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.

Regardless of the different braiding methods, the performance of the same braiding machine type primarily depends on the motion system.

In [

Motion system of the braiding machine.

Nonlinear dynamics of gear system has been studied for many years. Blankenship and Kahraman [

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.

To get the dynamic model, a schematic illustration is shown in Figure

Dynamic model of gear system.

According to Newton’s laws of motion, the differential equations of the gear system dynamics can be deduced as

Meanwhile,

As is known, fundamental frequency influences the model most. Therefore, neglecting the influence of phase angle,

For convenience and to simplify the equations, the following parameters are defined:

To analyze the equations, let

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, let ^{2},

The bifurcation diagram of system without random perturbation, which is generated with

System response without random perturbation.

Vibrational bifurcation diagram

Maximum Lyapunov exponent curve

Phase trajectories

Poincare maps

Phase trajectories

Poincare maps

Phase trajectories

Poincare maps

Phase trajectories

Poincare maps

Phase trajectories

Poincare maps

Phase trajectories

Poincare maps

On one hand, from the bifurcation diagram in Figure

On the other hand, from the maximum Lyapunov exponent curve graph in Figure

When

While observing system with random perturbation, the following parameters are assigned:

Meanwhile, values of other parameters are the same as in Section

The Poincare maps and corresponding phase trajectories are drawn from Figures

Phase trajectories

Poincare maps

Phase trajectories

Poincare maps

Phase trajectories

Poincare maps

Phase trajectories

Poincare maps

Phase trajectories

Poincare maps

Phase trajectories

Poincare maps

To obtain more detail of the system under random perturbation, power spectrum, time-process diagrams, and so forth are drawn in the following. Here, take

Time-process diagrams,

Without random perturbation

With random perturbation

Power spectrum,

Without random perturbation

With random perturbation

Lyapunov exponents,

Without random perturbation

With random perturbation

However, if the value of

Bifurcation diagrams with random perturbation.

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 [

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 of

Although the theoretical speed is very high, it has no regard for random perturbation. If

None of the authors have a conflict of interests.

This work was supported by the National Natural Science Foundation of China under Grant [51475091]; Program of Shanghai Leading Talent under Grant [20141032].