# Three-Dimensional Modeling and Structured Vibration Modes of Two-Stage Helical Planetary Gears Used in Cranes

Authors: Lina Zhang; Yong Wang; Kai Wu; Ruoyu Sheng
Journal: Shock and Vibration
 (2017)

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## Abstract

The dynamic investigation of helical planetary gears plays an important role in structure design as the vibration and noise are perceived negatively to the transmission quality. With consideration of the axial deformations of members, the gyroscopic effects, the time-variant meshing stiffness, and the coupling amongst stages, a three-dimensional dynamic model of the two-stage helical planetary gears is established by using of the lumped-parameter method in this paper. The model is applicable to variant number of planets in two stages, different planet phasing, and spacing configurations. Numerical simulation is conducted to detect the structured vibration modes of the equally spaced systems. Furthermore, the unique properties of these vibration modes are mathematically proved. Results show that the vibration modes of the two-stage helical planetary gears can be categorized as five classes: the rigid body mode, the axial translational-rotational mode, the radical translational mode, and the 1st-stage and the 2nd-stage planet mode.


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## Body

## 1. Introduction

Planetary gears are widely used in aerospace and wind turbine industries because of their high reliability and compactness. As the higher load carrying capacity and higher reduction ratio compared with spur planetary gear sets, the helical planetary gear sets are the norm for all automotive applications [1]. The high-frequency dynamic meshing forces generated by the operated planetary gears are typically larger than the quasi-static forces. Besides, vibration and noise induced by the dynamic forces are potentially hazard for the reliability and durability of the gears and bearings. Hence, the dynamic modeling and analysis of planetary system have great significance for designers.Many researches are conducted on the dynamics of planetary gear to reduce the vibration and noise of the system. Kahraman [2] extended a three-dimensional dynamic model of a single-stage planetary gear train to obtain the natural modes. Lin and Parker [3–5] established a translational-rotational dynamic model of a single-stage spur planetary gear and investigated the free vibration characteristics of systems with equally spaced and diametrically opposed planets, respectively. Eritenel and Parker [6] formulated the three-dimensional model of the single-stage helical planetary gears and mathematically categorized the structured modal properties. Sheng et al. [7] investigated the vibration modal properties of the double-helical planetary gear system with three-dimensional motion using numerical and analytical approach. Qian et al. [8] identified the vibration structures of a particular planetary gear used in a coal shearer and validated the model by comparing with the finite element model. Abousleiman and Velex [9] proposed an original hybrid FE/lumped-parameter model of planetary gear sets to simulate the three-dimensional dynamic behavior of planetary gears with tooth modifications and errors. Zhang et al. [10] investigated the vibration characteristics of the closed-form spur planetary gears. Most above dynamic models belong to lumped-parameter models [1–7, 10–19] and finite element models [8, 9, 17, 20–28]. The long computation times of the finite element model make lumped-parameter model more favorable. The precise modeling of the planetary gears gains more insights in the dynamics of the system. With these models, Al-Shyyab and Kahraman [29–31] solved the equations of motion semianalytically using multiterm harmonic balance method (HBM) in conjunction with inverse discrete Fourier transform and Newton–Rapson method. Chen et al. [32] investigated the dynamic response of the planetary gears with consideration of the flexibility of the internal ring gears. Chapron et al. [33] analyzed optimum profile modifications in helical and double-helical planetary gears (PGTs) with regard to dynamic mesh forces. Ericson and Parker [34] proved the accuracy of lumped-parameter and finite element models by experimentation and highlighted several design and modeling characteristics of planetary gears.Compared with the spur planetary gears, the helical planetary gears (HPGs) are more competent and smooth-operated. Nevertheless, for a HPG system, the axial dynamic forces generated by gear meshing exist ubiquitously. Thus, the axial deformations of gears should be considered in the dynamic behavior investigations. Also, numerous analytical models on single-stage helical gear dynamics can be found in the literature while research on multistage helical gears is quiet limited. The coupling stiffness amongst the adjacent stages should be considered.In the present paper, the main contributions of this paper contain two aspects. Firstly, the equations of motion of two-stage HPG used in cranes with consideration of axial deformations are derived. And secondly, the vibration characteristics of two-stage systems are investigated. The lumped-parameter model with three-dimensional motions is proposed to investigate the free vibration characteristics of HPGs. Modal properties are observed and categorized through the numerical simulation. Characteristics of each type of vibration modes are summarized. The unique properties of these vibration modes are then mathematically proved.


## 2. Three-Dimensional Dynamic Model of Planetary Gear

A two-stage HPG shown in Figure1 is considered in this study. It is composed of a 2K-H differential planetary gear train and a quasi-planetary gear set. The input power is split into two paths when it is transmitted to the differential gear set. One is the ring of the differential gear set and the other is the sun gear of the quasi-planetary gear set. Finally, the power is output by rings of two stages. The closed-form structure effectively extends the range of transmission ratio and efficiency.Figure 1
The discrete model of example HPG.The three-dimensional discrete model of the example system is proposed. Component bearings, gear meshes, and interactions between two stages are all modeled by linear springs. Three-dimensional translation and the axial rotation are considered for any member as shown in Figure2. The damping is not shown in the model. By choosing reasonable stiffness, any stage of the system in Figure 1 can be represented by Figure 2. The stiffness which equals a bearing or spline support stiffness is dependent on the specific kinematic configuration of the stage i (i=1,2) [35].Figure 2
The three-dimensional lumped-parameter model of the single-stage HPG.To obtain the relative displacements of members accurately, two kinds of three-dimensional coordinate systems shown in Figure3 are created. One is the three-dimensional absolute coordinate system OXYZ, and the other is the dynamic coordinate system oξηυ that rotate about the absolute coordinate. O and o are the theoretical installation center of sun gear and carrier, respectively. These two points superpose each other when the installation error is ignored. The unit vectors of the absolute coordinate system and the dynamic coordinate system are signed as {i′,j′,k′} and {i,j,k}, respectively. According to the coordinate transformation theory, three-dimensional translation relationships of members can be drawn from Figure 3. (1)XYZ=cos⁡Ωct-sin⁡Ωct0sin⁡Ωctcos⁡Ωct0001ξηυ,where XYZT is the three-dimensional component of the displacement vector in the absolute coordinate system and ξηυT is the component of the displacement vector in the dynamic coordinate system. Ωc is the angular speed.Figure 3
Schematic representation of coordinate systems.The corresponding acceleration of members in those two coordinate systems is(2)ξ¨η¨υ¨=cos⁡Ωctsin⁡Ωct0-sin⁡Ωctcos⁡Ωct0001X¨Y¨Z¨.The component of absolute acceleration of planet in the dynamic coordinate system is(3)ax=ξ¨-2Ωcη˙-Ωc2ξ,ay=η¨-2Ωcξ˙-Ωc2η,az=υ¨.


### 2.1. Formulation of the Equivalent Displacements

The structural member is denoted by the subscript of physical symbols and the superscripti denotes the structural member that is in stage i. The following analyses coincide with the rules that the torsional displacement is positive when it is counterclockwise and the relative displacements for gear meshes are defined positive when compressed.The relative displacement of the sun-planet mesh normal to tooth surface is(4)δsni=usi-xsisin⁡ψsni+ysicos⁡ψsni--uni+ξnisin⁡αi+ηnicos⁡αicos⁡βi+zsi-υnisin⁡βi.Similarly, the relative displacement of the ring-planet mesh normal to tooth surface is(5)δrni=uri-xrisin⁡ψrni+yricos⁡ψrni-uni-ξnisin⁡αi+ηnicos⁡αicos⁡βi+υni-zrisin⁡βi.The relative compression between planet and carrier along the line ofxc(i), yc(i), zc(i), and uc(i) is defined as(6)δcnxi=xci-ucisin⁡ψni-ξnicos⁡ψni+ηnisin⁡ψni,δcnyi=yci+ucicos⁡ψni-ξnisin⁡ψni-ηnicos⁡ψni,δcnzi=zci-υni,δcnui=uci-ηni-xcisin⁡ψni+ycicos⁡ψni.Besides, the relative compression between planet and carrier along the line ofξni and ηni is given as(7)δpnxi=ξni-xcicos⁡ψni-ycisin⁡ψni,δpnyi=ηni-uci+xcisin⁡ψni-ycicos⁡ψni.The ring gear of the 1st-stage PGT and the ring gear of 2nd-stage PGT are both a part of the shell, and they are coupled by the shell. The meshing position of each stage is different. Because the rotation speeds of planets of each stage differ, there should be a relative torsional displacement between the contact positions of rings in two stages. Similarly, the carrier of the 1st-stage PGT and the sun gear of the 2nd PGT are linked by the spline. The 3D model of the two-stage planetary gear set is shown in Figure4. Relative displacements of the adjacent members should be analyzed. Thus, the equivalent displacements of members in adjacent stages are written as(8)Xr1,2=xr1-xr2,Yr1,2=yr1-yr2Zr1,2=zr1-zr2,δr1,2=ur1-ur2Xcs1,2=xc1-xs2,Ycs1,2=yc1-ys2Zcs1,2=zc1-zs2,δcs1,2=uc1-us2.Figure 4
The 3D model of the two-stage HPG.


### 2.2. Formulation of Meshing Forces

The normal meshing forces of sun-planet and ring-planet in stagei are (9a)Fsni=ksnifδsni,bi+csniδ˙sni,(9b)Frni=krnifδrni,bi+crniδ˙rni,where fδsn(i),bi and fδrn(i),bi are the piecewise linear backlash function of sun-planet and ring-planet, respectively, and they are defined as (10)fδjni,bi=δjni-bi,δjni>bi,0,-bi≤δjni≤bi,δjni+bi,δjni<-bi,j=c,s.


### 2.3. Dynamic Equations of Motion

The displacement vector is written as(11)q=qc1︸c,qr1︸r,qs1︸s,q11︸planet  1,…,qN11︸planetN1︷stage  1,qc2︸c,qr2︸r,qs2︸s,q12︸planet  1,…,qN22︸planetN2︷stage  2T,where qj(i)=xj(i),yj(i),zj(i),uj(i), j=c,r,s; qn(i)=ξn(i),ηn(i),υn(i),un(i).The equations of motion of the two-stage helical planetary gear set under consideration of the rotational, translational, and axial displacements (a total of24+4(N1+N2) degrees of freedom) are derived by Newton’s second law and the Theorem of Moment of Momentum. The dynamic equations of motion of the example system are listed as follows.(1) Equations of the carrier in the 1st stage are(12a)mc1x¨c1-2Ωc1y˙c1-Ωc12xc1+∑cp1δ˙cnx1+∑kp1δcnx1+ccx1x˙c1+kcx1xc1+ccsx1,2X˙cs1,2+kcsx1,2Xcs1,2=0,(12b)mc1y¨c1+2Ωc1x˙c1-Ωc12yc1+∑cp1δ˙cny1+∑kp1δcny1+ccy1y˙c1+kcy1yc1+ccsy1,2Y˙cs1,2+kcsy1,2Ycs1,2=0,(12c)mc1z¨c1+∑cp1δ˙cnz1+∑kpz1δcnz1+ccz1z˙c1+kcz1zc1+ccsz1,2Z˙cs1,2+kcsz1,2Zcs1,2=0,(12d)Ic1rc12u¨c1+∑cp1δ˙cnu1+∑kp1δcnu1+ccu1u˙c1+kcu1uc1+ccsu1,2δ˙cs1,2+kcsu1,2δcs1,2=Tc1rc1.(2) Equations of the ring gear in the 1st stage are(13a)mr1x¨r1-2Ωc1y˙r1-Ωc12xr1-∑Frn1cos⁡β1sin⁡ψrn1+crx1x˙r1+krx1xr1+crx1,2X˙r1,2+krx1,2Xr1,2=0,(13b)mr1y¨r1+2Ωc1x˙r1-Ωc12yr1+∑Frn1cos⁡β1cos⁡ψrn1+cry1y˙r1+kry1yr1+cry1,2Y˙r1,2+kry1,2Yr1,2=0,(13c)mr1z¨r1+∑Frn1sin⁡β1+crz1z˙r1+krz1zr1+crz1,2Z˙r1,2+krz1,2Zr1,2=0,(13d)Ir1rr12u¨r1+∑Frn1cos⁡β1+cru1u˙r1+kru1ur1+cru1,2δ˙r1,2+kru1,2δr1,2=Tr1rr1.(3) Equations of the sun gear in the 1st stage are(14a)ms1x¨s1-2Ωc1y˙s1-Ωc12xs1-∑Fsn1sin⁡ψsn1cos⁡β1+csx1x˙s1+ksx1xs1=0,(14b)ms1y¨s1+2Ωc1x˙s1-Ωc12ys1+∑Fsn1cos⁡ψsn1cos⁡β1+csy1y˙s1+ksy1ys1=0,(14c)ms1z¨s1+∑Fsn1sin⁡β1+csz1z˙s1+ksz1zs1=0,(14d)Is1rs12u¨s1+∑Fsn1cos⁡β1+csu1u˙s1+ksu1us1=Ts1rs1.(4) Equations of planets in the 1st stage are(15a)mp1ξ¨n1-2Ωc1η˙n1-Ωc12ξn1-Fsn1cos⁡β1sin⁡αsp1+Frn1cos⁡β1sin⁡αrp1+cpx1δ˙pnx1+kpx1δpnx1=0,(15b)mp1η¨n1+2Ωc1ξ˙n1-Ωc12ηn1-Fsn1cos⁡β1cos⁡αsp1-Frn1cos⁡β1cos⁡αrp1+cpy1δ˙pny1+kpy1δpny1=0,(15c)mp1υ¨n1-Fsn1sin⁡β1-Frn1sin⁡β1-cpz1δ˙cnz1-kpz1δcnz1=0,(15d)Ip1rp12u¨p1+Fsn1cos⁡β1-Frn1cos⁡β1=0,n=1,…,N1.(5) Equations of the carrier in the 2nd stage are(16a)mc2x¨c2-2Ωc2y˙c2-Ωc22xc2+∑cp2δ˙cnxi+∑kp2δcnx2+ccx2x˙c2+kcx2xc2=0,(16b)mc2y¨c2-2Ωc2x˙c2-Ωc22yc2+∑cp2δ˙cny2+∑kp2δcny2+ccy2y˙c2+kcy2yc2=0,(16c)mc2z¨c2+∑cp2δ˙cnz2+∑kpz2δcnz2+ccz2z˙c2+kcz2zc2=0,(16d)Ic2rc22u¨ci+∑cp2δ˙cnu2+∑kp2δcnu2+ccu2u˙c2+kcu2uc2=Tc2rc2.(6) Equations of the ring gear in the 2nd stage are(17a)mr2x¨r2-2Ωc2y˙r2-Ωc22xr2-∑Frn2cos⁡β2sin⁡ψrn2+crx2x˙r2+krx2xr1-crx1,2X˙r1,2-krx1,2Xr1,2=0,(17b)mr2y¨r2+2Ωc2x˙r2-Ωc22yr1+∑Frn1cos⁡β1cos⁡ψrn1+cry1y˙r1+kry1yr1-cry1,2Y˙r1,2-kry1,2Yr1,2=0,(17c)mr2z¨r2+∑Frn2sin⁡β2+crz2z˙r2+krz2zr2-crz1,2Z˙r1,2-krz1,2Zr1,2=0,(17d)Ir2rr22u¨r2+∑Frn2cos⁡β2+cru2u˙r2+kru2ur2-cru1,2δ˙r1,2-kru1,2δr1,2=Tr1rr2.(7) Equations of the sun gear in the 2nd stage are(18a)ms2x¨s2-2Ωc2y˙s2-Ωc22xs2-∑Fsn2sin⁡ψsn2cos⁡β2+cs2x˙s2+ks2xs2-ccsx1,2X˙cs1,2-kcsx1,2Xcs1,2=0,(18b)ms2y¨s2+2Ωc2x˙s2-Ωc22ys2+∑Fsn2cos⁡ψsn2cos⁡β2+cs2y˙s2+ks2ys2-ccsy1,2Y˙cs1,2-kcsy1,2Ycs1,2=0,(18c)msiz¨si+∑Fsnisin⁡βi+csziz˙si+kszizsi-ccsz1,2Z˙cs1,2-kcsz1,2Zcs1,2=0,(18d)Is2rs22u¨s2+∑Fsn2cos⁡β2+c˙su2u˙s2+ksu2us2-ccsu1,2δ˙cs1,2-kcsu1,2δcs1,2=Ts2rs2.(8) Equations of the planet in the 2nd stage are(19a)mp2ξ¨n2-2Ωc2η˙n2-Ωc22ξn2-Fsn2cos⁡β2sin⁡αsp2+Frn2cos⁡β2sin⁡αrp2+cpx2δ˙pnx2+kpx2δpnx2=0(19b)mp2η¨n2+2Ωc2ξ˙n2-Ωc22ηn2-Fsn2cos⁡β2cos⁡αsp2-Frn2cos⁡β2cos⁡αrp2+cpy2δ˙pny2+kpy2δpny2=0(19c)mp2υ¨n2-Fsn2sin⁡β2-Frn2sin⁡β2-cpz2δ˙cnz2-kpz2δcnz2=0(19d)Ip2rp22u¨p2+Fsn2cos⁡β2-Frn2cos⁡β2=0,n=1,…,N2,where the summation index n ranges from 1 to Ni in equations of stage i.Assembling the dynamic equations in matrix form yields(20)Mq¨+cb+cm+ωcGq˙+Kb+Km-ωc2KΩq=Tt,whereM is the positive definite mass matrix and ωc is the velocity matrix, Kb is the bearing stiffness matrix, and Km is the meshing stiffness matrix. Cb and Cm are the corresponding damping terms.G is the gyroscopic matrix and KΩ is the stiffness matrix induced by gyroscopic effect. The matrix components are given in Appendix.


## 2.1. Formulation of the Equivalent Displacements

The structural member is denoted by the subscript of physical symbols and the superscripti denotes the structural member that is in stage i. The following analyses coincide with the rules that the torsional displacement is positive when it is counterclockwise and the relative displacements for gear meshes are defined positive when compressed.The relative displacement of the sun-planet mesh normal to tooth surface is(4)δsni=usi-xsisin⁡ψsni+ysicos⁡ψsni--uni+ξnisin⁡αi+ηnicos⁡αicos⁡βi+zsi-υnisin⁡βi.Similarly, the relative displacement of the ring-planet mesh normal to tooth surface is(5)δrni=uri-xrisin⁡ψrni+yricos⁡ψrni-uni-ξnisin⁡αi+ηnicos⁡αicos⁡βi+υni-zrisin⁡βi.The relative compression between planet and carrier along the line ofxc(i), yc(i), zc(i), and uc(i) is defined as(6)δcnxi=xci-ucisin⁡ψni-ξnicos⁡ψni+ηnisin⁡ψni,δcnyi=yci+ucicos⁡ψni-ξnisin⁡ψni-ηnicos⁡ψni,δcnzi=zci-υni,δcnui=uci-ηni-xcisin⁡ψni+ycicos⁡ψni.Besides, the relative compression between planet and carrier along the line ofξni and ηni is given as(7)δpnxi=ξni-xcicos⁡ψni-ycisin⁡ψni,δpnyi=ηni-uci+xcisin⁡ψni-ycicos⁡ψni.The ring gear of the 1st-stage PGT and the ring gear of 2nd-stage PGT are both a part of the shell, and they are coupled by the shell. The meshing position of each stage is different. Because the rotation speeds of planets of each stage differ, there should be a relative torsional displacement between the contact positions of rings in two stages. Similarly, the carrier of the 1st-stage PGT and the sun gear of the 2nd PGT are linked by the spline. The 3D model of the two-stage planetary gear set is shown in Figure4. Relative displacements of the adjacent members should be analyzed. Thus, the equivalent displacements of members in adjacent stages are written as(8)Xr1,2=xr1-xr2,Yr1,2=yr1-yr2Zr1,2=zr1-zr2,δr1,2=ur1-ur2Xcs1,2=xc1-xs2,Ycs1,2=yc1-ys2Zcs1,2=zc1-zs2,δcs1,2=uc1-us2.Figure 4
The 3D model of the two-stage HPG.


## 2.2. Formulation of Meshing Forces

The normal meshing forces of sun-planet and ring-planet in stagei are (9a)Fsni=ksnifδsni,bi+csniδ˙sni,(9b)Frni=krnifδrni,bi+crniδ˙rni,where fδsn(i),bi and fδrn(i),bi are the piecewise linear backlash function of sun-planet and ring-planet, respectively, and they are defined as (10)fδjni,bi=δjni-bi,δjni>bi,0,-bi≤δjni≤bi,δjni+bi,δjni<-bi,j=c,s.


## 2.3. Dynamic Equations of Motion

The displacement vector is written as(11)q=qc1︸c,qr1︸r,qs1︸s,q11︸planet  1,…,qN11︸planetN1︷stage  1,qc2︸c,qr2︸r,qs2︸s,q12︸planet  1,…,qN22︸planetN2︷stage  2T,where qj(i)=xj(i),yj(i),zj(i),uj(i), j=c,r,s; qn(i)=ξn(i),ηn(i),υn(i),un(i).The equations of motion of the two-stage helical planetary gear set under consideration of the rotational, translational, and axial displacements (a total of24+4(N1+N2) degrees of freedom) are derived by Newton’s second law and the Theorem of Moment of Momentum. The dynamic equations of motion of the example system are listed as follows.(1) Equations of the carrier in the 1st stage are(12a)mc1x¨c1-2Ωc1y˙c1-Ωc12xc1+∑cp1δ˙cnx1+∑kp1δcnx1+ccx1x˙c1+kcx1xc1+ccsx1,2X˙cs1,2+kcsx1,2Xcs1,2=0,(12b)mc1y¨c1+2Ωc1x˙c1-Ωc12yc1+∑cp1δ˙cny1+∑kp1δcny1+ccy1y˙c1+kcy1yc1+ccsy1,2Y˙cs1,2+kcsy1,2Ycs1,2=0,(12c)mc1z¨c1+∑cp1δ˙cnz1+∑kpz1δcnz1+ccz1z˙c1+kcz1zc1+ccsz1,2Z˙cs1,2+kcsz1,2Zcs1,2=0,(12d)Ic1rc12u¨c1+∑cp1δ˙cnu1+∑kp1δcnu1+ccu1u˙c1+kcu1uc1+ccsu1,2δ˙cs1,2+kcsu1,2δcs1,2=Tc1rc1.(2) Equations of the ring gear in the 1st stage are(13a)mr1x¨r1-2Ωc1y˙r1-Ωc12xr1-∑Frn1cos⁡β1sin⁡ψrn1+crx1x˙r1+krx1xr1+crx1,2X˙r1,2+krx1,2Xr1,2=0,(13b)mr1y¨r1+2Ωc1x˙r1-Ωc12yr1+∑Frn1cos⁡β1cos⁡ψrn1+cry1y˙r1+kry1yr1+cry1,2Y˙r1,2+kry1,2Yr1,2=0,(13c)mr1z¨r1+∑Frn1sin⁡β1+crz1z˙r1+krz1zr1+crz1,2Z˙r1,2+krz1,2Zr1,2=0,(13d)Ir1rr12u¨r1+∑Frn1cos⁡β1+cru1u˙r1+kru1ur1+cru1,2δ˙r1,2+kru1,2δr1,2=Tr1rr1.(3) Equations of the sun gear in the 1st stage are(14a)ms1x¨s1-2Ωc1y˙s1-Ωc12xs1-∑Fsn1sin⁡ψsn1cos⁡β1+csx1x˙s1+ksx1xs1=0,(14b)ms1y¨s1+2Ωc1x˙s1-Ωc12ys1+∑Fsn1cos⁡ψsn1cos⁡β1+csy1y˙s1+ksy1ys1=0,(14c)ms1z¨s1+∑Fsn1sin⁡β1+csz1z˙s1+ksz1zs1=0,(14d)Is1rs12u¨s1+∑Fsn1cos⁡β1+csu1u˙s1+ksu1us1=Ts1rs1.(4) Equations of planets in the 1st stage are(15a)mp1ξ¨n1-2Ωc1η˙n1-Ωc12ξn1-Fsn1cos⁡β1sin⁡αsp1+Frn1cos⁡β1sin⁡αrp1+cpx1δ˙pnx1+kpx1δpnx1=0,(15b)mp1η¨n1+2Ωc1ξ˙n1-Ωc12ηn1-Fsn1cos⁡β1cos⁡αsp1-Frn1cos⁡β1cos⁡αrp1+cpy1δ˙pny1+kpy1δpny1=0,(15c)mp1υ¨n1-Fsn1sin⁡β1-Frn1sin⁡β1-cpz1δ˙cnz1-kpz1δcnz1=0,(15d)Ip1rp12u¨p1+Fsn1cos⁡β1-Frn1cos⁡β1=0,n=1,…,N1.(5) Equations of the carrier in the 2nd stage are(16a)mc2x¨c2-2Ωc2y˙c2-Ωc22xc2+∑cp2δ˙cnxi+∑kp2δcnx2+ccx2x˙c2+kcx2xc2=0,(16b)mc2y¨c2-2Ωc2x˙c2-Ωc22yc2+∑cp2δ˙cny2+∑kp2δcny2+ccy2y˙c2+kcy2yc2=0,(16c)mc2z¨c2+∑cp2δ˙cnz2+∑kpz2δcnz2+ccz2z˙c2+kcz2zc2=0,(16d)Ic2rc22u¨ci+∑cp2δ˙cnu2+∑kp2δcnu2+ccu2u˙c2+kcu2uc2=Tc2rc2.(6) Equations of the ring gear in the 2nd stage are(17a)mr2x¨r2-2Ωc2y˙r2-Ωc22xr2-∑Frn2cos⁡β2sin⁡ψrn2+crx2x˙r2+krx2xr1-crx1,2X˙r1,2-krx1,2Xr1,2=0,(17b)mr2y¨r2+2Ωc2x˙r2-Ωc22yr1+∑Frn1cos⁡β1cos⁡ψrn1+cry1y˙r1+kry1yr1-cry1,2Y˙r1,2-kry1,2Yr1,2=0,(17c)mr2z¨r2+∑Frn2sin⁡β2+crz2z˙r2+krz2zr2-crz1,2Z˙r1,2-krz1,2Zr1,2=0,(17d)Ir2rr22u¨r2+∑Frn2cos⁡β2+cru2u˙r2+kru2ur2-cru1,2δ˙r1,2-kru1,2δr1,2=Tr1rr2.(7) Equations of the sun gear in the 2nd stage are(18a)ms2x¨s2-2Ωc2y˙s2-Ωc22xs2-∑Fsn2sin⁡ψsn2cos⁡β2+cs2x˙s2+ks2xs2-ccsx1,2X˙cs1,2-kcsx1,2Xcs1,2=0,(18b)ms2y¨s2+2Ωc2x˙s2-Ωc22ys2+∑Fsn2cos⁡ψsn2cos⁡β2+cs2y˙s2+ks2ys2-ccsy1,2Y˙cs1,2-kcsy1,2Ycs1,2=0,(18c)msiz¨si+∑Fsnisin⁡βi+csziz˙si+kszizsi-ccsz1,2Z˙cs1,2-kcsz1,2Zcs1,2=0,(18d)Is2rs22u¨s2+∑Fsn2cos⁡β2+c˙su2u˙s2+ksu2us2-ccsu1,2δ˙cs1,2-kcsu1,2δcs1,2=Ts2rs2.(8) Equations of the planet in the 2nd stage are(19a)mp2ξ¨n2-2Ωc2η˙n2-Ωc22ξn2-Fsn2cos⁡β2sin⁡αsp2+Frn2cos⁡β2sin⁡αrp2+cpx2δ˙pnx2+kpx2δpnx2=0(19b)mp2η¨n2+2Ωc2ξ˙n2-Ωc22ηn2-Fsn2cos⁡β2cos⁡αsp2-Frn2cos⁡β2cos⁡αrp2+cpy2δ˙pny2+kpy2δpny2=0(19c)mp2υ¨n2-Fsn2sin⁡β2-Frn2sin⁡β2-cpz2δ˙cnz2-kpz2δcnz2=0(19d)Ip2rp22u¨p2+Fsn2cos⁡β2-Frn2cos⁡β2=0,n=1,…,N2,where the summation index n ranges from 1 to Ni in equations of stage i.Assembling the dynamic equations in matrix form yields(20)Mq¨+cb+cm+ωcGq˙+Kb+Km-ωc2KΩq=Tt,whereM is the positive definite mass matrix and ωc is the velocity matrix, Kb is the bearing stiffness matrix, and Km is the meshing stiffness matrix. Cb and Cm are the corresponding damping terms.G is the gyroscopic matrix and KΩ is the stiffness matrix induced by gyroscopic effect. The matrix components are given in Appendix.


## 3. Vibration Modes of the Example System

The damping and external excitation are ignored to investigate the free vibration characteristics of the system. Besides, Coriolis acceleration of gears is neglected and all gyroscopic effects are excluded when the carrier speed is low [3]. These assumptions are valid for the example system, since the system is used in cranes with low velocity of carriers. The simplified equations of motion and associated eigenvalue equations are (21)Mq¨+Kb+Kmq=0,(22)Kb+Km-ωi2Mϕi=0,where ωi is the natural frequency and the corresponding modal vectors ϕi are expressed as ϕi=[pc1,pr1,ps1,p11,…,pN11,∣pc2,pr2,ps2,p12,…,pN2(2)]. The mesh stiffness of the system is calculated by Ishikawa tooth deformation calculation formula and stiffness calculation formula. Because rings of two stages are coupled by the shell and the carrier of the 1st-stage PGT and the sun gear of the 2nd PGT are linked by the spline. The coupling stiffness amongst rings of two stages equals the stiffness of the shell between two rings and stiffness between the carrier of the 1st-stage PGT and the sun gear of the 2nd-stage PGT is dependent on the spline. The system parameters are listed in Table 1.Table 1
Parameters of the example systems (i=1,2).
Parameter High-speed stage Ring Low-speed stage Sun Planet Carrier Sun Planet Mass (Kg) 0.92 0.77 4.04 19 1.73 0.94  I (Kg·mm2) 83.6 507.65 7814.7 156997 491.2 524.43  Rotational radius (mm) 11 32.25 57.8 120 21.5 30  Pressure angle 20° 20°  Helix angle 5° 9°  Mesh stiffness (N/m) k s n 1 = k r n 1 = 6.5 × 1 0 8 k s n 2 = k r n 2 = 8 × 1 0 8  Bearing stiffness (N/m) k j x i = k j y i = 1.2 × 1 0 9 (j=c,s) krxi=kryi=1.3×109 kpxi=kpyi=1×109 kjzi=1.08×109j=c,r,s  Torsional stiffness (N/m) k j u i = 0   j = s , r; kcu1=0, kcu2=1×109  Coupling stiffness (N/m) k r x 1,2 = k r y 1,2 = 2 × 1 0 9, krz1,2=2.3×1010, kru1,2=1.5×1010 kcsx1,2=kcsy1,2=2×109, krz1,2=3.5×1010, kcsu1,2=1.2×1010It is assumed that planets in both stages are equally spaced. The systems withN1=3 and N2 = 3~9 are simulated here to illustrate the structured vibration modes of two-stage HPGs. According to the modal theory, natural modes are grouped according to the multiplicity of the natural frequencies. The natural frequencies and mode types of the system are shown in Table 2.Table 2
Natural frequencies (Hz) and mode types for the example HPGs.
Multiplicity N 2 Mode type 3 4 5 6 7 8 9 m = 1 0 0 0 0 0 0 0 Axial translational-rotational mode 496.02 509.75 517.49 521.95 524.43 525.61 525.91 962.35 958.11 953.4 948.5 943.54 938.59 933.7 1934.2 1871.8 1813.5 1758.6 1706.9 1658.6 1613.9 2060.1 1994.5 1910.6 1840.5 1781.7 1731.4 1687.3 2108.3 2068.5 2066.2 2065.2 2064.6 2064 2063.5 3106.9 3106.6 3106.6 3106.5 3106.4 3106.3 3106.2 3392 3393.6 3394.1 3394.3 3394.3 3394.3 3394.2 4063.9 4111.3 4121.1 4124.6 4126.3 4127.4 4128 4180.5 4238.7 4283.3 4312.6 4333.4 4349.2 4361.8 4853.5 5138.5 5415.4 5664.8 5883.1 6063.3 6127.6 6140.1 6140.8 6141.6 6142.8 6145.7 6159.9 6266.9 6425 6425 6425 6425 6425 6425 6425 7622 7811.8 8008.5 8213.8 8427.1 8646.2 8867.7 8209.9 8748.5 9065.7 9130.9 9146.6 9153.9 9159.3 8934.8 9159 9305.8 9497 9507.9 9513.7 9520.8 9214.5 9345.8 9506 9583.4 9838.5 10086 10277 9498.4 9501.9 9728.5 10203 10277 10277 10324 10277 10277 10277 10280 10701 11186 11658 25977 25981 25985 25990 25994 25999 26004  m = 2 1187.9 1179.4 1171 1162.6 1154.2 1146 1137.8 Radical translational mode 1525.6 1566.3 1592.7 1609.1 1618.5 1622.9 1623.8 1961.5 1954 1946.6 1939.4 1932.3 1925.4 1918.7 2191.6 2214.6 2236.1 2255.3 2272 2286 2297.5 3280.2 3280.8 3281.6 3282.4 3283.2 3284.1 3285 4144.1 4126.6 4110.7 4097.4 4087.2 4079.6 4074.5 4705.3 4754.8 4798.1 4831 4853.8 4868.7 4878.4 5214.7 5217.2 5244.9 5294.1 5359.8 5437 5521 5897.9 5898.1 5898.3 5898.6 5899 5899.5 5900.3 6164.4 6168.1 6170.5 6172.3 6173.8 6175.2 6176.5 6483.5 6489.4 6493 6495.4 6497.1 6498.5 6499.6 7241.6 7283.9 7324.9 7365.2 7404.7 7443.3 7481 7545.7 7798 7962.5 8036.5 8073.2 8098.7 8120.6 8131.9 8186.1 8310.4 8510.9 8737.3 8965.9 9190.4  m = N 2 - 3 — 4155.7 4155.7 4155.7 4155.7 4155.7 4155.7 Planet mode 5897.3 5897.3 5897.3 5897.3 5897.3 5897.3 7121.8 7121.8 7121.8 7121.8 7121.8 7121.8 8074.7 8074.7 8074.7 8074.7 8074.7 8074.7From Table2, the vibration modes can be recognized as 3 groups: the axial translational-rotational mode, the radical translational mode, and the second-stage planet mode (N2>3). Following results can be concluded.(a)
The first-order natural frequency isω1=0 and the associated vibration mode is the rigid body mode. The multiplicity of this mode is 1. As shown in Table 2, this mode can be concluded in the axial translational-rotational mode.(b)
Twenty natural frequencies always have multiplicitym=1 for different N1 and N2. The corresponding vibration mode is the axial translational-rotational mode. This mode type is mainly characterized by the motions of central members which only rotate and translate axially. Furthermore, deflections of planets in the same stage are identical. The typical axial translational-rotational mode with N1=3, N2=4, and ω=5138.5 Hz is shown in Figure 5.(c)
There are always fourteen pairs of natural frequencies with multiplicitym=2 for various N1 and N2, and the corresponding vibration mode is the radical translational mode. In these modes, all central members have pure translation radically. Figure 6 shows the mode shape of radical translational mode with N1=3, N2=4, and ω=2214.6 Hz.(d)
As shown in Table2, four degenerate natural frequencies with multiplicity m=N2-3 are always calculated when N2>3. Associated modes are the 2nd-stage planet modes. In these modes, all central members are stationary and only planets in the 2nd stage have movements. The planets move in all four degrees of freedom (DOFs), and their modal deflection is a scalar multiple to that of the arbitrarily selected first planet. Figure 7 illustrates the typical second-stage planet mode.(e)
It is concluded that the natural frequency varies monotonically as additional planets in stage two introduced.Figure 5
Mode shapes at 5138.5 Hz withN1=3 and N2=4.
(a)
Stage 1 (b)
Stage 2Figure 6
Mode shapes at 2214.6 Hz withN1=3 and N2=4.
(a)
Stage 1 (b)
Stage 2Figure 7
Mode shapes at 7121.8 Hz withN1=3 and N2=4.
(a)
Stage 1 (b)
Stage 2In most planetary gears, the number of planets in any stage is more than 3. This structure brings the systems distinguishing advantage of power-split. In the following discussion, the eigensolution properties summarized in the example are analytically shown to be true for general HPGs.


## 4. Analytical Characterization of Vibration Modes

The eigenvalue matrix equations (21) are expanded to (6+N1+N2) groups of equations associated with the individual members(23a)Kcb1+Kcs1c+∑Kc11n-ωi2Mc1pc1+∑Kc21npn1-Kcs2sps2=0,(23b)Krb1+Kr1r+∑Kr11n-ωi2Mr1pr1+∑Kr21npn1-Kr2rpr2=0,(23c)Ksb1+∑Ks11n-ωi2Ms1ps1+∑Ks21npn1=0,(23d)Kc21nTpc1-Kr21nTpr1+Ks21nTps1+Kpp1n-ωi2Mp1pn1=0,(24a)Kcb2+∑Kc12n-ωi2Mc2pc2+∑Kc22npn2=0,(24b)Krb2+Kr2r+∑Kr12n-ωi2Mr2pr2+∑Kr22npn2-Kr1rpr1=0,(24c)Ksb2+Kcs2s+∑Ks12n-ωi2Ms2ps2+∑Ks22npn1-Kcs1cpc1=0,(24d)Kc22nTpc2-Kr22nTpr2+Ks22nTps2+Kpp2n-ωi2Mp2pn2=0,where0 has dimension 4 × 1.


### 4.1. Axial Translational-Rotational Mode

According to the above vibration mode analysis, the axial translational-rotational mode has the following characteristics:(a) The corresponding multiplicity of natural frequency is 1.(b) The radical deflections of carrier, ring gear, and sun gear in two stages are zero(25)pji=xji,yji,zji,ujiT=0,0,zji,ujiT,j=c,r,s;i=1,2.(c) The modal deflections of planets in the same stage are identical(26)pni=ξni,ηni,υni,uniT=p1i=ξ1i,η1i,υ1i,u1iT,i=1,2.The eigenvectors of the 1st stage and 2nd stage can be expressed as(27a)ϕi1=pc1,pr1,ps1,p11,…,p11T(27b)ϕi2=pc2,pr2,ps2,p12,…,p12T.Inserting (25)–((27a) and (27b)) into (23a) yields(28a)kcz1+kcsz1,2+N1kpz1-ωi2Ic1rc12zc1-N1kpz1z11-kcsz1,2zs1=0,(28b)kcu1+kcsu1,2+N1kp1-ωi2Ic1rc12uc1-N1kp1u11-kcsu1,2us1=0.Similarly, inserting (25)–(27a) into (23b)-(23c) yields(29a)krz1+krz1,2+N1krn1s2β1-ωi2Ir1rr12zr1-N1sβ1cβ1ur1-krz1,2zr2+N1krn1-ζ11sβ1cβ1sα1+η11sβ1cβ1cα1-s2β1υ11+sβ1cβ1u11=0,(29b)kru1+kru1,2+N1krn1c2β1-ωi2Ir1rr12ur1-N1sβ1cβ1zr1-kru1,2ur2+N1krn1ζ11c2β1sα1-η11c2β1cα1+sβ1cβ1υ11-c2β1u11=0,(30a)ksz1+N1ksn1s2β1-ωi2Is1rs12zs1+N1sβ1cβ1us1-N1ksn1ζ11sβ1cβ1sα1+η11sβ1cβ1cα1+υ11s2β1-u11sβ1cβ1=0,(30b)ksu1+N1ksn1c2β1-ωi2Is1rs12us1+N1sβ1cβ1zs1-N1ksn1ζ11c2β1sα1+η11c2β1cα1+υ11sβ1cβ1-u11c2β1=0,where sψjn(i)=sin⁡ψjn(i), cψjn(i)=cos⁡ψjn(i), sα(i)=sin⁡α(i), cα(i)=cos⁡α(i), sβ(i)=sin⁡β(i), and cβ(i)=cos⁡β(i). Equation (23d) can be simplified as (31)Kc21nTpc1-Kr21nTpr1+Ks21nTps1+Kpp1n-ωi2Mp1pn1=Kc211Tpc1-Kr211Tpr1+Ks211Tps1+Kpp1-ωi1Mp1p11=0.Inserting (25)–(27b) into (24a), we get(32a)kcz2+N2kpz2-ωi2Ic2rc22zc2-N2kpz2z12=0,(32b)kcu2+N2kp2-ωi2Ic2rc22uc2-N2kp2u12=0.In the same way, (24b)-(24c) reduce to(33a)krz2+krz1,2+N2krn2s2β2-ωi2Ir2rr22zr2-N2sβ2cβ2ur2+N2krn2-ζ12sβ2cβ2sα2+η12sβ2cβ2cα2-s2β2υ12+sβ2cβ2u12-krz1,2zr1=0,(33b)kru2+kru1,2+N2krn2c2β2-ωi2Ir2rr22ur1-N2sβ2cβ2zr2+N2krn2ζ12c2β2sα2-η12c2β2cα2+sβ2cβ2υ12-c2β2u12-kru1,2ur1=0,(34a)ksz2+kcsz1,2+N2ksn2s2β2-ωi2Is2rs22zs2+N2sβ2cβ2us2-N2ksn2ζ12sβ2cβ2sα2+η12sβ2cβ2cα2+υ12s2β2-u12sβ2cβ2-kcsz1,2zc1=0,(34b)ksu2+kcsu1,2+N2ksn2c2β2-ωi2Is2rs22us1+N2sβ2cβ2zs2-N2ksn2ζ12c2β2sα2+η12c2β2cα2+υ12sβ2cβ2-u12c2β2-kcsu1,2uc1=0.Applying the mode characteristics (b) and (c), (24d) becomes(35)Kc22nTpc2-Kr22nTpr2+Ks22nTps2+Kpp2n-ωi2Mp2pn2=Kc221Tpc2-Kr221Tpr2+Ks221Tps1+Kpp2-ωi2Mp2p12=0.The above equations imply that (23a)–(24d) reduce to a 20-degree-of-freedom eigenvalue problem consisting of (28a)–(35). The modal vectorq is determined by the reduced eigenvalue (36)zc1,uc1,zr1,ur1,zs1,us1,ξ11,η11,υ11,u11,zc2,uc2,zr2,ur2,zs2,us2,ξ12,η12,υ12,u12T.Twenty eigensolutions are derived from this reduced eigenvalue problem. Therefore, there are 20 axial translational-rotational modes and the corresponding natural frequency is distinct.


### 4.2. Radical Translational Mode

To sum up, the radical translational mode has the following characteristics:(a)
The natural frequency multiplicity of the radical translational mode is 2. There is a pair of orthonormal vibration modes associated with each natural frequency(37)ϕi=pc1,pr1,ps1,p11,…,pN11,pc2,pr2,ps2,p12,…,pN22T,ϕ-i=p-c1,p-r1,p-s1,p-11,…,p-N11,p-c2,p-r2,p-s2,p-12,…,p-N22T.(b)
The axial translation and rotation of central members in both stages are zero. Besides, the relation of the modal deflections of central members in a pair of orthonormal vibration modes is(38)pji=xji,yji,zji,ujiT=xji,yji,0,0,T,p-ji=-yji,xji,zji,ujiT=-yji,xji,0,0,T,j=c,r,s;i=1,2.(c)
The modal deflections of planets of same stage in a pair of orthonormal vibration modes can be expressed as(39)pni=p1icos⁡ψni+p-1isin⁡ψnip-ni=-p1isin⁡ψn1+p-1icos⁡ψni,wherepn(i)=ξn(i),ηn(i),υn(i),un(i)T and p-n(i)=ξ-n(i),η-n(i),υ-n(i),u-n(i)T are the modal deflections of planets in a pair of orthonormal vibration modes.Thus, a pair of candidate orthonormal radical translational modes of the 1st stage is given by(40a)ϕi1=pc1,pr1,ps1,cos⁡ϕ11p11+sin⁡ϕ11p-11,…,cos⁡ϕN11p11+sin⁡ϕN11p-11T,(40b)ϕ-i1=p-c1,p-r1,p-s1,cos⁡ϕ11p-11-sin⁡ϕ11p11,…,cos⁡ϕN11p-11-sin⁡ϕN11p11T.Analogous to the analyses of the axial translational-rotational mode, substituting (38)–(40b) into (23a) yields(41a)kcx1+kcsx1,2+N1kp1-ωi2mc1xc1-N12kp1ξ11-η-11-kcsx1,2xs2=0,(41b)kcy1+kcsy1,2+N1kp1-ωi2mc1xc1-N12kp1ξ-11+η11-kcsx1,2xs2=0.Similarly, the following (42a)–(43b) are derived by inserting (38)–(40b) into (23b)-(23c).(42a)krx1+krx1,2+N12krn1-ωi2mr1xr1-krx1,2xr2-N12krn1ζ11s2α1+sα1cα1ζ-11-η11-η-11c2α1c2β1=0,(42b)kry1+kry1,2+N12krn1-ωi2mr1yr1-kry1,2yr2+N12krn1ζ-11s2α1+sα1cα1ζ11-η-11-η11c2α1c2β1=0,(43a)ksx1+N12ksn1-ωi2ms1xs1+N12ksn1ζ11s2α1+sα1cα1ζ-11+η11+η-11c2α1c2β1=0,(43b)ksy1+N12ksn1-ωi2ms1ys1-N12ksn1ζ-11s2α1+sα1cα1ζ11+η-11+η11c2α1c2β1=0.Insertingϕi(1) and ϕ-i(1) into (24d), we get(44a)Ln1=Kc21nTpc1-Kr21nTpr1+Ks21nTps1+Kpp1-ωi2Mp1pn1=0,(44b)L-n1=Kc21nTp-c1-Kr21nTp-r1+Ks21nTp-s1+Kpp1-ωi2Mp1p-n1=0,where Ln(1) and L-n(1) are the introduced notations. Applying the mode characteristics of the radical translational modes (b) and (c), the above equations can be expressed as(45a)Ln1=L11cos⁡ψn1+L-11sin⁡ψn1,(45b)L-n1=-L11sin⁡ψn1+L-11cos⁡ψn1.The eigenvalue problems of the 1st stage can be reduced to fourteen independent equations: six equations from (41a)–(43b) and eight equations from (45a) and (45b). Thus, seven natural frequencies with multiplicity two are obtained. The reduced eigenvector of the 1st stage can be expressed as xc(1),yc(1),xr(1),yr(1),xs(1),ys(1),ξ1(1),η1(1),υ1(1),u1(1),ξ-1(1),η-1(1),υ-1(1), u-1(1)T. With similar operations on the eigenvalue matrix equations (24a), (24b), (24c), and (24d), seven natural frequencies with multiplicity two also can be derived from the reduced-order eigenvalue problems of the 2nd stage and the reduced eigenvector of the 2nd stage can be simplified as xc(2),yc(2),xr(2),yr(2),xs(2),ys(2),ξ1(2),η1(2),υ1(2),u1(2),ξ-1(2),η-1(2),υ-1(2), u-1(2)T. To sum up, there are always fourteen natural frequencies with multiplicity 2 for the two-stage HPGs, and the corresponding vibration modes are the radical translational modes.


### 4.3. Planet Mode

The planet mode is composed of the 1st-stage planet mode and the 2nd-stage planet mode. Taking the 1st-stage planet mode as an example, we analytically investigate the unique characteristics of the planet mode.The typical characteristics of the 1st-stage planet mode are(a)
The natural mode multiplicity of the 1st-stage planet mode ism=N1-3.(b)
The modal deflections of central members in both stages and planets in the 2nd stage are zero. Thus, the displacement subvectors can be written as(46)pji=xji,yji,zji,ujiT=0,0,0,0T,j=c,r,s;i=1,2pn2=ξn2,ηn2,υn2,un2T=0,0,0,0T,n=1,2,…,N2.(c)
Only planets in the 1st stage have motions. The deflections of diametrically opposed planets are identical. Besides, the motions of each planet are a scalar multiple of the motions of an arbitrarily selected first planet. The displacement subvector of planet in the 1st stage is given by(47)pn1=ξn1,ηn1,υn1,un1T=wn1p11=wn1ξ11,η11,υ11,u11Tn=1,…,N1,wherewn(1) are constants and w1(1)=1.The eigenvectors of the 1st stage can be expressed as(48)ϕi1=0,0,0,p11,…,wn1p11T,where the zero vectors are 4 × 1.Substituting (46)-(47) into (23a), (23b), (23c), and (23d) gives(49)
k
p
1
∑
n
=
1
N
1
w
n
1
ξ
1
1
cos
⁡
ψ
n
1
-
η
1
1
sin
⁡
ψ
n
1
ξ
1
1
sin
⁡
ψ
n
1
+
η
1
1
cos
⁡
ψ
n
1
υ
1
1
k
p
z
1
k
p
1
η
1
1
=
0
,
(50)
k
r
p
1
ξ
1
1
sin
⁡
α
1
+
η
1
1
cos
⁡
α
1
+
υ
1
1
+
u
1
1
∑
n
=
1
N
1
w
n
1
c
o
s
2
β
1
sin
⁡
ψ
r
n
1
-
c
o
s
2
β
1
cos
⁡
ψ
r
n
1
sin
⁡
β
1
cos
⁡
β
1
-
c
o
s
2
β
1
=
0
,
(51)
k
s
p
1
ξ
1
1
sin
⁡
α
+
η
1
1
cos
⁡
α
+
υ
1
1
+
u
1
1
∑
n
=
1
N
1
w
n
1
c
o
s
2
β
1
sin
⁡
ψ
s
n
1
-
c
o
s
2
β
1
cos
⁡
ψ
s
n
1
-
sin
⁡
β
1
cos
⁡
β
1
-
c
o
s
2
β
1
=
0
(52)
K
p
p
1
-
ω
i
2
M
p
1
w
n
1
p
1
1
=
0
.To satisfy (49)–(51), three independent constraint equations can be obtained.(53)∑n=1N1wn1=0,∑n=1N1wn1cos⁡ψn1=0,∑n=1N1wn1sin⁡ψn1=0.It is obvious that there is no solution ifN1<3. Four natural frequencies are calculated from (52). According to (48), N1-3 independent solutions can be deduced with wn(1) determined from (53) only if N1>3.Similarly, in the 2nd-stage planet modes, only planets in stage two have modal deflections. Without loss of generality, we also analytically investigate the reduced-order eigenvalue problems of the 2nd-stage planet modes. With similar operations on the eigenvalue matrix equations of the 2nd-stage members, it is concluded that four natural frequencies with a multiplicity ofN2-3 are always obtained and the associated vibration modes are the 2nd-stage planet mode.


## 4.1. Axial Translational-Rotational Mode

According to the above vibration mode analysis, the axial translational-rotational mode has the following characteristics:(a) The corresponding multiplicity of natural frequency is 1.(b) The radical deflections of carrier, ring gear, and sun gear in two stages are zero(25)pji=xji,yji,zji,ujiT=0,0,zji,ujiT,j=c,r,s;i=1,2.(c) The modal deflections of planets in the same stage are identical(26)pni=ξni,ηni,υni,uniT=p1i=ξ1i,η1i,υ1i,u1iT,i=1,2.The eigenvectors of the 1st stage and 2nd stage can be expressed as(27a)ϕi1=pc1,pr1,ps1,p11,…,p11T(27b)ϕi2=pc2,pr2,ps2,p12,…,p12T.Inserting (25)–((27a) and (27b)) into (23a) yields(28a)kcz1+kcsz1,2+N1kpz1-ωi2Ic1rc12zc1-N1kpz1z11-kcsz1,2zs1=0,(28b)kcu1+kcsu1,2+N1kp1-ωi2Ic1rc12uc1-N1kp1u11-kcsu1,2us1=0.Similarly, inserting (25)–(27a) into (23b)-(23c) yields(29a)krz1+krz1,2+N1krn1s2β1-ωi2Ir1rr12zr1-N1sβ1cβ1ur1-krz1,2zr2+N1krn1-ζ11sβ1cβ1sα1+η11sβ1cβ1cα1-s2β1υ11+sβ1cβ1u11=0,(29b)kru1+kru1,2+N1krn1c2β1-ωi2Ir1rr12ur1-N1sβ1cβ1zr1-kru1,2ur2+N1krn1ζ11c2β1sα1-η11c2β1cα1+sβ1cβ1υ11-c2β1u11=0,(30a)ksz1+N1ksn1s2β1-ωi2Is1rs12zs1+N1sβ1cβ1us1-N1ksn1ζ11sβ1cβ1sα1+η11sβ1cβ1cα1+υ11s2β1-u11sβ1cβ1=0,(30b)ksu1+N1ksn1c2β1-ωi2Is1rs12us1+N1sβ1cβ1zs1-N1ksn1ζ11c2β1sα1+η11c2β1cα1+υ11sβ1cβ1-u11c2β1=0,where sψjn(i)=sin⁡ψjn(i), cψjn(i)=cos⁡ψjn(i), sα(i)=sin⁡α(i), cα(i)=cos⁡α(i), sβ(i)=sin⁡β(i), and cβ(i)=cos⁡β(i). Equation (23d) can be simplified as (31)Kc21nTpc1-Kr21nTpr1+Ks21nTps1+Kpp1n-ωi2Mp1pn1=Kc211Tpc1-Kr211Tpr1+Ks211Tps1+Kpp1-ωi1Mp1p11=0.Inserting (25)–(27b) into (24a), we get(32a)kcz2+N2kpz2-ωi2Ic2rc22zc2-N2kpz2z12=0,(32b)kcu2+N2kp2-ωi2Ic2rc22uc2-N2kp2u12=0.In the same way, (24b)-(24c) reduce to(33a)krz2+krz1,2+N2krn2s2β2-ωi2Ir2rr22zr2-N2sβ2cβ2ur2+N2krn2-ζ12sβ2cβ2sα2+η12sβ2cβ2cα2-s2β2υ12+sβ2cβ2u12-krz1,2zr1=0,(33b)kru2+kru1,2+N2krn2c2β2-ωi2Ir2rr22ur1-N2sβ2cβ2zr2+N2krn2ζ12c2β2sα2-η12c2β2cα2+sβ2cβ2υ12-c2β2u12-kru1,2ur1=0,(34a)ksz2+kcsz1,2+N2ksn2s2β2-ωi2Is2rs22zs2+N2sβ2cβ2us2-N2ksn2ζ12sβ2cβ2sα2+η12sβ2cβ2cα2+υ12s2β2-u12sβ2cβ2-kcsz1,2zc1=0,(34b)ksu2+kcsu1,2+N2ksn2c2β2-ωi2Is2rs22us1+N2sβ2cβ2zs2-N2ksn2ζ12c2β2sα2+η12c2β2cα2+υ12sβ2cβ2-u12c2β2-kcsu1,2uc1=0.Applying the mode characteristics (b) and (c), (24d) becomes(35)Kc22nTpc2-Kr22nTpr2+Ks22nTps2+Kpp2n-ωi2Mp2pn2=Kc221Tpc2-Kr221Tpr2+Ks221Tps1+Kpp2-ωi2Mp2p12=0.The above equations imply that (23a)–(24d) reduce to a 20-degree-of-freedom eigenvalue problem consisting of (28a)–(35). The modal vectorq is determined by the reduced eigenvalue (36)zc1,uc1,zr1,ur1,zs1,us1,ξ11,η11,υ11,u11,zc2,uc2,zr2,ur2,zs2,us2,ξ12,η12,υ12,u12T.Twenty eigensolutions are derived from this reduced eigenvalue problem. Therefore, there are 20 axial translational-rotational modes and the corresponding natural frequency is distinct.


## 4.2. Radical Translational Mode

To sum up, the radical translational mode has the following characteristics:(a)
The natural frequency multiplicity of the radical translational mode is 2. There is a pair of orthonormal vibration modes associated with each natural frequency(37)ϕi=pc1,pr1,ps1,p11,…,pN11,pc2,pr2,ps2,p12,…,pN22T,ϕ-i=p-c1,p-r1,p-s1,p-11,…,p-N11,p-c2,p-r2,p-s2,p-12,…,p-N22T.(b)
The axial translation and rotation of central members in both stages are zero. Besides, the relation of the modal deflections of central members in a pair of orthonormal vibration modes is(38)pji=xji,yji,zji,ujiT=xji,yji,0,0,T,p-ji=-yji,xji,zji,ujiT=-yji,xji,0,0,T,j=c,r,s;i=1,2.(c)
The modal deflections of planets of same stage in a pair of orthonormal vibration modes can be expressed as(39)pni=p1icos⁡ψni+p-1isin⁡ψnip-ni=-p1isin⁡ψn1+p-1icos⁡ψni,wherepn(i)=ξn(i),ηn(i),υn(i),un(i)T and p-n(i)=ξ-n(i),η-n(i),υ-n(i),u-n(i)T are the modal deflections of planets in a pair of orthonormal vibration modes.Thus, a pair of candidate orthonormal radical translational modes of the 1st stage is given by(40a)ϕi1=pc1,pr1,ps1,cos⁡ϕ11p11+sin⁡ϕ11p-11,…,cos⁡ϕN11p11+sin⁡ϕN11p-11T,(40b)ϕ-i1=p-c1,p-r1,p-s1,cos⁡ϕ11p-11-sin⁡ϕ11p11,…,cos⁡ϕN11p-11-sin⁡ϕN11p11T.Analogous to the analyses of the axial translational-rotational mode, substituting (38)–(40b) into (23a) yields(41a)kcx1+kcsx1,2+N1kp1-ωi2mc1xc1-N12kp1ξ11-η-11-kcsx1,2xs2=0,(41b)kcy1+kcsy1,2+N1kp1-ωi2mc1xc1-N12kp1ξ-11+η11-kcsx1,2xs2=0.Similarly, the following (42a)–(43b) are derived by inserting (38)–(40b) into (23b)-(23c).(42a)krx1+krx1,2+N12krn1-ωi2mr1xr1-krx1,2xr2-N12krn1ζ11s2α1+sα1cα1ζ-11-η11-η-11c2α1c2β1=0,(42b)kry1+kry1,2+N12krn1-ωi2mr1yr1-kry1,2yr2+N12krn1ζ-11s2α1+sα1cα1ζ11-η-11-η11c2α1c2β1=0,(43a)ksx1+N12ksn1-ωi2ms1xs1+N12ksn1ζ11s2α1+sα1cα1ζ-11+η11+η-11c2α1c2β1=0,(43b)ksy1+N12ksn1-ωi2ms1ys1-N12ksn1ζ-11s2α1+sα1cα1ζ11+η-11+η11c2α1c2β1=0.Insertingϕi(1) and ϕ-i(1) into (24d), we get(44a)Ln1=Kc21nTpc1-Kr21nTpr1+Ks21nTps1+Kpp1-ωi2Mp1pn1=0,(44b)L-n1=Kc21nTp-c1-Kr21nTp-r1+Ks21nTp-s1+Kpp1-ωi2Mp1p-n1=0,where Ln(1) and L-n(1) are the introduced notations. Applying the mode characteristics of the radical translational modes (b) and (c), the above equations can be expressed as(45a)Ln1=L11cos⁡ψn1+L-11sin⁡ψn1,(45b)L-n1=-L11sin⁡ψn1+L-11cos⁡ψn1.The eigenvalue problems of the 1st stage can be reduced to fourteen independent equations: six equations from (41a)–(43b) and eight equations from (45a) and (45b). Thus, seven natural frequencies with multiplicity two are obtained. The reduced eigenvector of the 1st stage can be expressed as xc(1),yc(1),xr(1),yr(1),xs(1),ys(1),ξ1(1),η1(1),υ1(1),u1(1),ξ-1(1),η-1(1),υ-1(1), u-1(1)T. With similar operations on the eigenvalue matrix equations (24a), (24b), (24c), and (24d), seven natural frequencies with multiplicity two also can be derived from the reduced-order eigenvalue problems of the 2nd stage and the reduced eigenvector of the 2nd stage can be simplified as xc(2),yc(2),xr(2),yr(2),xs(2),ys(2),ξ1(2),η1(2),υ1(2),u1(2),ξ-1(2),η-1(2),υ-1(2), u-1(2)T. To sum up, there are always fourteen natural frequencies with multiplicity 2 for the two-stage HPGs, and the corresponding vibration modes are the radical translational modes.


## 4.3. Planet Mode

The planet mode is composed of the 1st-stage planet mode and the 2nd-stage planet mode. Taking the 1st-stage planet mode as an example, we analytically investigate the unique characteristics of the planet mode.The typical characteristics of the 1st-stage planet mode are(a)
The natural mode multiplicity of the 1st-stage planet mode ism=N1-3.(b)
The modal deflections of central members in both stages and planets in the 2nd stage are zero. Thus, the displacement subvectors can be written as(46)pji=xji,yji,zji,ujiT=0,0,0,0T,j=c,r,s;i=1,2pn2=ξn2,ηn2,υn2,un2T=0,0,0,0T,n=1,2,…,N2.(c)
Only planets in the 1st stage have motions. The deflections of diametrically opposed planets are identical. Besides, the motions of each planet are a scalar multiple of the motions of an arbitrarily selected first planet. The displacement subvector of planet in the 1st stage is given by(47)pn1=ξn1,ηn1,υn1,un1T=wn1p11=wn1ξ11,η11,υ11,u11Tn=1,…,N1,wherewn(1) are constants and w1(1)=1.The eigenvectors of the 1st stage can be expressed as(48)ϕi1=0,0,0,p11,…,wn1p11T,where the zero vectors are 4 × 1.Substituting (46)-(47) into (23a), (23b), (23c), and (23d) gives(49)
k
p
1
∑
n
=
1
N
1
w
n
1
ξ
1
1
cos
⁡
ψ
n
1
-
η
1
1
sin
⁡
ψ
n
1
ξ
1
1
sin
⁡
ψ
n
1
+
η
1
1
cos
⁡
ψ
n
1
υ
1
1
k
p
z
1
k
p
1
η
1
1
=
0
,
(50)
k
r
p
1
ξ
1
1
sin
⁡
α
1
+
η
1
1
cos
⁡
α
1
+
υ
1
1
+
u
1
1
∑
n
=
1
N
1
w
n
1
c
o
s
2
β
1
sin
⁡
ψ
r
n
1
-
c
o
s
2
β
1
cos
⁡
ψ
r
n
1
sin
⁡
β
1
cos
⁡
β
1
-
c
o
s
2
β
1
=
0
,
(51)
k
s
p
1
ξ
1
1
sin
⁡
α
+
η
1
1
cos
⁡
α
+
υ
1
1
+
u
1
1
∑
n
=
1
N
1
w
n
1
c
o
s
2
β
1
sin
⁡
ψ
s
n
1
-
c
o
s
2
β
1
cos
⁡
ψ
s
n
1
-
sin
⁡
β
1
cos
⁡
β
1
-
c
o
s
2
β
1
=
0
(52)
K
p
p
1
-
ω
i
2
M
p
1
w
n
1
p
1
1
=
0
.To satisfy (49)–(51), three independent constraint equations can be obtained.(53)∑n=1N1wn1=0,∑n=1N1wn1cos⁡ψn1=0,∑n=1N1wn1sin⁡ψn1=0.It is obvious that there is no solution ifN1<3. Four natural frequencies are calculated from (52). According to (48), N1-3 independent solutions can be deduced with wn(1) determined from (53) only if N1>3.Similarly, in the 2nd-stage planet modes, only planets in stage two have modal deflections. Without loss of generality, we also analytically investigate the reduced-order eigenvalue problems of the 2nd-stage planet modes. With similar operations on the eigenvalue matrix equations of the 2nd-stage members, it is concluded that four natural frequencies with a multiplicity ofN2-3 are always obtained and the associated vibration modes are the 2nd-stage planet mode.


## 5. Conclusions

In this study, a three-dimensional dynamic model of the two-stage HPGs with consideration of axial and radical deformations is established. Two three-dimensional coordinate systems are created to analyze the relative displacements of members. Dynamic equations of motion which can be used to analyze HPGs with variant number of planets in two stages, different planet phasing, and spacing configurations are derived by Newton’s second law and the Theorem of Moment of Momentum. Subsequently, the natural frequencies and vibration modes of the two-stage HPGs with equally spaced planets are simulated numerically. Finally, the unique properties of these vibration modes are mathematically proved. The main properties of the two-stage HPGs are summarized below:(1)
Vibration modes of the two-stage HPGs can be categorized as five classes: the rigid body mode, the axial translational-rotational mode, the radical translational mode, and the 1st-stage and the 2nd-stage planet mode.(2)
Axial translational-rotational modes: twenty natural frequencies always have multiplicitym=1 for various N1 and N2. The associated mode is the axial translational-rotational mode. All central members only rotate and translate axially in this vibration mode. Deflections of planets in the same stage are identical.(3)
Radical translational modes: there are always fourteen pairs of natural frequencies with multiplicitym=2 for different N1 and N2, and the corresponding vibration mode is the radical translational mode. In this mode, all central members have pure radical translations.(4)
Planet modes: the planet modes of the two-stage HPGs are composed of two types, the 1st-stage planet mode and the 2nd-stage planet mode. Four natural frequencies with multiplicitym=Ni-3 are calculated only if Ni>3. The associated vibration modes are the ith stage planet mode in which only planets in the ith stage have motions while other members remain stationary.(5)
The natural frequency with multiplicitym=1,2 varies monotonically as additional planets in both stages are introduced.



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*Source: 9864959-2017-02-27.xml*