# Chaos and Stability of Spur Gear Transmission System for Locomotive Based on Energy Method and Floquet Theory

Authors: Junguo Wang; Bo Lv; Yongxiang Zhao
Journal: Shock and Vibration
 (2018)

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## Abstract

Considering the internal and external excitations such as time-varying mesh stiffness (TVMS), backlash, transmission error, torque of the traction motor, and load torque of the wheel/rail, a lumped mass model of the spur gear drive system for a railway locomotive is established. Based on Ma models in the relevant literatures, TVMS is calculated by simplifying a gear tooth as a cantilever beam on the root circle, taking into account the effects of extended tooth contact as well as revised foundation stiffness. The bifurcation diagrams and Lyapunov exponent curves of the model parameters are drawn by the numerical method, and the mechanism of chaos evolution of the gear transmission system is analyzed. According to the Floquet theory, variation curves of the maximum Floquet multiplier with pinion speed and support stiffness ratio are drawn by numerical methods. Combined with the bifurcation diagram of the system, the influences of model parameter on the stability of the system are analyzed, and the evolution laws of periodic motion and bifurcation phenomenon are gained. These research results provide the theoretical evidence of model parameter design of the locomotive transmission system.


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## Body

## 1. Introduction

As an important device of rotating machinery, the gear transmission system has found extensive applications in many fields such as high-speed locomotive, petrochemical, and power generation industries. Due to the complicated structure, harsh operation environment, and flexible excitations, the geared system is prone to damage to some degree and recognized as the vulnerable part in most cases. At present, there is a growing demand for development of reliable transmission systems, with higher requirements of torque, speed, and compactness. To satisfy these demands, engineers and researchers need to further understand the function of nonlinear phenomena involved in gear transmission. As a consequence, it is imperative to investigate dynamic behavior and evolution mechanism by developing a reliable dynamic model of the gear system [1] and ensure the safe and stable operation for the rotating equipment.With the development of nonlinear dynamics theory, the nonlinear characteristics of the geared system including stability, bifurcations, and chaos have become the most interesting research areas [2, 3]. Over the past decades, the characteristic analysis above is practically based on the modelling, and thus, modelling of the gear system is considered as a fundamental problem which is still the research hotspot [4, 5]. The factors such as backlash [6, 7], time-varying mesh stiffness (TVMS) [8, 9], and transmission error and lateral-torsional coupled vibration [10, 11] will directly affect the dynamics including the stability and reliability of the gear system. Various approaches were successively adopted to illustrate the nonlinear characteristics including stability of the periodic solution, bifurcation, and chaos. In these gear models, TVMS is usually considered a simple sinusoidal function, a Fourier series, or a rectangular waveform function expression. An effective way to evaluate the TVMS is essential to understand the dynamic properties. In general, the finite-element methods (FEMs) and analytical methods are two main ways to calculate TVMS. FEM is time-consuming because every gear pair requires modelling, while analytical method can offer a general approach to evaluate TVMS. The relative contribution of individual components, such as bending, shear, and Hertzian contact stiffness, can be analyzed separately. Considering various impact factors including extended tooth contact, revised fillet-foundation stiffness, and nonlinear contact stiffness, Ma et al. [12] developed an analytical model for TVMS calculation. Chen and Shao [13] proposed a TVMS model with tooth root crack propagating along both tooth width and crack depth. In [14], the potential energy method was applied to evaluate TVMS as a planetary gear set. Based on single-point estimation of TVMS, Wan et al. [15] obtained the TVMS of a helical gear pair and verified the simulated results by the FEM and ISO standard method.In [16], Wei et al. developed a six-degree-of-freedom dynamic model coupled flexional, torsional, and axial motion of the helical gear transmission system, which includes TVMS, bearing support stiffness, mesh damping, and backlash. The effects on dynamic transmission errors and stabilities by contact ratio, support stiffness, and mesh damping as well as backlash were analyzed. Farshidianfar and Saghafi [17] studied the global homoclinic bifurcation and transition to chaotic behavior of a nonlinear gear system by means of the Melnikov method. The research results of Seyranian et al. [18] show that Floquet theory is more suitable for analyzing the stability of the multiparameter periodic system and multi-degree-of-freedom gear system. Based on the nonlinear dynamic model of the three-degree-of-freedom gear system, Liu et al. [19] studied the system stability and the type of bifurcation by Floquet theory and obtained the stable and unstable periodic orbits. Considering TVMS, variable damping, and sliding friction, Vaishya and Singh [20] built a spur gear system model and discussed the stability of the gear transmission system on the basis of Floquet theory.The gear transmission system of railway locomotive is a kind of typical nonlinear system with a complex structure and service environment. Due to the complex meshing relationship and wheel/rail excitation, the key components of the transmission system are prone to complex dynamic behaviors. At present, the research of chaos and stability is mostly limited to the general mechanical transmission system, and very limited works have addressed the effect on the gear dynamic response for railway locomotive. On account of the harsh operation environment including high speed and heavy load, the driving system of locomotive often bears a high-frequency alternating load generated by single- and double-gear meshing in the process of operation. For this reason, a three-degree-of-freedom spur gear drive system model is derived for a typical locomotive, in which the parameters including TVMS, transmission error, mesh damping, and wheel/rail adhesion torque are considered as uncertain but bounded parameters. The effects of those uncertain parameters on the chaos are discussed. And the influences of model parameter changes on the stability of the system are also analyzed on the basis of Floquet theory. Some results are considered as references for the parameters design or vibration control of the gear transmission system for railway locomotive.


## 2. Dynamic Model of the Geared System for Locomotive

### 2.1. Gear Transmission System Model

As illustrated in Figure1, the spur gear transmission system is mainly composed of a motor, spur gear pair, and wheel set. The pinion is directly installed on the armature axis of the motor, which provides driving torque of the transmission system, and a gear is installed on the wheel set, which is considered to be the load applied on the geared system. The gear transmission system can be simplified as seen in Figure 2, in which gear meshing is described by backlash, stiffness, and damping elements along the line-of-action (LOA) direction.Figure 1
Schematic of the spur gear transmission system for railway locomotive.Figure 2
Simplified model of the spur gear transmission system for railway locomotive.Letm1 and m2 be the mass of the pinion and gear, I1 and I2 the mass moment of inertia of the pinion and gear, R1 and R2 the base circle radius of the pinion and gear, k1 and k2 the bearing support stiffness of the pinion and gear, c1 and c2 the bearing damping of the pinion and gear, k (t) the time-varying mesh stiffness, cm the damping coefficient of the gear mesh, y1 and y2 the translational displacement of the pinion and gear, θ1 and θ2 the rotational displacement of the pinion and gear, T1 the drive torque provided by the motor under the rated power, and T2 the load torque caused by the wheel/rail. Hence, the model of the geared system can be derived as follows:(1)m1y1¨+c1y1.+k1y1=F0t,I1θ1¨=T1−R1F0t,m2y2¨+c2y2.+k2y2=−F0t,I2θ2¨=−T2+R2F0t,where meshing force F0t=ktfR1θ1−R2θ2+y1−y2−et+cmR1θ1˙−R2θ2˙+y˙1−y˙2−e˙t. Let y=R1θ1−R2θ2+y1−y2−et and et be the static transmission error, then the model above can be rewritten as follows:(2)m1y¨1+c1y.1+k1y1=cmy.+ktfy,m2y¨2+c2y.2+k2y2=−cmy.−ktfy,mey¨−mey¨1+mey¨2+cmy.+ktfy=Fet+Fht,where ω is the meshing frequency and me is the equivalent mass of the gear pair. If the total backlash is 2b, introducing natural frequency ωn=kt/me, nondimensional time τ=ωnt, w1=y1/b, w2=y2/b, and w3=y/b; hence, the model (2) can be simplified as follows:(3)w¨1+c11w˙1+k11w1=c13w˙+k13fw3,w¨2+c22w˙2+k22w2=−c23w˙3−k23fw3,w¨3−w¨1+w¨2+c33w˙3+k33fw3=Pm+PacosΩτ.The relevant variables are defined as follows:(4)c11=c1m1ωn,c22=c2m2ωn,c33=cmmeωn,c13=cmm1ωn,c23=cmm2ωn,k11=k1m1ωn2,k22=k2m2ωn2,k33=ktmeωn2,k23=kmmeωn2,Fm=cmR1θ1.−R2θ2.+y1.−y2.−e.,Pm=Fmbkm,Pa=eaΩ2b,fw3=w3−1,w3>1,0,−1<w3<1,w3+1,w3<−1.


### 2.2. Wheel/Rail Adhesion Torque

To express the slippage in a more physical sense, the slip rate of locomotive transmission dynamics can be defined as follows [21]:(5)s=v−ωwRv,where v is the instantaneous speed of the wheel set, ωw the instantaneous angular velocity, and R the radius of the wheel. If the number of gear teeth is Z2, ωw=2πω/Z2. The wheel/rail tangential force coefficient is inversely related to the slip after the peak of adhesion, and the simplified wheel/rail adhesion curve can be indicated by a sectional linear function [21], as shown in Figure 3:(6)F=fmssc,0≤s≤sc,fm+kfs−sc,sc<s,where fm is the maximum tangential force coefficient, sc the critical slip rate, and kf the negative slope on sliding. For the simplified model, the adhesion state under positive slope is considered to stick, the adhesion state under negative slope is considered to slip, but the dispersion of adhesion is not considered. When the motor runs at rated power, sc is 0.01, and s is changed from 0 to 0.03.Figure 3
(a) Simplified curve of the tangential force coefficient of locomotive. (b) Mechanism curve of slippery vibration.
(a)
(b)Let the axle load of locomotive bemg, the wheel/rail adhesion torque can be obtained as follows:(7)T2=FrR=mgRfmssc,0≤s≤sc,mgRfm+mgRkfs−sc,sc<s.


## 2.1. Gear Transmission System Model

As illustrated in Figure1, the spur gear transmission system is mainly composed of a motor, spur gear pair, and wheel set. The pinion is directly installed on the armature axis of the motor, which provides driving torque of the transmission system, and a gear is installed on the wheel set, which is considered to be the load applied on the geared system. The gear transmission system can be simplified as seen in Figure 2, in which gear meshing is described by backlash, stiffness, and damping elements along the line-of-action (LOA) direction.Figure 1
Schematic of the spur gear transmission system for railway locomotive.Figure 2
Simplified model of the spur gear transmission system for railway locomotive.Letm1 and m2 be the mass of the pinion and gear, I1 and I2 the mass moment of inertia of the pinion and gear, R1 and R2 the base circle radius of the pinion and gear, k1 and k2 the bearing support stiffness of the pinion and gear, c1 and c2 the bearing damping of the pinion and gear, k (t) the time-varying mesh stiffness, cm the damping coefficient of the gear mesh, y1 and y2 the translational displacement of the pinion and gear, θ1 and θ2 the rotational displacement of the pinion and gear, T1 the drive torque provided by the motor under the rated power, and T2 the load torque caused by the wheel/rail. Hence, the model of the geared system can be derived as follows:(1)m1y1¨+c1y1.+k1y1=F0t,I1θ1¨=T1−R1F0t,m2y2¨+c2y2.+k2y2=−F0t,I2θ2¨=−T2+R2F0t,where meshing force F0t=ktfR1θ1−R2θ2+y1−y2−et+cmR1θ1˙−R2θ2˙+y˙1−y˙2−e˙t. Let y=R1θ1−R2θ2+y1−y2−et and et be the static transmission error, then the model above can be rewritten as follows:(2)m1y¨1+c1y.1+k1y1=cmy.+ktfy,m2y¨2+c2y.2+k2y2=−cmy.−ktfy,mey¨−mey¨1+mey¨2+cmy.+ktfy=Fet+Fht,where ω is the meshing frequency and me is the equivalent mass of the gear pair. If the total backlash is 2b, introducing natural frequency ωn=kt/me, nondimensional time τ=ωnt, w1=y1/b, w2=y2/b, and w3=y/b; hence, the model (2) can be simplified as follows:(3)w¨1+c11w˙1+k11w1=c13w˙+k13fw3,w¨2+c22w˙2+k22w2=−c23w˙3−k23fw3,w¨3−w¨1+w¨2+c33w˙3+k33fw3=Pm+PacosΩτ.The relevant variables are defined as follows:(4)c11=c1m1ωn,c22=c2m2ωn,c33=cmmeωn,c13=cmm1ωn,c23=cmm2ωn,k11=k1m1ωn2,k22=k2m2ωn2,k33=ktmeωn2,k23=kmmeωn2,Fm=cmR1θ1.−R2θ2.+y1.−y2.−e.,Pm=Fmbkm,Pa=eaΩ2b,fw3=w3−1,w3>1,0,−1<w3<1,w3+1,w3<−1.


## 2.2. Wheel/Rail Adhesion Torque

To express the slippage in a more physical sense, the slip rate of locomotive transmission dynamics can be defined as follows [21]:(5)s=v−ωwRv,where v is the instantaneous speed of the wheel set, ωw the instantaneous angular velocity, and R the radius of the wheel. If the number of gear teeth is Z2, ωw=2πω/Z2. The wheel/rail tangential force coefficient is inversely related to the slip after the peak of adhesion, and the simplified wheel/rail adhesion curve can be indicated by a sectional linear function [21], as shown in Figure 3:(6)F=fmssc,0≤s≤sc,fm+kfs−sc,sc<s,where fm is the maximum tangential force coefficient, sc the critical slip rate, and kf the negative slope on sliding. For the simplified model, the adhesion state under positive slope is considered to stick, the adhesion state under negative slope is considered to slip, but the dispersion of adhesion is not considered. When the motor runs at rated power, sc is 0.01, and s is changed from 0 to 0.03.Figure 3
(a) Simplified curve of the tangential force coefficient of locomotive. (b) Mechanism curve of slippery vibration.
(a)
(b)Let the axle load of locomotive bemg, the wheel/rail adhesion torque can be obtained as follows:(7)T2=FrR=mgRfmssc,0≤s≤sc,mgRfm+mgRkfs−sc,sc<s.


## 3. Calculation of Time-Varying Mesh Stiffness

The mesh stiffness is a time-varying parameter, as the number of engaged teeth varies according to the contact ratio. Considering the misalignment of the gear root circle and base circle, Ma et al. [22] proposed a TVMS model for a healthy gear pair; thereafter, they presented an improved method for mesh stiffness of spur gears with tip relief [23, 24]. Based on ANSYS software, Wang [25] presented an approach to predict TVMS by establishing contact elements. Saxena et al. [26] developed a method to calculate TVMS for different spall shapes, sizes, and locations. In the traditional method, the gear tooth is mostly modelled as a nonuniform cantilever beam on the base circle. In this study, in order to improve the analytical method, considering the effects of extended tooth contact and revised foundation stiffness, an improved analytical TVMS calculation method is given according to Ma models and methods above.


### 3.1. Determination of the Tooth Profile Model

Deflections of a spur gear tooth can be determined by considering it as a nonuniform cantilever beam with an effective lengthd displayed in Figure 4. Here, rint is the radius of the gear inside holes, rc the radius of the tip fillet, β the operating pressure angle, θb the half tooth angle on the base circle of the gear, θc the angle corresponding to the point c, xβ the distance between contact point and central line of the tooth, and yβ the distance between contact point and original point:Figure 4
Modelling of the spur gear tooth as a nonuniform cantilever beam.(8)
r
b
=
m
Z
cos
α
2
,
r
f
=
m
Z
2
−
h
a
∗
+
c
∗
m
.
.For an involute curve, there exists the following geometric relationship:(9)x=risinφ,y=ricosφ,φ=π2Z−invαi−invα.The transition curve is the tooth profile between involute starting point and root circle, and it is cut out by the cutter tip. Hence, the transition curve equations can be written as follows:(10)x=rsinψ−a1sinγ+rρcosγ−ψ,y=rcosψ−a1sinγ+rρsinγ−ψ,ψ=a1/tanγ+b1r,and(11)a1=ha∗+c∗m−rρ,rρ=c∗×m1−sinα,α≤γ≤π/2,b1=πm4+ha∗mtanα+rρcosα.


### 3.2. An Energy Method for TVMS Calculation

Based on elastic mechanics, Hertzian contact stiffness can be written as follows [23–27]:(12)kh=πEL41−v2,kf=ELcos2βL∗uf/Sf2+M∗uf/Sf+P1+Q∗tan2β,where the parameters uf, Sf, L∗, M∗, P∗, and Q∗ can be found in [27]. The Hertzian energy, bending, shear, fillet foundation, and axial compressive energies can be written as follows [22]:(13)Uh=F22kh,Ub=F22kb,Us=F22ks,Uf=F22kf,Ua=F22ka.Based on the beam theory, the potential energy stored in a meshing gear tooth is obtained by the following equation [22]:(14)Ub=∫yDyCM122EIy1dy1+∫yCyβM222EIy2dy2,Us=∫yDyC1.2Fb22GAy1dy1+∫yCyβ1.2Fb22GAy2dy2,Ua=∫yDyCFa22EAy1dy1+∫yCyβFa22EAy2dy2,and(15)M1=Fbyβ−y1−Faxβ,M2=Fbyβ−y2−Faxβ,G=E21+v,xβ=rbβ+θbcosβ−sinβ,yβ=rbβ+θbsinβ+cosβ,Iy1=2x13L12,Iy2=2x23L12,Ay1=2x1L,Ay2=2x2L,x1=rsinψ−a1sinγ+rρcosγ−ψ,x2=rbτ+θbcosτ−sinτ,y1=rcosψ−a1sinγ+rρsinγ−ψ,y2=rbτ+θbsinτ+cosτ,where M1 and M2 are the bending moments of Fb and Fa on the part of transition and involute curves, y1 and y2 the horizontal coordinates, and G the shear modulus. Hence, the bending, shear, and axial compressive stiffness can be obtained as follows:(16)1kb=∫π/2αcosβyβ−y1−xβsinβ2EIy1dy1dγdγ+∫τcβcosβyβ−y2−xβsinβ2EIy2dy2dτdτ,1ks=∫π/2α1.2cos2βGAy1dy1dγdγ+∫τcβ1.2cos2βGAy2dy2dτdτ,1ka=∫π/2αsin2βEAy1dy1dγdγ+∫τcβsin2βEAy2dy2dτdτ,with(17)dy1dγ=a1sinψ1+tan2γtan2γ+a1cosγsinγ−ψsin2γ−a1sinγ+rρcosγ−ψ1+a11+tan2γrtan2γ,dy2dτ=rbτ+θbcosτ.Let subscripts 1 and 2 be pinion and gear, the single-tooth-pair mesh stiffness can be defined as follows [23, 24]:(18)ki=11/kh+1/kt1+1/kf1+1/kt2+1/kf2,and(19)kt=11/kb+1/ks+1/ka.IfN is the number of meshing tooth pairs, the TVMS can be calculated as follows:(20)k=∑i=1Nki.However, in fact, two meshing tooth pairs share the one gear body, which leads to a mesh stiffness higher than the actual value in double-tooth engagement. Thus, an improved method is adopted where the total mesh stiffness is expressed as follows [23, 24]:(21)k=11/λ1Nkf1+1/ktooth+1/λ2Nkf2λ11=λ21=1,whenN=1,ktooth=∑i=1Nktoothi,ktoothi=11/khi+1/kt1+1/kt2,where λ represents the correction coefficient of the fillet-foundation stiffness, ktooth the total mesh stiffness for N meshing tooth pairs, and ktoothi the mesh stiffness of the ith tooth pair.The correction coefficients of the fillet-foundation stiffness need to be determined using the FE method, and the detailed calculation is given in [23, 24]. Hence, the stiffness caused by tooth deformation during double-tooth engagement can be calculated by the following equation:(22)ktooth=FEr2−EPiwithEpi=minEp1,Ep2.The total mesh stiffness can be obtained from Equation (20). In addition, mesh stiffness of a single-tooth pair is a function of tooth contact deformation Ed and meshing position j, which can be expressed as follows [23, 24]:(23)k=ki,Edi>0,0,Edi≤0,where the calculation of correction coefficients is shown in the relevant literatures.The parameters of the gear pair are chosen as follows: the number of teeth on the pinion and gear is 23 and 120, radius of the pinion and gear is 86 and 450 mm, module is 8 mm, teeth width is 140 mm, coincidence degree is 1.7315, elastic modulus is 209 GPa, and Poisson’s ratio is 0.3. The TVMS curve which changes with dimensionless time is displayed in Figure5. With the help of this figure, the curve-fitted stiffness equation is derived as follows:Figure 5
The curve of mesh stiffness in a mesh period.(24)
k
τ
=
1
.
55
×
10
9
+
5.31
×
10
8
τ
−
1.06
×
10
9
τ
2
,
0
<
τ
≤
1.463
π
,
5.72
×
10
8
+
9.93
×
10
8
τ
−
8.25
×
10
8
τ
2
,
1.463
×
π
<
τ
≤
2
π
.


## 3.1. Determination of the Tooth Profile Model

Deflections of a spur gear tooth can be determined by considering it as a nonuniform cantilever beam with an effective lengthd displayed in Figure 4. Here, rint is the radius of the gear inside holes, rc the radius of the tip fillet, β the operating pressure angle, θb the half tooth angle on the base circle of the gear, θc the angle corresponding to the point c, xβ the distance between contact point and central line of the tooth, and yβ the distance between contact point and original point:Figure 4
Modelling of the spur gear tooth as a nonuniform cantilever beam.(8)
r
b
=
m
Z
cos
α
2
,
r
f
=
m
Z
2
−
h
a
∗
+
c
∗
m
.
.For an involute curve, there exists the following geometric relationship:(9)x=risinφ,y=ricosφ,φ=π2Z−invαi−invα.The transition curve is the tooth profile between involute starting point and root circle, and it is cut out by the cutter tip. Hence, the transition curve equations can be written as follows:(10)x=rsinψ−a1sinγ+rρcosγ−ψ,y=rcosψ−a1sinγ+rρsinγ−ψ,ψ=a1/tanγ+b1r,and(11)a1=ha∗+c∗m−rρ,rρ=c∗×m1−sinα,α≤γ≤π/2,b1=πm4+ha∗mtanα+rρcosα.


## 3.2. An Energy Method for TVMS Calculation

Based on elastic mechanics, Hertzian contact stiffness can be written as follows [23–27]:(12)kh=πEL41−v2,kf=ELcos2βL∗uf/Sf2+M∗uf/Sf+P1+Q∗tan2β,where the parameters uf, Sf, L∗, M∗, P∗, and Q∗ can be found in [27]. The Hertzian energy, bending, shear, fillet foundation, and axial compressive energies can be written as follows [22]:(13)Uh=F22kh,Ub=F22kb,Us=F22ks,Uf=F22kf,Ua=F22ka.Based on the beam theory, the potential energy stored in a meshing gear tooth is obtained by the following equation [22]:(14)Ub=∫yDyCM122EIy1dy1+∫yCyβM222EIy2dy2,Us=∫yDyC1.2Fb22GAy1dy1+∫yCyβ1.2Fb22GAy2dy2,Ua=∫yDyCFa22EAy1dy1+∫yCyβFa22EAy2dy2,and(15)M1=Fbyβ−y1−Faxβ,M2=Fbyβ−y2−Faxβ,G=E21+v,xβ=rbβ+θbcosβ−sinβ,yβ=rbβ+θbsinβ+cosβ,Iy1=2x13L12,Iy2=2x23L12,Ay1=2x1L,Ay2=2x2L,x1=rsinψ−a1sinγ+rρcosγ−ψ,x2=rbτ+θbcosτ−sinτ,y1=rcosψ−a1sinγ+rρsinγ−ψ,y2=rbτ+θbsinτ+cosτ,where M1 and M2 are the bending moments of Fb and Fa on the part of transition and involute curves, y1 and y2 the horizontal coordinates, and G the shear modulus. Hence, the bending, shear, and axial compressive stiffness can be obtained as follows:(16)1kb=∫π/2αcosβyβ−y1−xβsinβ2EIy1dy1dγdγ+∫τcβcosβyβ−y2−xβsinβ2EIy2dy2dτdτ,1ks=∫π/2α1.2cos2βGAy1dy1dγdγ+∫τcβ1.2cos2βGAy2dy2dτdτ,1ka=∫π/2αsin2βEAy1dy1dγdγ+∫τcβsin2βEAy2dy2dτdτ,with(17)dy1dγ=a1sinψ1+tan2γtan2γ+a1cosγsinγ−ψsin2γ−a1sinγ+rρcosγ−ψ1+a11+tan2γrtan2γ,dy2dτ=rbτ+θbcosτ.Let subscripts 1 and 2 be pinion and gear, the single-tooth-pair mesh stiffness can be defined as follows [23, 24]:(18)ki=11/kh+1/kt1+1/kf1+1/kt2+1/kf2,and(19)kt=11/kb+1/ks+1/ka.IfN is the number of meshing tooth pairs, the TVMS can be calculated as follows:(20)k=∑i=1Nki.However, in fact, two meshing tooth pairs share the one gear body, which leads to a mesh stiffness higher than the actual value in double-tooth engagement. Thus, an improved method is adopted where the total mesh stiffness is expressed as follows [23, 24]:(21)k=11/λ1Nkf1+1/ktooth+1/λ2Nkf2λ11=λ21=1,whenN=1,ktooth=∑i=1Nktoothi,ktoothi=11/khi+1/kt1+1/kt2,where λ represents the correction coefficient of the fillet-foundation stiffness, ktooth the total mesh stiffness for N meshing tooth pairs, and ktoothi the mesh stiffness of the ith tooth pair.The correction coefficients of the fillet-foundation stiffness need to be determined using the FE method, and the detailed calculation is given in [23, 24]. Hence, the stiffness caused by tooth deformation during double-tooth engagement can be calculated by the following equation:(22)ktooth=FEr2−EPiwithEpi=minEp1,Ep2.The total mesh stiffness can be obtained from Equation (20). In addition, mesh stiffness of a single-tooth pair is a function of tooth contact deformation Ed and meshing position j, which can be expressed as follows [23, 24]:(23)k=ki,Edi>0,0,Edi≤0,where the calculation of correction coefficients is shown in the relevant literatures.The parameters of the gear pair are chosen as follows: the number of teeth on the pinion and gear is 23 and 120, radius of the pinion and gear is 86 and 450 mm, module is 8 mm, teeth width is 140 mm, coincidence degree is 1.7315, elastic modulus is 209 GPa, and Poisson’s ratio is 0.3. The TVMS curve which changes with dimensionless time is displayed in Figure5. With the help of this figure, the curve-fitted stiffness equation is derived as follows:Figure 5
The curve of mesh stiffness in a mesh period.(24)
k
τ
=
1
.
55
×
10
9
+
5.31
×
10
8
τ
−
1.06
×
10
9
τ
2
,
0
<
τ
≤
1.463
π
,
5.72
×
10
8
+
9.93
×
10
8
τ
−
8.25
×
10
8
τ
2
,
1.463
×
π
<
τ
≤
2
π
.


## 4. Chaos and Stability Analysis of Locomotive Gear Transmission System

In this section, a case study is analyzed to demonstrate the effect of model parameters on the chaos and stability responses. For the subsequent numerical simulations, some parameters are as follows:k1=4.5×109N/m, k2=9×109N/m, c1=c2=500Ns/m, k33τ=kτ, and c33=0.05.


### 4.1. Effect of Pinion Speed on Chaotic Characteristics

Figure6 shows the bifurcation diagram of the gear system in dimensionless displacement using the pinion speed Ω as the control parameter. It can be seen that various kinds of periodic, multiperiodic, and chaotic responses exist in the system. When Ω is less than 1070 rpm, the dynamic behavior is synchronous with periodic-1 motion; subsequently, it comes into periodic-2 and periodic-4 motion at district of (1080–1160) and (1170–1240) rpm, respectively. After a short chaos at district of (1250–1270) rpm, the system enters periodic-1 motion of (1280–1430) rpm. With the increasing pinion speed, it enters chaos at district of (1440–1830) rpm. Thereafter, it comes into periodic-1 and periodic-2 motion between (1840–1990) and (2000–2110) rpm, respectively, and then it enters chaos again within (2120–2220) rpm. When Ω is higher than 2230 rpm, the system enters periodic-4, chaotic, and periodic-2 motion in sequence, in which the intervals are (2230–2400), (2410–2670), and (2680–2830) rpm, respectively. Once Ω is higher than 2840 rpm, the gear system eventually returns to the periodic-1 motion state.Figure 6
Bifurcation diagram of displacement with the pinion speed changes.The Lyapunov exponent is an important parameter for studying and determining whether the dynamic system is chaotic motion. Figure7 presents the Lyapunov exponent diagram of the geared system when pinion speed is changed from (500–3000) rpm. It is obvious that the Lyapunov exponents are greater than zero when chaos emerges in the same region of speed change. To accurately determine chaotic motion of the geared system, the phase-plane diagrams, Poincaré maps, time-domain response diagrams, and amplitude-frequency spectrums are listed to illustrate dynamic characteristics of the system in Figures 8–10.Figure 7
Lyapunov exponent diagram of the system with the gear speed changes.Figure 8
The dynamic characteristic curve of the system atΩ = 1000 rpm. (a) Phase-plane diagram. (b) Poincaré map. (c) Time-domain response diagram. (d) Amplitude-frequency spectrum.
(a)
(b)
(c)
(d)Figure 9
The dynamic characteristic curve of the system atΩ = 1150 rpm. (a) Phase-plane diagram. (b) Poincaré map. (c) Time-domain response diagram. (d) Amplitude-frequency spectrum.
(a)
(b)
(c)
(d)Figure 10
The dynamic characteristic curve of the system atΩ =1600 rpm. (a) Phase-plane diagram. (b) Poincaré map. (c) Time-domain response diagram (d) Amplitude-frequency spectrum.
(a)
(b)
(c)
(d)WhenΩ = 1000 rpm, there is a single point in the Poincaré map, the phase-plane diagram is only one closed circle, the time-domain response diagram shows a sine wave, and the amplitude-frequency spectrum has one-peak amplitude. It means that the system has periodic-1 motion. When Ω = 1150, there are two closed circles in the phase-plane diagram, the Poincaré map shows two isolated points, and the amplitude-frequency spectrum has harmonics 1/2, 2/2, and 3/2, signifying the presence of a period-2 unstable ghost attractor, as indicated in Figure 9. When Ω = 1600 rpm, it can be seen that the phase-plane diagram is disordered, the Poincaré map shows many discrete points, the time-domain response diagram also shows nonperiodic motion, and the amplitude-frequency spectrum is continuous. All the dynamic characteristics reveal that the system is in a state of chaos.


### 4.2. Effect of Support Stiffness Ratio on Chaotic Characteristics

Support stiffness of the pinion and gear is the key parameter that affects the dynamic behavior of the gear system. To study the effects,support stiffness ratio K, namely, support stiffness of the pinion to gear, is incorporated in the paper, where support stiffness of the pinion k1=4.5×109N/m. Figure 11 shows the bifurcation diagram of displacement using the support stiffness ratio K as a bifurcation parameter. When K is less than 4.5, the system is in the state of periodic-2 motion; after a short periodic-1 movement, the system enters chaotic motion at the district of 5-6. Once K is higher than 6, the gear transmission system returns to synchronous vibration with period-1.Figure 11
Bifurcation diagram of the displacement with support stiffness ratio changes.Figure12 also shows the Lyapunov exponent diagram of the geared system when the support stiffness ratio K is changed from 1 to 10, in which Lyapunov exponents are greater than zero in the same region compared to the bifurcation diagram, and the chaotic regions are basically the same as those in Figure 11. To accurately determine chaotic motion of the proposed system, the dynamic characteristic curves are also listed in Figures 13–15. The dynamic characteristics have obvious chaotic characteristics as seen in these figures, so the gear system is in chaotic motion in these stiffness ranges.Figure 12
Lyapunov exponent diagram of the system with support stiffness ratio changes.Figure 13
The dynamic characteristic curve of the system at support stiffness ratioK = 2.5. (a) Phase-plane diagram. (b) Poincaré map. (c) Time-domain response diagram. (d) Amplitude-frequency spectrum.
(a)
(b)
(c)
(d)Figure 14
The dynamic characteristic curve of the system at support stiffness ratioK = 5.5. (a) Phase-plane diagram. (b) Poincaré map. (c) Time-domain response diagram. (d) Amplitude-frequency spectrum.
(a)
(b)
(c)
(d)Figure 15
The dynamic characteristic curve of the system at support stiffness ratioK = 7.5. (a) Phase-plane diagram. (b) Poincaré map. (c) Time-domain response diagram. (d) Amplitude-frequency spectrum.
(a)
(b)
(c)
(d)


### 4.3. Stability Analysis of Locomotive Gear Transmission System

#### 4.3.1. Floquet Stability Theory

For the perturbation equation of a nonlinear system [15],(25)X˙=AtX,At+T=At,where T represents a system of periodic motion and At+T is the Jacobi matrix of the periodic solution of the steady state. Let Xt be the basic solution matrix of the equation, there is a constant matrix D and a nonsingular periodic T matrix φt=φt+T and Xt=φt⋅expt⋅D.SincedXt/dt=AtXt, it follows that(26)dXt+Tdt=At+TXt+T=AtXt+T,and(27)Xt+T=φt+T⋅expt+T⋅D=φt⋅expt⋅D⋅expT⋅D=Xt⋅expT⋅D,where the constant matrix expT⋅D is called the system state transfer matrix, and the eigenvalue of the matrix is called the Floquet multiplier λi.The research on the stability of periodic motion of nonlinear systems can be carried out in the following steps:(1)
Transform differential equation of the original model into a matrix form:(28)x¨+ctx˙+ktx¨=Ft.(2)
Transform the matrix equation into the state equation form:(29)y=x,x˙,y˙=Aty+Bt,with(30)At=0I−kt−ct,Bt=0Ft.(3)
The value of the following matrix equation at periodicT is solved by the numerical integration method:(31)y˙=Aty,y0=In.(4)
The Floquet multiplier of the state transition matrix is obtained from the QR algorithm based on the eigenvalue of the matrix.(5)
According to the location of the Floquet multiplier relative to the unit circle displayed in Figure16, the stability and bifurcation characteristics of the system can be judged as follows [28]:(i)
Whenλi < 1 (i = 1, 2, … , n), the stable periodic solution is asymptotically stable.(ii)
If there isλj which passes the unit circle outwards through the point of −1 and other λi < 1 (i = 1, 2, … , n, i ≠ j), the stable periodic solution will have period-doubling bifurcation.(iii)
If there isλj which passes the unit circle outwards through the point of +1 and other λi < 1 (i = 1, 2, … , n, i ≠ j), the stable periodic solution will have the saddle-node bifurcation.(iv)
If there is a pair of conjugate complex characteristic multipliers,λj = a ± jb, which passes the unit circle outward, and other λi < 1 (i = 1, 2, … , n, i ≠ j), the stable periodic solution will have the Hopf bifurcation, and the bifurcation will lead to an invariant torus.Figure 16
Schematic of the Floquet multiplier to generate bifurcation.


#### 4.3.2. Effect of Pinion Speed on Stability

The variation diagram of the maximum Floquet multipliersλi against the pinion speed of the gear system is displayed in Figure 17. It can be seen that when Ω is less than 1160 rpm, λi is less than 1, as shown in Figures 18(a) and 18(b), in which the maximum Floquet multipliers are in the unit circle, and the stable periodic solution is asymptotically stable according to the Floquet theory. But when the speed is changed at district of (1250–1270) or (1440–1830) rpm, the maximum Floquet multiplier is away from the unit circle in the form of a pair of conjugate complex eigenvalues. It means that the period-doubling bifurcation occurs, as shown in Figure 18(c).Figure 17
Diagram of the maximum Floquet multiplier with pinion speed changes.Figure 18
Position distribution of the maximum Floquet multiplier with pinion speed. (a)Ω = 650 rpm. (b) Ω = 1000 rpm. (c) Ω = 1450 rpm. (d) Ω = 2500 rpm. (e) Ω =2250 rpm. (f) Ω =2850 rpm.
(a)
(b)
(c)
(d)
(e)
(f)WhenΩ is changed at the district of (1840–2110) rpm, λi is less than 1, and the gear system returns to stable periodic-1 state. With the increasing pinion speed, λi is greater than 1, and the system leaves periodic-1 motion and enters chaos within (2410–2670) rpm, as shown in Figure 18(d). Once the speed is higher than 2680 rpm, the system leaves chaos and enters periodic-2 and periodic-1 motion, as shown in Figures 18(e) and 18(f).


#### 4.3.3. Effect of Support Stiffness Ratio on Stability

The variation diagram of maximum Floquet multipliersλi against the support stiffness ratio K of the system is displayed in Figure 19. It can be seen that when K is less than 5, λi is less than 1, and the stable periodic solution of the system is asymptotically stable, as shown in Figures 20(a) and 20(b). With the increasing K, the system leaves periodic-1 motion and enters chaotic motion at the district of 5∼6, in which the maximum Floquet multipliers are away from the unit circle in the form of complex eigenvalues, as shown in Figures 20(c) and 20(d). Once K is higher than 6, λi is less than 1, and the gear system returns to synchronous vibration with period-1.Figure 19
Diagram of the maximum Floquet multiplier with support stiffness ratio.Figure 20
Position distribution of the maximum Floquet multiplier with support stiffness ratio. (a)K = 3. (b) K = 4.7. (c) K = 5.4. (d) K = 5.9.
(a)
(b)
(c)
(d)


## 4.1. Effect of Pinion Speed on Chaotic Characteristics

Figure6 shows the bifurcation diagram of the gear system in dimensionless displacement using the pinion speed Ω as the control parameter. It can be seen that various kinds of periodic, multiperiodic, and chaotic responses exist in the system. When Ω is less than 1070 rpm, the dynamic behavior is synchronous with periodic-1 motion; subsequently, it comes into periodic-2 and periodic-4 motion at district of (1080–1160) and (1170–1240) rpm, respectively. After a short chaos at district of (1250–1270) rpm, the system enters periodic-1 motion of (1280–1430) rpm. With the increasing pinion speed, it enters chaos at district of (1440–1830) rpm. Thereafter, it comes into periodic-1 and periodic-2 motion between (1840–1990) and (2000–2110) rpm, respectively, and then it enters chaos again within (2120–2220) rpm. When Ω is higher than 2230 rpm, the system enters periodic-4, chaotic, and periodic-2 motion in sequence, in which the intervals are (2230–2400), (2410–2670), and (2680–2830) rpm, respectively. Once Ω is higher than 2840 rpm, the gear system eventually returns to the periodic-1 motion state.Figure 6
Bifurcation diagram of displacement with the pinion speed changes.The Lyapunov exponent is an important parameter for studying and determining whether the dynamic system is chaotic motion. Figure7 presents the Lyapunov exponent diagram of the geared system when pinion speed is changed from (500–3000) rpm. It is obvious that the Lyapunov exponents are greater than zero when chaos emerges in the same region of speed change. To accurately determine chaotic motion of the geared system, the phase-plane diagrams, Poincaré maps, time-domain response diagrams, and amplitude-frequency spectrums are listed to illustrate dynamic characteristics of the system in Figures 8–10.Figure 7
Lyapunov exponent diagram of the system with the gear speed changes.Figure 8
The dynamic characteristic curve of the system atΩ = 1000 rpm. (a) Phase-plane diagram. (b) Poincaré map. (c) Time-domain response diagram. (d) Amplitude-frequency spectrum.
(a)
(b)
(c)
(d)Figure 9
The dynamic characteristic curve of the system atΩ = 1150 rpm. (a) Phase-plane diagram. (b) Poincaré map. (c) Time-domain response diagram. (d) Amplitude-frequency spectrum.
(a)
(b)
(c)
(d)Figure 10
The dynamic characteristic curve of the system atΩ =1600 rpm. (a) Phase-plane diagram. (b) Poincaré map. (c) Time-domain response diagram (d) Amplitude-frequency spectrum.
(a)
(b)
(c)
(d)WhenΩ = 1000 rpm, there is a single point in the Poincaré map, the phase-plane diagram is only one closed circle, the time-domain response diagram shows a sine wave, and the amplitude-frequency spectrum has one-peak amplitude. It means that the system has periodic-1 motion. When Ω = 1150, there are two closed circles in the phase-plane diagram, the Poincaré map shows two isolated points, and the amplitude-frequency spectrum has harmonics 1/2, 2/2, and 3/2, signifying the presence of a period-2 unstable ghost attractor, as indicated in Figure 9. When Ω = 1600 rpm, it can be seen that the phase-plane diagram is disordered, the Poincaré map shows many discrete points, the time-domain response diagram also shows nonperiodic motion, and the amplitude-frequency spectrum is continuous. All the dynamic characteristics reveal that the system is in a state of chaos.


## 4.2. Effect of Support Stiffness Ratio on Chaotic Characteristics

Support stiffness of the pinion and gear is the key parameter that affects the dynamic behavior of the gear system. To study the effects,support stiffness ratio K, namely, support stiffness of the pinion to gear, is incorporated in the paper, where support stiffness of the pinion k1=4.5×109N/m. Figure 11 shows the bifurcation diagram of displacement using the support stiffness ratio K as a bifurcation parameter. When K is less than 4.5, the system is in the state of periodic-2 motion; after a short periodic-1 movement, the system enters chaotic motion at the district of 5-6. Once K is higher than 6, the gear transmission system returns to synchronous vibration with period-1.Figure 11
Bifurcation diagram of the displacement with support stiffness ratio changes.Figure12 also shows the Lyapunov exponent diagram of the geared system when the support stiffness ratio K is changed from 1 to 10, in which Lyapunov exponents are greater than zero in the same region compared to the bifurcation diagram, and the chaotic regions are basically the same as those in Figure 11. To accurately determine chaotic motion of the proposed system, the dynamic characteristic curves are also listed in Figures 13–15. The dynamic characteristics have obvious chaotic characteristics as seen in these figures, so the gear system is in chaotic motion in these stiffness ranges.Figure 12
Lyapunov exponent diagram of the system with support stiffness ratio changes.Figure 13
The dynamic characteristic curve of the system at support stiffness ratioK = 2.5. (a) Phase-plane diagram. (b) Poincaré map. (c) Time-domain response diagram. (d) Amplitude-frequency spectrum.
(a)
(b)
(c)
(d)Figure 14
The dynamic characteristic curve of the system at support stiffness ratioK = 5.5. (a) Phase-plane diagram. (b) Poincaré map. (c) Time-domain response diagram. (d) Amplitude-frequency spectrum.
(a)
(b)
(c)
(d)Figure 15
The dynamic characteristic curve of the system at support stiffness ratioK = 7.5. (a) Phase-plane diagram. (b) Poincaré map. (c) Time-domain response diagram. (d) Amplitude-frequency spectrum.
(a)
(b)
(c)
(d)


## 4.3. Stability Analysis of Locomotive Gear Transmission System

### 4.3.1. Floquet Stability Theory

For the perturbation equation of a nonlinear system [15],(25)X˙=AtX,At+T=At,where T represents a system of periodic motion and At+T is the Jacobi matrix of the periodic solution of the steady state. Let Xt be the basic solution matrix of the equation, there is a constant matrix D and a nonsingular periodic T matrix φt=φt+T and Xt=φt⋅expt⋅D.SincedXt/dt=AtXt, it follows that(26)dXt+Tdt=At+TXt+T=AtXt+T,and(27)Xt+T=φt+T⋅expt+T⋅D=φt⋅expt⋅D⋅expT⋅D=Xt⋅expT⋅D,where the constant matrix expT⋅D is called the system state transfer matrix, and the eigenvalue of the matrix is called the Floquet multiplier λi.The research on the stability of periodic motion of nonlinear systems can be carried out in the following steps:(1)
Transform differential equation of the original model into a matrix form:(28)x¨+ctx˙+ktx¨=Ft.(2)
Transform the matrix equation into the state equation form:(29)y=x,x˙,y˙=Aty+Bt,with(30)At=0I−kt−ct,Bt=0Ft.(3)
The value of the following matrix equation at periodicT is solved by the numerical integration method:(31)y˙=Aty,y0=In.(4)
The Floquet multiplier of the state transition matrix is obtained from the QR algorithm based on the eigenvalue of the matrix.(5)
According to the location of the Floquet multiplier relative to the unit circle displayed in Figure16, the stability and bifurcation characteristics of the system can be judged as follows [28]:(i)
Whenλi < 1 (i = 1, 2, … , n), the stable periodic solution is asymptotically stable.(ii)
If there isλj which passes the unit circle outwards through the point of −1 and other λi < 1 (i = 1, 2, … , n, i ≠ j), the stable periodic solution will have period-doubling bifurcation.(iii)
If there isλj which passes the unit circle outwards through the point of +1 and other λi < 1 (i = 1, 2, … , n, i ≠ j), the stable periodic solution will have the saddle-node bifurcation.(iv)
If there is a pair of conjugate complex characteristic multipliers,λj = a ± jb, which passes the unit circle outward, and other λi < 1 (i = 1, 2, … , n, i ≠ j), the stable periodic solution will have the Hopf bifurcation, and the bifurcation will lead to an invariant torus.Figure 16
Schematic of the Floquet multiplier to generate bifurcation.


### 4.3.2. Effect of Pinion Speed on Stability

The variation diagram of the maximum Floquet multipliersλi against the pinion speed of the gear system is displayed in Figure 17. It can be seen that when Ω is less than 1160 rpm, λi is less than 1, as shown in Figures 18(a) and 18(b), in which the maximum Floquet multipliers are in the unit circle, and the stable periodic solution is asymptotically stable according to the Floquet theory. But when the speed is changed at district of (1250–1270) or (1440–1830) rpm, the maximum Floquet multiplier is away from the unit circle in the form of a pair of conjugate complex eigenvalues. It means that the period-doubling bifurcation occurs, as shown in Figure 18(c).Figure 17
Diagram of the maximum Floquet multiplier with pinion speed changes.Figure 18
Position distribution of the maximum Floquet multiplier with pinion speed. (a)Ω = 650 rpm. (b) Ω = 1000 rpm. (c) Ω = 1450 rpm. (d) Ω = 2500 rpm. (e) Ω =2250 rpm. (f) Ω =2850 rpm.
(a)
(b)
(c)
(d)
(e)
(f)WhenΩ is changed at the district of (1840–2110) rpm, λi is less than 1, and the gear system returns to stable periodic-1 state. With the increasing pinion speed, λi is greater than 1, and the system leaves periodic-1 motion and enters chaos within (2410–2670) rpm, as shown in Figure 18(d). Once the speed is higher than 2680 rpm, the system leaves chaos and enters periodic-2 and periodic-1 motion, as shown in Figures 18(e) and 18(f).


### 4.3.3. Effect of Support Stiffness Ratio on Stability

The variation diagram of maximum Floquet multipliersλi against the support stiffness ratio K of the system is displayed in Figure 19. It can be seen that when K is less than 5, λi is less than 1, and the stable periodic solution of the system is asymptotically stable, as shown in Figures 20(a) and 20(b). With the increasing K, the system leaves periodic-1 motion and enters chaotic motion at the district of 5∼6, in which the maximum Floquet multipliers are away from the unit circle in the form of complex eigenvalues, as shown in Figures 20(c) and 20(d). Once K is higher than 6, λi is less than 1, and the gear system returns to synchronous vibration with period-1.Figure 19
Diagram of the maximum Floquet multiplier with support stiffness ratio.Figure 20
Position distribution of the maximum Floquet multiplier with support stiffness ratio. (a)K = 3. (b) K = 4.7. (c) K = 5.4. (d) K = 5.9.
(a)
(b)
(c)
(d)


## 4.3.1. Floquet Stability Theory

For the perturbation equation of a nonlinear system [15],(25)X˙=AtX,At+T=At,where T represents a system of periodic motion and At+T is the Jacobi matrix of the periodic solution of the steady state. Let Xt be the basic solution matrix of the equation, there is a constant matrix D and a nonsingular periodic T matrix φt=φt+T and Xt=φt⋅expt⋅D.SincedXt/dt=AtXt, it follows that(26)dXt+Tdt=At+TXt+T=AtXt+T,and(27)Xt+T=φt+T⋅expt+T⋅D=φt⋅expt⋅D⋅expT⋅D=Xt⋅expT⋅D,where the constant matrix expT⋅D is called the system state transfer matrix, and the eigenvalue of the matrix is called the Floquet multiplier λi.The research on the stability of periodic motion of nonlinear systems can be carried out in the following steps:(1)
Transform differential equation of the original model into a matrix form:(28)x¨+ctx˙+ktx¨=Ft.(2)
Transform the matrix equation into the state equation form:(29)y=x,x˙,y˙=Aty+Bt,with(30)At=0I−kt−ct,Bt=0Ft.(3)
The value of the following matrix equation at periodicT is solved by the numerical integration method:(31)y˙=Aty,y0=In.(4)
The Floquet multiplier of the state transition matrix is obtained from the QR algorithm based on the eigenvalue of the matrix.(5)
According to the location of the Floquet multiplier relative to the unit circle displayed in Figure16, the stability and bifurcation characteristics of the system can be judged as follows [28]:(i)
Whenλi < 1 (i = 1, 2, … , n), the stable periodic solution is asymptotically stable.(ii)
If there isλj which passes the unit circle outwards through the point of −1 and other λi < 1 (i = 1, 2, … , n, i ≠ j), the stable periodic solution will have period-doubling bifurcation.(iii)
If there isλj which passes the unit circle outwards through the point of +1 and other λi < 1 (i = 1, 2, … , n, i ≠ j), the stable periodic solution will have the saddle-node bifurcation.(iv)
If there is a pair of conjugate complex characteristic multipliers,λj = a ± jb, which passes the unit circle outward, and other λi < 1 (i = 1, 2, … , n, i ≠ j), the stable periodic solution will have the Hopf bifurcation, and the bifurcation will lead to an invariant torus.Figure 16
Schematic of the Floquet multiplier to generate bifurcation.


## 4.3.2. Effect of Pinion Speed on Stability

The variation diagram of the maximum Floquet multipliersλi against the pinion speed of the gear system is displayed in Figure 17. It can be seen that when Ω is less than 1160 rpm, λi is less than 1, as shown in Figures 18(a) and 18(b), in which the maximum Floquet multipliers are in the unit circle, and the stable periodic solution is asymptotically stable according to the Floquet theory. But when the speed is changed at district of (1250–1270) or (1440–1830) rpm, the maximum Floquet multiplier is away from the unit circle in the form of a pair of conjugate complex eigenvalues. It means that the period-doubling bifurcation occurs, as shown in Figure 18(c).Figure 17
Diagram of the maximum Floquet multiplier with pinion speed changes.Figure 18
Position distribution of the maximum Floquet multiplier with pinion speed. (a)Ω = 650 rpm. (b) Ω = 1000 rpm. (c) Ω = 1450 rpm. (d) Ω = 2500 rpm. (e) Ω =2250 rpm. (f) Ω =2850 rpm.
(a)
(b)
(c)
(d)
(e)
(f)WhenΩ is changed at the district of (1840–2110) rpm, λi is less than 1, and the gear system returns to stable periodic-1 state. With the increasing pinion speed, λi is greater than 1, and the system leaves periodic-1 motion and enters chaos within (2410–2670) rpm, as shown in Figure 18(d). Once the speed is higher than 2680 rpm, the system leaves chaos and enters periodic-2 and periodic-1 motion, as shown in Figures 18(e) and 18(f).


## 4.3.3. Effect of Support Stiffness Ratio on Stability

The variation diagram of maximum Floquet multipliersλi against the support stiffness ratio K of the system is displayed in Figure 19. It can be seen that when K is less than 5, λi is less than 1, and the stable periodic solution of the system is asymptotically stable, as shown in Figures 20(a) and 20(b). With the increasing K, the system leaves periodic-1 motion and enters chaotic motion at the district of 5∼6, in which the maximum Floquet multipliers are away from the unit circle in the form of complex eigenvalues, as shown in Figures 20(c) and 20(d). Once K is higher than 6, λi is less than 1, and the gear system returns to synchronous vibration with period-1.Figure 19
Diagram of the maximum Floquet multiplier with support stiffness ratio.Figure 20
Position distribution of the maximum Floquet multiplier with support stiffness ratio. (a)K = 3. (b) K = 4.7. (c) K = 5.4. (d) K = 5.9.
(a)
(b)
(c)
(d)


## 5. Conclusions

Taking the locomotive gear system as the object, the chaotic characteristics of the transmission system are analyzed when the pinion speed and stiffness are changed. At the same time, the parameter stability of the system and the evolution of periodic motion are studied based on the Floquet theory. The following conclusions can be drawn from the study:(1)
The system is relatively stable when the locomotive gear system runs at low speed; however, with the increasing speed of the traction motor (pinion), the responses of the gear system enter the unstable state of the chaotic motion through a variety of ways, which has an effect on the stable operation of the locomotive. Therefore, when the railway locomotive is operated, the speed of the traction motor can be increased selectively so that the transmission system can leave an unstable area of the speed as soon as possible. The analysis results show that the speed of the traction motor (pinion) is nearly 1500 rpm (rated speed), which ensures stable and safe operation of locomotives.(2)
With the increase of the support stiffness ratio of the pinion to gear, the stability of the system will fluctuate. Considering the manufacturing cost, the pinion stiffness of the gear system can be selectively designed. For instance, when the stiffness is varied from4.5×109 to 4.9×109 N/m, the gear system reaches the stable motion with period-1. Therefore, the interval above is the most suitable range of selection in the stiffness design of the transmission system.



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*Source: 5691892-2018-11-01.xml*