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Abstract

In the multi-ring braiding equipment, gear transmission system is an extremely important part, its operation process will be affected by a variety of factors. Considering the influence of contact temperature on the dynamic behavior of the gear transmission system, a mathematical model of the gear system is established in order to observe the nonlinear characteristics of the system. And the bifurcation diagrams, the maximum Lyapunov exponents diagrams, phase diagrams, Poincaré mapping diagrams, and the power spectrum diagrams are analyzed. Using the final iteration results to study the nonlinear characteristics of the gear system and the model is evaluated to show the key regions of sensitivity. The results show that $k_{w}$ must be controlled $k_{w} < 0.32$ .The amplitude response of the system is strong when $200 \leq ω \leq 240$ . Compared to the bifurcation diagram, the system exhibits chaotic behavior within the range of this frequency, and as $k_{w}$ increase, the strong amplitude response of the system increases at $200 \leq ω \leq 240$ .The nonlinear characteristics are greatly influenced by the incentive factors such as the contact temperature, and the analysis results are beneficial to the design and control of the braiding machine.

Keywords

Multi-ring braiding machine nonlinear characteristic contact temperature

Introduction

With the rapid development of technology, new material manufacturing technologies have become a focus of attention in various fields. Meanwhile, three-dimensional braiding technology has been widely applied in many industries due to its unique advantages. However, despite the potential of this technology, it still faces many challenges in practical applications, including research on new composite materials, optimization of structural design, and the control of braiding process.

Stig et al.¹ studied fully interlaced 3D fabrics composed of carbon fibers and tested the mechanical properties such as tensile and compressive strength of this new composite material. Zhang et al.² conducted tests on three-dimensional braided composite materials and traditional two-dimensional composite materials. The results show that three-dimensional braided composite materials with eight different braiding structures have excellent mechanical properties. Liu³ et al designed a new braiding technology based on a self-made braiding machine, an improved sleeve positioning system. Based on this system, the influence of the braiding structure of three-dimensional braided reinforced composite materials on the bending performance and failure mode of three-dimensional fabrics were been studied. Gereke et al.⁴ designed mathematical models with different metrics to study the relationship between the structure of braided fabrics and the mechanical properties of materials. Based on the three-dimensional braided composite fabric model, the results obtained through parameterization research were discussed. Zhao⁵ et al measured the thermal conductivity of braided composite materials using the transient hot-wire method. The research results indicate that the thermal conductivity of three-dimensional braided composite materials is influenced by braiding parameters such as fabric structure and fiber volume fraction. Dahale⁶ et al investigated the effect of truncation density (braiding parameters) on the mechanical properties such as tension and compression of three-dimensional braided composite structures.

The stability control of the braiding process in a 3D braiding machine is crucial for determining whether the mechanical properties of 3D braided composite components have changed. During this process, various internal and external incentive factors will more or less affect the stability of braiding. Bigaud et al⁷ investigated the effects of yarn to yarn interactions and consolidation parameters during the braiding process on the stiffness and strength properties of composite materials.

The vibration situation of the gear system is also extremely important for the three-dimensional braiding process. Gan et al⁸ analyzed the gear transmission error and vibration of the head structure of the bevel gear transmission system in a large horizontal three-dimensional braiding equipment, meanwhile, finite element, modal, and response analyses were conducted after modeling.

The gear transmission system is an important part of the braiding process in various braiding machines, and the study of gear dynamics originated in earlier studies. In the 1950s, Tuplin⁹ established a mass-spring model and proposed the concept of a gear dynamics model. Nonlinear factors, such as tooth profile error and time-varying mesh stiffness, are gradually being applied to the modeling of gear system dynamics. Howard¹⁰ used a finite element analysis method and combined it with the variation in gear torsional meshing stiffness to introduce a simplified gear dynamics model in detail. This provides ideas for the establishment and innovation of subsequent gear-dynamics models. Jiang Chunlei¹¹ established an equivalent contact model and a calculation model for the meshing stiffness of double involute gears, and studied the influence of tooth surface wear on the dynamic characteristics of double involute gears. Li¹² proposed a lubrication and friction dynamics model to fully couple the frictional and dynamic behaviors of spur gear pairs. Chen¹³ proposed a spur gear dynamic model in which the deformation of a single tooth was considered, and a comparative verification was conducted. Dadon¹⁴ proposed a gear dynamics model that can be solved numerically by combining the most advanced gear modeling methods. Margielewicz¹⁵ established a gear simulation model considering the time-varying stiffness and clearance of meshing, and bifurcation diagrams, Lyapunov exponents, amplitude frequency distributions, and Poincaré diagrams were obtained. Yang¹⁶ used the principle of motion synthesis to establish a gear dynamics model under variable load excitation, and derived a motion differential equation to obtain the influence of time-varying stiffness on response frequency and vibration. Chen¹⁷ studied the first pair of gears in the gear transmission system of a coal mining machine, considered the coupling effect of multiple excitations under thermal deformation, andestablished a nonlinear dynamic model of a gear system with three degrees of freedom . Liu¹⁸ studied the dynamic response characteristics of spur gear systems, including lateral and torsional responses, which combine various internal and external coupling excitations, such as power, gear eccentricity, and backlash. Chen¹⁹ established a nonlinear dynamic model of a gear system considering time-varying stiffness and backlash and solved it using numerical methods. Through the result analysis, the dynamic characteristics of gear vibration displacement and dynamic meshing force were obtained. Bruzzone²⁰ analyzed the excitation sources in cylindrical gears, such as fatigue wear, noise vibration, and frictional lubrication, and discussed various dynamic models. Zhou²¹ comprehensively considered nonlinear characteristics such as time-varying mesh stiffness, backlash, and transmission error, and established a nonlinear coupled dynamic model of a gear-rotor-bearing transmission system. The analysis results are helpful for the design and research of gear systems.

In summary, there is no mention of factors such as contact temperature, which also has a significant impact on the dynamic behavior of the gear system in the braiding machine and cannot be ignored. The effect of contact temperature was considered, and a mathematical model of the gear transmission system was established in this study. The analysis results have important research significance for establishing a dynamic model to improve braiding quality.

Dynamic modeling

Braiding process

A mathematical model of the gear system were established considering the contact temperature to study the nonlinear characteristics of the gear system. Using bifurcation diagrams, maximum Lyapunov exponent diagrams, phase diagrams, Poincaré mapping diagrams, and power spectra diagrams, the nonlinear characteristics of the gear system considering the contact temperature were studied. The numerical results indicated that factors such as temperature have a significant impact on the nonlinear characteristics of the braiding equipment, and the model was evaluated. The analysis of relevant parameters is helpful for the design and control of braiding equipment.

The multi-ring braiding equipment transmits motion by continuous meshing gears, and the braiding chassis of the multi-ring braiding equipment is partially shown in Figure 1. Consider a pair of gears in multi-ring braiding equipment for research, and the meshing situation of the other gears is the same. When the spindle of the multi-ring braiding equipment is $4 \times 88$ , the tension of each yarn is measured by a tension meter as be 1N, and the amplitude of the tension disturbance is $δ_{F}^{*} = 4 \times 88 \times 1 = 352 N$ .

Figure 1.

The diagram of the braiding chassis in multi-ring braiding equipment.

Dynamic modeling of the gear system

To obtain the dynamic equation of the gear system, the schematic diagram of the gear system dynamic model is established as shown in Figure 2. From Figure 2, p represents the driving gear and g represents the passive gear; the subscript p represents the variable of the driving gear and g represents the variable of the passive gear; $m_{p}$ and $m_{g}$ are the masses of the gears; $m_{e}$ is the effective mass; $c_{p x}$ , $c_{p y}$ , $c_{g x}$ and $c_{g y}$ are the equivalent damping of bearing; $δ_{p x}$ is the random disturbance of $c_{p x}$ , $δ_{p y}$ is the random disturbance of $c_{p y}$ , $δ_{g x}$ is the random disturbance of $c_{g x}$ , $δ_{g y}$ is the random disturbance of $c_{g y}$ ; $k_{p x}$ , $k_{p y}$ , $k_{g x}$ , $k_{g y}$ are the equivalent stiffness of the bearing; $f_{p x}$ , $f_{p y}$ , $f_{g x}$ , $f_{g y}$ are the displacement function of the bearing; $f_{h}$ is the meshing displacement function of the bearing; $F_{p x}$ , $F_{p y}$ , $F_{g x}$ , $F_{g y}$ are the force transmitted by the bearings; $F_{e 1}$ , $F_{e 2}$ are the eccentric force; $φ_{p}$ , $φ_{g}$ are the angular displacement of gears; $ϕ_{p} (t)$ , $ϕ_{g} (t)$ are the phase angle of the eccentric force; $c_{h}$ is the meshing damping coefficient of the gear; $δ_{c h}$ is the random interference of $c_{h}$ ; $R_{p}$ , $R_{g}$ are the base circle radius of gears; $x_{p}$ , $x_{g}$ , $y_{p}$ , $y_{g}$ are the center displacement of gears;e(t) is the static transmission error; $k_{h} (t)$ is the time-varying meshing stiffness coefficient; $J_{p}$ is the rotational inertia of the driving gear; $T_{p}$ is the driving torque of the active gear; $J_{g}$ is the rotational inertia of the passive gear; $e_{p, g}$ represents the eccentricity; $ω_{p}$ , $ω_{g}$ are the angular velocity of gears, $T_{g}$ is the load torque of the passive gears; $δ$ is the relative torsional displacement of the gear pair; $k_{w}$ is the amplitude of time-varying stiffness fluctuations caused by temperature changes; $δ_{F}^{*}$ is the random disturbance of the load.

Figure 2.

The schematic diagram of gear system dynamic model.

According to Newton’s law, the dynamic equation of the gear system in braiding equipment considering internal excitation and temperature factors is established as

{\begin{cases} m_{p} \cdot {\ddot{x}}_{p} + (c_{p x} + δ_{p x}) \cdot {\dot{x}}_{p} + k_{p x} \cdot f_{p x} (x_{p}) + F_{p x} + F_{e 1} \cdot \cos [φ_{p} (τ)] = 0 \\ \begin{array}{l} m_{p} \cdot {\ddot{y}}_{p} + (c_{p y} + δ_{p y}) \cdot {\dot{y}}_{p} + k_{p y} \cdot f_{p y} (y_{p}) + (c_{h} + δ_{c h}) \cdot [R_{p} \cdot {\dot{θ}}_{p} - R_{g} \cdot {\dot{θ}}_{g} + {\dot{y}}_{p} - {\dot{y}}_{g} - \dot{e} (τ)] + \\ (k_{h} (t) + k_{w} (t)) \cdot f_{h} [R_{p} \cdot θ_{p} - R_{g} \cdot θ_{g} + y_{p} - y_{g} - e (τ)] + F_{p y} + F_{e 1} \cdot \sin [φ_{p} (τ)] = 0 \end{array} \\ \begin{array}{l} J_{p} \cdot {\ddot{θ}}_{p} + R_{p} \cdot (c_{h} + δ_{c h}) \cdot [R_{p} \cdot {\dot{θ}}_{p} - R_{g} \cdot {\dot{θ}}_{g} + {\dot{y}}_{p} - {\dot{y}}_{g} - \dot{e} (τ)] + \\ R_{p} \cdot (k_{h} (t) + k_{w} (t)) \cdot f_{h} [R_{p} \cdot θ_{p} - R_{g} \cdot θ_{g} + y_{p} - y_{g} - e (τ)] = T_{p} + δ_{F}^{*} \end{array} \\ m_{g} \cdot {\ddot{x}}_{g} + (c_{g x} + δ_{g x}) \cdot {\dot{x}}_{g} + k_{g x} \cdot f_{g x} (x_{g}) - F_{g x} - F_{e 2} \cdot \cos [φ_{g} (τ)] = 0 \\ \begin{array}{l} m_{g} \cdot {\ddot{y}}_{g} + (c_{g y} + δ_{g y}) \cdot {\dot{y}}_{g} + k_{g y} \cdot f_{g y} (y_{g}) - (c_{h} + δ_{c h}) \cdot [R_{p} \cdot {\dot{θ}}_{p} - R_{g} \cdot {\dot{θ}}_{g} + {\dot{y}}_{p} - {\dot{y}}_{g} - \dot{e} (τ)] - \\ (k_{h} (t) + k_{w} (t)) \cdot f_{h} [R_{p} \cdot θ_{p} - R_{g} \cdot θ_{g} + y_{p} - y_{g} - e (τ)] - F_{g y} - F_{e 2} \cdot \sin [φ_{g} (τ)] = 0 \end{array} \\ \begin{array}{l} J_{g} \cdot {\ddot{θ}}_{g} - R_{g} \cdot (c_{h} + δ_{c h}) \cdot [R_{p} \cdot {\dot{θ}}_{p} - R_{g} \cdot {\dot{θ}}_{g} + {\dot{y}}_{p} - {\dot{y}}_{g} - \dot{e} (τ)] - \\ R_{g} \cdot (k_{h} (t) + k_{w} (t)) \cdot [R_{p} \cdot θ_{p} - R_{g} \cdot θ_{g} + y_{p} - y_{g} - e (τ)] = - T_{g} + δ_{F}^{*} \end{array} \end{cases}

(1)

In equation (1),

δ = R_{p} \cdot θ_{p} - R_{g} \cdot θ_{g} + y_{p} - y_{g} - e (t),

m_{e} = \frac{J_{p} \cdot J_{g}}{J_{p} \cdot R_{g}^{2} + J_{g} \cdot R_{p}^{2}},

f_{p x} (u), f_{p y} (u), f_{g x} (u), f_{g y} (u), f_{h} (u) = {\begin{cases} u - b, u > b \\ 0, - b \leq u \leq b \\ u + b, u < - b \end{cases} u = y_{p}, y_{g}, x_{p}, x_{g}, δ .

(2)

F_{e 1} = m_{p} \cdot ρ_{p} \cdot ω_{p}^{2}, F_{e 2} = m_{g} \cdot ρ_{g} \cdot ω_{g}^{2} .

After the Fourier series is expanded, take the first order components of $k_{h} (t)$ and $e (t)$ , and simplify to

k_{h} (t) = k_{0} + k_{v} \cdot \cos [(ω_{h} + δ_{ω_{h}}) t]

(3)

e (t) = e_{v} \cdot \sin (ω_{h} \cdot t)

(4)

In equations (3) and (4), $k_{0}$ is the average meshing stiffness, $k_{v}$ is the change of meshing stiffness, $ω_{h}$ is the driving frequency of the gear pair; $δ_{ω_{h}}$ is the random interference of $ω_{h}$ ; $e_{v}$ is the amplitude of transmission error.

The frequency of gear pairs is ω_{n} = \sqrt{\frac{k_{0}}{m_{e}}}

(5a)

The frequency of gear p is ω_{p} = \sqrt{\frac{k_{p y}}{m_{p}}}

(5b)

The frequency of gear g is ω_{g} = \sqrt{\frac{k_{g y}}{m_{g}}}

(5c)

The backlash ratio is b^{*} = b / b_{c}

(5d)

The stiffness ratio is ε = \frac{k_{h v}}{k_{h 0}}

(5e)

where

b

is the actual backlash,

b_{c}

is the standard backlash,

b

b_{c}

k_{h v}

is the amplitude of time-varying meshing stiffness, and

k_{h 0}

is the average stiffness,

ε

is the dimensionless meshing stiffness of the gear.

For convenience in solving, a dimensionless transformation is made:

{\begin{cases} t = ω_{n} τ y_{p}^{*} = y_{p} / b_{c} y_{g}^{*} = y_{g} / b_{c} δ^{*} = δ / b_{c} \\ k_{p x, y}^{*} = k_{p x, y} / b_{c} k_{g x, y}^{*} = k_{g x, y} / b_{c} k_{0}^{*} = k_{0} / b_{c} k_{v}^{*} = k_{v} / b_{c} \\ ω_{h}^{*} = ω_{h} / ω_{n} ω_{p}^{*} = ω_{p} / ω_{n} ω_{g}^{*} = ω_{g} / ω_{n} \end{cases}

(6)

For ease in programming, a variable substitution is made:

{\begin{cases} x_{1} = y_{p}^{*} x_{2} = y_{p}^{*} x_{3} = x_{p}^{*} x_{4} = x_{p}^{*} x_{5} = y_{g}^{*} \\ x_{6} = y_{g}^{*} x_{7} = x_{g}^{*} x_{8} = x_{g}^{*} x_{9} = δ^{*} x_{10} = δ^{*} \end{cases}

(7)

Substituting equations (2)∼(7) into (1) gives

x_{1} = x_{2}

(8)

\begin{array}{l} x_{2} = & - 2 \cdot (γ_{p y} + δ_{p y}) \cdot x_{2} - η_{p y} \cdot f_{p y} (x_{1}) - 2 \cdot (γ_{h 1} + δ_{c h 1}) \cdot x_{10} - \\ η_{11} \cdot f_{h} (x_{9}) - F_{p y}^{*} - ρ_{p}^{*} \cdot {(ω_{h}^{*} + δ_{ω_{h}^{*}})}^{2} / z_{1}^{2} \cdot \sin [(ω_{h}^{*} + δ_{ω_{h}^{*}}) t / z_{1}] \end{array}

(9)

x_{3} = x_{4}

(10)

{\dot{x}}_{4} = - 2 \cdot (γ_{p x} + δ_{p x}) \cdot x_{4} - η_{p x} \cdot f_{p x} (x_{3}) - F_{p x}^{*} - ρ_{p}^{*} \cdot {(ω_{h}^{*} + δ_{ω_{h}^{*}})}^{2} / z_{1}^{2} \cdot \cos [(ω_{h}^{*} + δ_{ω_{h}^{*}}) t / z_{1}]

(11)

x_{5} = x_{6}

(12)

\begin{array}{l} x_{6} = & - 2 \cdot (γ_{g y} + δ_{gy}) \cdot x_{6} - η_{g y} \cdot f_{g y} (x_{5}) + 2 \cdot (γ_{h 2} + δ_{c h 2}) \cdot x_{10} + \\ η_{22} \cdot f_{h} (x_{9}) + F_{gy}^{*} + ρ_{g}^{*} \cdot {(ω_{h}^{*} + δ_{ω_{h}^{*}})}^{2} / z_{2}^{2} \cdot \sin [(ω_{h}^{*} + δ_{ω_{h}^{*}}) t / z_{2}] \end{array}

(13)

x_{7} = x_{8}

(14)

{\dot{x}}_{8} = - 2 \cdot (γ_{g x} + δ_{g x}) \cdot x_{8} - η_{g x} \cdot f_{g x} (x_{7}) + F_{g x}^{*} + ρ_{g}^{*} \cdot {(ω_{h}^{*} + δ_{ω_{h}^{*}})}^{2} / z_{2}^{2} \cdot \cos [(ω_{h}^{*} + δ_{ω_{h}^{*}}) t / z_{2}]

(15)

x_{9} = x_{10}

(16)

{\dot{x}}_{10} = {\dot{x}}_{2} - {\dot{x}}_{6} - 2 \cdot (γ_{h 3} + δ_{c h 3}) \cdot x_{10} - η_{33} \cdot f_{h} (x_{9}) + F_{0}^{*} + e_{v}^{*} \cdot {(ω_{h}^{*} + δ_{ω_{h}^{*}})}^{2} \cdot \sin [(ω_{h}^{*} + δ_{ω_{h}^{*}}) t] + δ_{F}^{*}

(17)

In equations (8)∼(17), $γ_{p y} = \frac{c_{p y}}{2 \cdot m_{p} \cdot ω_{n}}, η_{p y} = \frac{k_{p y}}{m_{p} \cdot ω_{n}^{2}}, γ_{h 1} = \frac{c_{h}}{2 \cdot m_{p} \cdot ω_{n}}$ , $γ_{p x} = \frac{c_{p x}}{2 \cdot m_{p} \cdot ω_{n}}, η_{11} = 2 {1 - (ε + δ_{ε}) \cdot \cos [(ω_{h}^{*} + δ_{ω_{h}^{*}}) t] + k_{w}}, F_{p y}^{*} = \frac{F_{p y}}{m_{p} \cdot b_{c} \cdot ω_{n}^{2}}$ , $η_{p x} = \frac{k_{p x}}{m_{p} \cdot ω_{n}^{2}}, F_{p x}^{*} = \frac{F_{p x}}{m_{p} \cdot b_{c} \cdot ω_{n}^{2}}, γ_{h 3} = \frac{c_{h}}{2 \cdot m_{e} \cdot ω_{n}}, γ_{g y} = \frac{c_{g y}}{2 \cdot m_{g} \cdot ω_{n}}$ , $η_{g y} = \frac{k_{g y}}{m_{g} \cdot ω_{n}^{2}}, γ_{h 2} = \frac{c_{h}}{2 \cdot m_{g} \cdot ω_{n}}, γ_{g x} = \frac{c_{g x}}{2 \cdot m_{g} \cdot ω_{n}}, η_{g x} = \frac{k_{g x}}{m_{g} \cdot ω_{n}^{2}}$ , $η_{22} = 2 {1 - (ε + δ_{ε}) \cdot \cos [(ω_{h}^{*} + δ_{ω_{h}^{*}}] t + k_{w}}, F_{g y}^{*} = \frac{F_{g y}}{m_{g} \cdot b_{c} \cdot ω_{n}^{2}}, F_{g x}^{*} = \frac{F_{g x}}{m_{g} \cdot b_{c} \cdot ω_{n}^{2}}$ , $F_{0}^{*} = \frac{J_{g} \cdot R_{p} \cdot T_{p} + J_{p} \cdot R_{g} \cdot T_{g}}{J_{p} \cdot J_{g} \cdot b_{c} \cdot ω_{n}^{2}}, η_{33} = 2 {1 - (ε + δ_{ε}) \cdot \cos [(ω_{h}^{*} + δ_{ω_{h}^{*}}) t] + k_{w}}$ .

Results and discussion

Using iterative method to solve equations (8)–(17). According to the step $t = 0.05$ , the number of gear teeth $z_{1} = z_{2} = 30$ , the masses of gears $m_{p} = m_{q} = m_{e} = 0.456 kg$ , the rotational inertia of the gear $J_{p} = J_{q} = 6 \times 10^{- 6} kg \cdot m^{2}$ , the standard backlash $b_{c} = 0.068$ mm, the amplitude of transmission error $e_{v} = 0.034$ mm, the average meshing stiffness $k_{0} = 2.23 \times 10^{6}$ N/m, the dimensionless amplitude of transmission error $e_{v}^{*} = 0.05$ , the dimensionless force transmitted by the x-direction bearing of the driving gear $F_{p x}^{*} = 0.2$ , the dimensionless force transmitted by the y-direction bearing of the driving gear $F_{p y}^{*} = 0.2$ , the dimensionless force transmitted by the x-direction bearing of the passive gear $F_{g x}^{*} = 0.2$ , the dimensionless force transmitted by the y-direction bearing of the passive gear $F_{g y}^{*} = - 0.2$ , the dimensionless time-varying meshing stiffness $η_{11} = 1.1$ , the dimensionless time-varying meshing stiffness $η_{22} = 1.1$ , the dimensionless time-varying meshing stiffness $η_{33} = 1.1$ , the dimensionless equivalent damping of bearings in the x-direction of the driving gear $γ_{p x} = 0.01$ , the dimensionless equivalent damping of bearings in the y-direction of the driving gear $γ_{p y} = 0.01$ , the dimensionless equivalent damping of bearings in the x-direction of the passive gear $γ_{gx} = 0.01$ , the dimensionless equivalent damping of bearings in the y-direction of the passive gear $γ_{gy} = 0.01$ , the dimensionless meshing damping coefficient of the gear $γ_{h 1} = 0.012$ , the dimensionless meshing damping coefficient of the gear $γ_{h 2} = 0.012$ , the dimensionless meshing damping coefficient of the gear $γ_{h 3} = 0.05$ , the dimensionless meshing stiffness of the gear $ε = 0.18$ , the dimensionless eccentricity $ρ_{p}^{*} = 0.1$ , the dimensionless eccentricity $ρ_{g}^{*} = 0.1$ , the expansion angle of the driving gear $α_{A} = 0.324$ rad, the expansion angle of the driving gear $α_{B} = 0.453$ rad, the expansion angle of the driving gear $α_{C} = 0.413$ rad, the expansion angle of the driving gear $α_{D} = 0.717$ rad, the expansion angle of the driving gear $α_{E} = 0.845$ rad, the base circle radius of gears $R_{p} = R_{g} = 0.06$ m, the frictional coefficient $S_{avg} = 0.991$ μm.

Initial conditions: $x_{1} (0) = 0$ , $x_{2} (0) = - 0.1$ , $x_{3} (0) = 0$ , $x_{4} (0) = - 0.1$ , $x_{5} (0) = 0$ , $x_{6} (0) = - 0.1$ , $x_{7} (0) = 0$ , $x_{8} (0) = - 0.1$ , $x_{9} (0) = 0$ , $x_{10} (0) = - 0.1$ .

Analysis of gear systems without disturbance

The vibration bifurcation diagram of the gear transmission system for braiding equipment without disturbance is shown in Figure 3(a), the maximum Lyapunov exponent diagram of the system is shown in Figure 3(b), and the Poincaré mapping diagram and corresponding phase trajectories of the system are shown in Figure 4∼9.

Figure 3.

The vibration response curve of the gear pair. (a) The vibration bifurcation diagram (b). The maximum Lyapunov exponent diagram.

Figure 4.

The phase trajectory and Poincaré mapping diagram of $ω = 110$ . (a) Phase trajectory (b). The Poincaré mapping diagram.

Figure 5.

The phase trajectory and Poincaré mapping diagram of $ω = 170$ . (a)Phase trajectory (b) The Poincaré mapping diagram.

Figure 6.

The phase trajectory and Poincaré mapping diagram of $ω = 200$ . (a) Phase trajectory. (b) The Poincaré mapping diagram.

Figure 7.

The phase trajectory and Poincaré mapping diagram of $ω = 232$ . (a) Phase trajectory. (b) The Poincaré mapping diagram.

Figure 8.

The phase trajectory and Poincaré mapping diagram of $ω = 243$ . (a) Phase trajectory. (b) The Poincaré mapping diagram.

Figure 9.

The phase trajectory and Poincaré mapping diagram of $ω = 265$ . (a)Phase trajectory. (b) The Poincaré mapping diagram.

The nonlinear vibration characteristics of the system and the relationship are shown in Figure 3(a). It can be seen that the system had a periodic point when $ω < 163.3$ in Figure 3(a) and 4 shows the Poincaré mapping and corresponding phase trajectory; at that time, the system had four periodic points when $163.3 < ω < 186.5$ , and Figure 5 shows the Poincaré mapping and corresponding phase trajectories when $ω = 170$ ; as it increases, the number of periodic points changes when $186.5 < ω < 190$ ; At that time, the system had nine periodic points when $190 < ω < 213$ , and Figure 6 shows the Poincaré mapping and corresponding phase trajectories; At that time, chaos occurred in the system when $213 < ω < 240$ , and Figure 7 shows the Poincaré mapping and corresponding phase trajectory when $ω = 232$ ; At that time, chaos occurred in the system when $240 < ω < 264.7$ , and Figure 8 shows the Poincaré mapping and corresponding phase trajectory when $ω > 264.7$ ; At that time, the system became divergent and uncontrollable, as shown in Figure 9 with Poincaré mapping and corresponding phase trajectories when $ω = 265$ .

As shown in Figure 3(b), the maximum Lyapunov exponent is less than zero or fluctuates around zero when. According to the Lyapunov theorem, it indicates that the system is stable; The maximum $ω < 213$ Lyapunov exponent is positive when $ω \geq 213$ , it indicates that the system undergoes chaos or even eventually diverges and becomes uncontrollable.

Analysis of gear systems considering temperature factors

Analysis of system bifurcation and chaotic behavior

The nonlinear characteristics of the gear transmission system in the braiding machine under the consideration of $k_{w}$ (stiffness changes caused by temperature changes) is shown as Figure 17(a) and (b). The vibration bifurcation diagram of the system considering $k_{w} = 0.248$ is shown in Figure 10(a). The maximum Lyapunov diagram of the system considering $k_{w} = 0.248$ is shown in Figure 10(b). The Poincaré mapping and the corresponding phase trajectories are shown in Figures 11–16. Meanwhile, the vibration bifurcation diagram of the system considering $k_{w} = 0.32$ is shown in Figure 17(a). The maximum Lyapunov diagram of the system considering $k_{w} = 0.32$ is shown in Figure 17(b).

Figure 10.

The vibration response curve of the gear pair under temperature excitation. (a) The vibration bifurcation diagram considering $k_{w} = 0.248$ . (b) The maximum Lyapunov exponent diagram considering $k_{w} = 0.248$ .

Figure 11.

The phase trajectory and Poincaré mapping diagram of $ω = 110$ , $k_{w} = 0.248$ under temperature excitation. (a) Phase trajectory. (b) The Poincaré mapping diagram.

Figure 12.

The phase trajectory and Poincaré mapping diagram of $ω = 170$ , $k_{w} = 0.248$ under temperature excitation. (a) Phase trajectory. (b) The Poincaré mapping diagram.

Figure 13.

The phase trajectory and Poincaré mapping diagram of $ω = 192$ , $k_{w} = 0.248$ under temperature excitation. (a) Phase trajectory. (b) The Poincaré mapping diagram.

Figure 14.

The phase trajectory and Poincaré mapping diagram of $ω = 214$ , $k_{w} = 0.248$ under temperature excitation. (a) Phase trajectory. (b) The Poincaré mapping diagram.

Figure 15.

The phase trajectory and Poincaré mapping diagram of $ω = 234$ , $k_{w} = 0.248$ under temperature excitation. (a) Phase trajectory. (b) The Poincaré mapping diagram.

Figure 16.

The phase trajectory and Poincaré mapping diagram of $ω = 240$ , $k_{w} = 0.248$ under temperature excitation. (a) Phase trajectory. (b) The Poincaré mapping diagram.

Figure 17.

The vibration response curve of the gear pair under temperature excitation. (a) The vibration bifurcation diagram considering $k_{w} = 0.32$ . (b) The maximum Lyapunov exponent diagram considering $k_{w} = 0.32$ .

From Figure 10(a) and 17(a), it can be seen that $k_{w}$ have a significant impact on the nonlinear characteristics of the gear system. If $k_{w}$ is relatively small, such as $k_{w} = 0.248$ , it has a small impact on the nonlinear characteristics of the gear system. The vibration bifurcation diagram, maximum Lyapunov diagram, Poincaré mapping diagram, and corresponding system phase trajectories of the system considering $k_{w} = 0.248$ are shown in Figure 10(a)–16.

If $k_{w}$ is relatively small, such as $k_{w} = 0.248$ , the nonlinear vibration characteristics of the system and the relationship between $ω$ and $x_{10}$ are shown in Figure 10(a). From Figure 10(a), it can be seen that the system had a periodic point when $ω < 163.3$ , and Poincaré mapping and the corresponding phase trajectory when $ω = 110$ are shown in Figure 11. The system had four periodic points when $163.3 < ω < 186.5$ , and Poincaré mapping and the corresponding phase trajectory when $ω = 170$ are shown in Figure 12. As $ω$ increases, the number of periodic points changes when $186.5 < ω < 190$ . The system had nine periodic points when $190 < ω < 200$ , and Poincaré mapping and the corresponding phase trajectory when $ω = 192$ are shown in Figure 13. Chaos occurs in the system when $200 < ω < 240$ , and Poincaré mapping and the corresponding phase trajectory when $ω = 214$ are shown in Figure 14. Poincaré mapping and the corresponding phase trajectory when $ω = 234$ are shown in Figure 15. The system became divergent and uncontrollable when $ω \geq 240$ , and Poincaré mapping and the corresponding phase trajectory when $ω = 240$ are shown in Figure 16.

If $k_{w}$ is relatively large, such as $k_{w} = 0.32$ , it has a significant impact on the nonlinear characteristics of the gear transmission system. The vibration bifurcation diagram and the maximum Lyapunov diagram considering $k_{w} = 0.32$ are shown as Figure 17(a) and (b). From Figure 17(b), it can be seen that $k_{w}$ is relatively large, such as $k_{w} = 0.32$ , the maximum Lyapunov exponent is positive and the system undergoes chaos or even eventually becomes divergent and uncontrollable. The Poincaré mapping and corresponding phase trajectories of the system are omitted here.

Analysis of inherent characteristics of the system

The contact temperature of the tooth surface $Δ_{B} (t)$ is

Δ_{B} (t) = Δ_{M} + Δ_{f} (t)

(18)

According to Blok’s flash temperature theory,²² the flash temperature of the tooth surface $Δ_{f} (t)$ is

Δ_{f} (t) = \frac{u f_{m} f_{e} | v_{1} (t) - v_{2} (t) |}{(\sqrt{g_{1} ρ_{1} c_{1} v_{1} (t)} + \sqrt{g_{2} ρ_{2} c_{2} v_{2} (t)}) \sqrt{B (t)}}

(19)

In equation (19), the interpretation of each parameter is based on the Blok flash temperature theory.

According to equation (19), the trend chart of tooth surface flash temperature variation with meshing point is shown in Figure 18, and maximum flash temperature of tooth surface under coupling of load and frequency is shown in Figure 19.

Figure 18.

The trend chart of tooth surface flash temperature variation with meshing point.

Figure 19.

The maximum flash temperature of tooth surface under coupling of load and frequency.

From Figure 18, it can be observed that the flash temperature of the tooth surface is the smallest at the node, and the value is the highest when the tooth top enters meshing and the tooth root exits meshing. From Figure 19, it can be seen that the vertical axis represents the load change, and the load F_n = 5000–5800N was selected. The horizontal axis represents the frequency change, and the frequency range was (120,280). Different colors represent the maximum flash temperature of the corresponding tooth surface. From Figure 19, it can be observed that as the load and frequency increase, the maximum flash temperature of the tooth surfaces also increases during gear meshing.

Frequency and load affect the flash temperature of the tooth surface. The contact temperature of the tooth surface is equal to the flash temperature of the tooth surface and body temperature. The change in contact temperature causes a change in the amplitude of the stiffness fluctuation k_w, which leads to a change in the amplitude of vibration. At the same time, the change in body temperature T has a small impact on the amplitude of vibration; that is, the flash temperature on the tooth surface causes a change in the amplitude of vibration.

The relationship between the system amplitude on the frequency w-body temperature T parameter plane at k_w = 0.1, k_w = 0.2, k_w = 0.3 is shown in Figures 20–22. From Figures 20–22, it can be seen that the amplitude response of the system is strong when $200 < ω < 240$ . Compared with the bifurcation diagram, the system exhibits chaotic behavior within the range of this frequency. Simultaneously, as k_w increases, the strong amplitude response of the system increases at $200 < ω < 240$ .

Figure 20.

Cloud image with amplitude on the w-T plane at k_w = 0.1.

Figure 21.

Cloud image with amplitude on the w-T plane at k_w = 0.2.

Figure 22.

Cloud image with amplitude on the w-T plane at k_w = 0.3.

Conclusions

This section investigates the nonlinear characteristics of the gear transmission system in braiding equipment under various influencing factors. The results indicate that the nonlinear characteristics of the gear transmission system under various influencing factors conform to engineering practice. The main conclusions are summarized as follows:

(1) According to the bifurcation diagram of the system without random disturbances, chaos occurs when $ω > 213$ . Therefore, considering the yarn tension and safety factors, the safety speed of the braiding equipment is less than 31.25 $r / s$ , that is, $1.875 \times 10^{3} r / \min$ when $ω < 213$ .

(2) Considering the bifurcation of the system due to $k_{w}$ , $k_{w}$ must be controlled $k_{w} < 0.32$ . The amplitude response of the system was strong when $200 \leq ω \leq 240$ . Compared with the bifurcation diagram, the system exhibits chaotic behavior within the range of this frequency, and as $k_{w}$ increases, the strong amplitude response of the system increases at $200 \leq ω \leq 240$ .

(3) If $k_{w}$ reaches the above limit value, it will lead to instability of the braiding system, interference during the braiding process, interweaving, or even yarn breakage.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project was supported by the Applied Basic Research Programs of Changzhou (grant number CJ20235047), Major Program of Natural Science Research in Higher Education Institutions in Jiangsu Province (grant no. 21KJA460004), the China Postdoctoral Science Foundation [2023M740554] and Key Project of Shaoxing University [13011001005/132].

ORCID iDs

Lingling Yao

Chengjie Du

Yapeng Lu

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Investigation of nonlinear characteristics of a gear transmission system in a multi-ring braiding machine considering contact temperature

Abstract

Keywords

Introduction

Dynamic modeling

Braiding process

Dynamic modeling of the gear system

Results and discussion

Analysis of gear systems without disturbance

Analysis of gear systems considering temperature factors

Analysis of system bifurcation and chaotic behavior

Analysis of inherent characteristics of the system

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References