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

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-offreedom 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.


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.
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 superharmonic 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-offreedom 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.

Dynamic Modeling
To get the dynamic model, a schematic illustration is shown in Figure 2. Here, the influence from two directions displacement to the stiffness coefficient of bearings is considered. 1 ,  2 are the mass of gear;   is effective mass;  1 ,  2 are the moment of inertia;  1 ,  2 are the base circle radius of gear;  1 ,  2 ,  1 , and  2 are the center displacement of gear;  1 ,  1 ,  2 , and  2 are the force transmitted from bearing;  ℎ is the damping coefficient of gear mesh;  1 ,  1 ,  2 , and  2 are the damping coefficient of bearing;  1 ,  2 are the eccentric force;  1 ,  2 are the angular displacement of gear;  1 (),  2 () are the phase angle of eccentric force;  1 ,  1 ,  2 , and  2 are the displacement function of bearing; () is the static transmission error between gears;  1 ,  1 ,  2 , and  2 are the stiffness coefficient of bearing;  ℎ () is the time-varying mesh stiffness coefficient;  0 is the mean force of input torque;  V is the variable amplitude of input torque;  Δ is the random disturbance of relevant parameter (subscript Δ stands for parameter);  1,2 is the mass eccentric arm;  1 ,  2 are the angular velocity of gear.
According to Newton's laws of motion, the differential equations of the gear system dynamics can be deduced as where Meanwhile,  ℎ () and () are time-varying functions.They can be expanded into the following equations by Fourier transform: where  0 is the mean mesh stiffness;  V is the variable amplitude of mesh stiffness;  V is the amplitude of transmission error;  ℎ is the driving frequency;  ℎ ,  ℎ are the phase angle of each harmonic term.
As is known, fundamental frequency influences the model most.Therefore, neglecting the influence of phase angle,  ℎ () and () can be written as For convenience and to simplify the equations, the following parameters are defined: where  is the relative torsional displacement of gear system;  1 ,  2 are equivalent frequency;  is half of total gear backlash;   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, let ,  9 =  * , and  10 = α * , assign  =   , and neglect initial phase angle of  1 ,  2 .By using the defined dimensionless parameters above, the state equations (1) of the system can be turned into

Analysis of System without Random Perturbation.
The bifurcation diagram of system without random perturbation, which is generated with  and  10 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.
On one hand, from the bifurcation diagram in Figure 3(a), it can be observed that the system presents a nonlinear vibration periodically.(1) When  < 153.5, the system only has one periodic point, which is shown in Figure 3(a).From the phase trajectory of  = 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  increases from 153.70 to 174.18, the number of periodic points varies.(3) Nonetheless, when 174.19 <  < 179.15, there are 7 periodic points according to Figure 3(a).And a case of  = 175 is shown in Figure 5. (4) During 179.16 <  < 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  = 180.
(5) In Figure 3(a), while 180.06 <  < 220, the system exhibits a chaotic behavior.And Figure 7 is an illustration of  = 200.(6) By observing Figures 3(a) and 8, when 220 <  < 261.17, the periodic points form a limit-circle.(7) Until  is above 261.18 in Figure 3(a), with amplitudes expanding, the vibration and noise increase rapidly.At last, in Figure 9, while  = 262 system becomes divergent and uncontrollable.
On the other hand, from the maximum Lyapunov exponent curve graph in Figure 3  Lyapunov exponents are calculated by using Jacobi matrix.Meanwhile, Lyapunov exponent is defined as When  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  keeps increasing, the maximum Lyapunov exponents always stay positive.Chaotic behavior occurs, and after that, system becomes diverging gradually.

Analysis of System with
Random Perturbation.While observing system with random perturbation, the following parameters are assigned: Meanwhile, values of other parameters are the same as in Section 3.1.
The Poincare maps and corresponding phase trajectories are drawn from Figures 10-15.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  = 180, the system advanced exhibits a chaotic behavior in Figure 12.By comparison, the system without random perturbation only has 18 periodic points.
To obtain more detail of the system under random perturbation, power spectrum, time-process diagrams, and   However, if the value of  *  fluctuates more violently, the dynamical property of the system will be changed.(1) When  *  ∼ (0, 0.01 2 ), the bifurcation of the system in Figure 19(a) is not clearly seen.( 2) When  *  ∼ (0, 0.1 2 ), the bifurcation of the system disappears completely, and the system becomes uncontrollable in Figure 19(b).

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:

Figure 1 :
Figure 1: Motion system of the braiding machine.

Figure 2 :
Figure 2: Dynamic model of gear system.
(b), the nonlinear phenomenon also can be deduced.Here, the maximum