Lyapunov exponents, Poincare maps, phase trajectories, and so forth are used to analyze the dynamic behavior of the system. In the bifurcation diagrams, it is shown that the bifurcation diagram of the system is similar to that of Clifford system, but more complex. As there are more nonlinear terms than those in Clifford system, the number of bifurcation points increases, and more periodic points can be seen. These causes the phenomenon of period doubling bifurcations cannot be distinguished significantly.The results of numerical analysis have shown that, accompanied by the increasing ofω, periodic bifurcations take place. When ω=180.06, chaotic behavior can be observed. In other words, the dimensionless parameter ω ought to be controlled under 180.06, which is equal to a speed of 21125 r/s. If the spur transferred from tension of yarns and security coefficient is taken into account, the safe speed in this braiding machine might be under about 250 r/s, that is, 1.5×104 r/min.Although the theoretical speed is very high, it has no regard for random perturbation. IfδF∗~N(0,0.12), the system of the braiding machine becomes uncontrollable. In other words, the dimensionless parameter δF∗ ought to be controlled under an amplitude of 0.1, which is equal to about 2.2 kN. Since there are hundreds of gears forming a long transmission chain, with many other excitations existing, it is very likely to reach the value. When this situation happens, the system might be out of order, which would result in the interruption of braiding process, or the breaking of yarns. Therefore, the values obtained above might be a reference during the design of this type of braiding machines.



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*Source: 8457645-2016-01-11.xml*