# Frequency and Vibration Characteristics of High-Speed Gear-Rotor-Bearing System with Tooth Root Crack considering Compound Dynamic Backlash

Authors: Jie Liu; Weiqiang Zhao; Weiwei Liu
Journal: Shock and Vibration
 (2019)

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## Abstract

Considering the microstructure of tooth surface and the dynamic characteristics of the vibration responses, a compound dynamic backlash model is employed for the gear transmission system. Based on the fractal theory and dynamic center distance, respectively, the dynamic backlash is presented, and the potential energy method is applied to compute the time-varying meshing stiffness, including the healthy gear system and the crack fault gear system. Then, a 16-DOF coupled lateral-torsional gear-rotor-bearing transmission system with the crack fault is established. The fault characteristics in the time-domain waveform and frequency response and statistics data are described. The effect of crack on the time-varying meshing stiffness is analyzed. The vibration response of three backlash models is compared. The dynamic response of the system is explored with the increase in crack depth in detail. The results show that the fault features of countershaft are more obvious. Obvious fluctuations are presented in the time-domain waveform, and sidebands can be found in the frequency domain responses when the tooth root crack appears. The effect of compound dynamic backlash on the system is more obvious than fixed backlash and backlash with changing center distance. The vibration displacement along meshing direction and dynamic meshing force increases with the increase in crack depth. Backlash and variation of center distance show different tendencies with increasing crack depth under different rotational speeds. Amplitude of the sidebands increases with crack depth increasing. The amplitude of multiplication frequency of rotational frequency has an obvious variation with growing crack depth. The sidebands of the multiplication frequency of meshing frequency show more details on the system with complex backlash and crack fault.


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## Body

## 1. Introduction

The gear transmission systems are widely used in many machinery fields, which are employed for transmitting power and changing speed. The steady operation is a necessary condition for the gear system. Therefore, the mechanism and dynamic characteristics of gear failure are a major subject in the dynamic field.The gearbox failure is closely related to gear tooth, including spalling, pitting, crack, and broken tooth, which potentially lead to a complete damage. In order to clearly detect and diagnose in the gear transmission system, the vibration responses and fault features of the gear transmission system with fault are developed [1–3].The typical kinds of damage are crack failure and spalling failure. The vibration responses of the gear system are closely related to the time-varying meshing stiffness (TVMS) of the gear pair. Yang and Lin [4] calculated TVMS by the potential energy method considering axial compressive energy, Hertzian energy, and bending energy. Tian [5] further investigated TVMS of a gear pair with crack. Govind et al. [6] studied TVMS, crack propagation behaviour, and the vibration responses of spur gear. Chen et al. [7] presented two improved calculation models for gear tooth fillet-foundation stiffness. When a gear pair appeared a large crack length, the computational accuracy of the two models was compared. Chen and Shao [8] developed a gear pair meshing stiffness calculation method considering tooth profile modification and tooth root crack.Many outstanding scholars have made great contributions to the research of healthy gear system. Zhou et al. [9] presented a coupled lateral-torsional 16-DOF gear-rotor-bearing system considering piecewise periodic stiffness, friction, and eccentricity, and the motion states and frequency variations were analyzed in detail. In addition, the backlash is an important factor to ensure smooth and reliable operation. Chen et al. [10] investigated the dynamic backlash with the fractal feature and the effect of the dynamic backlash on the 2-DOF gear system. Li et al. [11] researched TVMS that was affected by tooth profile changes and developed bifurcation characteristics of the gear system. Xiang and Gao [12] calculated the differential equation of the gear transmission model with dynamic friction, TVMS, and dynamic backlash on the basis of the 4th order Runge–Kutta method and observed the effect of dynamic backlash and gear eccentricity under different rotational speeds.Many scholars had made a mountain of work for the gearbox with crack fault. Saxena et al. [13] studied the influences of the meshing stiffness and damping on the flexible rotor-shaft system. They explored vibration responses and frequency characteristics in detail. Mohammed and Rantatalo [14] established a 6-DOF dynamic gear model to study the natural frequency and time-frequency with different crack sizes. Saeed et al. [15] researched vibration responses in the presence of single- and multi-tooth simultaneous crack using the finite element method (FEM). Ma et al. [16] established a perforated gear system with different degree cracks. The gear crack propagation paths, frequency components, and frequency values were analyzed in detail. Ma et al. [17] built a meshing stiffness model for a cracked gear system, and the influences of the crack depth and initial position were analyzed by FEM. Chen et al. [18] presented a meshing stiffness calculation method for the nonuniform tooth root crack along the tooth width. The meshing stiffness and vibration response calculated by three different algorithms were compared. Ma and Chen [19] established a 4-DOF gear system to investigate the dynamic characteristics and vibration responses with local failure and explore the failure mechanism. Hu et al. [20] presented a finite element node dynamic model for gear with crack. The RMS (root-mean-square) of the gear system parameters, vibration waveform, and frequency spectra were analyzed. Chen and Shao [21] investigated the effect of crack on a planetary gear system. Liu et al. [22] studied vibration characteristics of the planetary gear system when sun gear appeared a tooth root crack.For the research of crack, some scientists devoted to establish a simple gear model, such as simplifying backlash as a fix value, employing a periodic meshing stiffness, and establishing a few-DOF model. In this paper, a 16-DOF coupled lateral-torsional gear-rotor-bearing transmission system considering compound-type dynamic backlash, TVMS, eccentricity, and friction force is established. The compound dynamic backlash on the basis of the microstructure characteristics and vibration features is defined. TVMS is calculated by the potential energy method. This complex model is employed for investigating the effect of crack faults and researching crack failure features. The vibration responses of the gear system are researched by analyzing influences of the rotational speed and crack depth value variation. The differential equations of motion are solved using the Runge–Kutta numerical method. The simulation results, including time-domain waveforms, frequency-domain spectra, and waterfall plots, are shown. We focus on the effect of the crack depth growing on a complex gear system. The amplitude statistics of sideband supply an effective reference for the condition monitoring and fault diagnosis of the gear-rotor-bearing system.


## 2. Dynamic Model of the Gear-Rotor-Bearing System

### 2.1. Lumped Mass Model of the Gear System

The lumped mass model of the gear system is consisted of input, gears, shafts, bearing, and output. This model is described in Figure1. O1(x1, y1) and O2(x2, y2) are the centers of two gears. G1xg1,yg1 and G2xg2,yg2 are the barycenters of two gears. mi (i = 1, 2) are quality of gears. J1 and J2 represent the moments of gear inertial. mbi (i = 1, 2, 3, 4) represent quality of bearing. Jd and Jg are equivalent rotational inertia of input and output terminals. The torsional stiffness and damping of shaft are given by kti and cti (i = 1, 2). ksi and csi (i = 1, 2) are, respectively, lateral stiffness and damping. cbi (i = 1, 2, 3, 4) are equivalent bearing damping, including x direction and y direction. In the x and y directions, Fxi and Fyi (i = 1, 2, 3, 4) are defined as nonlinear bearing forces. ρi (i = 1, 2) represent the eccentricity. φi (i=1,2,d,g) are, respectively, angle displacements of gear, pinion, input, and output, and the angle displacements are composed of an angle displacement ωit (i = 1, 2) and microscopic displacement θi(t). According to the geometrical relationship, the equations for the displacement of gears and input/output are given by(1)φ1=ω1t+θ1,φ2=ω2t+θ2,φd=ω1t+θd,φg=ω2t+θg.Figure 1
The gear system model.Because of the eccentricity of gears, the rotating geometrical centersO1(x1, y1) and O2(x2, y2) and the barycenters G1xg1,yg1 and G2xg2,yg2 are at different positions. The relationship equations are found as(2)xg1=x1+ρ1cosφ1,yg1=y1+ρ1sinφ1,xg2=x2+ρ2cosφ2,yg2=y2+ρ2sinφ2.In thexoy plane, elastic deformation of the shafts can be determined as(3)δx1=x1−ξ1xb1−ξ2xb2,δy1=y1−ξ1yb1−ξ2yb2,δx2=x2−ξ3xb4−ξ4xb3,δy2=y2−ξ3yb4−ξ4yb3,where ξi can be computed as(4)ξi=lbilj,i=1,2,3,4;j=1,2,where lbi represents the distances between the gear’s centers and centers of bearing and lj is the length of shafts. The deformation between two gears along the meshing direction δ(t) is given by(5)δt=rb1φ1−rb2φ2+yg1−yg2−et,=rb1θ1−rb2φ2+y1−y2+ρ1sinφ1−ρ2sinφ2−et,where static transmission error e(t) is caused by manufacture and assembly, and rb1 and rb2 are gear base circle radii. This error can be treated as a sinusoidal function e(t) = e0 + ersin(ωmt + φm) [9], where e0 and er severally represent mean and amplitude of the error, and φm and ωm are, respectively, initial phase and meshing frequency. ωm meshing frequency is expressed as ωm = 2πn1z1/60(2πn2z2/60). z1 and z2 are the gears’ teeth. n1 and n2 are the gears’ rotating velocity.The dynamic meshing force (DMF)Fm (j = 1, 2) can be determined as(6)Fm=∑i=1jFmi=∑i=1jktifδt+cmδ˙t,where the state is the single-tooth meshing at j = 1, while the state is double-tooth meshing at j = 2. kti (i = 1, 2) and cm are TVMS and meshing damping, respectively. f(δ) is the backlash function in the gear system, which is given by [23](7)fδ=δt−1−ψbht,δt>bht,ψbht,−bht<δt<bht,0≤ψ≤1,δt+1−ψbht,δt<−bht,cm=2ξmkt1/m1+1/m2,where bh(t) is the gear dynamic backlash and ψ is a nonlinear coefficient. Meshing damping is given by cm. bh(t) is presented in Section 2.3.


### 2.2. Time-Varying Meshing Stiffness for Gears with Tooth Crack

The meshing stiffness model of a gear pair with a tooth root crack was described by Tian in 2004. The energy method was used to solve the mathematical modeling problem of the gear meshing stiffness. On the basis of the research, we further investigated the gear system with a tooth root crack with time-varying meshing stiffness. The schematic graph of tooth crack is shown in Figures2(a) and 2(b).Figure 2
(a, b) Schematic graph of tooth crack.
(a)
(b)The potential energy of the meshing gear teeth is shown as follows [21]:(8)Ub=∫0dFbd−x−Fah22EIxdx,Us=∫0d1.2Fb22GAxdx,Ua=∫0dFa22EAxdx.I
x and Ax are equivalent to the area moment of inertia and the area of the section, respectively [5], and G represents the shear modulus. They are determined as(9)Ix=1122hx3L,Ax=2hxL,G=E21+υ,where hx is the distance between the point on the tooth’s curve and the tooth’s central line [5].For the single-tooth-pair meshing process, the overall meshing stiffness for the gears with tooth crack could be given by(10)kt=11/kh+1/kb1+1/ks1+1/ka1+1/kb2+1/ks2+1/ka2,where kh is the Hertzian contact stiffness, ks is the shear stiffness, kb is the bending stiffness, and ka is the axial compression stiffness. And kh can be given by(11)kh=πEL41−v2.where E is Young’s modulus and L is the tooth face width. Ff=μλvSFm represents Poisson’s ratio. In addition, kb, ks, and ka can be expressed as(12)1kb=∫−α1α231+cosα1α2−αsinα−cosα3α2−αcosαE2Lsinα+α2−αcosα3dα,1ks=∫−α1α21.21+vα2−αcosαcos2α1ELsinα+α2−αcosαdα,1ka=∫−α1α2α2−vcosαsin2α1E2Lsinα+α2−αcosαdα.For the double-tooth-pair meshing process, the overall meshing stiffness for the gears with tooth crack could be given by(13)kt=kt1+kt2=∑i=1211/kh,i+1/kb1,i+1/ks1,i+1/ka1,i+1/kb2,i+1/ks2,i+1/ka2,i,where i (i = 1, 2) are the first or second pair of meshing teeth.When the gear pair appears a tooth root crack,Ixc and Axc are expressed as(14)Ixc=112hc+hx3L,ifx≤gc,1122hx3L=23hx3L,ifx>gc,Axc=hc+hxL,ifx≤gc,2hxL,ifx>gc,where hc represents the distance between the root the crack and the central line of the tooth. The other parameters and formulas are given in [4, 5, 21].The overall effective meshing stiffness is given by the equations. Crack position is a tooth root of pinion. Equation (14) is input into equation (8), and then TVMS can be calculated. The time-varying meshing stiffness can be provided as shown in Figure 3.Figure 3
The time-varying meshing stiffness.


### 2.3. Dynamic Backlash

In the gear mesh process, the backlash is one of the most important factors. The backlash is the distance between two tooth profiles. In order to prevent the gears from getting stuck and ensure lubrication of gears, the backlash must exist. Rough tooth surfaces are not considered as an inherent characteristic. In the section, the compound dynamic backlash model is established considering rough tooth surface and vibration characteristic of the gear system. The sketch of backlash is shown in Figure4. The relative displacement is generated when two rough teeth surface gears mesh.Figure 4
Sketch of backlash.The dynamic backlashbh(t) is consisted of the microscopic backlash Δbt and vibration variation backlash b(t). In order to approximate rough tooth surfaces, the W-M fractal function is employed. The microscopic backlash Δbt, which represents the inherent characteristic of the rough tooth surface, can be determined as [24–26](15)Δbt=LsGcLsD−1∑n=0nmaxcos2πγnt/Lsγ2−Dn,1<D<2,γ>1,where Ls is the sampling width, γn represents space frequency of flank profile, Gc is the characteristic scale coefficient, and D represents the fractal dimension controlling the complexity of the fractal curve. The relationship between Gc and Rq is given by(16)Rq=Gc2D−12lnγ14−2D1ωk4−2D1ωh4−2D0.5,where ωk and ωh are, respectively, equivalent to lower cutoff frequency and upper cutoff frequency. When the gear system appeared vibration, the center distance of gears is changed. By changing the center distance of gears, the backlash must be generated with volatility. The vibration variation backlash b(t) and dynamic center distance a can be determined as [27, 28](17)a=a0cosα0+x2−x12+y2−y1−a0sinα02,bt=a−a0tanφ1+b0,where a0 represents initial center distance, α0 is the pressure angle, and b0 represents fixed backlash. The simplified center distance model is shown in Figure 5(a).Figure 5
(a) Simplified center distance model and (b) simplified sketch of meshing.
(a)
(b)When crack occurs on the teeth surfaces, the dynamic center distancea is increased. The dynamic backlash with crack is given by(18)bht=bt+Δbt.


### 2.4. The Friction Force

Due to the meshing contact points located in different positions, the direction of friction force must be different. Thus, the frictional force changes periodically with the changing of contact points. Friction is given by(19)Ff=∑i=1jμλiνsFmi,j=1,2,λνs=1,νs>0,0,νs=0,−1,νs<0,where the state is the single tooth meshing at j = 1, while the state is double tooth meshing at j = 2. λvs is the direction coefficient of the friction force, μ is the friction coefficient of the gear surface, and Fmi is the dynamic meshing force.v
s represents relative sliding velocity of the meshing point M of the tooth pairs in Figure 5(b), which can be determined as(20)νS=νM1sinφ1−νM2sinφ2,where vM1 and vM2 are, respectively, on the meshing point M and are described as follows:(21)νM1=ω1O1M¯,νM2=ω1O2M¯.Friction torque is determined as(22)Tf1=FfN1M¯,Tf2=FfN2M¯,where N1M¯ and N2M¯ are, respectively, the friction arms and are defined as(23)N1M¯=rb1+rb2tanα−ra22−rb22+rb1ω1t,N2M¯=rb1+rb2tanα−N1M¯.


### 2.5. Ball Bearing Model

In Figure6, the ball bearing model is shown. The outer ring is fastened to the bearing chock. The inner ring is fastened to the shaft. The rolling elements are evenly distributed between inner and outer rings. vi and vo representing the contact point velocities between the rolling elements and inner/outer rings are given by(24)νi=ωi⋅r,νo=ωo⋅R,where R and r are radii of inner and outer rings, and the angular velocities of inner and outer rings are ωi and ωo. The velocity of cage is expressed as(25)vb=vo+vi2=ωoR+ωir2.Figure 6
Rolling bearing model.Due toωi = ω and ωo = 0, the angular velocity of cage ωb is given by(26)ωb=2vbR+r=ωi⋅rR+r.The momentary angular displacementφi1 of the ith rolling element is given by(27)φi1=ωb⋅t+2πi−1Nb,i=1,2,3,…,Nb,where Nb is the number of rolling balls and ωb is the angular velocity of the cage.The deformation of theith rolling ball can be described by(28)δi1=xcosφi1+ysinφi1−γ0,where γ0 represents bearing clearance. The Hertz contact theory is employed for computing contact force. Contact force of the ith rolling element fi can be determined as(29)fi=Kbxcosφi1+ysinφi1−γ03/2⋅Hxcosφi1+ysinφi1−γ0,where Kb is the Hertz contact stiffness and H(x) is the Heaviside function. So, the bearing forces are given by(30)Fxj=∑i=1Nbficosφi1=Fcosφi1,Fyj=∑i=1Nbfisinφi1=Fsinφi1,j=1,2,3,4.


### 2.6. Mathematical Model of the GRBS

The nonlinear differential equation (33) of the GRBS are evolved from Lagrange’s equation.ct
1, ct2, cs1, and cs2 are torsion and bending damping, respectively, and are given by(31)ct1=2ξtkt11/Id+1/I1,ct2=2ξtkt21/I2+1/Ig,(32)cs1=2ξsks11/mb1+1/m1,cs2=2ξsks21/mb3+1/m2,(33)Jdθ¨d+ct1θ˙d−θ˙1+kt1θd−θ1=Td,m1x¨1+cs1x˙1−ξ2x˙b1−ξ1x˙b2+ks1x1−ξ2xb1−ξ1x˙b2=m1ρ1θ¨1sinω1t+θ1+m1ρ1ω1+θ˙12cosω1t+θ1−Ff,m1y¨1+cs1y˙11−ξ2y˙b1−ξ1y˙b2+ks1y1−ξ2yb1−ξ1yb2=m1ρ1ω1+θ˙12sinω1t+θ1−m1ρ1θ¨1cosω1t+θ1−Fm−m1g,J1+m1ρ12θ¨1+ct1θ˙1−θ˙d+kt1θ1−θd=m1ρ1sinω1t+θ1x¨1−m1ρ1cosω1t+θ1y¨1−Fmrb1+Tf1,m2x¨2+cs2x˙2−ξ4x˙b3−ξ3x˙b4+ks2x2−ξ4xb3−ξ3xb4=m2ρ2θ¨2sinω2t+θ2+m2ρ2ω2+θ˙22cosω2t+θ2+Ff,m2y¨2+cS2y˙2−ξ4y˙b3−ξ3y˙b4+ks1y2−ξ4yb3−ξ3yb4=m2ρ2ω2+θ˙22sinω2t+θ2−m2ρ2θ¨2cosω2t+θ2−Fm−m2g,J2+m2ρ22θ¨2+ct2θ˙2−θ˙g+kt2θ2−θg=m2ρ2sinω2t+θ2x¨2−m2ρ2cosω2t+θ2y¨2−Fmrb2−Tf2,mb1x¨b1+cs1ξ2−x˙1+ξ2x˙b1+ξ1x˙b2+cbx1x˙b1+ks1ξ2−x1+ξ2xb1+ξ1xb2=Fx1,mb1y¨b1+cs1ξ2−y˙1+ξ2y˙b1+ξ1y˙b2+cby1y˙b1+ks1ξ2−y1+ξ2yb1+ξ1yb2=Fy1−mb1g,mb2x¨b2+cs1ξ1−x˙1+ξ2x˙b1+ξ1x˙b2+cbx2x˙b2+ks1ξ1−x1+ξ2xb1+ξ1xb2=Fx2,mb2y¨b2+cs1ξ1−y˙1+ξ2y˙b1+ξ1y˙b2+cby2y˙b1+ks1ξ1−y1+ξ2yb1+ξ1yb2=Fy2−mb2g,mb3x¨b3+cs2ξ4−x˙2+ξ4x˙b3+ξ3x˙b4+cbx3x˙b3+ks2ξ4−x2+ξ4xb3+ξ3xb4=Fx3,mb3y¨b3+cs2ξ4−y˙2+ξ4y˙b3+ξ3y˙b4+cby3y˙b3+ks2ξ4−y2+ξ4yb3+ξ3yb4=Fy3−mb3g,mb4x¨b4+cs2ξ3−x˙2+ξ4x˙b3+ξ3x˙b4+cbx4x˙b4+ks2ξ3−x2+ξ4xb3+ξ3xb4=Fx4,mb4y¨b4+cs2ξ3−y˙2+ξ4y˙b3+ξ3y˙b4+cby4y˙b4+ks2ξ3−y2+ξ4yb3+ξ3yb4=Fy4−mb4g,Jgθ¨g+ct2θ˙g−θ˙2+kt2θg−θ2=−Tg,where ks1, ks2, kt1, and kt2 represent bending stiffness and torsion stiffness of the shafts. cbij (i = x, y) (j = 1, 2, 3, 4) are equivalent bearing damping in the directions x and y.The gear-rotor-bearing model with strong nonlinear and time-varying characteristics is established. The dynamic backlash and time-varying meshing stiffness with crack are included in the system. The main parameters are shown in Tables1 and 2.Table 1
Parameters of the gear-bearing system.
Parameter
Symbol
Numerical value
Pressure angle
α
20°
Moment of inertia
J
1/J2
7.1 × 10−3/1.39 × 10−4 kg·m2
Moment of inertia of input/output
J
d
/
J
g
1.25 × 10−4/5.75 × 10−4 kg·m2
Mass
m
1/m2
2.6/0.9 kg
Stiffness of bearings
k
1/k2
2/2 × 108 N·m−1
Teeth
z
1/z2
30/20
Module
m
4 mm
Tooth width
L
0.03 m
Mean/amplitude of the transmission error
e
0/er
2 × 10−5/3 × 10−5 m
Eccentricity
ρ
1/ρ2
3 × 10−5/2 × 10−5 m
Driving torque
T
l
270 N·m
Loading torque
T
d
160 N·m
Damping ratio of gear meshing
ξ
m
0.02
Damping ratio of shaft
ξ
s/ξt
0.1/0.1Table 2
Model parameters of the bearing.
Parameter
Symbol
Numerical value
Outer ring radius of bearing
R
1/R2
0.031/0.021 m
Inner ring radius of bearing
r
1/r2
0.02/0.015 m
Contact stiffness
K
b1/Kb2
13.34 × 109/10.56 × 109 Nm3/2
Bearing clearance
γ
01/γ02
3 × 10−5/2 × 10−5 m
Ball number
N
1/N2
14/18


## 2.1. Lumped Mass Model of the Gear System

The lumped mass model of the gear system is consisted of input, gears, shafts, bearing, and output. This model is described in Figure1. O1(x1, y1) and O2(x2, y2) are the centers of two gears. G1xg1,yg1 and G2xg2,yg2 are the barycenters of two gears. mi (i = 1, 2) are quality of gears. J1 and J2 represent the moments of gear inertial. mbi (i = 1, 2, 3, 4) represent quality of bearing. Jd and Jg are equivalent rotational inertia of input and output terminals. The torsional stiffness and damping of shaft are given by kti and cti (i = 1, 2). ksi and csi (i = 1, 2) are, respectively, lateral stiffness and damping. cbi (i = 1, 2, 3, 4) are equivalent bearing damping, including x direction and y direction. In the x and y directions, Fxi and Fyi (i = 1, 2, 3, 4) are defined as nonlinear bearing forces. ρi (i = 1, 2) represent the eccentricity. φi (i=1,2,d,g) are, respectively, angle displacements of gear, pinion, input, and output, and the angle displacements are composed of an angle displacement ωit (i = 1, 2) and microscopic displacement θi(t). According to the geometrical relationship, the equations for the displacement of gears and input/output are given by(1)φ1=ω1t+θ1,φ2=ω2t+θ2,φd=ω1t+θd,φg=ω2t+θg.Figure 1
The gear system model.Because of the eccentricity of gears, the rotating geometrical centersO1(x1, y1) and O2(x2, y2) and the barycenters G1xg1,yg1 and G2xg2,yg2 are at different positions. The relationship equations are found as(2)xg1=x1+ρ1cosφ1,yg1=y1+ρ1sinφ1,xg2=x2+ρ2cosφ2,yg2=y2+ρ2sinφ2.In thexoy plane, elastic deformation of the shafts can be determined as(3)δx1=x1−ξ1xb1−ξ2xb2,δy1=y1−ξ1yb1−ξ2yb2,δx2=x2−ξ3xb4−ξ4xb3,δy2=y2−ξ3yb4−ξ4yb3,where ξi can be computed as(4)ξi=lbilj,i=1,2,3,4;j=1,2,where lbi represents the distances between the gear’s centers and centers of bearing and lj is the length of shafts. The deformation between two gears along the meshing direction δ(t) is given by(5)δt=rb1φ1−rb2φ2+yg1−yg2−et,=rb1θ1−rb2φ2+y1−y2+ρ1sinφ1−ρ2sinφ2−et,where static transmission error e(t) is caused by manufacture and assembly, and rb1 and rb2 are gear base circle radii. This error can be treated as a sinusoidal function e(t) = e0 + ersin(ωmt + φm) [9], where e0 and er severally represent mean and amplitude of the error, and φm and ωm are, respectively, initial phase and meshing frequency. ωm meshing frequency is expressed as ωm = 2πn1z1/60(2πn2z2/60). z1 and z2 are the gears’ teeth. n1 and n2 are the gears’ rotating velocity.The dynamic meshing force (DMF)Fm (j = 1, 2) can be determined as(6)Fm=∑i=1jFmi=∑i=1jktifδt+cmδ˙t,where the state is the single-tooth meshing at j = 1, while the state is double-tooth meshing at j = 2. kti (i = 1, 2) and cm are TVMS and meshing damping, respectively. f(δ) is the backlash function in the gear system, which is given by [23](7)fδ=δt−1−ψbht,δt>bht,ψbht,−bht<δt<bht,0≤ψ≤1,δt+1−ψbht,δt<−bht,cm=2ξmkt1/m1+1/m2,where bh(t) is the gear dynamic backlash and ψ is a nonlinear coefficient. Meshing damping is given by cm. bh(t) is presented in Section 2.3.


## 2.2. Time-Varying Meshing Stiffness for Gears with Tooth Crack

The meshing stiffness model of a gear pair with a tooth root crack was described by Tian in 2004. The energy method was used to solve the mathematical modeling problem of the gear meshing stiffness. On the basis of the research, we further investigated the gear system with a tooth root crack with time-varying meshing stiffness. The schematic graph of tooth crack is shown in Figures2(a) and 2(b).Figure 2
(a, b) Schematic graph of tooth crack.
(a)
(b)The potential energy of the meshing gear teeth is shown as follows [21]:(8)Ub=∫0dFbd−x−Fah22EIxdx,Us=∫0d1.2Fb22GAxdx,Ua=∫0dFa22EAxdx.I
x and Ax are equivalent to the area moment of inertia and the area of the section, respectively [5], and G represents the shear modulus. They are determined as(9)Ix=1122hx3L,Ax=2hxL,G=E21+υ,where hx is the distance between the point on the tooth’s curve and the tooth’s central line [5].For the single-tooth-pair meshing process, the overall meshing stiffness for the gears with tooth crack could be given by(10)kt=11/kh+1/kb1+1/ks1+1/ka1+1/kb2+1/ks2+1/ka2,where kh is the Hertzian contact stiffness, ks is the shear stiffness, kb is the bending stiffness, and ka is the axial compression stiffness. And kh can be given by(11)kh=πEL41−v2.where E is Young’s modulus and L is the tooth face width. Ff=μλvSFm represents Poisson’s ratio. In addition, kb, ks, and ka can be expressed as(12)1kb=∫−α1α231+cosα1α2−αsinα−cosα3α2−αcosαE2Lsinα+α2−αcosα3dα,1ks=∫−α1α21.21+vα2−αcosαcos2α1ELsinα+α2−αcosαdα,1ka=∫−α1α2α2−vcosαsin2α1E2Lsinα+α2−αcosαdα.For the double-tooth-pair meshing process, the overall meshing stiffness for the gears with tooth crack could be given by(13)kt=kt1+kt2=∑i=1211/kh,i+1/kb1,i+1/ks1,i+1/ka1,i+1/kb2,i+1/ks2,i+1/ka2,i,where i (i = 1, 2) are the first or second pair of meshing teeth.When the gear pair appears a tooth root crack,Ixc and Axc are expressed as(14)Ixc=112hc+hx3L,ifx≤gc,1122hx3L=23hx3L,ifx>gc,Axc=hc+hxL,ifx≤gc,2hxL,ifx>gc,where hc represents the distance between the root the crack and the central line of the tooth. The other parameters and formulas are given in [4, 5, 21].The overall effective meshing stiffness is given by the equations. Crack position is a tooth root of pinion. Equation (14) is input into equation (8), and then TVMS can be calculated. The time-varying meshing stiffness can be provided as shown in Figure 3.Figure 3
The time-varying meshing stiffness.


## 2.3. Dynamic Backlash

In the gear mesh process, the backlash is one of the most important factors. The backlash is the distance between two tooth profiles. In order to prevent the gears from getting stuck and ensure lubrication of gears, the backlash must exist. Rough tooth surfaces are not considered as an inherent characteristic. In the section, the compound dynamic backlash model is established considering rough tooth surface and vibration characteristic of the gear system. The sketch of backlash is shown in Figure4. The relative displacement is generated when two rough teeth surface gears mesh.Figure 4
Sketch of backlash.The dynamic backlashbh(t) is consisted of the microscopic backlash Δbt and vibration variation backlash b(t). In order to approximate rough tooth surfaces, the W-M fractal function is employed. The microscopic backlash Δbt, which represents the inherent characteristic of the rough tooth surface, can be determined as [24–26](15)Δbt=LsGcLsD−1∑n=0nmaxcos2πγnt/Lsγ2−Dn,1<D<2,γ>1,where Ls is the sampling width, γn represents space frequency of flank profile, Gc is the characteristic scale coefficient, and D represents the fractal dimension controlling the complexity of the fractal curve. The relationship between Gc and Rq is given by(16)Rq=Gc2D−12lnγ14−2D1ωk4−2D1ωh4−2D0.5,where ωk and ωh are, respectively, equivalent to lower cutoff frequency and upper cutoff frequency. When the gear system appeared vibration, the center distance of gears is changed. By changing the center distance of gears, the backlash must be generated with volatility. The vibration variation backlash b(t) and dynamic center distance a can be determined as [27, 28](17)a=a0cosα0+x2−x12+y2−y1−a0sinα02,bt=a−a0tanφ1+b0,where a0 represents initial center distance, α0 is the pressure angle, and b0 represents fixed backlash. The simplified center distance model is shown in Figure 5(a).Figure 5
(a) Simplified center distance model and (b) simplified sketch of meshing.
(a)
(b)When crack occurs on the teeth surfaces, the dynamic center distancea is increased. The dynamic backlash with crack is given by(18)bht=bt+Δbt.


## 2.4. The Friction Force

Due to the meshing contact points located in different positions, the direction of friction force must be different. Thus, the frictional force changes periodically with the changing of contact points. Friction is given by(19)Ff=∑i=1jμλiνsFmi,j=1,2,λνs=1,νs>0,0,νs=0,−1,νs<0,where the state is the single tooth meshing at j = 1, while the state is double tooth meshing at j = 2. λvs is the direction coefficient of the friction force, μ is the friction coefficient of the gear surface, and Fmi is the dynamic meshing force.v
s represents relative sliding velocity of the meshing point M of the tooth pairs in Figure 5(b), which can be determined as(20)νS=νM1sinφ1−νM2sinφ2,where vM1 and vM2 are, respectively, on the meshing point M and are described as follows:(21)νM1=ω1O1M¯,νM2=ω1O2M¯.Friction torque is determined as(22)Tf1=FfN1M¯,Tf2=FfN2M¯,where N1M¯ and N2M¯ are, respectively, the friction arms and are defined as(23)N1M¯=rb1+rb2tanα−ra22−rb22+rb1ω1t,N2M¯=rb1+rb2tanα−N1M¯.


## 2.5. Ball Bearing Model

In Figure6, the ball bearing model is shown. The outer ring is fastened to the bearing chock. The inner ring is fastened to the shaft. The rolling elements are evenly distributed between inner and outer rings. vi and vo representing the contact point velocities between the rolling elements and inner/outer rings are given by(24)νi=ωi⋅r,νo=ωo⋅R,where R and r are radii of inner and outer rings, and the angular velocities of inner and outer rings are ωi and ωo. The velocity of cage is expressed as(25)vb=vo+vi2=ωoR+ωir2.Figure 6
Rolling bearing model.Due toωi = ω and ωo = 0, the angular velocity of cage ωb is given by(26)ωb=2vbR+r=ωi⋅rR+r.The momentary angular displacementφi1 of the ith rolling element is given by(27)φi1=ωb⋅t+2πi−1Nb,i=1,2,3,…,Nb,where Nb is the number of rolling balls and ωb is the angular velocity of the cage.The deformation of theith rolling ball can be described by(28)δi1=xcosφi1+ysinφi1−γ0,where γ0 represents bearing clearance. The Hertz contact theory is employed for computing contact force. Contact force of the ith rolling element fi can be determined as(29)fi=Kbxcosφi1+ysinφi1−γ03/2⋅Hxcosφi1+ysinφi1−γ0,where Kb is the Hertz contact stiffness and H(x) is the Heaviside function. So, the bearing forces are given by(30)Fxj=∑i=1Nbficosφi1=Fcosφi1,Fyj=∑i=1Nbfisinφi1=Fsinφi1,j=1,2,3,4.


## 2.6. Mathematical Model of the GRBS

The nonlinear differential equation (33) of the GRBS are evolved from Lagrange’s equation.ct
1, ct2, cs1, and cs2 are torsion and bending damping, respectively, and are given by(31)ct1=2ξtkt11/Id+1/I1,ct2=2ξtkt21/I2+1/Ig,(32)cs1=2ξsks11/mb1+1/m1,cs2=2ξsks21/mb3+1/m2,(33)Jdθ¨d+ct1θ˙d−θ˙1+kt1θd−θ1=Td,m1x¨1+cs1x˙1−ξ2x˙b1−ξ1x˙b2+ks1x1−ξ2xb1−ξ1x˙b2=m1ρ1θ¨1sinω1t+θ1+m1ρ1ω1+θ˙12cosω1t+θ1−Ff,m1y¨1+cs1y˙11−ξ2y˙b1−ξ1y˙b2+ks1y1−ξ2yb1−ξ1yb2=m1ρ1ω1+θ˙12sinω1t+θ1−m1ρ1θ¨1cosω1t+θ1−Fm−m1g,J1+m1ρ12θ¨1+ct1θ˙1−θ˙d+kt1θ1−θd=m1ρ1sinω1t+θ1x¨1−m1ρ1cosω1t+θ1y¨1−Fmrb1+Tf1,m2x¨2+cs2x˙2−ξ4x˙b3−ξ3x˙b4+ks2x2−ξ4xb3−ξ3xb4=m2ρ2θ¨2sinω2t+θ2+m2ρ2ω2+θ˙22cosω2t+θ2+Ff,m2y¨2+cS2y˙2−ξ4y˙b3−ξ3y˙b4+ks1y2−ξ4yb3−ξ3yb4=m2ρ2ω2+θ˙22sinω2t+θ2−m2ρ2θ¨2cosω2t+θ2−Fm−m2g,J2+m2ρ22θ¨2+ct2θ˙2−θ˙g+kt2θ2−θg=m2ρ2sinω2t+θ2x¨2−m2ρ2cosω2t+θ2y¨2−Fmrb2−Tf2,mb1x¨b1+cs1ξ2−x˙1+ξ2x˙b1+ξ1x˙b2+cbx1x˙b1+ks1ξ2−x1+ξ2xb1+ξ1xb2=Fx1,mb1y¨b1+cs1ξ2−y˙1+ξ2y˙b1+ξ1y˙b2+cby1y˙b1+ks1ξ2−y1+ξ2yb1+ξ1yb2=Fy1−mb1g,mb2x¨b2+cs1ξ1−x˙1+ξ2x˙b1+ξ1x˙b2+cbx2x˙b2+ks1ξ1−x1+ξ2xb1+ξ1xb2=Fx2,mb2y¨b2+cs1ξ1−y˙1+ξ2y˙b1+ξ1y˙b2+cby2y˙b1+ks1ξ1−y1+ξ2yb1+ξ1yb2=Fy2−mb2g,mb3x¨b3+cs2ξ4−x˙2+ξ4x˙b3+ξ3x˙b4+cbx3x˙b3+ks2ξ4−x2+ξ4xb3+ξ3xb4=Fx3,mb3y¨b3+cs2ξ4−y˙2+ξ4y˙b3+ξ3y˙b4+cby3y˙b3+ks2ξ4−y2+ξ4yb3+ξ3yb4=Fy3−mb3g,mb4x¨b4+cs2ξ3−x˙2+ξ4x˙b3+ξ3x˙b4+cbx4x˙b4+ks2ξ3−x2+ξ4xb3+ξ3xb4=Fx4,mb4y¨b4+cs2ξ3−y˙2+ξ4y˙b3+ξ3y˙b4+cby4y˙b4+ks2ξ3−y2+ξ4yb3+ξ3yb4=Fy4−mb4g,Jgθ¨g+ct2θ˙g−θ˙2+kt2θg−θ2=−Tg,where ks1, ks2, kt1, and kt2 represent bending stiffness and torsion stiffness of the shafts. cbij (i = x, y) (j = 1, 2, 3, 4) are equivalent bearing damping in the directions x and y.The gear-rotor-bearing model with strong nonlinear and time-varying characteristics is established. The dynamic backlash and time-varying meshing stiffness with crack are included in the system. The main parameters are shown in Tables1 and 2.Table 1
Parameters of the gear-bearing system.
Parameter
Symbol
Numerical value
Pressure angle
α
20°
Moment of inertia
J
1/J2
7.1 × 10−3/1.39 × 10−4 kg·m2
Moment of inertia of input/output
J
d
/
J
g
1.25 × 10−4/5.75 × 10−4 kg·m2
Mass
m
1/m2
2.6/0.9 kg
Stiffness of bearings
k
1/k2
2/2 × 108 N·m−1
Teeth
z
1/z2
30/20
Module
m
4 mm
Tooth width
L
0.03 m
Mean/amplitude of the transmission error
e
0/er
2 × 10−5/3 × 10−5 m
Eccentricity
ρ
1/ρ2
3 × 10−5/2 × 10−5 m
Driving torque
T
l
270 N·m
Loading torque
T
d
160 N·m
Damping ratio of gear meshing
ξ
m
0.02
Damping ratio of shaft
ξ
s/ξt
0.1/0.1Table 2
Model parameters of the bearing.
Parameter
Symbol
Numerical value
Outer ring radius of bearing
R
1/R2
0.031/0.021 m
Inner ring radius of bearing
r
1/r2
0.02/0.015 m
Contact stiffness
K
b1/Kb2
13.34 × 109/10.56 × 109 Nm3/2
Bearing clearance
γ
01/γ02
3 × 10−5/2 × 10−5 m
Ball number
N
1/N2
14/18


## 3. Dynamic Response and Discussion

### 3.1. Comparison of Three Backlash Models

In the section, the three models about backlash are explored by the same parameters. The time-domain waveforms and frequency spectra are shown in Figures7 and 8. The waveform of fixed backlash is stable, as is shown in Figure 7(a). The vibration waveform of dynamic backlash based on the dynamic center distance model produces significant fluctuations in Figure 7(b). It is clearly seen that the vibration displacement fluctuations of the compound dynamic backlash model are the biggest in Figure 7(c). In addition, the amplitude of 10fr1 has an obvious difference among Figures 8(a)–8(c). The frequency components of compound dynamic backlash are more than the others. It is distinct that the waveforms, frequency components, and amplitude are most different for the three models.Figure 7
Vibration displacement inθ1 direction: (a) fixed backlash model, (b) backlash based on dynamic center distance model, and (c) compound dynamic backlash model (in this paper).
(a)
(b)
(c)Figure 8
Frequency spectra inθ1 direction: (a) fixed backlash model, (b) backlash based on dynamic center distance model, and (c) compound dynamic backlash model (in this paper).
(a)
(b)
(c)


### 3.2. Dynamic Response of the Gear System with a Tooth Root Crack

On the basis of the gear transmission system model equation (30), the vibration responses of the gear system with different crack depths are listed to investigate the effect on the system. Fractal dimension D is assumed to be 1.5, and friction coefficient μ = 0.05. The crack depth is set from zero to forty percent of the tooth thickness. Considering the different spindle speeds, such as 600 r/min, 1800 r/min, and 3000 r/min, the response characteristics are investigated under the different crack depths. In Figures 9(a)–9(d), the RMS of each significant parameter is presented, including the dynamic meshing force Fm, variation of dynamic center distance of gears Δa, the comprehensive elastic deformation δ(t), and the backlash bh(t) of the gear. Apparently, in Figure 7(a), it is stationary for the 600 r/min of the blue line with increasing crack depth. Within the limits of 0–30%, the RMS green curve of bh(t) is little different. When the rotational speed is 3000 r/min, the maximum value of the red line of bh(t) is crack 22%.Figure 9
RMS variations of nonlinear parameters: (a) dynamic backlash, (b) meshing force, (c) comprehensive elastic deformation, and (d) variation of center distance.
(a)
(b)
(c)
(d)The dynamic meshing forceFm increases with the increase in crack depth in the range of 0% and 30% in Figure 9(b), setting to 1800 r/min or 3000 r/min. It indicates that Fm increases with crack depth within the limits of high speed. But, it slightly decreases in the range of low speed. When the crack is severe in the range of 30% and 40%, Fm of 600 r/min and 3000 r/min has a rapid, fluctuating increase. However, Fm of 1800 r/min is a rapid fall.The RMS of comprehensive elastic deformationδ(t) speed increases with the increase in crack depth in Figure 9(c). But it is a fall with the increase in speed. The variation of center distance ΔaΔa=a−a0 is shown in Figures 9(d) and 10(a)–10(c). The larger the speed set, the smaller the variation value of center distance. An enlarged view of each speed is shown in Figure 10.Figure 10
Partial enlarged drawing of variation of center distance: (a) 600 r/min, (b) 1800 r/min, and (c) 3000 r/min.
(a)
(b)
(c)When the speed is low, the RMS ofΔa slowly increases in the range of 0% and 30%, and it grows rapidly within the limit of 30% and 40%. With increasing speed, an increase in volatility is shown in Figures 10(b) and 10(c). The value corresponding to the crack 22% is the minimum value. The value corresponding to the crack 40% is the maximum value.


### 3.3. Vibration Responses and Frequency Characteristics with Different Crack Depths

In order to study the effect of crack on dynamic responses of the system, the time-domain waveform and frequency-domain response are shown with different crack depths. The rotational speed is set to 3000 r/min. The crack depth is set to 0%, 10%, and 30% of the tooth thickness. Figures11(a)–11(e), 12(a)–12(e), and 13(a)–13(e) display the vibration displacements of the gears and the bearings, which are, respectively, at 0%, 10%, and 30% of the tooth thickness. Figures (14)–(16) show the frequency domain responses at the different crack depths.Figure 11
Time-domain waveforms of the healthy gear system in (a)y1, (b) θ1, (c) x2, (d) xb1, and (e) yb1 directions.
(a)
(b)
(c)
(d)
(e)Figure 12
Time-domain waveforms of crack 10% in (a)y1, (b) θ1, (c) x2, (d) xb1, and (e) yb1 directions.
(a)
(b)
(c)
(d)
(e)Figure 13
Time-domain waveforms of crack 30% in (a)y1, (b) θ1, (c) x2, (d) xb1, and (e) yb1 directions.
(a)
(b)
(c)
(d)
(e)Figure 14
Frequency spectra of the healthy gear in (a)y1, (b) θ1, (c) x2, (d) θ1, (e) y1, (f) xb1, (g) yb1, (h) xb3, and (i) yb3 directions.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figure 15
Frequency of the gear with crack 10% in (a)y1, (b) θ1, (c) θ1, (d) x2, (e) x2, (f) xb1, (g) yb1, (h) xb3, and (i) yb3 directions.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figure 16
Frequency of the gear with crack 30% in (a)y1, (b) θ1, (c) θ1, (d) x2, (e) x2, (f) xb1, (g) yb1, (h) xb3, and (i) yb3 directions.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)For the healthy gear system shown in Figure11(b), the fluctuation of the vibration displacement in torsional direction (θ1) is more obvious than others. It indicates that the vibration response in torsional direction (θ1) is more sensitive to nonlinear factors and excitation. In Figure 11(e), those factors have less impact on the vibration of the bearing.The time-domain waveform will exist as a pulse in theory when a tooth root crack appears. Though the variation of waveform in all directions is obvious in Figures12 and 13, a pulse is covered with the fluctuation caused by other excitation factors. So, the pulse form that should have been generated is not so obvious. The amplitude of the pinion in x2 direction is distinctly increased because of the crack fault, and the change of waveform is also obvious in Figures 12(c) and 13(c). In addition, the amplitude of the other directions has a slight increase, as is shown in Figures 12(a), 12(b), 13(a), and 13(b). The vibration response of the bearing is among those least affected in Figures 12(d), 12(e), 13(d), and 13(e).Obviously, the effect of a tooth root crack on vibration response of the gear system components is different. In the case of vibration response inx2 direction, the statistic datum from the results shows that the amplitude of crack 10% increases by 1.428 times than the healthy system. The amplitude increases by 1.735 times than the healthy system. But the rate of increase of vibration amplitude on the bearing does not exceed 15%. In addition, the time-domain waveform fluctuates more intense in x2 direction. For transverse (xb1) and longitudinal (yb1) vibrations of the bearing, the vibration waveforms become more disordered because a tooth root crack appears, as is shown in Figures 12(d), 12(e), 13(d), and 13(e). But the amplitudes of vibration are less changed.Figures(14)–(16) display, respectively, the frequency spectra of the healthy gear system, gear system with crack 10%, and gear system with crack 30%. The vibration frequency on the healthy gear system is explained in Figures 14(a)–14(i). In the healthy frequency spectra, it is obviously shown that the rotating frequency fr1 (fr1 = n1/60 = 50 Hz) and the mesh frequency fm (fm = n1z1/60 = 1500 Hz) are presented in Figure 14. However, besides fr1 and fm, the rotating frequency fr2 (fr2 = n2/60 = 75 Hz), variable stiffness frequencies fb1 (fb1 = N1r1n1/60(R1 + r1) = 274.3 Hz) and fb2 (fb1 = N2r2n2/60(R2 + r2) = 563.4 Hz), and combination frequency 0.5fm(10fr2), 2fm, 3fm, fm − fb1, fm + fb1, and fm − fb2 can be observed in the frequency spectra.The meshing frequencyfm is the highest amplitude and the (0.5fm)10fr1 is the second highest in y1 and θ1 direction, as is shown in Figures 14(a) and 14(b). Besides, multiplication frequency of meshing frequency fm can be seen in Figures 14(d) and 14(e), including 2fm, 3fm, 4fm, 5fm, 6fm, 7fm, 8fm, 9fm, 10fm, and 11fm. In order to further explore the bearing vibration frequency, Figures 14(f)–14(i) are presented. It is clearly observed that fm are the highest amplitude frequency. But the frequency components of the transverse (xb1, xb3) are more abundant than longitudinal (yb1, yb3).For crack 10%, the sidebands near multiplication frequency components can be found in Figures15(b) and 15(d). The range of the whole sideband of 8fm is 20fr2. In addition, it is a significant phenomenon for the early cracked gear system that obvious multiplication frequency of fr2 (75 Hz) is observed in Figures 15(f)–15(i). This phenomenon is not found from the healthy gear system. In bearing frequency spectra, sidebands in xb1 and xb3 directions are more obvious than in yb1 and yb3 directions. In addition, 0.5fm(10fr2) amplitude increases in all directions.As the crack depth increases, the changes in the amplitudes of meshing frequency and the multiplication frequency are not obvious. But sidebands of the multiplication frequency offm are distinctly changed by the crack. In Figures 15(c), 15(e), 16(c), and 16(e), the amplitude of the sideband gradually increases with the increase in crack depth. For crack 30%, the frequency component becomes more complex than early crack (10%) and healthy system. 0.5fm(10fr2) and fm are the dominant frequencies in all directions.The statistics data of the sidebands are shown in Tables3 and 4. In Tables 3 and 4, the amplitudes of sideband sharply increase when the crack depth changes from 10% to 20%. From 20% to 30%, the variations will become a slow increase or decrease.Table 3
Amplitude of the sideband inθ1 direction.
Crack depth (%)
8fm − 6fr2 (Hz)
Amplitude (×10−6 m)
8fm − 3fr2 (Hz)(a)
Amplitude (×10−6 m)
8fm + 3fr2 (Hz)(a)
Amplitude (×10−6 m)
8fm + 6fr2 (Hz)a
Amplitude (×10−6 m)
0
11550
≈0
11775
≈0
12225
≈0
12450
≈0
10
230.4%16.2%
0.0913
172.0%10.0%
0.1103
159.6%−24.5%
0.1489
130.0%−43.3%
0.1741
20
0.3017
0.3009
0.3866
0.4004
30
0.3503
0.3309
0.2918
0.2270
aThe first line is the frequency of sideband, respectively, and the other lines indicate the growth rate when every crack 10% increases.Table 4
Amplitude of the sideband inx2 direction.
Crack depth (%)
8fm − 6fr2 (Hz)
Amplitude (×10−8 m)
8fm − 3fr2 (Hz)(a)
Amplitude (×10−8 m)
8fm + 3fr2 (Hz)(a)
Amplitude (×10−8 m)
8fm + 3fr2 (Hz)a
Amplitude (×10−8 m)
0
11550
≈0
11775
≈0
12225
≈0
12450
≈0
10
192.9%34.3%
0.0183
132.4%38.7%
0.0238
189.7%9.0%
0.0292
192.7%−6.7%
0.0358
20
0.0536
0.0553
0.0846
0.1048
30
0.0720
0.0767
0.0922
0.0978
aThe first line is the frequency of sideband, respectively, and the other lines indicate the growth rate when every crack 10% increases.In order to further research the influences of a tooth root crack, waterfall maps of different crack depths in all directions can be seen in Figure17. The amplitude of 0.5fm(10fr2) in y1 direction increases in the range of (0%, 22%) in Figure 17(a). When the crack depth is 22%, the amplitude of 0.5fm(10fr2) is the biggest. Then the amplitude decreases in the range of (24%, 40%). The amplitude of 10fr1 only appears a slight fluctuation near crack 22%. The amplitude of fm shows less variation. In Figure 17(b), the multiplication frequency of fm is shown, including 5fm, 6fm, …, 10fm. The sidebands appear near multiplication frequency. These sidebands are irregular, and the amplitude of sidebands is different. The change rules of 0.5fm(10fr2), fm, and 10fr1 in θ1 direction are similar to those in y1 direction, as is shown in Figure 17(c). In addition, some noise frequency occurs in the lower frequency. In x2 direction, fr2 and its multiplication frequency 2fr2, 3fr2, …, 10fr2 appear and some compound frequencies fb1+ fb2, fm − fb2 also exist in Figure 17(e). The multiplication frequency of fm in x2 direction is different from that in y1 and θ1 directions (compare Figure 17(f) with Figures 17(b) and 17(d)). The amplitude of sidebands increases with crack growing in all directions, as is shown in Figures 17(b), 17(d), and 17(f).Figure 17
The waterfall maps in (a)y1, (b) y1, (c) θ1, (d) θ1, (e) x2, and (f) x2 directions.
(a)
(b)
(c)
(d)
(e)
(f)Influences of crack growing on the bearing is shown in Figures18(a)–18(d). It can be seen that the frequency changes of transverse (xb1, xb3) vibrations are more violent than those of the longitudinal (yb1, yb3) vibrations. fb1, fb2, 0.5fm, and fm are found from the bearings in xb3 direction, and the amplitude of 0.5fm(10fr2) is changed as the crack depth increases, as is shown in Figure 18(a). The number of noise frequencies in the spectra of transverse vibration is more than that in the frequency spectra of longitudinal vibration.Figure 18
The waterfall maps in (a)xb1, (b) yb1, (c) xb3, (d) yb3 directions.
(a)
(b)
(c)
(d)In Figures18(b) and 18(d), the frequency components include 0.5fm(10fr2), 10fr1, and fm.For the comprehensive comparison, the impact in the transverse vibration is more dramatic than that in the longitudinal vibration from the change of the crack, whether the bearings or the gears. Moreover, besides the sidebands, 0.5fm(10fr2) and the noise frequency, the amplitude variation of other frequency components is inconspicuous, as shown in Figure 18.


## 3.1. Comparison of Three Backlash Models

In the section, the three models about backlash are explored by the same parameters. The time-domain waveforms and frequency spectra are shown in Figures7 and 8. The waveform of fixed backlash is stable, as is shown in Figure 7(a). The vibration waveform of dynamic backlash based on the dynamic center distance model produces significant fluctuations in Figure 7(b). It is clearly seen that the vibration displacement fluctuations of the compound dynamic backlash model are the biggest in Figure 7(c). In addition, the amplitude of 10fr1 has an obvious difference among Figures 8(a)–8(c). The frequency components of compound dynamic backlash are more than the others. It is distinct that the waveforms, frequency components, and amplitude are most different for the three models.Figure 7
Vibration displacement inθ1 direction: (a) fixed backlash model, (b) backlash based on dynamic center distance model, and (c) compound dynamic backlash model (in this paper).
(a)
(b)
(c)Figure 8
Frequency spectra inθ1 direction: (a) fixed backlash model, (b) backlash based on dynamic center distance model, and (c) compound dynamic backlash model (in this paper).
(a)
(b)
(c)


## 3.2. Dynamic Response of the Gear System with a Tooth Root Crack

On the basis of the gear transmission system model equation (30), the vibration responses of the gear system with different crack depths are listed to investigate the effect on the system. Fractal dimension D is assumed to be 1.5, and friction coefficient μ = 0.05. The crack depth is set from zero to forty percent of the tooth thickness. Considering the different spindle speeds, such as 600 r/min, 1800 r/min, and 3000 r/min, the response characteristics are investigated under the different crack depths. In Figures 9(a)–9(d), the RMS of each significant parameter is presented, including the dynamic meshing force Fm, variation of dynamic center distance of gears Δa, the comprehensive elastic deformation δ(t), and the backlash bh(t) of the gear. Apparently, in Figure 7(a), it is stationary for the 600 r/min of the blue line with increasing crack depth. Within the limits of 0–30%, the RMS green curve of bh(t) is little different. When the rotational speed is 3000 r/min, the maximum value of the red line of bh(t) is crack 22%.Figure 9
RMS variations of nonlinear parameters: (a) dynamic backlash, (b) meshing force, (c) comprehensive elastic deformation, and (d) variation of center distance.
(a)
(b)
(c)
(d)The dynamic meshing forceFm increases with the increase in crack depth in the range of 0% and 30% in Figure 9(b), setting to 1800 r/min or 3000 r/min. It indicates that Fm increases with crack depth within the limits of high speed. But, it slightly decreases in the range of low speed. When the crack is severe in the range of 30% and 40%, Fm of 600 r/min and 3000 r/min has a rapid, fluctuating increase. However, Fm of 1800 r/min is a rapid fall.The RMS of comprehensive elastic deformationδ(t) speed increases with the increase in crack depth in Figure 9(c). But it is a fall with the increase in speed. The variation of center distance ΔaΔa=a−a0 is shown in Figures 9(d) and 10(a)–10(c). The larger the speed set, the smaller the variation value of center distance. An enlarged view of each speed is shown in Figure 10.Figure 10
Partial enlarged drawing of variation of center distance: (a) 600 r/min, (b) 1800 r/min, and (c) 3000 r/min.
(a)
(b)
(c)When the speed is low, the RMS ofΔa slowly increases in the range of 0% and 30%, and it grows rapidly within the limit of 30% and 40%. With increasing speed, an increase in volatility is shown in Figures 10(b) and 10(c). The value corresponding to the crack 22% is the minimum value. The value corresponding to the crack 40% is the maximum value.


## 3.3. Vibration Responses and Frequency Characteristics with Different Crack Depths

In order to study the effect of crack on dynamic responses of the system, the time-domain waveform and frequency-domain response are shown with different crack depths. The rotational speed is set to 3000 r/min. The crack depth is set to 0%, 10%, and 30% of the tooth thickness. Figures11(a)–11(e), 12(a)–12(e), and 13(a)–13(e) display the vibration displacements of the gears and the bearings, which are, respectively, at 0%, 10%, and 30% of the tooth thickness. Figures (14)–(16) show the frequency domain responses at the different crack depths.Figure 11
Time-domain waveforms of the healthy gear system in (a)y1, (b) θ1, (c) x2, (d) xb1, and (e) yb1 directions.
(a)
(b)
(c)
(d)
(e)Figure 12
Time-domain waveforms of crack 10% in (a)y1, (b) θ1, (c) x2, (d) xb1, and (e) yb1 directions.
(a)
(b)
(c)
(d)
(e)Figure 13
Time-domain waveforms of crack 30% in (a)y1, (b) θ1, (c) x2, (d) xb1, and (e) yb1 directions.
(a)
(b)
(c)
(d)
(e)Figure 14
Frequency spectra of the healthy gear in (a)y1, (b) θ1, (c) x2, (d) θ1, (e) y1, (f) xb1, (g) yb1, (h) xb3, and (i) yb3 directions.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figure 15
Frequency of the gear with crack 10% in (a)y1, (b) θ1, (c) θ1, (d) x2, (e) x2, (f) xb1, (g) yb1, (h) xb3, and (i) yb3 directions.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figure 16
Frequency of the gear with crack 30% in (a)y1, (b) θ1, (c) θ1, (d) x2, (e) x2, (f) xb1, (g) yb1, (h) xb3, and (i) yb3 directions.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)For the healthy gear system shown in Figure11(b), the fluctuation of the vibration displacement in torsional direction (θ1) is more obvious than others. It indicates that the vibration response in torsional direction (θ1) is more sensitive to nonlinear factors and excitation. In Figure 11(e), those factors have less impact on the vibration of the bearing.The time-domain waveform will exist as a pulse in theory when a tooth root crack appears. Though the variation of waveform in all directions is obvious in Figures12 and 13, a pulse is covered with the fluctuation caused by other excitation factors. So, the pulse form that should have been generated is not so obvious. The amplitude of the pinion in x2 direction is distinctly increased because of the crack fault, and the change of waveform is also obvious in Figures 12(c) and 13(c). In addition, the amplitude of the other directions has a slight increase, as is shown in Figures 12(a), 12(b), 13(a), and 13(b). The vibration response of the bearing is among those least affected in Figures 12(d), 12(e), 13(d), and 13(e).Obviously, the effect of a tooth root crack on vibration response of the gear system components is different. In the case of vibration response inx2 direction, the statistic datum from the results shows that the amplitude of crack 10% increases by 1.428 times than the healthy system. The amplitude increases by 1.735 times than the healthy system. But the rate of increase of vibration amplitude on the bearing does not exceed 15%. In addition, the time-domain waveform fluctuates more intense in x2 direction. For transverse (xb1) and longitudinal (yb1) vibrations of the bearing, the vibration waveforms become more disordered because a tooth root crack appears, as is shown in Figures 12(d), 12(e), 13(d), and 13(e). But the amplitudes of vibration are less changed.Figures(14)–(16) display, respectively, the frequency spectra of the healthy gear system, gear system with crack 10%, and gear system with crack 30%. The vibration frequency on the healthy gear system is explained in Figures 14(a)–14(i). In the healthy frequency spectra, it is obviously shown that the rotating frequency fr1 (fr1 = n1/60 = 50 Hz) and the mesh frequency fm (fm = n1z1/60 = 1500 Hz) are presented in Figure 14. However, besides fr1 and fm, the rotating frequency fr2 (fr2 = n2/60 = 75 Hz), variable stiffness frequencies fb1 (fb1 = N1r1n1/60(R1 + r1) = 274.3 Hz) and fb2 (fb1 = N2r2n2/60(R2 + r2) = 563.4 Hz), and combination frequency 0.5fm(10fr2), 2fm, 3fm, fm − fb1, fm + fb1, and fm − fb2 can be observed in the frequency spectra.The meshing frequencyfm is the highest amplitude and the (0.5fm)10fr1 is the second highest in y1 and θ1 direction, as is shown in Figures 14(a) and 14(b). Besides, multiplication frequency of meshing frequency fm can be seen in Figures 14(d) and 14(e), including 2fm, 3fm, 4fm, 5fm, 6fm, 7fm, 8fm, 9fm, 10fm, and 11fm. In order to further explore the bearing vibration frequency, Figures 14(f)–14(i) are presented. It is clearly observed that fm are the highest amplitude frequency. But the frequency components of the transverse (xb1, xb3) are more abundant than longitudinal (yb1, yb3).For crack 10%, the sidebands near multiplication frequency components can be found in Figures15(b) and 15(d). The range of the whole sideband of 8fm is 20fr2. In addition, it is a significant phenomenon for the early cracked gear system that obvious multiplication frequency of fr2 (75 Hz) is observed in Figures 15(f)–15(i). This phenomenon is not found from the healthy gear system. In bearing frequency spectra, sidebands in xb1 and xb3 directions are more obvious than in yb1 and yb3 directions. In addition, 0.5fm(10fr2) amplitude increases in all directions.As the crack depth increases, the changes in the amplitudes of meshing frequency and the multiplication frequency are not obvious. But sidebands of the multiplication frequency offm are distinctly changed by the crack. In Figures 15(c), 15(e), 16(c), and 16(e), the amplitude of the sideband gradually increases with the increase in crack depth. For crack 30%, the frequency component becomes more complex than early crack (10%) and healthy system. 0.5fm(10fr2) and fm are the dominant frequencies in all directions.The statistics data of the sidebands are shown in Tables3 and 4. In Tables 3 and 4, the amplitudes of sideband sharply increase when the crack depth changes from 10% to 20%. From 20% to 30%, the variations will become a slow increase or decrease.Table 3
Amplitude of the sideband inθ1 direction.
Crack depth (%)
8fm − 6fr2 (Hz)
Amplitude (×10−6 m)
8fm − 3fr2 (Hz)(a)
Amplitude (×10−6 m)
8fm + 3fr2 (Hz)(a)
Amplitude (×10−6 m)
8fm + 6fr2 (Hz)a
Amplitude (×10−6 m)
0
11550
≈0
11775
≈0
12225
≈0
12450
≈0
10
230.4%16.2%
0.0913
172.0%10.0%
0.1103
159.6%−24.5%
0.1489
130.0%−43.3%
0.1741
20
0.3017
0.3009
0.3866
0.4004
30
0.3503
0.3309
0.2918
0.2270
aThe first line is the frequency of sideband, respectively, and the other lines indicate the growth rate when every crack 10% increases.Table 4
Amplitude of the sideband inx2 direction.
Crack depth (%)
8fm − 6fr2 (Hz)
Amplitude (×10−8 m)
8fm − 3fr2 (Hz)(a)
Amplitude (×10−8 m)
8fm + 3fr2 (Hz)(a)
Amplitude (×10−8 m)
8fm + 3fr2 (Hz)a
Amplitude (×10−8 m)
0
11550
≈0
11775
≈0
12225
≈0
12450
≈0
10
192.9%34.3%
0.0183
132.4%38.7%
0.0238
189.7%9.0%
0.0292
192.7%−6.7%
0.0358
20
0.0536
0.0553
0.0846
0.1048
30
0.0720
0.0767
0.0922
0.0978
aThe first line is the frequency of sideband, respectively, and the other lines indicate the growth rate when every crack 10% increases.In order to further research the influences of a tooth root crack, waterfall maps of different crack depths in all directions can be seen in Figure17. The amplitude of 0.5fm(10fr2) in y1 direction increases in the range of (0%, 22%) in Figure 17(a). When the crack depth is 22%, the amplitude of 0.5fm(10fr2) is the biggest. Then the amplitude decreases in the range of (24%, 40%). The amplitude of 10fr1 only appears a slight fluctuation near crack 22%. The amplitude of fm shows less variation. In Figure 17(b), the multiplication frequency of fm is shown, including 5fm, 6fm, …, 10fm. The sidebands appear near multiplication frequency. These sidebands are irregular, and the amplitude of sidebands is different. The change rules of 0.5fm(10fr2), fm, and 10fr1 in θ1 direction are similar to those in y1 direction, as is shown in Figure 17(c). In addition, some noise frequency occurs in the lower frequency. In x2 direction, fr2 and its multiplication frequency 2fr2, 3fr2, …, 10fr2 appear and some compound frequencies fb1+ fb2, fm − fb2 also exist in Figure 17(e). The multiplication frequency of fm in x2 direction is different from that in y1 and θ1 directions (compare Figure 17(f) with Figures 17(b) and 17(d)). The amplitude of sidebands increases with crack growing in all directions, as is shown in Figures 17(b), 17(d), and 17(f).Figure 17
The waterfall maps in (a)y1, (b) y1, (c) θ1, (d) θ1, (e) x2, and (f) x2 directions.
(a)
(b)
(c)
(d)
(e)
(f)Influences of crack growing on the bearing is shown in Figures18(a)–18(d). It can be seen that the frequency changes of transverse (xb1, xb3) vibrations are more violent than those of the longitudinal (yb1, yb3) vibrations. fb1, fb2, 0.5fm, and fm are found from the bearings in xb3 direction, and the amplitude of 0.5fm(10fr2) is changed as the crack depth increases, as is shown in Figure 18(a). The number of noise frequencies in the spectra of transverse vibration is more than that in the frequency spectra of longitudinal vibration.Figure 18
The waterfall maps in (a)xb1, (b) yb1, (c) xb3, (d) yb3 directions.
(a)
(b)
(c)
(d)In Figures18(b) and 18(d), the frequency components include 0.5fm(10fr2), 10fr1, and fm.For the comprehensive comparison, the impact in the transverse vibration is more dramatic than that in the longitudinal vibration from the change of the crack, whether the bearings or the gears. Moreover, besides the sidebands, 0.5fm(10fr2) and the noise frequency, the amplitude variation of other frequency components is inconspicuous, as shown in Figure 18.


## 4. Conclusion

A nonlinear model with the crack fault, compound backlash, and dynamic stiffness has been built to investigate the coupled lateral-torsional fault features of the 16-DOF gear-rotor-bearing system. The potential energy method is employed for analyzing effective meshing stiffness, and assumed that the dynamic backlash is time varying. This time-varying backlash is composed of two parts. One is the dynamic backlash with fractal characteristics based on the microstructure of the tooth surface. The other is time-varying center distance based on the gear vibration characteristics. The comparison between the healthy gear system and crack fault system, including time-domain waveforms and frequency-domain responses, is analyzed in detail. The analysis results are shown as follows:(1)
The effect of crack growing on meshing force of the gear system is distinct at high rotational speed. With the increase in crack, different crack degrees show different vibration responses. In addition, for high speed, the meshing force slowly increases in the range of crack (0, 30%). But the meshing force rapidly increases within the limits of 30% and 40% when the rotational speeds are 600 r/min and 3000 r/min. The meshing force declines in the range of 30% and 40% when the speed is 1800 r/min. The comprehensive elastic deformationδ(t) and the variation of center distance Δaare mainly affected by rotational speed. In addition, δ(t) and Δa slowly increase with growing crack depth. Moreover, the dynamic backlash RMS exhibits peak at crack 22% when the speed is 3000 r/min. The backlash is a main reason causing the amplitude variation of 0.5fm(10fr1).(2)
The shaft frequency and multiplication frequency of the dominant frequency can be found in the frequency domain spectra when the gear system appears crack fault. And the amplitudes of these sidebands and the amplitudes of part multiplication frequency go up with the growing of the crack depth, yet the amplitude of frequency components is not changed. For this spur gear model, the pinion and shaft 2 show more obvious fault characteristics due to which sideband effect and time-domain waveforms are more complex as the crack increases.



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*Source: 1854263-2019-11-19.xml*