Influence of Asymmetric Mesh Stiffness on Dynamics of Spiral Bevel Gear Transmission System

Abstract

An 8-DOF (degrees-of-freedom) nonlinear dynamic model of a spiral bevel gear pair which involves time-varying mesh stiffness, transmission error, backlash, and asymmetric mesh stiffness is established. The effect of the asymmetric mesh stiffness on vibration of spiral bevel gear transmission system is studied deliberately with numerical method. The results show that the mesh stiffness of drive side has more effect on dynamic response than those of the coast side. Only double-sided impact region is affected considerably by mesh stiffness of coast side while single-sided impact and no-impact regions are unchanged. In addition, the increase in the mesh stiffness of drive side tends to worsen the dynamic response of the transmission system especially for light-load case.

1. Introduction

Spiral bevel gear has been widely used in many power transmission systems due to its considerable technical advantages. Owing to backlash, time-varying mesh stiffness, and many other nonlinear factors, gear transmission system produces complex dynamic behavior which has aroused wide attention from scholars around the world. Since the first gear dynamic model was proposed by Tuplin in 1950s, considerable progress has been made in parallel axis gear dynamics [1–4]. However, the dynamics of a spiral bevel-geared system is lack of investigations when compared with the parallel system. Liangyu et al. [5] derived a twelve-degrees-of-freedom vibration model of a pair of Spiral bevel gears. Meanwhile an 8-DOF and a 2-DOF simplified model and a modified model with mesh error are given, which provides a theoretical foundation for studying dynamic response of Spiral bevel gears. Gosselin et al. [6] proposed a general formula and applied it to analyze the load distribution and transmission error of spiral bevel gear pair and hypoid gear pair. Bibel et al. [7] studied the three-dimensional stress of a spiral bevel gear pair with finite element method. Fujii et al. [8] carried out analysis of dynamic behaviors of straight bevel-geared shaft supported on tapered roller and angular contact bearings, respectively, and then they measured the torsional and bending vibrations of the geared rotor and the vibrating displacement in the axial direction under a constant transmitted torque by using straight and skew bevel gears with a power circulating-type bevel-gear testing machine [9]. Xu et al. [10] analyzed the coupled lateral-torsional vibration behavior of the rotors with the engagement of spiral bevel gears by means of the transfer matrix method, which neglected the relationships of the generalized displacements between two bevel gears. Donley et al. [11] developed a dynamic model of a hypoid gear set for use infinite element analysis of gearing systems. In their gear mesh model, the mesh point and line-of-action are time invariant. Fang [12] developed a lumped parameter vibration model of a spiral bevel gear transmission to compute dynamic load and gear response. His mesh model is based on the classical gear mesh force equations that produce a simple unidirectional gear-mesh-coupling vector. Cheng and Lim [13] proposed a hypoid gear dynamic model based on exact gear geometry for analyzing gear mesh mechanism and applied the corresponding linear dynamic model to study the hypoid gear pair dynamics with transmission error excitation. In recent years, spiral bevel gear dynamics has gained extensive attention from many scholars and many relevant investigations have been published [14–17].

Unlike spur or helical gears, the mesh couplings in spiral bevel gears are not symmetric, due to the complex curvilinear features of the spiral bevel gear tooth geometry. That is, their mesh parameters for the drive and coast sides are very different. Most of the previous studies on the dynamics of spiral bevel gear transmissions focus on backlash and time-varying mesh stiffness, assuming symmetric mesh parameters for simplicity.

In this paper, we focus mainly on the dynamics of high-speed, precision spiral bevel gear pairs often used in automotive and aerospace power transmission systems. An 8-DOF nonlinear dynamic model of a spiral bevel gear pair which involves time-varying mesh stiffness, transmission error, backlash, and mesh stiffness asymmetry is proposed. Compared to mesh stiffness, the mesh damping with time-varying and asymmetric has less effect on dynamic response. Therefore, the study focuses on the effect of the asymmetric mesh stiffness on the vibration characteristics of spiral bevel gear transmission system, and the mesh damping parameter is assumed to be constant as in references [13, 14].

2. Dynamic Model

The proposed nonlinear dynamic model of a spiral bevel gear pair is shown in Figure 1. Three-dimensional Cartesian coordinate is set up, which uses the theoretical intersection of the two bevel gear axis for the origin. The two gear bodies are considered as rigid cone disks and bending rotation can be neglected for the bearing layout in Figure 1. Then the model includes transverse and torsion coordinates as shown in Figure 1.

Figure 1 

Dynamic model of a spiral bevel gear pair.

The coordinate vector of the system can be expressed with [𝑋₁, 𝑌₁, 𝑍₁, 𝜃_1𝑥, 𝑋₂, 𝑌₂, 𝑍₂,𝜃_2𝑦]^𝑇.

The normal dynamic load of pinion and its components along axis can be calculated as𝐹_𝑛=𝑘_ℎ(𝜆_𝑛)𝑓(𝜆_𝑛)+𝑐_ℎ̇‌𝜆_𝑛,𝐹_𝑥=𝐹_𝑛(sin𝛼_𝑛sin𝛿₁+cos𝛼_𝑛sin𝛽_𝑚cos𝛿₁𝐹),_𝑦=−𝐹_𝑛(sin𝛼_𝑛cos𝛿₁−cos𝛼_𝑛sin𝛽_𝑚sin𝛿₁𝐹),_𝑧=−𝐹_𝑛cos𝛼_𝑛cos𝛽_𝑚,𝑘_ℎ(𝜆_𝑛)=𝑘_ℎ1,𝜆_𝑛>𝑏,0,−𝑏≤𝜆_𝑛≤𝑏,𝑘_ℎ2,𝜆_𝑛<−𝑏,𝑘_ℎ1=𝑘_ℎ𝑚1+^∞_𝑟=1𝑘_ℎ𝑎𝑟1cos(𝑟𝜔_ℎ𝑡+𝜙_ℎ𝑟1),𝑘_ℎ2=𝑘_ℎ𝑚2+^∞_𝑟=1𝑘_ℎ𝑎𝑟2cos(𝑟𝜔_ℎ𝑡+𝜙_ℎ𝑟2),𝑓(𝜆_𝑛)=𝜆_𝑛−𝑏,𝜆_𝑛>𝑏,0,−𝑏≤𝜆_𝑛≤𝑏,𝜆_𝑛+𝑏,𝜆_𝑛<−𝑏, (2.1)(2.2)

where 𝑘_ℎ1, 𝑘_ℎ2—mesh stiffness of drive side and coast side, respectively; 𝜆_𝑛—Dynamic transmission error; 𝑘_ℎ(𝜆_𝑛), 𝑓(𝜆_𝑛)—Nonlinear displacement function; 𝑐_ℎ—Mesh damping coefficient; 𝛼_𝑛—Normal surface pressure angle; 𝛿₁—Pitch cone angle of the pinion; 𝛽_𝑚—Helix angle at meshing point; 𝑘_ℎ𝑚1, 𝑘_ℎ𝑚2—Mean mesh stiffness of drive side and coast side, respectively; 𝑘_ℎ𝑎𝑟1, 𝑘_ℎ𝑎𝑟2—Fourier series coefficients; 𝑏—Half of the gear backlash.

Dynamic transmission error can be defined with𝜆_𝑛=(−𝑋₁+𝑋₂)𝑎₁+(𝑌₁−𝑌₂)𝑎₂+(𝑍₁−𝑍₂+𝑟_𝑚1𝜃_1𝑥−𝑟_𝑚2𝜃_2𝑦)𝑎₃+𝑒(𝑡), (2.3) where 𝑎₁=sin𝛼_𝑛sin𝛿₁+cos𝛼_𝑛sin𝛽_𝑚cos𝛿₁,𝑎₂=sin𝛼_𝑛cos𝛿₁−cos𝛼_𝑛sin𝛽_𝑚sin𝛿₁,𝑎₃=cos𝛼_𝑛cos𝛽_𝑚, (2.4)𝑒(𝑡) is the static transmission error on normal direction of meshing surface. It can be expressed in a Fourier series form: 𝑒(𝑡)=∑^∞_𝑟=1𝑒_𝑟sin(𝑟𝜔_ℎ𝑡+𝜙_𝑒𝑟), where 𝑟_𝑚1, 𝑟_𝑚2—mean radius at meshing point; 𝑒_𝑟—amplitude of the 𝑟-order harmonic; 𝜙_𝑒𝑟—phase angle of the 𝑟-order harmonic.

The equations of the dynamic model can be described as𝑚₁𝑋̈₁+𝐶_1𝑥𝑋̇₁+𝐾_1𝑥𝑋₁=𝐹_𝑥,𝑚₁𝑌̈₁+𝐶_1𝑦𝑌̇₁+𝐾_1𝑦𝑌₁=𝐹_𝑦,𝑚₁𝑍̈₁+𝐶_1𝑧𝑍̇₁+𝐾_1𝑧𝑍₁=𝐹_𝑧,𝐼_1𝑥𝜃̈_1𝑥=𝑇₁+𝐹_𝑧𝑟_𝑚1,𝑚₂𝑋̈₂+𝐶_2𝑥𝑋̇₂+𝐾_2𝑥𝑋₂=−𝐹_𝑥,𝑚₂𝑌̈₂+𝐶_2𝑦𝑌̇₂+𝐾_2𝑥𝑌₂=−𝐹_𝑦,𝑚₂𝑍̈₂+𝐶_2𝑧𝑍̇₂+𝐾_2𝑧𝑍₂=−𝐹_𝑧,𝐼_2𝑦𝜃̈_2𝑦=−𝑇₂−𝐹_𝑧𝑟_𝑚2,𝑇₁=𝑇_1𝑚+𝑇_1𝑣,𝑇₂=𝑇_2𝑚, (2.5)(2.6) where 𝑚₁, 𝑚₂—Mass of pinion and gear, respectively; 𝐼_1𝑥, 𝐼_2𝑦—Mass moments of inertias of pinion and gear, respectively; 𝐾_𝑖𝑗, 𝐶_𝑖𝑗—Bearing stiffness and damping along the three coordinate axis, respectively; 𝑇₁, 𝑇₂—Load torques on pinion and gear, respectively; 𝑇_1𝑚, 𝑇_2𝑚—Mean load torques on pinion and gear, respectively; 𝑇_1𝑣—Ripple torque on pinion.

Using 𝜆_𝑛 as a new freedom degree, the two rigid-body rotation equations of the original model can be changed into a single one: 𝑚_𝑒𝑎₁𝑋̈₁−𝑚_𝑒𝑎₂𝑌̈₁−𝑚_𝑒𝑎₃𝑍̈₁−𝑚_𝑒𝑎₁𝑋̈₂+𝑚_𝑒𝑎₂𝑌̈₂+𝑚_𝑒𝑎₃𝑍̈₂+𝑚_𝑒̈‌𝜆_𝑛+𝑐_ℎ𝑎²₃̇‌𝜆_𝑛+𝑘_ℎ(𝜆_𝑛)𝑎²₃𝑓(𝜆_𝑛)=𝑎₃(𝐹_1𝑚+𝐹_1𝑣)+𝑚_𝑒𝑒_𝑛̈‌(𝑚𝑡),_𝑒=𝐼_1𝑥𝐼_2𝑦(𝐼_1𝑥𝑟²_𝑚2+𝐼_2𝑦𝑟²_𝑚1),𝐹_1𝑚=𝑇_1𝑚𝑟_𝑚1=𝑇_2𝑚𝑟_𝑚2,𝐹_1𝑣=𝑇_1𝑣𝑟_𝑚1𝑚_𝑒𝐼_1𝑥=^∞_𝑟=1𝐹_𝑟cos(𝑟𝜔_𝑇1𝑡+𝜙_𝑇1𝑟), (2.7) where 𝑚_𝑒—Equivalent mass of the gear pair; 𝐹_1𝑣, 𝐹_1𝑚—Ripple and mean force on pinion, respectively.

Next, introducing the parameters𝑥_𝑖=𝑋_𝑖𝑙,𝑦_𝑖=𝑌_𝑖𝑙,𝑧_𝑖=𝑍_𝑖𝑙,𝑏=𝑏𝑙,𝜆_𝑛=𝜆_𝑛𝑙,𝜔_𝑛=𝑘_ℎ𝑚𝑚_𝑒,𝜔_𝑖𝑗=𝑘_𝑖𝑗𝑚_𝑖,𝜍_𝑖𝑗=𝑐_𝑖𝑗(2𝑚_𝑖𝜔_𝑛),𝜍_𝑖ℎ=𝑐_ℎ(2𝑚_𝑖𝜔_𝑛),𝑘_𝑖𝑗=𝜔²_𝑖𝑗𝜔²_𝑛,𝑘_𝑖ℎ=𝑘_ℎ(𝜆_𝑛)(𝑚_𝑖𝜔²_𝑛),𝑡=𝜔_𝑛𝑡,𝜔_ℎ=𝜔_ℎ𝜔_𝑛,𝜔_𝑇=𝜔_𝑇𝜔_𝑛,𝑘_ℎ𝑚=(𝑘_ℎ𝑚1+𝑘_ℎ𝑚2)2,𝑔₁=𝑘_ℎ1𝑘_ℎ𝑚=𝑘_ℎ𝑚1+^∞_𝑟=1𝑘_ℎ𝑟1cos(𝑟𝜔_ℎ𝑡+𝜙_ℎ𝑟1𝑔),₂=𝑘_ℎ2𝑘_ℎ𝑚=𝑘_ℎ𝑚2+^∞_𝑟=1𝑘_ℎ𝑟2cos(𝑟𝜔_ℎ𝑡+𝜙_ℎ𝑟2),𝑔(𝜆_𝑛𝑔)=¹,𝜆_𝑛>𝑏,0,−𝑏≤𝜆_𝑛𝑔≤𝑏,₂,𝜆_𝑛<−𝑏,𝑓(𝜆_𝑛𝜆)=^𝑛−𝑏,𝜆_𝑛>𝑏,0,−𝑏≤𝜆_𝑛𝜆≤𝑏,_𝑛+𝑏,𝜆_𝑛𝑓<−𝑏,_1𝑚=𝐹_1𝑚𝑚_𝑒𝑙𝜔²_𝑛,𝑓_1𝑣=𝐹_1𝑣𝑚_𝑒𝑙𝜔²_𝑛,𝑓_𝑒=^∞_𝑟=1𝐹_ℎ𝑟(𝑟𝜔_ℎ)²cos(𝑟𝜔_ℎ𝑡+𝜙_𝑒𝑟), (2.8) where 𝑖=1,2; 𝑗=𝑥,𝑦,𝑧.

The dimensionless form of (2.5) can be obtained as𝑥̈₁+2𝜍_1𝑥𝑥̇₁+𝑘_1𝑥𝑥₁−2𝑎₁𝜍_1ℎ𝜆̇_𝑛−𝑎₁𝑘_1ℎ𝑓(𝜆_𝑛𝑦̈)=0,₁+2𝜍_1𝑦𝑦̇₁+𝑘_1𝑦𝑦₁+2𝑎₂𝜍_1ℎ𝜆̇_𝑛+𝑎₂𝑘_1ℎ𝑓(𝜆_𝑛𝑧̈)=0,₁+2𝜍_1𝑧𝑧̇₁+𝑘_1𝑧𝑧₁+2𝑎₃𝜍_1ℎ𝜆̇_𝑛+𝑎₃𝑘_1ℎ𝑓(𝜆_𝑛𝑥̈)=0,₂+2𝜍_2𝑥𝑥̇₂+𝑘_2𝑥𝑥₂+2𝑎₂𝜍_2ℎ𝜆̇_𝑛+𝑎₂𝑘_2ℎ𝑓(𝜆_𝑛𝑦̈)=0,₂+2𝜍_2𝑦𝑦̇₂+𝑘_2𝑦𝑦₂−2𝑎₂𝜍_2ℎ𝜆̇_𝑛−𝑎₂𝑘_2ℎ𝑓(𝜆_𝑛𝑧̈)=0,₂+2𝜍_2𝑧𝑧̇₂+𝑘_2𝑧𝑧₂−2𝑎₃𝜍_2ℎ𝜆̇_𝑛−𝑎₃𝑘_2ℎ𝑓(𝜆_𝑛𝑎)=0,₁𝑥̈₁−𝑎₂𝑦̈₁−𝑎₃𝑧̈₁−𝑎₁𝑥̈₂+𝑎₂𝑦̈₂+𝑎₃𝑧̈₂+𝜆̈_𝑛+2𝑎²₃𝜍_ℎ𝜆̇_𝑛+𝑎²₃𝑔(𝜆_𝑛)𝑓(𝜆_𝑛)=𝑎₃𝑓_1𝑚+𝑎₃𝑓_1𝑣+𝑓_𝑒. (2.9)

3. Numerical Results

As there is no analytical method existing for (2.9), the equation is solved by applying the explicit Runge-Kutta integration routine with variable step that is generally applicable to strong nonlinear equation. The effect of tooth mesh stiffness asymmetry on dynamic response of spiral bevel gear system for both light and heavy loads is studied here. For subsequent numerical study, the baseline data used are𝑍₁=36,𝑍₂=40,𝛼_𝑛=20^∘,𝛽_𝑚=35^∘,𝑏=35𝜇m. (3.1)

In this part, it is focused on effect of mean mesh stiffness 𝑘_ℎ𝑚1, 𝑘_ℎ𝑚2, assuming that 𝑘_ℎ𝑟1=𝑘_ℎ𝑟1, 𝜙_ℎ𝑟1=𝜙_ℎ𝑟2 for simplicity.

3.1. Effect of Mesh Stiffness Asymmetry for Lightly Loaded Case

Dimensionless dynamic parameters set in the system are as follows:𝜍_𝑖𝑗=0.01,𝜍_𝑖ℎ=0.0125,𝜍_𝑖ℎ𝑘=0.0125,_𝑖ℎ=0.25,𝑘_ℎ𝑟1=𝑘_ℎ𝑟2𝜙=0.2,_ℎ𝑟1=𝜙_ℎ𝑟2𝑓=0,_1𝑚=0.25,𝑓_1𝑣=0𝜔_ℎ𝑓=0.7,_𝑒=0.5𝜔²_ℎcos𝜔_ℎ𝑡,𝑖=1,2,𝑗=𝑥,𝑦,𝑧. (3.2)

The effect of the mean mesh stiffness of coast side on dynamic response for lightly loaded case is shown in Figure 2. The horizontal axis represents excitation frequency 𝜔_ℎ and the vertical axis represents peak-peak value of the dynamic transmission error 𝐴. As shown in Figure 2(a), when 𝑘_ℎ𝑚2=0.5, single-sided impact response and double-sided impact response alternate in the low-frequency regions. And the max of 𝐴 occurs at 𝜔_ℎ=0.9. In the region 𝜔_ℎ∈[0.96,1.06], there are no impact responses, and the gear pair running smoothly as 𝑘_ℎ𝑚2 is increased to 1 in Figure 2(b); the max of 𝐴 8.5 moves to the frequency 𝜔_ℎ=0.52. The dynamic response in no-impact and single-sided impact regions 𝜔_ℎ∈[0.96,1.5] keeps unchanged. When 𝑘_ℎ𝑚2 is increased further to 1.5 and 2, the peak-peak value 𝐴 in double-sided impact region changes obviously while the location and vibration response of single-sided impact and no-impact regions are nearly unchanged. The above results show that the mesh stiffness of coast side only affects double-sided tooth impact region.

(a) khm2=0.5

(b) khm2=1

(c) khm2=1.5

(d) khm2=2

Figure 2 

Effect of mean mesh stiffness of coast side for lightly loaded case. $*$—single-sided impact °—no impact $+$—double-sided impact.

Figure 3 shows the effect of mean mesh stiffness of drive side for lightly loaded case. It can be seen from Figure 3(a) that double-sided impacts dominate the range 𝜔_ℎ∈[0.67,0.79]. And four response jump discontinuities can be seen at frequency 𝜔_ℎ= 0.67, 0.79, 0.97, 1.25. It can be observed that response jump discontinuities appear at these frequencies. As 𝑘_ℎ𝑚1 is increased to 1, it can be observed that more double-sided impacts appear and the location of each impact region shifts obviously. As 𝑘_ℎ𝑚1 is further increased to 1.5 and 2, both response jumps and single-sided impacts increased considerably, and response in the whole excitation frequency areas changes significantly. As can be seen in Figure 3(c), except the two narrow bands 𝜔_ℎ∈[1.03,1.17] and 𝜔_ℎ∈[1.36,1.5], the rest excitation frequencies are all double-sided impact regions. When 𝑘_ℎ𝑚1=2, width of double-sided tooth impact region is further increased.

(a) khm1=0.5

(b) khm1=1

(c) khm1=1.5

(d) khm1=2

Figure 3 

Effect of mean mesh stiffness of drive side for lightly loaded case. $*$—single-sided impact °—no impact $+$—double-sided impact.

To further explain the effect of the mean mesh stiffness of drive side on dynamic response, bifurcation diagram is introduced in Figure 4. For the current light-load case, it can be observed that, as 𝑘_ℎ𝑚1 is increased from 1 to 2, chaotic motion regions increase considerably. From above observations, it can be concluded that the mesh stiffness of drive side affects dynamic response greatly in the whole excitation frequency region. Furthermore, the increase of drive side mesh stiffness tends to worsen dynamic response.

(a) khm1=1

(b) khm1=2

Figure 4 

Bifurcation diagram for lightly loaded case.

3.2. Effect of Mesh Stiffness Asymmetry for Heavily Loaded Case

Dimensionless dynamic parameters set in this part are as follows:𝜍_𝑖𝑗=0.01,𝜍_𝑖ℎ=0.0125,𝜍_𝑖ℎ=0.0125,𝑘_𝑖ℎ=0.25,𝑘_ℎ𝑟1=𝑘_ℎ𝑟2𝜙=0.2,_ℎ𝑟1=𝜙_ℎ𝑟2=0,𝑓_1𝑚=1,𝑓_1𝑣=0,𝑓_𝑒=0.5𝜔²_ℎcos𝜔_ℎ𝑡,𝑖=1,2,𝑗=𝑥,𝑦,𝑧. (3.3)

When 𝑘_ℎ𝑚1=1, the effect of 𝑘_ℎ𝑚2 for heavy-load case is shown in Figure 5. No-impact region 𝜔_ℎ∈[0.74,1.18] is far wider than double-sided impact region 𝜔_ℎ∈[0.5,0.61] in Figure 5(a). And the max of 𝐴=15.9 which comes at 𝜔_ℎ=0.52 is much larger than that of 6 for light-load case. There are three single-sided impact regions 𝜔_ℎ∈[0.62,0.73], 𝜔_ℎ∈[1.19,1.27], and 𝜔_ℎ∈[1.32,1.5], and little response jump can be seen. As 𝑘_ℎ𝑚2 is increased to 1, dynamic response in double-sided impact region changes greatly while responses in single-sided impact and no-impact regions remain the same. And the max of 𝐴 is increased to 24. As 𝑘_ℎ𝑚2 is increased further to 1.5 and 2, only double-sided impact region is affected. The results show that double-sided impact region becomes obviously narrower and response jumps diminish considerably for heavily loaded case. Although the peak-peak value of dynamic transmission error in double-sided impact region is larger than that of light-load case, response jump discontinuities and double-sided tooth impacts for heavily loaded case are far fewer. As light-load condition, only double-sided impact region is affected considerably by mesh stiffness of coast side. This is because 𝑘_ℎ𝑚2 only takes effect for double-sided impact condition.

(a) khm2=0.5

(b) khm2=1

(c) khm2=1.5

(d) khm2=2

Figure 5 

Effect of mean mesh stiffness of coast side for heavily loaded case. $*$—single-sided impact °—no impact $+$—double-sided impact.

Figure 6 shows dynamic response for different mean mesh stiffness of drive side while leaving mean mesh stiffness of coast side unchanged. In Figure 6(a), there are three no-impact regions 𝜔_ℎ∈[0.98,1.33], 𝜔_ℎ∈[0.57,0.79], and 𝜔_ℎ∈[1.44,1.5], one double-sided impact region 𝜔_ℎ∈[0.8,0.91], and three response jump frequencies 𝜔_ℎ=0.8, 𝜔_ℎ=0.91, and 𝜔_ℎ=0.97. The max of 𝐴=32 occurs at 𝜔_ℎ=0.92. As 𝑘_ℎ𝑚1 is increased to 1, no-impact regions are reduced to two 𝜔_ℎ∈[0.74,1.18] and 𝜔_ℎ∈[1.28,1.31], and the location of each impact region changes significantly. For 𝑘_ℎ𝑚1=1.5 in Figure 6(c), the number of double-sided impact region is increased to two 𝜔_ℎ∈[0.57,0.76] and 𝜔_ℎ∈[1.31,1.41], and more response jumps appear. When 𝑘_ℎ𝑚1 is increased to 2, double-sided impact region becomes wider and response in double-sided impact region takes significant changes from Figure 6(c).

(a) khm1=0.5

(b) khm1=1

(c) khm1=1.5

(d) khm1=2

Figure 6 

Effect of mean mesh stiffness of drive side for heavily loaded case. $*$—single-sided impact °—no impact $+$—double-sided impact.

Figure 7 shows response bifurcation for different 𝑘_ℎ𝑚1, while 𝑘_ℎ𝑚2 is fixed to 1. For 𝑘_ℎ𝑚1=1 in Figure 7(a), no chaotic motion can be seen. However, as 𝑘_ℎ𝑚1 is increased to 2, there are two obvious chaotic motion areas. Comparing with Figure 4, it can be seen that chaotic motion areas diminish considerably in Figure 7. From these results we can conclude that mesh stiffness of drive side also affects dynamic response greatly for heavy-load case. In addition, the increase of 𝑘_ℎ𝑚1 will worsen the dynamic response of the system especially for light-load case. Compared with light-load condition, the degree of gear backlash nonlinearity is lower for heavy-load case. This is because mesh force under heavy-load condition is large and hence mesh teeth are difficult to separate.

(a) khm1=1

(b) khm1=2

Figure 7 

Bifurcation diagram for heavily loaded case.

4. Conclusions

A nonlinear dynamic model of a spiral bevel gear pair which involves time-varying mesh stiffness, transmission error, backlash, and mesh stiffness asymmetry is proposed. The effect of tooth mesh stiffness asymmetry on vibration of spiral bevel gear transmission system is studied deliberately. Some important conclusions are obtained.

Firstly, the mesh stiffness for drive side has more effect on dynamic response than those of the coast side. Only double-sided impact region is affected considerably by mesh stiffness of coast side while single-sided impact and no-impact regions are unchanged.

Secondly, the dynamic response of the system is very sensitive to mesh stiffness of drive side. The change of mesh stiffness of drive side will affect response in the whole excitation frequency areas. Furthermore, the increase of drive side mesh stiffness tends to worsen dynamic response especially for lightly loaded case.

Thirdly, the vibration characteristic for heavily loaded case is far better than that for lightly loaded case. And the mesh stiffness asymmetry affects the dynamic response more compared with heavy-load case.

Acknowledgments

This paper is supported by National Natural Science Foundation of China (Grant no. 50875270), Key Scientific Projects of Ministry of Education of China (Grant no. 108108) and Natural Science Foundation of Chongqing (CSTC, 2008BA6025).