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Abstract

A computational model is proposed for the calculation of thermal deformation in ring gear, and the ANSYS Workbench platform is utilized to simulate the steady-state temperature field and thermal deformation in a planetary gear system. The simulation results are utilized to validate the accuracy of the ring gear thermal deformation calculation model. Furthermore, based on the data obtained from the steady-state temperature field, this research investigates variations in gear backlash and coupling stiffness within planetary gear systems caused by thermal deformation under variable speed and torque conditions. Additionally, a nonlinear dynamic model incorporating thermal deformation factors is established for a planetary gear system. By employing the 4-5 order Runge-Kutta numerical method, bifurcation diagrams and Largest Lyapunov exponents of the system under different operating conditions are obtained. Combining Poincaré maps, phase diagram, and spectral diagrams, an analysis is conducted on tooth impact states as well as nonlinear dynamic characteristics of the system influenced by thermal deformation.

Keywords

Bifurcation chaos planet gear transmission system thermal deformation

Introduction

In high-speed and overloaded conditions, the elevated temperature resulting from friction between meshing tooth surfaces can lead to thermal deformation of gears. This thermal deformation significantly impacts the vibration, noise, and transmission accuracy of the system. In severe cases, it can even cause gear jamming and other failures that compromise operational safety and reliability. Therefore, in order to comprehensively analyze the nonlinear characteristics of the system and provide a theoretical basis for vibration reduction and noise control, it is imperative to investigate the nonlinear dynamic characteristics of the system under thermal deformation effects. Currently, there are two primary research methods employed for studying gear thermal deformation: analytical method and finite element simulation method. Li et al.^1,2 proposed a theoretical calculation model based on thermal expansion theory and heat conduction differential equations to analyze the thermal deformation of external involute gears under temperature effects. Luo et al.³ using an external meshing single-tooth finite element simulation model developed with Ansys APDL language, examined distribution patterns of tooth surface temperature as well as thermal deformations on gear surfaces. Chen et al.⁴ also utilizing APDL language, analyzed radial distribution patterns of thermal deformations on gears through steady-state temperature field analysis. Tian et al.⁵ through finite element modeling, investigated both steady-state temperature fields and thermal deformations in planetary gear systems. Wang et al.⁶ conducted an analysis on the influencing factors of thermal deformation in gear teeth using response surface methodology and sample point experiments, and concluded that there is a positive correlation between thermal deformation and rotational speed as well as torque.

Thermal deformation not only alters the geometry and clearance of gear teeth but also affects their stiffness. Sun et al.⁷ employed a finite element model to analyze the impact of temperature rise and load torque on the meshing stiffness of gears. In practical gear system operation, working conditions often vary, leading to different effects on the dynamic characteristics of the system.

Taking a two-degree-of-freedom external meshing single-stage spur gear as the research subject, Xia et al.⁸ incorporated friction factors into the dynamic model of the gear system and investigated its bifurcation characteristics with parameters such as rotational speed. Wang et al.⁹ analyzed the nonlinear characteristics of the system under low-speed heavy load conditions by considering the influence of temperature on tooth clearance. Lu et al.¹⁰ examined the nonlinear characteristics of the system by converting spatial temperature alternation effects into tooth clearance fluctuations. Sheng et al.¹¹ comprehensively considered friction and thermal deformation factors in their analysis of an external meshing gear reducer for coal mining machines, studying its nonlinear characteristics with parameters including meshing frequency and friction coefficient. Li et al.¹² analyzed the nonlinear characteristics of a spiral bevel gear system by taking into account thermal deformation and friction forces, using rotational speed as a bifurcation parameter. Pan et al.¹³ focusing on an external meshing gear-shaft-rotor system, established a dynamic model that considers contact temperature and studied its nonlinear characteristics with meshing frequency as a bifurcation parameter. Gou et al.¹⁴ analyzed the nonlinear characteristics of the system using parameters such as meshing frequency, tooth clearance, and stiffness coefficients for bifurcation analysis.

Taking planetary gear systems as the research subject, Adebayo et al.¹⁵ conducted an analysis on the vibration characteristics and fault diagnosis feature frequencies of a helicopter’s planetary gearbox under different speeds and torques. Wang et al.¹⁶ developed a nonlinear dynamic model of planetary gears considering tooth surface friction, and examined the system’s nonlinear characteristics by utilizing bifurcation parameters such as excitation frequency and torque. Xiang et al.¹⁷ formulated a dynamic model for a planetary-fixed axis gear set with multiple clearances, investigating the system’s nonlinear dynamic characteristics using meshing frequency, backlash, and damping ratio as bifurcation parameters. Zhou¹⁸ studied the impact of operating parameters on the vibration characteristics of high-speed light load planetary gears. Hu¹⁹ considered factors like meshing phase and manufacturing errors to analyze the overall transmission error in planetary gear systems. Wang²⁰ analyzed the nonlinear characteristics of a planetary gear system while taking into account tooth surface friction and extrusion oil film effects. Tang²¹ analyzed the dynamic properties of wind turbine planetary gears under variable speed conditions, while Li²² investigated bifurcation properties in variable power situations for planet gear systems. Xiang²³ conducted an analysis on nonlinear dynamics characteristics of two-stage increasing speed planetary gear box system using mesh frequency and damping ratio as bifurcation parameters.

To sum up, the existing research primarily focuses on the influence of thermal deformation on nonlinear dynamic characteristics in fixed-axis gear systems, with a lack of studies incorporating both internal and external meshing pairs' thermal deformations into the dynamic model of planetary gear systems to observe their effects on system nonlinear dynamics. Additionally, considering that thermal deformation is closely associated with operating parameters, so this paper comprehensively investigates the impact of thermal deformation on planetary gear systems and analyzes their nonlinear dynamic characteristics under various operating conditions.

Nonlinear dynamics model of planetary gear system

The torsional vibration model of the planetary gear is illustrated in Figure 1. In the diagram, $S$ , $P$ , $C$ , and $R$ represent the sun gear, planet gear, carrier, and ring gear respectively. The variables $u_{s}$ , $u_{p}$ , $u_{c}$ , and $u_{r}$ denote the torsional displacements of each component. Let i = SP, PR, indicate the sun-planet meshing pair and planet-ring meshing pair, respectively. The variables $k_{i} (t)$ , $c_{i} (t)$ , $e_{i} (t)$ correspond to the coupling stiffness, meshing damping and static transmission error of each meshing pair. The OXY coordinate system is a static reference frame with its origin fixed at the center of the sun gear. On the other hand, the oxy coordinate system serves as a servo reference frame that shares the same origin as the static coordinate system and aligns its X-axis along the central connection line between the sun gear and planet gear, rotating with the carrier’s motion. Lastly, $O_{n} x_{n} y_{n}$ represents a dynamic coordinate system centered around the planetary gear, where its X-axis runs parallel to that of $O x y$ .

Figure 1.

Nonlinear dynamics model of planetary gear system.

Based on Newton’s laws, the dynamic differential equations of a planetary gear system can be derived¹⁶

{\begin{cases} \frac{I_{s}}{{r_{s}}^{2}} \overset{\cdot \cdot}{u_{s}} + \sum_{n = 1}^{N} (k_{s p} (t) f (δ_{s p}) + c_{s p} f (\overset{\cdot}{δ_{s p}})) \\ - \sum_{n = 1}^{N} (k_{s p} (t) f (δ_{s p}) + c_{s p} f (\overset{\cdot}{δ_{s p}})) \frac{λ_{s i g s p} μ_{s p} L_{s p}}{r_{s}} = \frac{T_{i n}}{r_{s}} \\ \frac{I_{n}}{{r_{n}}^{2}} \overset{\cdot \cdot}{u_{n}} + (k_{s p} (t) f (δ_{s p}) + c_{s p} f (\overset{\cdot}{δ_{s p}})) (1 - \frac{λ_{s i g s p} μ_{s p} L_{s p}}{r_{s}}) \\ + (k_{pr} (t) f (δ_{pr}) + c_{pr} f (\overset{\cdot}{δ_{pr}})) (1 - \frac{λ_{s i g pr} μ_{pr} L_{pr}}{r_{r}}) = 0 \\ (\frac{I_{c}}{{r_{c}}^{2}} + N m_{n}) \overset{\cdot \cdot}{u_{c}} - \sum_{n = 1}^{N} (k_{pr} (t) f (δ_{pr}) + c_{pr} f (\overset{\cdot}{δ_{pr}})) \cos (α_{pr}) \\ - \sum_{n = 1}^{N} (k_{s p} (t) f (δ_{s p}) + c_{s p} f (\overset{\cdot}{δ_{s p}})) \cos (α_{s p}) = \frac{T_{o u t}}{r_{c}} \\ \frac{I_{r}}{{r_{r}}^{2}} \overset{\cdot \cdot}{u_{r}} + \sum_{n = 1}^{N} (k_{pr} (t) f (δ_{pr}) + c_{pr} f (\overset{\cdot}{δ_{pr}})) \\ - \sum_{n = 1}^{N} (k_{pr} (t) f (δ_{pr}) + c_{pr} f (\overset{\cdot}{δ_{pr}})) \frac{λ_{s i g pr} μ_{pr} L_{r p}}{r_{r}} = 0 \end{cases}

(1)

Where

δ_{i}

is the relative displacement of SP and PR meshing pairs, which can be expressed as¹⁶

\{\begin{cases} δ_{s p} = u_{s} - u_{c} \cos (α_{s p}) + u_{p} + e_{s p} \\ δ_{p r} = u_{r} - u_{c} \cos (α_{p r}) - u_{p} + e_{p r} \end{cases}

(2)

Where

e_{i}

is the static meshing error composed of gear base error and tooth profile error, which can be expressed as a periodic function of meshing frequency.¹⁶

e_{i} = E_{i} \sin (ω_{m} t + ϕ)

(3)

L_{S P}

and

L_{P S}

are the friction force arms of the sun gear and the planet gear in the SP meshing pair, respectively, which can be expressed as¹⁶

\{\begin{cases} L_{s p} (t) = \sqrt{{r_{a s}}^{2} - {r_{s}}^{2}} - ε_{s p} S_{s p} + ω_{s} r_{s} t \\ L_{p s} (t) = \sqrt{{r_{a p}}^{2} - {r_{p}}^{2}} - ε_{s p} S_{s p} + ω_{s} r_{s} t \end{cases}

(4)

L_{PR}

and

L_{R P}

are the friction force arms of the planet gear and the ring gear in the PR meshing pair, respectively, which can be expressed as¹⁶

\{\begin{cases} L_{p r} (t) = \sqrt{{r_{a p}}^{2} - {r_{p}}^{2}} - ε_{p r} S_{p b} + ω_{p} r_{p} t \\ L_{r p} (t) = \sqrt{{r_{a r}}^{2} - {r_{r}}^{2}} - ε_{p r} S_{p b} + ω_{p} r_{p} t \end{cases}

(5)

In the process of moving the meshing point along the meshing line, when the meshing point passes through the pitch node, the relative sliding speed of the meshing tooth surface changes in the direction, so the direction transformation coefficient is adopted to reflect this behavior, which can be expressed as

λ_{s i g} = s i g (ξ) = \{\begin{matrix} 1 & ξ > 0 \\ 0 & ξ = 0 \\ - 1 & ξ < 0 \end{matrix}

(6)

The tooth side clearance plays a crucial role in introducing significant nonlinearity into gear systems. During the manufacturing and assembly processes of gears, it is essential to maintain a specific gap between meshing teeth to ensure adequate lubrication and prevent issues such as gear jamming caused by thermal expansion. The tooth side clearance can be represented using a piecewise function.²⁰

f (δ_{i}) = \{\begin{matrix} δ_{i} - (b_{i} - ∆ b_{i}) & δ_{i} > (b_{i} - ∆ b_{i}) \\ 0 & |δ_{i}| ⩽ (b_{i} - ∆ b_{i}) \\ δ_{i} + (b_{i} - ∆ b_{i}) & δ_{i} < (∆ b_{i} - b_{i}) \end{matrix}

(7)

Where

Δ b_{i}

is the change of backlash caused by thermal deformation.

In order to facilitate subsequent solution and analysis, the vibration displacement in the dynamics equation is changed into relative displacement by formula (2), and dimensionless parameters in formula (8) are selected, and the system is transformed into dimensionless equation of state by formula (9), as shown in formula (10).²⁰

\{\begin{cases} M_{p} = I_{p} / {r_{p}}^{2} & M_{s} = I_{s} / {r_{s}}^{2} & Ω = ω_{m} / ω_{n} \\ \bar{δ_{i}} = δ_{i} / b_{c} & \bar{b_{i}} = b_{i} / b_{c} & \dot{\bar{δ_{i}}} = δ_{i} / b_{c} ω_{n} \\ \bar{η} = η / η_{0} & \ddot{\bar{δ_{i}}} = δ_{i} / b_{c} {ω_{n}}^{2} & ω_{n} = \sqrt{k_{s p a v e} (1 / M_{s} + 1 / M_{p})} \end{cases}

(8)

\{\begin{matrix} z_{1} = \bar{δ_{s p}} & z_{2} = \dot{\bar{δ_{s p}}} \\ z_{3} = \bar{δ_{p r}} & z_{4} = \dot{\bar{δ_{p r}}} \end{matrix}

(9)

\{\begin{cases} \dot{z_{1}} = z_{2} \\ \dot{z_{2}} = (- \frac{1}{M_{s} {ω_{n}}^{2}} + \frac{μ_{s p} λ_{s p} L_{s p}}{M_{s} {r_{s} ω_{n}}^{2}} - \frac{\cos^{2} (α_{s p})}{M_{c} {ω_{n}}^{2}}) \sum_{n = 1}^{N} \bar{k_{s p}} f (\bar{δ_{s p}}) \\ + (\frac{μ_{s p} λ_{s p} L_{s p}}{M_{s} r_{s} ω_{n}} - \frac{1}{M_{s} ω_{n}} - \frac{\cos^{2} (α_{s p})}{M_{c} ω_{n}}) \sum_{n = 1}^{N} c_{s p} \dot{\bar{δ_{s p}}} - \frac{\cos (α_{s p}) \cos (α_{p r})}{M_{c} {ω_{n}}^{2}} \bar{k_{s p}} f (\bar{δ_{s p}}) \\ - \frac{\cos (α_{s p}) \cos (α_{p r})}{M_{c} ω_{n}} \sum_{n = 1}^{N} c_{p r} \dot{\bar{δ_{p r}}} + (\frac{μ_{s p} λ_{s p} L_{p s}}{M_{p} r_{p} {ω_{n}}^{2}} - \frac{1}{M_{c} {ω_{n}}^{2}}) \bar{k_{s p}} f (\bar{δ_{s p}}) \\ + (\frac{μ_{s p} λ_{s p} L_{p s}}{M_{p} r_{p} ω_{n}} - \frac{1}{M_{p} ω_{n}}) c_{s p} \dot{\bar{δ_{s p}} + (\frac{1}{M_{p} {ω_{n}}^{2}} - \frac{μ_{p r} λ_{p r} L_{p r}}{M_{p} r_{p} {ω_{n}}^{2}})} \bar{k_{p r}} f (\bar{δ_{p r}}) \\ + (\frac{1}{M_{p} ω_{n}} - \frac{μ_{p r} λ_{p r} L_{p r}}{M_{p} r_{p} ω_{n}}) c_{p r} \dot{\bar{δ_{p r}}} + \frac{T_{i n}}{M_{s} {r_{s} ω_{n}}^{2} b_{c}} + \frac{T_{o u t} \cos (α_{s p})}{M_{c} {r_{c} ω_{n}}^{2} b_{c}} + \ddot{\bar{e_{s p}}} \\ \dot{z_{3}} = z_{4} \\ \dot{z_{4}} = (- \frac{1}{M_{r} {ω_{n}}^{2}} + \frac{μ_{p r} λ_{p r} L_{p r}}{M_{r} {r_{r} ω_{n}}^{2}} - \frac{\cos^{2} (α_{p r})}{M_{c} {ω_{n}}^{2}}) \sum_{n = 1}^{N} \bar{k_{p r}} f (\bar{δ_{p r}}) \\ + (\frac{μ_{p r} λ_{p r} L_{p r}}{M_{r} r_{r} ω_{n}} - \frac{1}{M_{r} ω_{n}} - \frac{\cos^{2} (α_{p r})}{M_{c} ω_{n}}) \sum_{n = 1}^{N} c_{p r} \dot{\bar{δ_{p r}}} - \frac{\cos (α_{s p}) \cos (α_{p r})}{M_{c} {ω_{n}}^{2}} \bar{k_{p r}} f (\bar{δ_{s p}}) \\ - \frac{\cos (α_{s p}) \cos (α_{p r})}{M_{c} ω_{n}} \sum_{n = 1}^{N} c_{s p} \dot{\bar{δ_{s p}}} + (\frac{1}{M_{p} {ω_{n}}^{2}} - \frac{μ_{s p} λ_{s p} L_{p s}}{M_{p} r_{p} {ω_{n}}^{2}}) \bar{k_{s p}} f (\bar{δ_{s p}}) \\ + (\frac{1}{M_{p} ω_{n}} - \frac{μ_{s p} λ_{s p} L_{p r}}{M_{p} r_{p} ω_{n}}) c_{s p} \dot{\bar{δ_{s p}}} + (\frac{μ_{p r} λ_{p r} L_{p r}}{M_{p} {r_{p} ω_{n}}^{2}} - \frac{1}{M_{p} {ω_{n}}^{2}}) \bar{k_{p r}} f (\bar{δ_{p r}}) \\ + (\frac{μ_{p r} λ_{p r} L_{p r}}{M_{p} {r_{p} ω_{n}}^{2}} - \frac{1}{M_{p} ω_{n}}) c_{p r} \dot{\bar{δ_{p r}}} + \frac{T_{o u t} \cos (α_{p r})}{M_{c} {r_{c} ω_{n}}^{2} b_{c}} + \ddot{\bar{e_{p r}}} \end{cases}

(10)

Steady-state temperature field of planetary gear system

Steady-state temperature field calculation model of planetary gear

In the process of gear meshing transmission, the main sources of heat are the elastic-plastic deformation of the teeth, rolling and relative sliding between the meshing tooth surfaces, with relative sliding accounting for a major portion.²⁴ The heat flux generated by sliding friction is related to contact stress on gear tooth mating surfaces, relative sliding velocity between meshing tooth surfaces, friction coefficient, thermal conversion coefficient, and load distribution factor. It can be expressed as⁵

Q = γ μ σ_{H} V_{s}

(11)

A pair of meshing gears can be equivalently represented as two contacting cylinders. In practical gear contact, there exists a contact zone between the meshing teeth, where the contact stress is distributed in an elliptical shape along the normal direction of the contact zone. For ease of calculation and analysis, the average contact stress in the contact area is adopted According to contact mechanics, the average contact stress at the point of meshing is⁵

σ_{H} = \frac{π}{4} \sqrt{\frac{ς w}{π b R} \cdot \frac{1}{\frac{1 - {P_{o 1}}^{2}}{E_{1}} + \frac{1 - {P_{o 2}}^{2}}{E_{2}}}}

(12)

ς_{i}

is the load distribution coefficient. During the meshing process, the number of engaged teeth alternates between 1 and 2. In the double-tooth meshing zone, the mesh load is shared by two pairs of teeth, while in the single-tooth meshing zone, it is borne by a single pair of teeth. Figure 2 shows the load distribution coefficient diagram for SP meshing pair and PR meshing pair. The horizontal axis represents relative displacement from the engagement point to pitch node, where

ζ_{i}

= 0 indicates node position. D1 and D2 represent the first and second double-tooth meshing zones during engagement, respectively, while S denotes the single-tooth meshing zone.

Figure 2.

Load distribution coefficient of SP and PR meshing pairs.

The contact stress of SP and PR meshing pair when the speed is 5000 rpm and the torque is 100 N·m is shown in Figure 3.

Figure 3.

The contact stress diagram of SP and PR meshing pair when the input speed is 5000 rpm and the torque is 100 N·m.

When the temperature changes, the elastic modulus of the gear material also changes, which can be expressed as²⁵

E (T) = \frac{{(1 + \int_{0}^{T r e f 1} l (t) d t)}^{3}}{{(1 + \int_{0}^{T} l (t) d t)}^{3}} \times {1 - \frac{T - T_{r e f 1}}{T_{r e f 2} - T_{r e f 1}} [1 - {(\frac{E (T_{r e f 2})}{E (T_{r e f 1})} \times \frac{{(1 + \int_{0}^{T r e f 2} l (t) d t)}^{3}}{{(1 + \int_{0}^{T r e f 1} l (t) d t)}^{3}})}^{2}]}^{\frac{1}{2}}

(13)

The friction coefficient between the meshing tooth surfaces can be represented as²⁶

μ = 0.002 {[\frac{2 σ_{H}}{v_{1} + v_{2}} \cdot \frac{R_{1} + R_{2}}{R_{1} R_{2}}]}^{0.2} {η_{m}}^{- 0.05} R_{a}

(14)

The frictional heat flux will be distributed to the driving and driven gear through the heat distribution coefficient, as expressed by equation.²⁷

Λ = \frac{\sqrt{λ_{1} ρ_{1} c_{1} v_{1}}}{\sqrt{λ_{1} ρ_{1} c_{1} v_{1}} + \sqrt{λ_{2} ρ_{2} c_{2} v_{2}}}

(15)

The frictional heat flux of the driving and driven gear can be expressed as

\{\begin{cases} Q_{1} = Λ Q \\ Q_{2} = (1 - Λ) Q \end{cases}

(16)

The heat transfer mechanism on the tooth surface of a straight cylindrical gear is similar to that of bevel gears.²⁸ The convective heat transfer coefficient on the tooth surface can be calculated for straight cylindrical gears using Handschuh’s convective heat transfer coefficient for bevel gears,⁵ as proposed by several scholars^5,29–31

α_{t} = 0.228 {R_{ε}}^{0.731} {P_{r}}^{0.333} K_{0} / d

(17)

Reynolds number and Prandtl number can be expressed as⁵

\{\begin{cases} R_{ε} = \frac{V h_{m}}{η_{m}} \\ P_{r} = \frac{ρ_{f} c_{f} η_{m}}{K_{0}} \end{cases}

(18)

The convective heat transfer of the gear face can be equivalent to the convective heat transfer coefficient of the disc when the disc rotates in the lubricating oil. The flow state of the lubricating oil on the surface of the disc can be divided into laminar flow (Re < 1) and turbulent flow (Re ≥ 1), and different Reynolds number ranges correspond to the convective heat transfer coefficients under different flow states.³²

\{\begin{cases} α_{s} = 0.308 K_{0} {(c + 2)}^{0.5} {P_{r}}^{0.5} {(\frac{ω}{η_{m}})}^{0.5}, R_{ε} = \frac{ω {r_{c}}^{2}}{η_{m}} \leq 2 ˜ 2.5 \times 10^{5} \\ α_{s} = 0.0197 K_{0} {(c + 2.6)}^{0.2} {P_{r}}^{0.6} {(\frac{ω}{η_{m}})}^{0.8} {r_{c}}^{0.6}, R_{ε} = \frac{ω {r_{c}}^{2}}{η_{m}} > 2 ˜ 2.5 \times 10^{5} \end{cases}

(19)

Simulation analysis of steady-state temperature field of planetary gear

In practical operating conditions, planetary gears often encounter diverse working scenarios. This article defines two sets of working conditions:

Condition 1 involves a fixed input torque of 100 N·m and a speed range from 1000 rpm to 20000 rpm.

Condition 2 entails a fixed input speed of 5000 rpm and a torque range from 100 N·m to 360 N·m.

Utilizing Ansys Workbench’s Steady-Thermal module, the mesh numbers for the sun gear, planet gear, and ring gear are respectively set as follows: 258891, 301380, and 556432, while the node numbers are designated as follows: 530474 for the sun gear, 616070 for the planet gear, and 1128068 for the ring gear.

The temperatures at the pitch circle, base circle, and aperture of the sun gear and planet gear are designated as $T e m_{a}$ , $T e m_{b}$ , and $T e m_{c}$ for subsequent thermal deformation calculations. Similarly, the temperatures at the gear ring pitch circle, tooth root circle, and outer ring are denoted as $T e m_{a}$ , $T e m_{b}$ , and $T e m_{c}$ . The temperature variation curves under two operating conditions are illustrated in the Figures 4 and 5.

Figure 4.

Temperature change curve of planetary gear system under variable speed condition.

Figure 5.

Temperature change curve of planetary gear system under variable torque condition.

From Figures 4 and 5, it can be observed that under the same operating conditions, the temperature gradually decreases from the sun gear to the ring gear. Under variable speed conditions, the temperature at different positions increases with increasing speed, but the rate of increase gradually diminishes, following a logarithmic pattern. Under variable torque conditions, temperatures in different parts of the system increase linearly with increasing torque. This is because as speed increases, the convective heat transfer coefficient of the system also increases, resulting in more heat exchange with the surroundings and thus reducing the magnitude of temperature rise. On the other hand, an increase in torque does not affect the convective heat transfer coefficient of the system; hence, there is no change in magnitude of temperature rise.

Calculation model for thermal deformation of planetary gear system

Flash temperature of tooth surface

The temperature of the meshing tooth surface consists of two parts: body temperature and flash temperature. The body temperature remains unchanged after the gear reaches a steady state. Flash temperature is a transient increase in tooth surface temperature caused by energy loss due to mutual friction between meshing tooth surfaces.³³ Flash temperature is determined by the relative sliding speed, load, and material parameters of the meshing tooth surfaces, and can be expressed as³⁴

T e m_{f l a s h i} (t) = \frac{0.83 μ_{i} w_{i} |v_{i 1} (t) - v_{i 2} (t)|}{(\sqrt{λ_{j} ρ_{j} c_{j} v_{i 1} (t)} + \sqrt{λ_{j} ρ_{i} c_{i} v_{i 2} (t)}) \sqrt{B_{i} (t)}}

(20)

v_{1}

and

v_{2}

represents the distance from the point of engagement to the center of the gear. For internal gear pairs, it can be expressed as³⁵

v_{i j} (t) = ω_{i j} r_{k i j} (t) \sin (a r \cos (\frac{r_{k i j} (t) \cos (α)}{r_{k i j} (t)}))

(21)

r_{k i j} (t)

represents the distance from the point of engagement to the center of the gear. For internal gear pairs, it can be expressed as³⁵

\{\begin{cases} r_{k p r 1} (t) = \sqrt{(\sqrt{{r_{a 2}}^{2} - {r_{b 2}}^{2}} - a r n \cdot \sin {(α) + r_{b 1} ω_{1} t)}^{2} + {r_{b 1}}^{2}} \\ r_{k p r 2} (t) = \sqrt{(\sqrt{{r_{a 2}}^{2} - {r_{b 2}}^{2}} {+ r_{b 1} ω_{1} t)}^{2} + {r_{b 2}}^{2}} \end{cases}

(22)

The half-width $B_{i} (t)$ of the meshing secondary contact zone can be expressed as³⁵

B_{i} (t) = 1.128 \sqrt{w_{i} (\frac{1 - {P_{o i 1}}^{2}}{E_{i 1}} + \frac{1 - {P_{o i 2}}^{2}}{E_{i 2}}) (\frac{R_{i 1} (t) R_{i 2} (t)}{R_{i 1} (t) + R_{i 2} (t)})}

(23)

The curvature radius $R_{i j} (t)$ of the contact point can be expressed as³⁵

R_{i j} (t) = r_{k i j} (t) \sin (α_{p i j} (t))

(24)

The pressure angle at the point of engagement can be expressed as³⁵

α_{p i j} (t) = a r \cos (\frac{r_{b i j}}{r_{k i j} (t)})

(25)

Thermal deformation calculation model

The thermal deformation calculation principle of external meshing gear has been elaborated in literature,^1,2 and its thermal deformation calculation model is as follows:

σ_{i j} (t) = \frac{- ∆ T_{i j} (t) λ r_{b j} (r_{b j} + u_{b j}) (\frac{S_{k j}}{r_{b j}} - 2 (i n v (α_{k j}) - i n v (α)))}{2 (r_{b j} + u_{b j} \cos (α_{k j}) + ∆ T_{i j} (t) λ r_{b j} (1 - \cos (α_{k j})))}

(26)

Where the thermal deformation of the base circle

u_{b j}

u_{b j} = λ r_{b j} T e m_{j c} + \frac{λ r_{b j} (1 + P_{o}) ({r_{b j}}^{2} (1 - 2 P_{o}) - {r_{o j}}^{2}) (T e m_{b j} - T e m_{c j})}{(1 - P_{o}) ({r_{b j}}^{2} - {r_{o j}}^{2})}

(27)

For the ring gear, its thermal deformation is shown in the Figure 6.

Figure 6.

Ring gear thermal deformation diagram.

Thermal deformation of the ring gear can be regarded as comprising two components: circumferential deformation and radial deformation. The circumferential deformation, which occurs along the direction of tooth thickness after heating, can be mathematically expressed as

∆ S_{k} = \frac{λ (T e m_{k} - T e m_{0})}{2} [\frac{π m r_{k}}{2 r} - 2 r_{k} (i n v (α_{k}) - i n v (α))]

(28)

The temperature of the contact point $T e m_{k}$ can be represented as

T e m_{k} = T e m_{f l a s h} + T e m_{a}

(29)

r_{k}

represents the distance from the point of engagement to the center of the gear before thermal deformation occurs

r_{k} = \frac{r_{b}}{\cos (α_{k})}

(30)

The change in the angle of extension caused by thermal deformation $Δ φ_{k}$ can be expressed as

∆ φ_{k} = \frac{∆ S_{k}}{{r_{k}}^{'}}

(31)

The radial deformation of the gear can be regarded as comprising two components: the radial deformation of the gear body and the radial deformation of the teeth. The radial deformation of the gear body can be mathematically represented as

u_{b r} = λ r_{f r} T e m_{b r} - \frac{2 λ P_{o} r_{f r} {r_{o r}}^{2} (1 + P_{o}) (T e m_{b r} - T e m_{c r})}{(1 - P_{o}) ({r_{o r}}^{2} - {r_{f r}}^{2})}

(32)

The radial deformation of the gear tooth can be represented as

∆ r_{k} = u_{b r} + λ (T e m_{k r} - T e m_{0}) (r_{k r} - r_{b})

(33)

The distance from the contact point to the center of the gear after thermal deformation can be expressed as

{r_{k}}^{'} = r_{k} - ∆ r_{k}

(34)

The comprehensive thermal deformation of the gear can be formulated as

\{\begin{cases} σ_{P R r} = \overset{⇀}{∆ S_{k}} + \overset{⇀}{∆ r_{k}} \\ σ_{P R r} = {r_{k}}^{'} \sin (2 φ_{k} + ∆ φ_{k}) - r_{k} \sin (2 φ_{k}) \end{cases}

(35)

Verification of thermal deformation calculation model of ring gear

In order to validate the accuracy of the proposed gear thermal deformation calculation model, a working condition with a rotational speed of 5000 rpm and torque of 100 N·m was specifically chosen. The steady-state temperature field values of the gear were utilized as loads in ANSYS static structural module to obtain simulated data on thermal deformation of gear teeth. A comparison was made between the simulation value and calculated value. The obtained results are illustrated in the Figure 7.

Figure 7.

Comparison of calculated and simulated thermal deformation values of gear ring.

The trend of the theoretical calculated values is observed to be consistent with the simulated values along the meshing line, as depicted in Figure 7. The tooth deformation of the gear gradually increases from the root to the tip, which aligns with previous studies.^5,7 However, a comparison between the results reveals that the calculated values are slightly larger than the simulated values. This discrepancy can be attributed to neglecting other directions influenced by material continuity, such as tooth width direction, in the theoretical calculation model where only radial and circumferential thermal deformations were considered. Additionally, in simulation calculations, fixed support was applied at the outer wall of the ring gear (i.e., zero thermal deformation), leading to slightly larger calculated values compared to simulated ones. In conclusion, numerical models yield similar trends and exhibit minor differences when compared to simulated values, thereby verifying the accuracy of our numerical calculation model.

Analysis of system backlash variation under the influence of thermal deformation

The change in backlash of the SP meshing pair and PR meshing pair caused by thermal deformation can be expressed as

\{\begin{cases} ∆ b_{S P} = \sum_{n = 1}^{2} σ_{s p n} (t) \\ ∆ b_{P R} = \sum_{n = 1}^{2} σ_{p r n} (t) \end{cases}

(36)

The changes in backlash of the SP meshing pair and PR meshing pair under variable speed conditions and variable torque conditions are shown in Figures 8 and 9, respectively.

Figure 8.

Backlash variation of planetary gear system under variable speed conditions. (a) Backlash variation of SP and (b) Backlash variation of PR.

Figure 9.

Backlash variation of planetary gear system under variable torque conditions. (a) Backlash variation of SP and (b) Backlash variation of PR.

The variation of backlash gradually increases along the line of engagement, as observed from Figures 8 and 9. The maximum value of backlash variation for the SP meshing pair is found near the tooth tip, while for the PR meshing pair, it occurs at the tooth top position. Under similar operating conditions, the backlash variation is greater for SP meshing compared to PR meshing. Figure 8 indicates that under variable speed conditions, the contour lines of backlash variation become more widely spaced with increasing speed, suggesting a gradual decrease in magnitude of backlash variation. Conversely, Figure 9 demonstrates that there is a constant spacing between contour lines of backlash variation with increasing torque, indicating a linear relationship between backlash variation and torque.

Analysis of system coupling stiffness under the influence of thermal deformation

The thermal stiffness of gear teeth caused by thermal deformation can be expressed as

k_{i n} (t) = w_{i} / (b_{i n} σ_{i n})

(37)

The thermal stiffness of the meshing pair, derived from the tooth contact thermal stiffness, can be expressed as

k_{w i} = k_{w i 1} k_{w i 2} / (k_{w i 1} + k_{w i 2})

(38)

The coupled meshing stiffness obtained after the coupling of thermal stiffness and meshing stiffness can be represented as

k_{i} = k_{w 1} k_{t i} / (k_{w 1} + k_{t i})

(39)

The variations in coupling meshing stiffness of the gear pair are illustrated in Figures 10 and 11 under conditions of variable speed and torque.

Figure 10.

Coupling stiffness of planetary gear system under variable speed conditions. (a) Coupling stiffness of SP and (b) Coupling stiffness of PR.

Figure 11.

Coupling stiffness of planetary gear system under variable torque conditions. (a) Coupling stiffness of SP and (b) Coupling stiffness of PR.

The coupling mesh stiffness of the SP is observed to be smaller than that of the PR under identical operating conditions, as evident from Figures 10 and 11. When subjected to variable speed conditions, the coupling stiffness decreases with an increase in rotational speed. Conversely, under variable torque conditions, the coupling stiffness increases with higher torque.

Analysis of nonlinear dynamical characteristics of the system

Nonlinear dynamic analysis of system under variable speed conditions

Choose the dimensionless meshing frequency as the bifurcation parameter and assign values to parameters $k = 0.2$ , $z_{n} = 0.85$ , $e_{a m p} = 3$ , $T_{i n} = 100$ .

The bifurcation diagram and LLE graph of the system are presented in Figure 12, respectively. The maximum Lyapunov exponent (LLE) serves as a crucial indicator for evaluating the stability of the system, with its values being associated with different states of the system as demonstrated in Table 1. Specifically, a negative, zero, or positive LLE corresponds to periodic motion state ( $P_{p}$ ), quasi-periodic motion state ( $P_{q}$ ), and chaotic motion state ( $P_{c}$ ) of the system, respectively.³⁵

Figure 12.

Bifurcation diagram and LLE diagram of system under variable speed condition.

Table 1.

LLE value range in different motion states.

The motion state of system	The value range of LLE
Periodic motion state	<0
Quasi-periodic motion state	= 0
Chaotic motion state	>0

When $Ω$ ∈[0.01,0.97], LLE<0, indicating that the system is in a state of periodic motion. By selecting specific values for $Ω$ = 0.54, 0.82, and 0.91, the system exhibits distinct motion states as P1, P2, and P4, respectively, as depicted in Figures 13–15. The poincaré map of the system manifests as discrete points while its phase trajectory forms closed loops in the phase plane with characteristic patterns of periodic encirclement. Furthermore, the frequency spectrum of the system comprises discrete frequencies.

Figure 13.

P1 motion state of the System when $Ω$ = 0.54.

Figure 14.

P2 motion state of the System when $Ω$ = 0.82.

Figure 15.

P4 motion state of the System when $Ω$ = 0.91.

When $Ω$ ∈[0.98,0.99], the system undergoes a transition from periodic motion to quasi-periodic motion with LLE = 0. At $Ω$ = 0.98, as depicted in Figure 16, the system displays four distinct sets of discrete points in its poincaré map and its phase trajectory takes on an approximately non-closed shape as trajectories wind around in the phase plane. The frequency spectrum reveals subharmonics that defy rationalization.

Figure 16.

Quasi-P4 motion state of the system when $Ω$ = 0.98.

When $Ω$ ∈[1,1.13], the system transitions from quasi-periodic motion to periodic motion. At $Ω$ = 1.07, the system is in P2 motion state as shown in Figure 17. When $Ω$ = 1.14, the system transitions from periodic motion to quasi-periodic motion, and at $Ω$ = 1.15, it enters the Quasi-P2 motion state as depicted in Figure 18. For $Ω$ ∈[1.16,1.21], the system switches back from quasi-periodic motion to periodic motion again. At $Ω$ = 1.19, it exhibits P12 motion state as illustrated in Figure 19.

Figure 17.

P2 motion state of the system when $Ω$ = 1.07.

Figure 18.

Quasi-P2 motion state of the system when $Ω$ = 1.15.

Figure 19.

P12 motion state of the system when $Ω$ = 1.19.

When $Ω$ ∈[1.22, 1.62], the system transitions from periodic motion to chaotic motion, with LLE>0. When $Ω$ = 1.30, as shown in Figure 20, the system’s Poincaré map appears as an irregular set of discrete points and its phase trajectory forms an unclosed shape with non-periodic trajectories entangled in the phase plane. The frequency spectrum exhibits continuous subharmonics.

Figure 20.

Chaotic motion state of the system when $Ω$ = 1.30.

When $Ω$ ∈[1.63,4], the system transitions from chaotic motion to periodic motion. This is illustrated in Figures 21–23, where the system is in states P4, P8, and P1, respectively, for $Ω$ = 2.2, 2.47, and 3.5.

Figure 21.

P4 motion state of the system when $Ω$ = 2.2.

Figure 22.

P8 motion state of the system when $Ω$ = 2.47.

Figure 23.

P1 motion state of the system when $Ω$ = 3.5.

Analysis of the influence of thermal deformation on the system under variable speed conditions

Influence on the meshing impact state of the tooth surface

During the meshing process, there are impact types on the meshing tooth surface, such as non-impact state, tangential state, unilateral impact state, and bilateral impact state. The relationship between impact state and dimensionless relative displacement is as follows³⁵

The original system (without considering thermal deformation) undergoes a transition from a non-impact state to a unilateral impact state at $Ω$ = 0.56, and transitions back to a non-impact state at $Ω$ = 3.9. The phase diagrams at $Ω$ = 0.55, 0.56, 3.00, and 3.90 are shown in Figure 24, respectively. At $Ω$ = 0.55, which $z_{1 \min}$ is greater than the boundary value of 1, the system remains in a non-impact state. However, as $Ω$ = 0.56, the system enters into tangential state contact and subsequently transitions into a unilateral impact state. As $z_{1 \min}$ increases gradually towards $z_{1 \min}$ = 1 and eventually reaches at $Ω$ = 3.90, the system returns back to an non-impact state. Tables 2 and 3.

Figure 24.

Phase trajectory diagram of the original system when $Ω$ = 0.55, 0.56, 3.00, 3.90. (a) $Ω$ =0.55, (b) $Ω$ =0.56, (c) $Ω$ =3.00, and (d) $Ω$ =3.90.

Table 2.

Dimensionless relative displacement under different impact states.

Impact state of meshing tooth surface	$z_{1 \min}$
Non-impact state	$z_{1 \min} > 1$
Tangential state	$\| z_{1 m i n} \| = 1$
Unilateral impact state	$\| z_{1 \min} \| < 1$
Bilateral impact state	$z_{1 \min} < - 1$

Table 3.

Nomenclature.

Parameters	Sun gear	Planet gear	Ring gear	Planet carrier
Number of teeth, $Z$	20	29	79	—
Module, $m$ /mm	2.25
Base circle diameter, $d$ /mm	42.29	61.31	167.03	113
Moment of inertia, $I$ /(kg.mm²)	146	297	17440	11178
Pressure angle, $α$ /(°)	20
Tooth width, $b$ /mm	25	28	25	25
Mass, $M$ /kg	0.53	0.35	1.85	4.5
Poisson’s ratio, $P_{o}$	0.3
Addendum coefficient, $h_{a}$	1
Top backlash coefficient, $c_{b}$	0.25
Density of gear, $ρ$ /kg/m³	7833
Linear expansion coefficient, $l (t)$	1.16 $\times$ 10⁻⁵
Reference temperature 1, $T_{r e f 1}$ /°C	20
Reference temperature 2, $T_{r e f 2}$ /°C	300
Reference elastic modulus 1, $E (T_{r e f 1})$ /GPA	210
Reference elastic modulus 2, $E (T_{r e f 2})$ /GPA	186
Heat transfer coefficients of gear, $λ$ /J/m·s·°C	46.47
Specific heat capacity of the gear, $c$ /J/kg·°C	481.47
Dynamic viscosity of lubricating oil, $η_{m}$ /m²/s	1.2 $\times$ 10⁻⁴
Thermal conductivity of lubricating oil, $K_{0}$	0.14
Density of lubricating oil, $ρ_{f}$ /kg/m³	870
Specific heat capacity of lubricating oil, $c_{f}$ /J/kg·k	2000
$i$ = $S P$ , $PR$	SP: Sun-planet gear meshing pair PR: Planet-ring gear meshing pair
$j$ = $s$ , $p$ , $c$ , $r$	Sun gear, planet gear, carrier, ring gear
$n$ = 1,2	The driving and driven gears in the meshing pair
$r_{a j}$	Addendum radius of the gear
$r_{j}$	Reference circle radius of the gear
$ε_{i}$	Contact ratio of meshing pair
arn	The center distance of the PR
$S_{p b}$	Pitch of the base circle
$Q$	Frictional heat flux (W/m²)
$γ$	Heat energy conversion coefficient (0.9∼0.95)
$μ$	Friction coefficient
$ω$	Rotational speed
$S_{k}$	Tooth thickness of meshing point

Selecting the phase diagram at $Ω$ = 0.55, 0.56, 3.00, 3.90 in the thermal system (the system considering thermal deformation), as shown in Figure 25. It is known that the system is always in a unilateral impact state in $Ω$ ∈(0,4).

Figure 25.

Phase trajectory diagram of the thermal system when $Ω$ = 0.55, 0.56, 3.00, 3.90. (a) $Ω$ =0.55, (b) $Ω$ =0.56, (c) $Ω$ =3.00, and (d) $Ω$ =3.90.

Influence on the system motion status

When $Ω$ = 0.85, the thermal system is in a P4 motion state as shown in Figure 26, while the original system is in a P2 motion state as shown in Figure 27.

Figure 26.

P4 motion state of the thermal system when $Ω$ = 0.85.

Figure 27.

P2 motion state of the original system when $Ω$ = 0.85.

When $Ω$ = 0.94, the thermal system is in a P4 motion state as shown in Figure 28, while the original system is in a quasi-P2 motion state as shown in Figure 29.

Figure 28.

P4 motion state of the thermal system when $Ω$ = 0.94.

Figure 29.

Quasi-P2 motion state of the original system when $Ω$ = 0.94.

When $Ω$ = 1.21, the thermal system is still in periodic motion, as shown in Figure 30, while the original system enters chaotic motion earlier than the thermal system, as shown in Figure 31.

Figure 30.

P24 motion state of the thermal system when $Ω$ = 1.21.

Figure 31.

Chaotic motion state of the original system when $Ω$ = 1.21.

When $Ω$ = 1.63, the thermal system transitions from chaotic motion to periodic motion, as shown in Figure 32, while the original system remains in a state of chaotic motion, as depicted in Figure 33.

Figure 32.

P10 motion state of the thermal system when $Ω$ = 1.63.

Figure 33.

Chaotic motion state of the original system when $Ω$ = 1.63.

When $Ω$ = 2.26, the thermal system is in a P1 motion state, as shown in Figure 34, while the original system is in a P2 motion state, as shown in Figure 35.

Figure 34.

P1 motion state of the thermal system when $Ω$ = 2.26.

Figure 35.

P2 motion state of the original system when $Ω$ = 2.26.

When $Ω$ = 2.33, the thermal system is in a P4 motion state, as shown in Figure 36, while the original system is in a P16 motion state, as shown in Figure 37.

Figure 36.

P4 motion state of the thermal system when $Ω$ = 2.33.

Figure 37.

P16 motion state of the original system when $Ω$ = 2.33.

When $Ω$ = 2.43, the thermal system is in a P8 motion state, as shown in Figure 38, while the original system is in a quasi-P4 motion state, as shown in Figure 39.

Figure 38.

P8 motion state of the thermal system when $Ω$ = 2.43.

Figure 39.

Quasi-P4 motion state of the original system when $Ω$ = 2.43.

Nonlinear dynamic analysis of system under variable torque conditions

Taking the input torque $T_{i n}$ as the bifurcation parameter, with parameters set as $k$ = 0.2, $z_{n}$ = 0.85, $e_{a m p}$ = 3, and $Ω$ = 1, the bifurcation diagram and LLE of the system are shown in Figure 40.

Figure 40.

Bifurcation diagram and LLE diagram of the system under variable torque condition.

When $T_{i n}$ = 100 N·m, the system is in a state of P4 motion, as shown in Figure 41.

Figure 41.

P4 motion state of the system when $T_{i n}$ = 100 N·m.

The system exhibits quasi-periodic motion when $T_{i n}$ ∈[101,103]. Figures 42–44 illustrate this behavior for $T_{i n}$ = 101, 102, 103.

Figure 42.

Quasi-P4 motion state of the system when $T_{i n}$ = 101 N·m.

Figure 43.

Quasi-P4 motion state of the system when $T_{i n}$ = 102 N·m.

Figure 44.

Quasi-P4 motion state of the system when $T_{i n}$ = 103 N·m.

The system exhibits periodic motion when $T_{i n}$ ∈[104,118]. Taking $T_{i n}$ = 106 as an example, as illustrated in Figure 45

Figure 45.

P4 motion state of the system when $T_{i n}$ = 106 N·m.

When $T_{i n}$ ∈[119,125], the system is in a quasi-periodic motion state. The system state at $T_{i n}$ = 120 is shown in Figure 46

Figure 46.

Quasi-P2 motion state of the system when $T_{i n}$ = 120 N·m.

When $T_{i n}$ ∈[126,300], the system is in a periodic motion state. Taking $T_{i n}$ = 126 N·m, 138 N·m, and 280 N·m, the system states are shown in Figures 47–49 respectively.

Figure 47.

P4 motion state of the system when $T_{i n}$ = 126 N·m.

Figure 48.

P2 motion state of the system when $T_{i n}$ = 138 N·m.

Figure 49.

P1 motion state of the system when $T_{i n}$ = 280N·m.

Analysis of the influence of thermal deformation on the system under variable torque conditions

Influence on the meshing impact state of the tooth surface

The original system undergoes a transition from a unidirectional impact state to a no-impact state when $T_{i n}$ = 138 N·m. Figure 50 illustrates the states at $T_{i n}$ = 120 N·m, 138 N·m, and 280 N·m. At $T_{i n}$ = 120 N·m, the system remains in a unidirectional impact state until it experiences grazing motion at $T_{i n}$ = 138 N·m, eventually leading to its transition into a no-impact state.

Figure 50.

Phase trajectory diagram of the original system when $T_{i n}$ = 120 N·m, $T_{i n}$ = 138 N·m, $T_{i n}$ = 280 N·m. (a) $T_{i n}$ =120 N·m, (b) $T_{i n}$ =138 N·m, and (c) $T_{i n}$ =280 N·m.

By selecting the thermal system with $T_{i n}$ = 120 N·m, 138 N·m, and 280 N·m as shown in Figure 51, it can be observed that the thermal system remains in a unidirectional shock state throughout the range of $T_{i n}$ ∈[100,300].

Figure 51.

Phase trajectory diagram of the thermal system when $T_{i n}$ = 120 N·m, $T_{i n}$ = 138 N·m, $T_{i n}$ = 280 N·m. (a) $T_{i n}$ =120 N·m, (b) $T_{i n}$ =138 N·m, and (c) $T_{i n}$ =280 N·m.

Influence on the motion state of the system

When $T_{i n}$ = 100 N·m, the thermal system undergoes a P4 motion as shown in Figure 41, while the original system at this time undergoes a multiple-period motion as shown in Figure 52.

Figure 52.

Multi-period motion state of the original system when $T_{i n}$ = 100 N·m.

When $T_{i n}$ = 101 N·m, the thermal system exhibits quasi-periodic motion as shown in Figure 42, while the original system undergoes P12 motions as depicted in Figure 53.

Figure 53.

P12 motion state of the original system when $T_{i n}$ = 101 N·m.

The thermal system undergoes a P4 motion, as depicted in Figure 54, when $T_{i n}$ = 104 N·m. In contrast, the original system exhibits quasi-P4 motion, as illustrated in Figure 55.

Figure 54.

P4 motion state of the thermal system when $T_{i n}$ = 104 N·m.

Figure 55.

Quasi-P4 motion state of the original system when $T_{i n}$ = 104 N·m.

The thermal system exhibits quasi-P2 motion, as depicted in Figure 46, when $T_{i n}$ = 120 N·m. In contrast, the original system exhibits P4 motion, as illustrated in Figure 56.

Figure 56.

P4 motion state of the original system when $T_{i n}$ = 120 N·m.

When $T_{i n}$ = 126 N·m, the thermal system is in a P4 motion, as shown in Figure 47, while the original system is in a P2 motion, as shown in Figure 57.

Figure 57.

P2 motion state of the original system when $T_{i n}$ = 126 N·m.

Conclusion

1. In a planetary gear system where the sun gear serves as the input component and the planet carrier acts as the output component, there is a gradual decrease in temperatures observed for the sun gear, planet gears, and ring gear. Under variable speed conditions, an increase in speed logarithmically raises temperatures across different parts of each component. Similarly, under variable torque conditions, an increase in torque linearly elevates temperatures across different parts of each component.

2. Under variable speed conditions, with an increasing meshing frequency (speed), the planetary gear system exhibits periodic motion initially followed by quasi-periodic motion and chaotic motion states with transitions between these types of motions. Thermal deformation affects nonlinear dynamics characteristics of the system in two ways:

(1) It can change meshing contact from non-impact to unilateral impact state impact state which deteriorates meshing impact condition;

(2) The influence of thermal deformation on system motion states is more pronounced in high meshing frequency region (high temperature region) compared to low meshing frequency region (medium-low temperature region). This leads to higher multiple-periodic motions degrading into lower multiples and quasi-periodic motions transitioning into periodic motions. Additionally, thermal deformation delays entry into chaos zone while advancing exit from chaos zone thereby shortening chaotic interval of the system.

3. Under variable torque conditions, the system exhibits periodic and quasi-periodic motion states as the torque increases. Thermal deformation also impacts the system in two ways:

(1) It transforms the meshing tooth surfaces from a non-impact state to a unilateral impact state impact state, thereby deteriorating the meshing impact condition;

(2) The influence of thermal deformation on the system’s motion state is more pronounced in the low torque region (medium-low temperature region) compared to the high torque region (high torque region). Thermal deformation causes degradation of multi-periodic motion into low-periodic motion, but there is also an occurrence where periodic motion of the system deteriorates into quasi-periodic motion.

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: The authors gratefully acknowledge the support of project funded by the State Key Laboratory of Traction Power (TPL2310), Liaoning BaiQianWan Talents Program (2020921031), Natural Science Foundation of Liaoning Province of China (2020-MS-216).

ORCID iD

Jingyue Wang

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Research on bifurcation and chaos characteristics of planet gear transmission system with thermal deformation

Abstract

Keywords

Introduction

Nonlinear dynamics model of planetary gear system

Steady-state temperature field of planetary gear system

Steady-state temperature field calculation model of planetary gear

Simulation analysis of steady-state temperature field of planetary gear

Calculation model for thermal deformation of planetary gear system

Flash temperature of tooth surface

Thermal deformation calculation model

Verification of thermal deformation calculation model of ring gear

Analysis of system backlash variation under the influence of thermal deformation

Analysis of system coupling stiffness under the influence of thermal deformation

Analysis of nonlinear dynamical characteristics of the system

Nonlinear dynamic analysis of system under variable speed conditions

Analysis of the influence of thermal deformation on the system under variable speed conditions

Influence on the meshing impact state of the tooth surface

Influence on the system motion status

Nonlinear dynamic analysis of system under variable torque conditions

Analysis of the influence of thermal deformation on the system under variable torque conditions

Influence on the meshing impact state of the tooth surface

Influence on the motion state of the system

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References