Couple-Stress Fluid Improves Dynamic Response of Gear-Pair System Supported by Journal Bearings

Abstract

A systematic analysis of the dynamic behavior of a gear-bearing system with nonlinear suspension, couple-stress fluid flow effect, nonlinear oil-film force, and nonlinear gear mesh force is performed in the present study. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless rotational speed ratio as a control parameter. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents and fractal dimension of the gear-bearing system. The numerical results reveal that the system exhibits a diverse range of periodic, subharmonic, quasiperiodic, and chaotic behaviors. The couple-stress fluid would be a useful lubricating fluid to suppress nonlinear dynamic responses and improve the steady of the systems. The results presented in this study provide some useful insights into the design and development of a gear-bearing system for rotating machinery that operates in highly rotational speed and highly nonlinear regimes.

1. Introduction

Lubricants with additives are often considered as non-Newtonian fluids and are found to be able to used to improve the properties of lubricants such as the viscosity of lubricants in many studies and also many applications. According to the results of experiments, it evidenced that the use of small amounts of suitable additives can improve lubricant properties. It can make the viscosity of the lubricant to be independent of the temperature. With the development of modern machine elements, the increasing use of complex fluids as lubricants has been emphasized. Oliver [1] had found that the presence of dissolved polymer in the lubricant could increase the load-carrying capacity and decrease the friction. Spikes [2] showed that the base oil blending with additives could reduce the friction and the surface damage in elastohydrodynamic contacts. Because of the behavior of the complex fluids violates the linear shear-stress-rate relationship, it fails to describe the rheological behaviour of the non-Newtonian fluid. Therefore, a different microcontinuum theory has been developed to better describe the rheological behaviour of the non-Newtonian fluid. Shehawey and Mekheimer [3] applied the couple-stress model to analyze the peristalsis problem for its relative mathematical simplicity. Das [4] proposed the analysis of elastohydrodynamic theory of line contacts. Das [5] studied the slider bearing lubricated with couple-stress fluids in magnetic field and observed that both the values of the maximum load capacity, and corresponding inlet-outlet film ratio depend on couple stress, magnetic parameters and the shape of bearings. Elsharkawy and Guedouar [6] proposed an efficient numerical scheme to solve the direct lubrication problem for journal bearings lubricated with couple stress fluids, which consists of the modified Reynolds equation, the film thickness equation, and the boundary conditions for the pressure field. Hsu et al. [7] studied the short journal bearings lubricated with the non-Newtonian fluid which is combined with the effects of couple-stresses and surface roughness. It was found that the combined effects of couple stress and surface roughness can improve the load-carrying capacity and decrease the attitude angle and friction parameters. Lahmar [8] also found that the lubricants with couple stress would increase the load-carrying capacity and stability and decrease the friction factor and the attitude angle. The above researches are all about the applications of the couple-stress fluids, and all the results of their studies emphasized that the couple-stress fluids are more superior and more stable than the traditional Newtonian fluids. Chang-Jian and Chen [9] found couple-stress fluid can be used to suppress nonperiodic or even chaotic dynamic response especially for higher rotating speeds in rotor-bearing systems.

Gears have played important roles in power transmission, and therefore the study of gear dynamics is necessary. Many studies have focused on analyzing gear dynamics or relative researches. Vedmar and Andersson [10] presented a method to calculate dynamic gear tooth force and bearing forces and the bearing model was under elastic bearings assumption. The simulation results of elastic model were somewhat different comparing with stiff one. The gear mesh model used with constant was studied by a lot of people, such as Kahraman and Singh [11], Lin et al. [12], Yoon and Rao [13], and Ichimaru and Hirano [14]. Amabili and Fregolent [15] introduced a method based on the measurement of the gear torsional vibrations to identify natural frequency, damping parameters, and equivalent gear error of a spur-gear-pair model. Özgüven and Houser [16, 17] performed dynamic analysis on gears with the effects of variable mesh stiffness, damping, gear errors profile modification, and backlash. Cai and Hayashi [18] calculated the optimum profile modification to obtain a zero vibration of the gear pair. Umezawa et al. [19] analyzed a single DOF numerical gear pair model and compared their numerical results with experimental dynamic transmission errors. McFadden and Toozhy [20] used the high-frequency technique combined with synchronous averaging to detect the failure in rolling element bearings. Litvin et al. [21] proposed a modified geometry of an asymmetric spur-gear drive designed as a favorable shape of transmission errors of reduced magnitude and also reduced contact and bending stresses for an asymmetric spur-gear drive. Guan et al. [22] performed finite-element method to simulate the geared rotor system constructed from beam and lumped mass/stiffness elements and compared the required actuation effort, control robustness, and implementation cost. Giagopulos et al. [23] presented an analysis on the nonlinear dynamics of a gear-pair system supported on rolling element bearings and used a suitable genetic algorithm to measure noise and model error. Theodossiades and Natsiavas [24] investigated dynamic responses and stability characteristics of rotordynamic systems interconnected with gear pairs, supported on oil journal bearings. They found many nonperiodic dynamic behaviors. They [25] also analyzed the motor-driven gear-pair systems with backlash and found periodic and chaotic dynamics in this system.

Although virtually all physical phenomena in the real world can be regarded as nonlinear, most of these phenomena can be simplified to a linear form given a sufficiently precise linearization technique. However, this simplification is inappropriate for high-power, high-rotational-speed gear system, and its application during the design and analysis stage may result in a flawed or potentially dangerous operation. As a result, nonlinear analysis methods are generally preferred within engineering and academic circles. The current study performs a nonlinear analysis of the dynamic behavior of a gear-pair system equipped with couple-stress fluid lubricated journal bearings under nonlinear suspension. The nondimensional equation of the gear-bearing system is then solved using the Runge-Kutta method. The nonperiodic behavior of this system is characterized by using phase diagrams, power spectra, Poincaré maps, bifurcation diagrams, Lyapunov exponents, and the fractal dimension of the system.

2. Mathematical Modeling

2.1. Nonlinear Dynamic Models

Some assumptions are presented to simplify dynamic models to be able to simulate the gear-bearing system, that is, nonlinear suspension (hard spring effect), short journal bearing, nonlinear gear meshing force, and nonlinear couple-stress fluid film force effect are established. Figure 1 shows the gear-baring model presented in this study. The equations of motion used to describe geometric centers of gear and pinion (𝑂_𝑔(𝑋_𝑔,𝑌_𝑔) and 𝑂_𝑝(𝑋_𝑝,𝑌_𝑝)) can be written as𝑚_𝑝𝑚_𝑝𝑋̈_𝑝+𝐶𝑋̇_𝑝+𝐾𝑋_𝑝−𝑋_𝑗1=𝑊_𝑐𝑥,𝑌_𝑝̈+𝑚𝐶𝑌_𝑝̇+𝐾𝑌_𝑝−𝑌_𝑗1=𝐿_𝑝𝑦−𝐺_𝑝𝑦−𝑊_𝑐𝑦−𝑚_𝑝𝑔,_𝑔𝑚_𝑔𝑋̈_𝑔+𝐶𝑋̇_𝑔+𝐾𝑋_𝑔−𝑋_𝑗𝑔=−𝑊_𝑐𝑥,𝑌_𝑔̈+𝐶𝑌_𝑔̇+𝐾𝑌_𝑔−𝑌_𝑗2=𝐿_𝑝𝑦−𝐺_𝑝𝑦+𝑊_𝑐𝑦−𝑚_𝑔𝑔,(2.1) where 𝐿_𝑝𝑦 and 𝐿_𝑔𝑦 are the centrifugal forces in the vertical gear mesh direction for pinion and gear, 𝐺_𝑝𝑦 and 𝐺_𝑔𝑦 are the inertia forces in the vertical gear mesh direction for pinion and gear, 𝑊_𝑐𝑥 is the dynamic gear mesh force in the horizontal direction, and 𝑊_𝑐𝑦 is the dynamic gear mesh force in the vertical direction.

Figure 1 

Schematic illustration of the gear-bearing system under nonlinear suspension and the model of force diagram for pinion and gear.

The equations of motion of the center of bearing 1 (𝑋₁,𝑌₁) and the center of bearing 2 (𝑋₂,𝑌₂) under the assumption of nonlinear suspension can be written as𝑚₁𝑚₁𝑋̈₁+𝑐₁𝑋̇₁+𝑘₁₁𝑋₁+𝑘₁₂𝑋³¹=𝐹_𝑥1,𝑌𝑚₁̈₂+𝑐₁𝑌₁̈+𝑘₁₁𝑌₁+𝑘₁₂𝑌³¹=−𝑚₁𝑔+𝐹_𝑦1,𝑚₂𝑋̈₂+𝑐₂𝑋̇₂+𝑘₂₁𝑋₂+𝑘₂₂𝑋³²=𝐹_𝑥2,𝑌₂̈+𝑐₂𝑌₂̈+𝑘₂₁𝑌₂+𝑘₂₂𝑌³²=−𝑚₂𝑔+𝐹_𝑦2.(2.2) According to the principles of force equilibrium, the forces acting at the center of journal 1, that is, 𝑂_𝑗1(𝑋_𝑗1,𝑌_𝑗1) and center of journal 2, that is, 𝑂_𝑗2(𝑋_𝑗2,𝑌_𝑗2) are given by𝐹_𝑥1=𝑓_𝑒1cos𝜑₁+𝑓_𝜑1sin𝜑₁=𝐾^𝑝1𝑋_𝑝−𝑋_𝑗1𝐹_𝑦12,=𝑓_𝑒1sin𝜑₁−𝑓_𝜑1cos𝜑₁=𝐾^𝑝1𝑌_𝑝−𝑌_𝑗1𝐹_𝑥22,=𝑓_𝑒2cos𝜑₂+𝑓_𝜑2sin𝜑₂=𝐾^𝑝2𝑋_𝑔−𝑋_𝑗2𝐹_𝑦22,=𝑓_𝑒2sin𝜑₂−𝑓_𝜑2cos𝜑₂=𝐾^𝑝2𝑌_𝑔−𝑌_𝑗22,(2.3) in which 𝑓_𝑒1 and 𝑓_𝜑1 are the viscous damping forces in the radial and tangential directions for the center of journal 1, respectively, and 𝑓_𝑒2 and 𝑓_𝜑2 are the viscous damping forces in the radial and tangential directions for the center of journal 2, respectively.

2.2. Nonlinear Couple-Stress Fluid Film Force

The non-Newtonian Reynolds-type equation can be performed as𝜕𝜕𝑥𝑔(ℎ,𝑙)𝜕𝜕𝑥𝑝+𝜕𝜕𝑧𝑔(ℎ,𝑙)𝜕𝜕𝑧𝑝=6𝜇𝑈𝜕𝜕ℎ𝑥+12𝜇𝜕𝜕ℎ𝑡,(2.4) where 𝑔(ℎ,𝑙)=ℎ³−12𝑙²ℎ+24𝑙³tanℎ(ℎ/2𝑙), 𝜕ℎ/𝜕x=−(𝑐𝜀/𝑅)sin𝜃, 𝜕ℎ/𝜕𝑡=𝑐𝜀̇cos𝜃+𝑐𝜀𝜑̇sin𝜃, 𝑥=𝑅𝜃, 𝑈=𝑅𝜔, 𝜀=𝑒/𝑐, and ℎ=𝑐(1+𝜀cos(𝛾−𝜑(𝑡)))=𝑐(1+𝜀cos𝜃). Thus 𝑔(ℎ,𝑙) can also be performed as 𝑔(ℎ,𝑙)=𝑐³(1+𝜀cos𝜃)³−12𝑙²𝑐(1+𝜀cos𝜃)+24𝑙³tanℎ(𝑐(1+𝜀cos𝜃)/2𝑙), where 𝑙=(𝜂/𝜇)^1/2, in which 𝜇 is the classical viscosity parameter and 𝜂 is a new material constant peculiar to fluids with couple stresses and Reynolds equation that can be rewritten as𝜕𝑅𝜕𝜃𝑔(ℎ,𝑙)𝑅𝜕𝜕𝑝𝜃+𝜕𝜕𝑧𝑔(ℎ,𝑙)𝜕𝜕𝑧𝑝=−6𝜇𝜔𝑐𝜀sin𝜃+12𝜇(𝑐𝜀̇cos𝜃+𝑐𝜀𝜑̇sin𝜃).(2.5) Using the “short bearing approximation” (𝐿/𝐷<0.25, 𝜕𝑝/𝜕𝜃≪𝜕𝑝/𝜕𝑧), then we can set 𝜕𝑝/𝜕𝜃=0. The following equation can be introduced:𝜕²𝑝𝜕𝑧²=−6𝜇𝜔𝑐𝜀sin𝜃+12𝜇(𝑐𝜀̇cos𝜃+𝑐𝜀𝜑̇sin𝜃)𝑔(ℎ,𝑙).(2.6)

The resulting damping forces about the journal center in the radial and tangential directions are determined by integrating pressure distribution over the area of the journal sleeve:𝑓𝑓_𝑒=𝐹cos𝜑=^𝐿0𝑟^02𝜋𝑝(𝜃)cos𝜃𝑑𝜃𝑑𝑧,(2.7)_𝜑=𝐹sin𝜑=^𝐿0𝑟^02𝜋𝑝(𝜃)sin𝜃𝑑𝜃𝑑𝑧.(2.8) Substituting (2.7) and (2.8) into (2.2)–(2.3) enables the values of 𝐹_𝑥1, 𝐹_𝑥2, 𝐹_𝑦1, and 𝐹_𝑦2 to be obtained.

2.3. Nonlinear Gear Meshing Force

Figure 2 presents a schematic illustration of the dynamic model considered between gear and pinion. 𝐿_𝑝𝑦 and 𝐿_𝑔𝑦 are the centrifugal forces in the vertical gear mesh direction for pinion and gear, 𝐺_𝑝𝑦 and 𝐺_𝑔𝑦 are the inertia forces in the vertical gear mesh direction for pinion and gear, 𝑊_𝑐𝑥 is the dynamic gear mesh force in the horizontal direction, and 𝑊_𝑐𝑦 is the dynamic gear mesh force in the vertical direction. Thus, 𝐿_𝑝𝑦, 𝐿_𝑔𝑦, 𝐺_𝑝𝑦, 𝐺_𝑔𝑦, 𝑊_𝑐𝑥, and 𝑊_𝑐𝑦 can be performed as𝐿𝐿_{𝑝𝑦𝑔𝑦}=𝑚_𝑝𝑒_𝑝𝜔^2𝑝sin𝜃₁,𝐺_𝑝𝑦=𝑚_𝑔𝑒_𝑔𝜔^2𝑔sin𝜃₂,𝐺_𝑔𝑦=𝑚_𝑝𝑒_𝑝𝜃̈₁cos𝜃₁,𝑊_𝑐𝑥=𝑚_𝑔𝑒_𝑔𝜃̈₂cos𝜃₂,𝑊=𝐶_𝑚𝑋̇_𝑝−𝑋̇_𝑔−𝑒_𝑝Ωsin(Ω𝑡)+𝐾_𝑚𝑋_𝑝−𝑋_𝑔−𝑒_𝑝cos(Ω𝑡)_𝑐𝑦,=𝐶_𝑚𝑌_𝑝̇−𝑌_𝑔̇−𝑒_𝑝Ωcos(Ω𝑡)+𝐾_𝑚𝑌_𝑝−𝑌_𝑔−𝑒_𝑝sin(Ω𝑡).(2.9) Substituting (2.9) into (2.1) enables the values of 𝐿_𝑝𝑦, 𝐿_𝑔𝑦, 𝐺_𝑝𝑦, 𝐺_𝑔𝑦, 𝑊_𝑐𝑥, and 𝑊_𝑐𝑦 to be obtained.

(a)

(b)

Figure 2 

Bifurcation diagrams of gear-bearing system using dimensionless rotational speed ratio, 𝑠, as bifurcation parameter for non-Newtonian lubricating fluid with 𝑙∗=0.0.

2.4. Dimensionless of Nonlinear Dynamic Models

Couple-stress fluid film force and gear meshing force would be introduced in Equations (2.1)–(2.3) and can be nondimensionalized to yield the following equations: 𝜀^1=𝛽¹𝑐𝐾_𝑝𝑦_𝑝−𝑦₁−𝜀₁sin𝜑₁cos𝜑₁−𝑥_𝑝−𝑥₁−𝜀₁cos𝜑₁sin𝜑₁−𝛿₁4𝛼₁𝛾₁𝛿₁−𝛽₁₂𝜔𝑐𝐾_𝑝𝑥_𝑝−𝑥₁−𝜀₁cos𝜑₁cos𝜑₁+𝑦_𝑝−𝑦₁−𝜀₁sin𝜑₁sin𝜑₁𝜑_14𝛼₁𝛾₁𝛿₁−𝛽₁₂𝜔,=12−𝑐𝐾^𝑝𝑦_𝑝−𝑦₁−𝜀₁sin𝜑₁cos𝜑₁−𝑥_𝑝−𝑥₁−𝜀₁cos𝜑₁sin𝜑₁−𝛽₁₂4𝛼₁𝛿₁𝜀₁𝜔𝑐𝐾_𝑝𝑦_𝑝−𝑦₁−𝜀₁sin𝜑₁cos𝜑₁−𝑥_𝑝−𝑥₁−𝜀₁cos𝜑₁sin𝜑₁−𝛽₁4𝛼₁𝜀₁𝛿₁𝛾₁𝛿₁−𝛽₁₂𝜔𝛿₁𝑐𝐾_𝑝𝑥_𝑝−𝑥₁−𝜀₁cos𝜑₁cos𝜑₁+𝑦_𝑝−𝑦₁−𝜀₁sin𝜑₁sin𝜑₁𝜀_24𝛼₁𝜀₁𝛿₁𝛾₁𝛿₁−𝛽₁₂𝜔,=𝛽²𝑐𝐾_𝑝𝑦_𝑔−𝑦₂−𝜀₂sin𝜑₂cos𝜑₂−𝑥_𝑔−𝑥₂−𝜀₂cos𝜑₂sin𝜑₂−𝛿₂4𝛼₂𝛾₂𝛿₂−𝛽₂₂𝜔𝑐𝐾_𝑝𝑥_𝑔−𝑥₂−𝜀₂cos𝜑₂cos𝜑₂+𝑦_𝑔−𝑦₂−𝜀₂sin𝜑₂sin𝜑₂𝜑_24𝛼₂𝛾₂𝛿₂−𝛽₂₂𝜔,=12−𝑐𝐾^𝑝𝑦_𝑔−𝑦₂−𝜀₂sin𝜑₂cos𝜑₂−𝑥_𝑔−𝑥₂−𝜀₂cos𝜑₂sin𝜑₂−𝛽₂₂4𝛼₂𝛿₂𝜀₂𝜔𝑐𝐾_𝑝𝑦_𝑔−𝑦₂−𝜀₂sin𝜑₂cos𝜑₂−𝑥_𝑔−𝑥₂−𝜀₂cos𝜑₂sin𝜑₂−𝛽₂4𝛼₂𝜀₂𝛿₂𝛾₂𝛿₂−𝛽₂₂𝜔𝛿₂𝑐𝐾_𝑝𝑥_𝑔−𝑥₂−𝜀₂cos𝜑₂cos𝜑₂+𝑦_𝑔−𝑦₂−𝜀₂sin𝜑₂sin𝜑₂𝑥_𝑝4𝛼₂𝜀₂𝛿₂𝛾₂𝛿₂−𝛽₂₂𝜔,=−2𝜉²𝑠𝑥^𝑝−1𝑠²−Λ𝑥_𝑝−𝑥₁−𝜀₁cos𝜑₁+𝛽cos𝜙4−2𝑠𝜉³𝑥^𝑝−𝑥^𝑔−𝐸_𝑝sin𝜙𝑠²𝑦𝑥_𝑝−𝑥_𝑔−𝐸_𝑝cos𝜙_𝑝,=−2𝜉²𝑠𝑦^𝑝−1𝑠²−Λ𝑦_𝑝−𝑦₁−𝜀₁sin𝜑₁+𝛽sin𝜙4−2𝑠𝜉³𝑦^𝑝−𝑦^𝑔−𝐸_𝑝cos𝜙𝑠²𝑦_𝑝−𝑦_𝑔−𝐸_𝑝sin𝜙−𝑓𝑠²𝑥_𝑔,=−2𝜉⁴𝑠𝑥^𝑔−1𝑠²−Λ𝑥_𝑔−𝑥₂−𝜀₂cos𝜑₂+𝛽_𝑔cos𝜙8+2𝑠𝜉⁵𝑥^𝑝−𝑥^𝑔−𝐸_𝑝sin𝜙_𝑔𝑠²𝑦𝑥_𝑝−𝑥_𝑔−𝐸_𝑝cos𝜙_𝑔,=−2𝜉⁴𝑠𝑦^𝑔−1𝑠²+Λ𝑦_𝑔−𝑦₂−𝜀₂sin𝜑₂+𝛽_𝑔sin𝜙8+2𝑠𝜉⁵𝑦^𝑝−𝑦^𝑔−𝐸_𝑝cos𝜙_𝑔𝑠²𝑦_𝑝−𝑦_𝑔−𝐸_𝑝sin𝜙−𝑓^𝑔𝑠²𝑥_1,+2𝜉¹𝑠₁𝑥_1+1𝑠₁₂𝑥₁+𝛼¹𝑠²𝑥₁₃−12𝐶_1𝑝𝑠²𝑦𝑥_𝑝−𝑥₁−𝜀₁cos𝜑₁=0,_1+2𝜉¹𝑠₁𝑦_1+1𝑠₁₂𝑦₁+𝛼¹𝑠²𝑦₁₃−12𝐶_𝑜𝑚𝑠²𝑦_𝑝−𝑦₁−𝜀₁sin𝜑₁+𝑓𝑠²𝑥=0,_2+2𝜉⁶𝑠₂𝑥_2+1𝑠₂₂𝑥₂+𝛼²𝑠²𝑥₂₃−12𝐶_2𝑝𝑠²𝑦𝑥_𝑔−𝑥₂−𝜀₂cos𝜑₂=0,_2+2𝜉⁶𝑠₂𝑦_2+1𝑠₂₂𝑦₂+𝛼²𝑠²𝑦₂₃−12𝐶_2𝑝𝑠²𝑦_𝑔−𝑦₂−𝜀₂sin𝜑₂+𝑓𝑠²=0,(2.10) where𝛼₁=−𝜇𝐿³𝑅2𝑐²𝛽₁,=^𝜋0sin𝜃cos𝜃𝛾1+𝜀¹cos𝜃³−12𝑙^∗²1+𝜀₁cos𝜃+24𝑙^∗³tanh1+𝜀₁cos𝜃/2𝑙^∗𝑑𝜃,₁=^𝜋0𝑐𝑜𝑠²𝜃𝛿1+𝜀¹cos𝜃³−12𝑙^∗²1+𝜀₁cos𝜃+24𝑙^∗³tanh1+𝜀₁cos𝜃/2𝑙^∗𝑑𝜃,₁=^𝜋0sin²𝜃1+𝜀¹cos𝜃³−12𝑙^∗²1𝛼+𝜀₁cos𝜃+24𝑙^∗³tanh1+𝜀₁cos𝜃/2𝑙^∗𝑑𝜃,₂=−𝜇𝐿³𝑅2𝑐²𝛽₂,=^𝜋0sin𝜃cos𝜃𝛾1+𝜀²cos𝜃³−12𝑙^∗²1+𝜀₂cos𝜃+24𝑙^∗³tanh1+𝜀₂cos𝜃/2𝑙^∗𝑑𝜃,₂=^𝜋0cos²𝜃𝛿1+𝜀²cos𝜃³−12𝑙^∗²1+𝜀₂cos𝜃+24𝑙^∗³tanh1+𝜀₂cos𝜃/2𝑙^∗𝑑𝜃,₂=^𝜋0sin²𝜃1+𝜀²cos𝜃³−12𝑙^∗²1+𝜀₂cos𝜃+24𝑙^∗³tanh1+𝜀₂cos𝜃/2𝑙^∗𝑑𝜃,(2.11) where 𝑙^∗=𝑙/𝑐 is the dimensionless parameter for 𝑙.

Equation (2.10) describes a nonlinear dynamic system. In the current study, the approximate solutions of these coupled non-linear differential equations are obtained using the fourth-order Runge-Kutta numerical scheme.

3. Analytical Tools for Observing Nonlinear Dynamics of Gear-Bearing System

In the present study, the nonlinear dynamics of the gear-bearing system shown in Figure 1 are analyzed using Poincaré maps, bifurcation diagrams, Lyapunov exponent, and fractal dimension. The dynamic trajectories of the gear-bearing system provide a basic indication as to whether the system behavior is periodic or nonperiodic. The projection of the Poincaré section on the 𝑦(𝑛𝑇) plane is referred to as the Poincaré map of the dynamic system. When the system performs quasiperiodic motion, the return points in the Poincaré map form a closed curve. For chaotic motion, the return points form a fractal structure comprising many irregularly distributed points. Finally, for 𝑛𝑇-periodic motion, the return points have the form of n discrete points. In this study, the spectrum components of the motion performed by the gear-bearing system are analyzed by using the Fast Fourier Transformation method to derive the power spectrum of the displacement of the dimensionless dynamic transmission error. In the analysis, the frequency axis of the power spectrum plot is normalized using the rotational speed, 𝜔.

A bifurcation diagram summarizes the essential dynamics of a gear-train system and is therefore a useful means of observing its nonlinear dynamic response. In the present analysis, the bifurcation diagrams are generated using the dimensionless rotational speed ratio, 𝑠, to be a control parameter. In each case, the bifurcation control parameter is varied with a constant step and the state variables at the end of one integration step are taken as the initial values for the next step. The corresponding variations of the 𝑦(𝑛𝑇) coordinates of the return points in the Poincaré map are then plotted to form the bifurcation diagram.

The Lyapunov exponent of a dynamic system characterizes the rate of separation of infinitesimally close trajectories and provides a useful test for the presence of chaos. In a chaotic system, the points of nearby trajectories starting initially within a sphere of radius 𝜀₀ form after time 𝑡 an approximately ellipsoidal distribution with semiaxes of length 𝜀_𝑗(𝑡). The Lyapunov exponents of a dynamic system are defined by 𝜆_𝑗=lim_𝑡→∞(1/𝑡)log(𝜀_𝑗(𝑡)/𝜀₀), where 𝜆_𝑗 denotes the rate of divergence of the nearby trajectories. The exponents of a system are usually ordered into a Lyapunov spectrum, that is, 𝜆₁>𝜆₂>⋯>𝜆_𝑚. A positive value of the maximum Lyapunov exponent (𝜆₁) is generally taken as an indication of chaotic motion.

The presence of chaotic vibration in a system is generally detected using either the Lyapunov exponent or the fractal dimension property. The Lyapunov exponent test can be used for both dissipative systems and nondissipative (i.e., conservative) systems, but it is not easily applied to the analysis of experimental data. Conversely, the fractal dimension test can only be used for dissipative systems, but it is easily applied to experimental data. In contrast to Fourier transform-based techniques and bifurcation diagrams, which provide only a general indication of the change from periodic motion to chaotic behavior, dimensional measures allow chaotic signals to be differentiated from random signals. Although many dimensional measures have been proposed, the most commonly applied measure is the correlation dimension 𝑑_𝐺 defined by Grassberger and Procaccia due to its computational speed and the consistency of its results. However, before the correlation dimension of a dynamic system flow can be evaluated, it is first necessary to generate a time series of one of the system variables using a time-delayed pseudo-phase-plane method. Assume an original time series of 𝑥_𝑖={𝑥(𝑖𝜏);𝑖=1,2,3,…𝑁}, where 𝜏 is the time delay (or sampling time). If the system is acted upon by an excitation force with a frequency 𝜔, the sampling time, 𝜏, is generally chosen such that it is much smaller than the driving period. The delay coordinates are then used to construct an n-dimensional vector 𝑋=(𝑥(𝑗𝜏), 𝑥[(𝑗+1)𝜏], 𝑥[(𝑗+2)𝜏],…, 𝑥[(𝑗+𝑛−1)𝜏]), where 𝑗=1,2,3,…(𝑁−𝑛+1). The resulting vector comprises a total of (𝑁−𝑛+1) vectors, which are then plotted in an n-dimensional embedding space. Importantly, the system flow in the reconstructed n-dimensional phase space retains the dynamic characteristics of the system in the original phase space. In other words, if the system flow has the form of a closed orbit in the original phase plane, it also forms a closed path in the n-dimensional embedding space. Similarly, if the system exhibits a chaotic behavior in the original phase plane, its path in the embedding space will also be chaotic. The characteristics of the attractor in the n-dimensional embedding space are generally tested using the function ∑^{𝑁𝑖,𝑗=1}𝐻(𝑟−|𝑥_𝑖−𝑥_𝑗|) to determine the number of pairs (𝑖,𝑗) lying within a distance |𝑥_𝑖−𝑥_𝑗|<𝑟 in {𝑥_𝑖}^𝑁𝑖=1, where 𝐻 denotes the Heaviside step function, 𝑁 represents the number of data points, and r is the radius of an n-dimensional hyper-sphere. For many attractors, this function exhibits a power law dependence on 𝑟 as 𝑟→0, that is, 𝑐(𝑟)∝𝑟^𝑑𝐺. Therefore, the correlation dimension, 𝑑_𝐺, can be determined from the slope of a plot of [log𝑐(𝑟)] versus [log𝑟]. Grassberger and Procaccia [26] showed that the correlation dimension represents the lower bound to the capacity or fractal dimension 𝑑_𝑐 and approaches its value asymptotically when the attracting set is distributed more uniformly in the embedding phase space. A set of points in the embedding space is said to be fractal if its dimension has a finite noninteger value. Otherwise, the attractor is referred to as a “strange attractor.” To establish the nature of the attractor, the embedding dimension is progressively increased, causing the slope of the characteristic curve to approach a steady-state value. This value is then used to determine whether the system has a fractal structure or a strange attractor structure. If the dimension of the system flow is found to be fractal (i.e., to have a noninteger value), the system is judged to be chaotic.

4. Numerical Results and Discussions

The nonlinear dynamic equations presented in (2.10) for the gear-bearing system with nonlinear suspension effects, couple-stress fluid effect, strongly nonlinear oil-film force, and nonlinear gear mesh force were solved using the fourth-order Runge-Kutta method. The time step in the iterative solution procedure was assigned a value of 𝜋/300 and the termination criterion was specified as an error tolerance of less than 0.0001. The time series data corresponding to the first 800 revolutions of the two gears were deliberately excluded from the dynamic analysis to ensure that the analyzed data related to steady-state conditions. The sampled data were used to generate the dynamic trajectories, Poincaré maps, and bifurcation diagrams of the spur-gear system in order to obtain a basic understanding of its dynamic behavior. The maximum Lyapunov exponent and the fractal dimension measure were then used to identify the onset of chaotic motion. For convenience, only the data of the displacements in the vertical direction were used to generate diagrams.

The rotational speed ratio 𝑠 is commonly used as a control parameter for bifurcation diagrams to analyze nonlinear behaviors of rotor-dynamics, gear-dynamics, bearing systems or machine tools. Accordingly, the dynamic behavior of the current gear-bearing system was examined using the dimensionless rotational speed ratio 𝑠 as a bifurcation control parameter. Figures 2, 3, 4 and 5 are the bifurcation diagrams for the gear-bearing system displacement against the dimensionless rotational speed ratio, 𝑠, for non-Newtonian lubricating fluid with 𝑙^∗=0.0,0.1,0.3, and 0.5 to describe corresponding dynamic responses. It can be observed that the dynamic behaviors of bearing center and gear center show abundant nonperiodic responses at low speeds. The dynamic behaviors become steady and come into subharmonic motions as rotating with high speed but would behave nonperiodic vibrations and higher amplitude for higher rotating speeds (i.e., 𝑠>3.0). It is said that the couple-stress fluid can improve lubricating conditions and therefore can enhance dynamic responses for turbomachineries. According to bifurcation results, we would find dynamic behaviors show different results with different 𝑙^∗. When 𝑙^∗=0.0, lubricating fluid as Newtonian fluid and the effect of couple-stress fluid would be greater as 𝑙^∗ becomes greater. The results show that the range of nonperiodic motions or subharmonic vibrations would be shortened and the amplitude of the systems would also be suppressed as 𝑙^∗ increases. Nevertheless, higher values of 𝑙^∗ could not suppress nonperiodic vibrations occurring at high speeds. We also find that the dynamic behaviors of bearing center and gear center are nonsynchronous and the couple-stress fluid can not also improve gear-dynamics as 𝑙^∗ increases.

(a)

(b)

Figure 3 

Bifurcation diagrams of gear-bearing system using dimensionless rotational speed ratio, 𝑠, as bifurcation parameter for non-Newtonian lubricating fluid with 𝑙∗=0.1.

(a)

(b)

Figure 4 

Bifurcation diagrams of gear-bearing system using dimensionless rotational speed ratio, 𝑠, as bifurcation parameter for non-Newtonian lubricating fluid with 𝑙∗=0.3.

(a)

(b)

Figure 5 

Bifurcation diagrams of gear-bearing system using dimensionless rotational speed ratio, 𝑠, as bifurcation parameter for non-Newtonian lubricating fluid with 𝑙∗=0.5.

Figures 6(a) and 6(b) present the bifurcation diagrams for the gear-bearing system displacement against the dimensionless rotational speed ratio, 𝑠, for non-Newtonian lubricating fluid with 𝑙^∗=0.3 and some dynamic trajectories and Poincaré maps (e.g., 𝑠=0.1, 0.6, 0.8, 1.3, 2.0, 3.4, and 5.0) are exemplified to describe corresponding dynamic responses. The bifurcation diagram of bearing center shows quasiperiodic responses at low speeds as 𝑠<0.15 and turns to nonperiodic or even chaotic motions as 𝑠>0.15. When rotating speed becomes greater (𝑠>1.39), the dynamic responses behave 𝑛𝑇-periodic motions, but dynamic responses of bearing center present strongly nonperiodic motions and the amplitude of the system is also greater than the amplitude of lower rotating speeds. Figures 7, 8, 9, 10, 11, and 12 are simulation results of phase diagrams, power spectra, Poincaré maps, Lyapunov exponent and fractal dimension with 𝑠=0.8, 1.0, 1.2, 3.6, 4.2, and 5.0 for 𝑦₁ (𝑙^∗=0.3). In the above case, the phase diagrams are highly disordered and the power spectra reveal numerous excitation frequencies. Furthermore, it can be seen that the return points in the Poincaré maps form geometrically fractal structures and the maximum Lyapunov exponent is positive in each case. In other words, the results presented in these figures all indicate that the gear center exhibits a chaotic behavior at those values.

(a)

(b)

Figure 6 

Bifurcation diagrams of gear-bearing system using dimensionless rotational speed ratio, 𝑠, as bifurcation parameter for non-Newtonian lubricating fluid with 𝑙∗=0.3.

Figure 7 

Simulation results obtained for gear-bearing system with 𝑠=0.8 for 𝑦1 (𝑙∗=0.3).

Figure 8 

Simulation results obtained for gear-bearing system with 𝑠=1.0 for 𝑦1 (𝑙∗=0.3).

Figure 9 

Simulation results obtained for gear-bearing system with 𝑠=4.2 for 𝑦1 (𝑙∗=0.3).

Figure 10 

Simulation results obtained for gear-bearing system with 𝑠=5.0 for 𝑦1 (𝑙∗=0.3).

Figure 11 

Simulation results obtained for gear-bearing system with 𝑠=1.2 for 𝑦1 (𝑙∗=0.3).

Figure 12 

Simulation results obtained for gear-bearing system with 𝑠=3.6 for 𝑦1 (𝑙∗=0.3).

5. Conclusions

This study has performed a numerical analysis of the nonlinear dynamic response of a gear-bearing system subject to nonlinear suspension effects, couple-stress fluid effect, nonlinear oil-film force, and nonlinear gear mesh force. The dynamics of the system have been analyzed by reference to its dynamic trajectories, power spectra, Poincaré maps, bifurcation diagrams, maximum Lyapunov exponents, and fractal dimensions. When rotating machineries are turned on, they must meet low rotating speeds and according to the results shown in the bifurcation diagrams, the dynamic behaviors are strongly nonperiodic responses at low rotating speeds. As 𝑙^∗ increases, we can find that the range of nonperiodic motions becomes shorter and amplitude of the systems is also suppressed. Thus, we may conclude that couple-stress fluid can be used to improve dynamics of systems and help the systems escaping nonperiodic responses to be steady.

Nomenclature

𝑐:	Radial clearance, 𝑐=𝑅−𝑟
𝐶:	Damping coefficient of the gear mesh
𝐶1:	Damping coefficients of the supported structure for bearing 1
𝐶2:	Damping coefficients of the supported structure for bearing 2
𝐶1𝑝:	Dimensionless parameter, 𝐶1𝑝=𝑚1/𝑚𝑝
𝐶01:	Dimensionless parameter, 𝐶01=𝐾/𝑘1
𝐶2𝑝:	Dimensionless parameter, 𝐶2𝑝=𝑚2/𝑚𝑝
𝐶02:	Dimensionless parameter, 𝐶02=𝐾/𝑘2
𝑒:	Static transmission error and varies as a function of time
𝑒𝑖:	Offset of the journal center of the rotor relative to the 𝑋-coordinate direction
𝑓:	Dimensionless parameter, 𝑓=𝑚𝑝𝑔/𝑑𝐾
𝑓𝑔:	Dimensionless parameter, 𝑓𝑔=𝐾𝑔/𝑑𝑚𝑝
𝑓𝑒, 𝑓𝜑:	Components of the fluid film force in radial and tangential directions
𝑔:	Acceleration of gravity
𝐾:	Stiffness coefficient of the gear mesh
𝐾11, 𝐾12:	Stiffness coefficients of the springs supporting the two bearing housings for bearing 1
𝐾21, 𝐾22:	Stiffness coefficients of the springs supporting the two bearing housings for bearing 2
𝐾𝑝1, 𝐾𝑝2:	Stiffness coefficients of the shafts
𝐿:	Bearing length
𝑚1:	Mass of the bearing housing for bearing 1
𝑚2:	Mass of the bearing housing for bearing 2
𝑚𝑝:	Mass of the pinion
𝑚𝑔:	Mass of the gear
𝑂1:	Geometric centers of the bearing 1
𝑂2:	Geometric centers of the bearing 2
𝑂𝑗1:	Geometric centers of the journal 1
𝑂𝑗2:	Geometric centers of the journal 2
𝑂𝑔:	Center of gravity of the gear
𝑂𝑝:	Center of gravity of the pinion
𝑝:	Pressure distribution in the fluid film
𝑅:	Inner radius of the bearing housing
𝑟:	Radius of the journal
𝑠:	Rotational speed ratio, 𝑠=(𝜔2/𝜔𝑛2)1/2
𝑠1:	Dimensionless parameter, 𝑠12=𝐶𝑜1𝐶1𝑝𝑠2
𝑠2:	Dimensionless parameter, 𝑠22=𝐶𝑜2𝐶2𝑝𝑠2
𝑋, 𝑌, 𝑍:	Horizontal, vertical and axial coordinates
𝑥𝑗, 𝑦𝑗:	𝑋𝑗/𝑐, 𝑌𝑗/𝑐,  𝑗=1,2,𝑗1,𝑗2,𝑝,𝑔
𝛼1:	Dimensionless parameter, 𝛼1=𝐾12𝑑2𝐾/𝑚1𝑚𝑝
𝛼2:	Dimensionless parameter, 𝛼2=𝐾22𝑑2𝐾/𝑚2𝑚𝑝
𝜉1:	Dimensionless parameter, 𝜉1=𝐶1/2√𝐾1𝑚1
𝜉2:	Dimensionless parameter, 𝜉2=𝐶/2√𝐾𝑚𝑝
𝜉3:	Dimensionless parameter, 𝜉3=𝐶𝑚/2√𝐾𝑚𝑝
𝜉4:	Dimensionless parameter, 𝜉4∶𝐶/2√𝐾/𝑚𝑝𝑚𝑔
𝜉5:	Dimensionless parameter, 𝜉5=𝐶𝑚√𝑚𝑝/2𝑚𝑔√𝐾
𝜉6:	Dimensionless parameter, 𝜉6=𝐶2/2√𝐾2𝑚2
Λ:	Dimensionless parameter, Λ=𝐾𝑚/𝐾
Λ𝑔:	Dimensionless parameter, Λ𝑔=𝐾𝑚𝑚2𝑝/𝑚𝑔𝐾2
𝜌:	Mass eccentricity of the rotor
𝜙:	Rotational angle, 𝜙=𝜔𝑡
𝜔:	Rotational speed of the shaft
𝜃:	The angular position
𝜇:	Oil dynamic viscosity
𝜀:	Eccentricity ratio, 𝜀=𝑒/𝑐
𝜔𝑛:	Natural frequency, 𝜔𝑛=√𝐾/𝑚𝑝
𝜔𝑔:	Dimensionless parameter, 𝜔𝑔=𝜔𝑛/8
𝜔𝑝:	Dimensionless parameter, 𝜔𝑝=𝜔𝑛/4
𝜑𝑖:	Attitude angle of the rotor relative to the 𝑋-coordinate direction.