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.
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 Figure 1. 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. 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; refers to the angle between one yarn and the Z-axis; refers to the angular velocity of the spindle; 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.
2.2. Dynamic Modeling of the Gear System
Figures 3(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.
(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 Figure 4 and the dynamic equations of the gear system areas are established in equation (1). Here, and are the masses of the gears; is the effective mass; , , , and are the equivalent dampings of bearing; is the random disturbance of ; is the random disturbance of ; is the random disturbance of ; is the random disturbance of ; , , , and are the equivalent stiffnesses of bearings; , , , , and are the displacement functions of the bearing; is the sign function; represents friction at the tooth surface; , , , and are the forces transmitted from the bearing; and are the eccentric forces; and are the angular displacements of the gear; and are the phase angles of eccentric force; is the damping coefficient of gear meshing; is the random disturbance of ; and are the base circle radii of the gear; , , , and are the center displacements of gears; is the static transmission error; is the time-varying meshing stiffness coefficient; is the rotational inertia of the initiative gear; is the driving torque of the initiative gear; is the rotational inertia of the passive gear; and are the masses of eccentric arms; and are the angular velocities of gears; is the load torque of the passive gear; is the relative torsional displacement of gear pairs; is the amplitude of the time-varying stiffness fluctuation caused by temperature variation; 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.
The dynamic equations of a gear transmission system in a radial braiding machine are established according to Newton’s laws as follows:
After Fourier series expansion, we take the first-order components of and , which are simplified as follows:
Here, is the average meshing stiffness; is the magnitude of variation of meshing stiffness; is the driving frequency of gear pairs; is the random disturbance of ; and is the amplitude of transmission error.
The frequency of gear pairs is
The frequency of gear is
The frequency of gear is
The backlash ratio is
The stiffness ratio iswhere is the actual backlash, is the standard backlash, , is the amplitude of time-varying meshing stiffness, and 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:
Here, is the prescribed friction coefficient, satisfies , is the time when the meshing point of the gear passes through a base pitch after meshing, , , , , and , 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:where , the smoothness, , is between 20 and 100, and is the overlap ratio.
For convenience in solving, a dimensionless transformation is made:
For ease in programming, a variable substitution is made:
Substituting equations (2)–(9) into (1) givesafter dimensionless processing, and with and as the tooth numbers of the gear.
Here, , , , , , , , , , , , , , , , , , , , , , and .
3. Results and Discussion
Equations (10)–(19) are solved by iterative methods with step size , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and initial conditions are as follows: .
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 Figures 6(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.
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
The nonlinear vibration characteristics of the system and the relationship between and are shown in Figure 6(a). Observations include the following:(1)From Figure 6(a), the system has one periodic point when . The Poincare maps and corresponding phase trajectories of the system at are shown in Figure 7, where it can be observed that the system converged rapidly to one periodic point at ; this suggests that the system was stable.(2)The system has four periodic points when and Poincare maps and corresponding phase trajectories of the system at are shown in Figure 8.(3)With the increase of , the number of periodic points changes when .(4)The system has nine periodic points when , and Poincare maps and corresponding phase trajectories of the system at are shown in Figure 9.(5)From Figure 6(a), chaos occurs in the system when . Poincare maps and corresponding phase trajectories of the system at are shown in Figure 10.(6)When , the corresponding phase trajectories of the system comprise a limit cycle, and Poincare maps and corresponding phase trajectories of the system at are shown in Figure 11.(7)When , the system becomes divergent and uncontrollable, and Poincare maps and corresponding phase trajectories of the system at 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 , , where . In Figure 6(b), the maximum Lyapunov exponent is negative or fluctuates near zero when , which indicates that the system is stable according to the Lyapunov theorem. The maximum Lyapunov exponent is positive when , which indicates that the system exhibits chaos and becomes divergent and uncontrollable.
The nonlinearity shown in Figure 6(b) is consistent with the analysis from Poincare maps and the corresponding phase trajectories of the system.
3.2. Systematic Analysis of
The nonlinear characteristics of a gear transmission system as a function of are analyzed in Figures 13(a)–20(b). The vibrational bifurcation diagram of the system with is shown in Figure 13(a), the maximum Lyapunov diagram of the system with is shown in Figure 13(b), and Poincare maps and corresponding phase trajectories of the system with are shown in Figures 14–19. In addition, the vibrational bifurcation diagram of a system with is shown in Figure 20(a), and the maximum Lyapunov diagram of the system with is shown in Figure 20(b). From Figures 13(a) and 20(a), it appears that exerts a significant influence on the nonlinear characteristics of the gear transmission system. If is relatively small, such as , 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 are shown in Figures 13(a)–19. Comparing Figures 6(a) and 13(a), it is observed that the vibration amplitude when is larger than the resulting when , and the system exhibits chaos earlier when as opposed to when . When , the system exhibits chaos with , but when , the system exhibits chaos with . When , the form of the solution with is similar to that of . When , the system exhibits chaos and Poincare maps and corresponding phase trajectories of the system at are shown in Figure 17. When , the corresponding phase trajectories of the system form a limit cycle. Poincare maps and corresponding phase trajectories of the system at are shown in Figure 18. When , the system becomes divergent and uncontrollable. Poincare maps and corresponding phase trajectories of the system at are shown in Figure 19. From Figure 13(b), the maximum Lyapunov exponent is negative or fluctuates near zero when , indicating that the system is stable according to the Lyapunov theorem. The maximum Lyapunov exponent is positive when , 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 is relatively large, such as , 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 are shown in Figures 20(a) and 20(b). From Figure 20(b), it is clear that the maximum Lyapunov exponent is positive if is relatively large, such as , indicating that the system exhibits chaos and becomes divergent and uncontrollable. Here, the Poincare maps and corresponding phase trajectories of the system are omitted.
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
3.3. Systematic Analysis of Friction on the Tooth Surface
The dynamic friction resulting from the Coulomb model is shown in Figure 21. Friction on the tooth surface exhibits an obvious periodic variation when and . Bifurcation diagrams, maximum Lyapunov exponents, phase diagrams, Poincare maps corresponding to Figure 21, and specific analyses are shown in Figures 22(a)–29. When , 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)When and , the system has one periodic point. Poincare maps and corresponding phase trajectories of system at are shown in Figure 23; this shows that the system rapidly converges to one periodic point, indicating that the system is stable.(2)When and , the system has four periodic points, and Poincare maps and corresponding phase trajectories of system at are shown in Figure 24.(3)With increasing , the number of periodic points changes when and .(4)When and , the system has nine periodic points, and Poincare maps and corresponding phase trajectories of the system at are shown in Figure 25.(5)From Figure 22(a), the system has a finite number of periodic points when and . Poincare maps and corresponding phase trajectories of the system at are shown in Figure 26.(6)From Figure 22(a), the phase trajectories of the system when and constitute a limit cycle. Poincare maps and corresponding phase trajectories of the system at are shown in Figure 27.(7)When , the system has 13 periodic points. Poincare maps and corresponding phase trajectories of the system at are shown in Figure 28.(8)When , the system becomes divergent and uncontrollable. Poincare maps and corresponding phase trajectories of the system at are shown in Figure 29.
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
From Figure 22(b), the maximum Lyapunov exponent is negative or fluctuates near zero when , indicating that the system is stable according to the Lyapunov theorem. The maximum Lyapunov exponent is positive when , 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 . 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). When, 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 , 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 , 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.
(a)
(b)
The dynamic friction obtained with the smoothed Coulomb model is shown in Figure 31. Friction on the tooth surface exhibits obvious periodic changes when and . 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 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)When and , the system has one periodic point. Poincare maps and corresponding phase trajectories of the system at are shown in Figure 33. In Figure 33, the system rapidly converges to one periodic point, indicating that the system is stable.(2)When and , the system has four periodic points. Poincare maps and corresponding phase trajectories of the system at are shown in Figure 34.(3)With the increase of , the number of periodic points changes when and .(4)When , the system has nine periodic points. Poincare maps and corresponding phase trajectories of the system at are shown in Figure 35.(5)When and , corresponding phase trajectories of the system constitute a limit cycle. Poincare maps and corresponding phase trajectories of the system at are shown in Figure 36.(7)When , the system has 13 periodic points. Poincare maps and corresponding phase trajectories of the system at are shown in Figure 37.(8)When , the system becomes divergent and uncontrollable. Poincare maps and corresponding phase trajectories of the system at are shown in Figure 38.
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
From Figure 32(b), the maximum Lyapunov exponent is negative or fluctuates near zero when , indicating that the system is stable according to the Lyapunov theorem. The maximum Lyapunov exponent is positive when , 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 . 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 , 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 , 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.
(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 Figures 40–48. The basic parameters are the same as those presented above, but some random perturbations are added here; these include , , , , , , , , , and . 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.
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
The trend of the power spectrum of with and without random perturbation is almost identical, but the fluctuation of the power spectrum of with random perturbation is greater than that without random perturbation.
The vibrational bifurcation diagrams of the system with , are shown in Figures 47 and 48. With the increase of , the dynamic characteristics of the system change from Figure 47 to 48. When increases to the extent that , 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 when . Therefore, must be controlled so , that is, the speed is 25561.25 . If factors such as yarn tension and safety are taken into account, the safe speed in this braiding machine might be under about 302.5, or .(2) has a great influence on the nonlinear characteristics of the gear transmission system. With an increase in , e.g., with , the system of the radial braiding machine with one layer always exhibits chaos. Therefore, must be controlled so that .(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 when with the Coulomb model used to predict the time-varying friction coefficient. Meanwhile, the system always exhibits chaos when , 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. When increases to the extent that , 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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Acknowledgments
This project was supported by the Special Fund of Scientific and Technological Achievements Transformation in Jiangsu Province (grant no. BA2018061), the Program of Jiangsu Innovative Talent (grant no. 20171122), the Fundamental Research Funds for the Central Universities, and Graduate Student Innovation Fund of Donghua University (grant number CUSF-DH-D-2019097).