dsaf

1. Dynamic modeling and vibration characteristics of a two-stage
closed-form planetary gear train
Lina Zhang a,b
, Yong Wanga,b,
⁎, Kai Wu c
, Ruoyu Shenga
, Qilin Huang a
a
School of Mechanical Engineering, Shandong University, Jinan, China
b
Key Laboratory of High-efficiency and Clean Mechanical Manufacture (Shandong University), Ministry of Education, China
c
Weichai Power Co., Ltd., Weifang, China
a r t i c l e i n f o a b s t r a c t
Article history:
Received 27 January 2015
Received in revised form 17 October 2015
Accepted 21 October 2015
Available online 22 November 2015
Planetary gears are widely used in modern machines as one of the most effective forms of
power transmission. However, the vibration resulting from gear meshing may shorten the machine’s
service life. In this paper, a translational–rotational dynamic model of a two-stage closed-form
planetary gear set under consideration of the rotational and translational displacements is
first presented to investigate the dynamic response and to avoid resonance. The dynamic
equations are formulated into matrix form for the calculation of the natural frequencies
and mode shapes. The corresponding 3D model is also proposed to accurately confirm the
physical parameters. The effect of planet number in each stage and coupling stiffness on natural
modes is then analyzed. Finally, the lumped-parameter model is compared with the finite element
simulation. The results show that mode types can be classified into five groups: a rigid body mode,
rotational modes, translational modes, the first-stage and the second-stage planet modes. Natural
frequencies of rotational and translational modes vary monotonically with planet number.
Coupling-twist stiffness has a significant impact on translational modes and the coupling-
translational stiffness only affects rotational modes.
© 2015 Elsevier Ltd. All rights reserved.
Keywords:
Planetary gear
Multi-stage
Vibration characteristics
Dynamics
Coupling stiffness
1. Introduction
Planetary gears are extensively used in a variety of industrial fields including aerospace, marine and automobiles. Their popularity
mainly comes from the substantial advantages such as high torque ratio, compactness and excellent bearing capacity [1]. As a form of
planetary gear, the closed-form planetary gears are more competent due to its power split and higher reduction ratio. However, vibration
and noise are still the key concerns that influence the planetary gears’ reliability and durability. Most researches on dynamic behaviors are
conducted experimentally and theoretically because there is a direct link between the dynamic behavior and noise. Besides, the free
vibration characteristics of planetary gear systems have a pronounced effect on the dynamic stability and parametric resonance
characteristics.
Before 1970, research was seldom conducted on planetary gear sets. Most dynamic models belonged to lumped-parameter models
[2–15] and finite element models [16–25]. Some models are linear time-invariant since the gear backlash and the time-varying mesh
stiffness are not taken into account [26,27]. These simplifications can make the equations adapted to calculation since eigenvalue and
modal summation formulations are applicable, but the explanations of the behavior at the gear mesh cannot be given. Earlier researches
about vibration mode structures of planetary gears were first conducted in lumped-parameter models. Cunliffe et al. [3] identified the
vibration characteristics of a specific thirteen DOF (degree-of-freedom) planetary gear with a translational–rotational model. Botman
Mechanism and Machine Theory 97 (2016) 12–28
⁎ Corresponding author at: School of Mechanical Engineering, Shandong University, Jin-Shi Road 17923, Jinan 250061, China.
E-mail addresses: zhanglina134679@126.com (L. Zhang), meywang@sdu.edu.cn (Y. Wang).
http://dx.doi.org/10.1016/j.mechmachtheory.2015.10.006
0094-114X/© 2015 Elsevier Ltd. All rights reserved.
Contents lists available at ScienceDirect
Mechanism and Machine Theory
journal homepage: www.elsevier.com/locate/mechmt

2. [4] established an eighteen DOF model and grouped the vibration modes as axisymmetric and nonaxisymmetric modes. Kahraman
[5,28,29] extended a three-dimensional dynamic model of a single-stage planetary gear train to obtain natural modes. He also derived
a purely torsional model to predict the natural frequencies and vibration modes and validated the accuracy of the purely torsional
model by comparison with a more sophisticated transverse-torsional model. Furthermore, Kahraman derived a family of torsional dy-
namic models of compound gear sets and predicted the free vibration characteristics for each power flow configuration separately. Final-
ly, he classified the natural modes into three groups. Wang et al. [14] established a rotational dynamic model for a 2K-H spur planetary
gear set and investigated the influence of the bearing stiffness on the vibration mode. Huang et al. [30] developed a purely torsional dy-
namic model of closed-form planetary gear set to investigate its natural frequency and free vibration modes. Lin and Parker [2,8,31] de-
veloped the translational–rotational coupling dynamic models of single stage planetary gears with equally spaced and diametrically
opposed planets and analytically investigated the natural frequency spectra and vibration modes to tune the system frequencies to
avoid resonances. In addition, they also investigated the natural frequency and vibration mode sensitivities to system parameters for
both tuned and mistuned systems. Eritenel and Parker [7] formulated the three-dimensional model of single-stage helical planetary
gears with equally spaced planets and mathematically categorized the structured modal properties. Guo and Parker [32,33] developed
a purely rotational model of general compound planetary gears and classified all vibration modes into two types and then studied the
sensitivity of general compound planetary gear natural frequencies and vibration modes to inertia and stiffness parameters. The
modal properties of a herringbone planetary gear train with journal bearings was determined by Bu et al. [34]. Qian et al. [35] identified
the vibration structures of a particular planetary gear used in a coal shearer and validated the model by comparing with the finite el-
ement model. Using these models, Shyyab and Kahraman [36–38] solved the nonlinear equations of motion with harmonic balance
method (HBM) in conjunction with inverse discrete Fourier transform and Newton–Raphson method, and showed the influence of
key gear design parameters on dynamic response. Bahk and Parker [39] examined the nonlinear dynamics of planetary gears by
multiple-scale method, and the accuracy of this method was evaluated by the harmonic balance method with arc-length continuation.
Huang [15] established the optimization mathematical model, which aims to minimize the vibration displacement of the low-speed
carrier and the total mass of the gear transmission system. Chen et al. [40] investigated the dynamic response of planetary gears
with the contribution of ring gear flexibility and tooth errors. Ericson and Parker [20] proved the accuracy of lumped-parameter and
finite element models by experimentation and highlighted several design and modeling characteristics of planetary gears.
However, the majority of studies published are about simple single-stage planetary gears while researches on multi-stage
planetary gears are rare. What’s more, it is essential to establish the lateral–torsional coupled dynamic model although more
DOFs are taken into consideration and the models have become more sophisticated when the bearing stiffness are not larger
than tenfold mesh stiffness [28]. There are very few literatures about multi-stage planetary gear dynamics modeled with rotational
and translational motions.
In this paper, we attempt to establish a translational–rotational coupled dynamic model of a two-stage closed-form planetary gear
set to predict the natural frequencies and vibration modes. The proposed model is linearly time-invariant in which torsional, bearing
and interstage coupling stiffnesses are considered. The main distinguishing point of this dynamic model is that the translational and
rotational motions of all members are modeled and the approach can be easily applied to multi-stage systems with similar structures.
The influence of system parameters on modal characteristics is analyzed. The modal properties of the two-stage system and single-
stage system are compared. The finite element mechanics model is proposed subsequently to validate that the results are in agreement
with the analytical model.
2. Dynamic model and equations of motion
A two-stage closed-form planetary gear set shown in Fig. 1 is considered in this study. The compound epicyclic gear train has
already been applied to cranes. It is composed of a differential planetary gear train and a quasi-planetary gear set. The interactions
Fig. 1. The discrete model of example planetary gear systems.
13L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

3. between gear pairs are modeled by linear springs acting along the line of action. Bearings and interactions between two stages are
also represented by linear springs.
2.1. Analysis of the system coordinates
As the purely rotational model cannot accurately predict the natural frequencies of planetary gear systems when bearing stiffness is
an order of magnitude lower than mesh stiffnesses [28], a two-stage translational–rotational dynamic model is developed. Fig. 2 shows a
single stage planetary gear set. By choosing reasonable stiffness, the model can represent any stage of the closed-form planetary gear set
in Fig. 1. The lumped-parameter model here was based on some assumptions [38]: each gear body in any stage is assumed to be rigid and
planets in the same stage are identical and equally spaced; the friction at the gear meshes is neglected; the radial and circumferential
error of planets and damping are also ignored in the system; bearings and shafts are assumed to be isotropic.
The deflections of each gear are described by radical and tangential coordinates and each component has three degrees of freedom:
two translations and one rotation. Two kinds of coordinate systems shown in Fig. 3 are created since the kinematic configuration is
complicated. One is the absolute coordinate system OXY, which is fixed on the theoretical installation center of the sun gear, and the
other is the dynamic coordinate system Onξnηn, which orbits around the origin of the absolute coordinate O with a constant carrier
angular speed Ωc. On is the theoretical installation center of planet n. The gyroscopic effect induced by the relative motion between
absolute coordinate system and dynamic coordinate system will cause centripetal accelerations of planets. The effect can be overlooked
when there is a low-speed carrier [2]. Fig. 3 shows the relationship between two coordinate systems.
P is the supposed centroid. The radius vector of P is signed as r′P in the absolute coordinate system {i′,j′,k′} and is signed as rP
in the dynamic coordinate system {i,j,k}. The radius vector of On is signed as rOn
in the absolute coordinate system and jrOn
j ¼ R. θ
is the initial angle between rOn
and i′. Equations can be drawn from Fig. 3
r
0
P ¼ rp þ rOn
ð1Þ
rOn
¼ R cos Ωct þ θð Þi
0
þ R sin Ωct þ θð Þj
0
ð2Þ
rp ¼ x cosΩct−y sinΩctð Þi
0
þ y cosΩct þ x sinΩctð Þj
0
ð3Þ
Substituting Eqs. (2) and (3) into Eq. (1) yields
r
0
P ¼ rOn
þ rP ¼ R cos Ωct þ θð Þ þ x cos Ωctð Þ−y sin Ωctð Þ½ Ši
0
þ R sin Ωct þ θð Þ þ y cos Ωctð Þ þ x sin Ωctð Þ½ Šj
0
¼ rx0 i
0
þ ry0 j
0
ð4Þ
Fig. 2. The lumped-parameter model of the single stage.
14 L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

4. The absolute acceleration of P is
a
→′ ¼ ax0 i
0
þ ay0 j
0
¼ €rx0 i
0
þ €ry0 j
0
ð5Þ
The component of absolute acceleration of P in the dynamic coordinate system is
a
!
¼ ax0 cosΩct þ ay0 sinΩct
i þ ay0 cosΩct−ax0 sinΩct
j
¼ €x−2Ωc y
−Ω
2
c x−RΩ
2
c cosθ
i þ €y−2Ωc x
−Ω
2
c y−RΩ
2
c sinθ
j
ð6Þ
where 2Ωcẋ and 2Ωcẏ denotes the Coriolis accelerations of P. Ωc
2
x and Ωc
2
y are centripetal accelerations with respect to the dynamic
coordinate system. RΩc
2
cos θ and RΩc
2
sin θ are centripetal accelerations with respect to the absolute coordinate system. Considering
that R = 0 when the component is a center member, RΩc
2
cos θ and RΩc
2
sin θ only exist when planets are studied.
2.2. Relative displacements of members in the same stage
To derive equations of the whole system, we should first illustrate the relative displacements in any stage i(i = 1,2). The number of
planets in stage i is Ni. As shown in Fig. 2, the corresponding structural member is denoted by the subscript of physical symbols and the
superscript i denotes the structural member is in stage i. xj
(i)
, yj
(i)
(j = c,s,r) are the translational displacements of carrier, sun and ring,
and ξn
(i)
, ηn
(i)
(n = 1,…, Ni) are planet translations. The rotational displacements are uj
(i)
= rj
(i)
θj
(i)
(j = c,s,r,1,2,…, Ni), where θj
(i)
is the
torsional displacement and rj
(i)
is the base circle radius for a gear, and the radius of the circle passing through planet centers for a carrier.
kjn
(i)
(j = s,r; n = 1,2,…, Ni) denotes mesh stiffness of the nth sun-planet or ring-planet. kju
(i)
(j = c,s,r) is the torsional stiffness and kjx
(i)
, kjy
(i)
(j = c,s,r, 1,2,…, Ni) are the bearing stiffness of carrier, sun, ring and planet. kjx
(1,2)
, kjy
(1,2)
, kju
(1,2)
(j = c,s,r) are the interstage coupling stiff-
ness. ψn
(i)
= 2π(n − 1)/Ni is the circumferential angle of the planet n which is defined positive when it is counterclockwise. αs
(i)
, αr
(i)
are the
pressure angle of sun-planet and ring-planet, respectively. 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 between members of stage i is shown in Fig. 4. The following formulas can be derived.
(1) The relative displacement of sun-planet mesh
δ
ið Þ
sn ¼ u
ið Þ
s −x
ið Þ
s sinψ
ið Þ
sn þ y
ið Þ
s cosψ
ið Þ
sn
− −u
ið Þ
n þ ξ
ið Þ
n sinα
ið Þ
þ η
ið Þ
n cosα
ið Þ
ð6Þ
(2) The relative displacement of ring-planet mesh
δ
ið Þ
rn ¼ u
ið Þ
r −x
ið Þ
r sinψ
ið Þ
rn þ y
ið Þ
r cosψ
ið Þ
rn
− u
ið Þ
n −ξ
ið Þ
n sinα
ið Þ
þ η
ið Þ
n cosα
ið Þ
ð7Þ
Fig. 3. Schematic drawing of coordinate systems.
15L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

5. (3) Radial compression between planet and carrier
δ
ið Þ
nrad ¼ x
ið Þ
c cosψ
ið Þ
n þ y
ið Þ
c sinψ
ið Þ
n −ξ
ið Þ
n ð8Þ
(4) Tangential compression between planet and carrier
δ
ið Þ
n tan ¼ y
ið Þ
c cosψ
ið Þ
n −x
ið Þ
c sinψ
ið Þ
n þ u
ið Þ
c −η
ið Þ
n ð9Þ
2.3. Relative displacements of members in adjacent stages
As illustrated in Fig. 1, the system investigated is composed of a differential planetary gear train and a quasi-planetary gear set.
It is linked by the carrier in the first stage and the sun gear in the second stage as well as the rings. The relative displacements
between rings of two stages are listed as follows:
δ
1;2ð Þ
r ¼ u
1ð Þ
r −u
2ð Þ
r ð10aÞ
X
1;2ð Þ
r ¼ x
1ð Þ
r −x
2ð Þ
r ð10bÞ
Y
1;2ð Þ
r ¼ y
1ð Þ
r −y
2ð Þ
r ð10cÞ
The compressions between carrier in the first stage and sun gear in the second stage are
δ
1;2ð Þ
cs ¼ u
1ð Þ
c −u
2ð Þ
s ð11aÞ
X
1;2ð Þ
cs ¼ x
1ð Þ
c −x
2ð Þ
s ð11bÞ
Y
1;2ð Þ
cs ¼ y
1ð Þ
c −y
2ð Þ
s ð11cÞ
Fig. 4. The relative displacement between members of the single stage.
16 L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

6. 2.4. Dynamic equations of motion and 3D modeling
According to the analyses above, the displacement vector is
q ¼ x
1ð Þ
c ; y
1ð Þ
c ; u
1ð Þ
c
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
c
; x
1ð Þ
r ; y
1ð Þ
r ; u
1ð Þ
r
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
r
; x
1ð Þ
s ; y
1ð Þ
s ; u
1ð Þ
s
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
s
; ξ
1ð Þ
1 ; η
1ð Þ
1 ; u
1ð Þ
1
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
planet 1
; ⋯; ξ
1ð Þ
np1
; η
1ð Þ
np1
; u
1ð Þ
np1
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
planet N1
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
stage 1
8
:
;
x
2ð Þ
c ; y
2ð Þ
c ; u
2ð Þ
c
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
c
; x
2ð Þ
r ; y
2ð Þ
r ; u
2ð Þ
r
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
r
; x
2ð Þ
s ; y
2ð Þ
s ; u
2ð Þ
s
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
s
; ξ
2ð Þ
1 ; η
2ð Þ
1 ; u
2ð Þ
1
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
planet 1
; ⋯; ξ
2ð Þ
np2
; η
2ð Þ
np2
; u
2ð Þ
np2
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
planet N2
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
stage 2
9
=
;
T
ð12Þ
Mass and mass moment of inertia are denoted as mg
(i)
, Ig
(i)
(g = c, r, s, p). Equations of motion of the model shown in Fig. 1 are
assembled in matrix form
M€q þ ωcG q
þ Kb þ Km−ω
2
c KΩ
h i
q ¼ T tð Þ ð13Þ
where M is the positive definite mass matrix and ωc is the velocity matrix determined by angular velocities of carriers, Kb is the bearing
stiffness matrix and Km is the meshing stiffness matrix. G is the gyroscopic matrix and KΩ is the stiffness matrix induced by gyroscopic
effect. The summation index n ranges from 1 to Ni in equations of stage i. Matrix components are given in Appendix A.
The analysis method we used can be easily applied to planetary gears of multi-stage. Meanwhile, the nonlinear time-varying
dynamic equations can be derived by modifying the linear equations with nonlinear factors.
The development of CAE technology can help us accurately confirm some parameters of the system. In this paper, we make use of
the Unigraphics NX software to establish the 3D model of the planetary gear set. The geometric parameters and component material
are set in accordance with the practical system. Thus, the magnitudes of physical parameters in Table 1, such as stiffness, mass and
inertia, are confirmed by the 3D system model with N1 = 3, N2 = 4 as shown in Fig. 5.
3. Results and discussion
Lin and Parker [7] investigated the natural frequency spectra and mode shapes of a single-stage planetary gear. The natural frequen-
cies and vibration modes of the two-stage planetary gear set are also obtained by solving the eigenvalue problem governed by Eq. (13). In
practice, our research achievement has been used in cranes with low velocity of carriers. So, Coriolis accelerations have little influence on
the vibration modes and dynamic response of the system, and the corresponding terms G and KΩ are neglected. The equations of motion
and characteristic equations are simplified as
M€q þ Kb þ Kmð Þq ¼ 0 ð14Þ
Kb þ Km−ω
2
i M
ϕi ¼ 0 ð15Þ
Natural frequencies ωi and modal vectors ϕi are calculated by solving Eq. (15). The modal vector is expressed as
ϕi ¼ p
1ð Þ
c ; p
1ð Þ
r ; p
1ð Þ
s ; p
1ð Þ
1 ; ⋯; p
1ð Þ
N1
; p
2ð Þ
c

9. ; p
2ð Þ
r ; p
2ð Þ
s ; p
2ð Þ
1 ; ⋯; p
2ð Þ
N2
h i
;
where pj
(i)
= [xj
(i)
, yj
(i)
, uj
(i)
]T
(j = c, r, s; i = 1, 2) is the sub-vector for carrier, ring and sun in stage i, and pn
(i)
= [ξn
(i)
, ηn
(i)
, un
(i)
]T
(n =
1, ⋯ Ni; i = 1, 2) is the sub-vector of planet n in stage i. Systems with the parameters listed in Tables 1 and 2 are considered as the example
cases.
Table 1
Physical parameters of the example systems (i = 1,2).
Parameter High-speed stage Ring Low-speed stage
Sun Planet Carrier Sun Planet
Mass (kg) 0.72 0.77 2.04 14 0.63 0.48
I (kg · mm2
) 43.6 800.85 6815.3 179097 291.2 424.83
Base diameter (mm) 11 32.25 57.8 120 21.5 30
Mesh stiffness (N/m) ksn
(1)
= krn
(1)
= 4.5 × 108
ksn
(2)
= krn
(2)
= 5 × 108
Bearing stiffness (N/m) kjx
(i)
= kjy
(i)
= 1.2 × 109
( j = c, s) krx
(i)
= kry
(i)
= 1.3 × 109
kpx
(i)
= kpy
(i)
= 1 × 109
Torsional stiffness (N/m) kju
(i)
= 0( j = s, r); kcu
(1)
= 0, kcu
(2)
= 1 × 109
Coupling stiffness (N/m) kcsx
(1,2)
= kcsy
(1,2)
= 2 × 109
, kcsu
(1,2)
= 1.2 × 1010
17L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

10. The natural modes are grouped according to the multiplicity of the natural frequencies. Systems with N1 = 3, N2 ranges from 3 to
10 are evaluated numerically here to illustrate the modal characteristics. Natural frequencies and vibration modes of the example gear
sets are given in Table 3. Typical vibration modes of each type are shown in Figs. 6–10. Several characteristics are revealed after a deep
comparison on the natural frequencies ωi and modal vectors ϕi.
(1) The first-order natural frequency is ω1 = 0, and the corresponding vibration mode is the rigid body mode. It is obvious that the
rigid body mode can be eliminated by removing the rigid-body motion.
(2) Twelve natural frequencies with multiplicity m = 1 for various N2 are calculated, and the values vary monotonically as N2
increases. Related vibration modes are rotational modes in which carriers, rings and suns all have pure rotation. In a rotational
mode, planets in the same stage all move in phase and have identical motions [ξ1
(i)
, η1
(i)
, u1
(i)
]T
. Fig. 6 shows the typical rotational
modes of the example system. The deflections of carriers and rings are not shown; the equilibrium positions are represented
by a solid black line and the deflected positions are shown by a dashed red line. Similarly, Figs. 7–10 all abide by these rules.
(3) There are twelve pairs of translational modes for any N2, and the corresponding natural frequency multiplicity are m = 2. Their
values also vary monotonically as N2 increases. In these modes, all central members have pure translation. So, those modes are
referred to as translational modes. Fig. 7 illustrates the typical translational mode of the example system.
(4) Three degenerate natural frequencies with multiplicity m = N2-3 exist only if N2 N 3. Associated modes are the second stage planet
modes because all central members are stationary and only planets in the second stage have modal deflection. The modal deflec-
tion of each planet is a scalar multiple of the modal deflection of an arbitrarily selected first planet. It should be noted that motions
of planets in the first stage are zero as N1 = 3. Movements of the diametrically opposed planets in the second stage are always in
phase. Fig. 8 illustrates the typical second stage planet mode. Similarly, in the first stage planet modes, only planets in stage one
have modal deflections.
(5) To further predict the rules of planet modes, systems with N2 = 5, N1 = 4 ~ 6 are investigated here. The results show that three
natural frequencies 4025.6 Hz, 5639.5 Hz and 6263.4 Hz always have multiplicity m = N2-3 and the corresponding vibration
modes are the second stage planet modes (only planets in stage two have motions). Other three natural frequencies 3265.7 Hz,
4635 Hz and 4945.2 Hz always have multiplicity m = N1-3 and associated vibration modes are the first stage planet modes
(all members remain stationary except for planets in stage one). Figs. 9 and 10 illustrate the vibration modes of the example
system with N1 = 6, N2 = 5 whose corresponding natural frequencies are 3265.7 Hz and 1162.6 Hz, respectively.
(6) The free vibration characteristics of the closed form planetary gear sets are greatly influenced by the number of planets in each
stage. Natural frequencies of example systems vary monotonically with the numbers of planets changing in a reasonable range,
and the trends of different orders are not all the same. The tendencies of the first four orders of natural frequencies with planets
ranges from 3 to 6 are shown in Fig. 11.
Fig. 5. 3D model of the planetary gear set.
Table 2
Geometric parameters of the example gear set.
Parameter High-speed stage Low-speed stage
Sun Planet Ring Sun Planet Ring
Number of teeth 16 38 92 16 24 62
Module (mm) 3 3
Pressure angle 20o
20o
Modification 0.3 0.12 0.54 0.475 0.471 0.309
Face width (mm) 22 20 20 34.5 30 32
18 L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

11. (7) Modal properties of the two-stage system are analogous to those of simple, single-stage planetary gears. Features of rotational and
translational modes are identical. The number of natural frequencies with multiplicity m = 1 or m = 2 in two-stage systems is
twice the number in the single-stage planetary gear set. The major difference is that planet modes in the two-stage system are
divided into two classes: the first stage planet mode (N1 N 3) and the second stage planet mode (N2 N 3). The corresponding
vibration modes are termed the i-th stage planet mode because only planets in the i-th stage have modal deflection.
4. Influence of coupling stiffness
It can be found from Eq. (14) that the modal characteristic is affected by the coupling stiffness. To detect the influence of coupling
stiffness on modal characteristics, gear systems with variable coupling stiffnesses are investigated. Table 4 lists some natural frequencies
with N1 = 3, N2 = 6, under different rotational stiffnesses between the carriers kru
(1,2)
.
Table 3
Natural frequencies (Hz) and their multiplicities m for the example systems.
N2 3 4 5 6 7 8 9 10 Vibration mode
Multiplicity
m = 1 0 0 0 0 0 0 0 0 Rotational mode
485 500 508 513 516 518 519.21 520
946 942 937 933 928 923 918.48 913.71
2061 1957 1869 1795 1731 1675 1626.9 1583.5
3115.7 3120.9 3122.1 3122.1 3121.7 3121.2 3120.7 3120.3
3703 3794 3811.5 3815 3816.2 3816.7 3817 3817.1
3840.7 3879.8 3957.1 4023.7 4076.2 4118.4 4152.9 4181.9
4655.8 4764.6 4861.1 4946.9 5024 5094.1 5158.7 5219
5599.6 5599.7 5599.8 5599.8 5599.9 5600 5600.2 5600.5
6638.4 6934.4 7233.6 7532 7828.1 8118.5 8402.6 8680
8309 8935 9523.3 10079 10211 10211 10211 10211
10211 10211 10211 10212 10608 11112 11594 12058
m = 2 1186.8 1178.1 1169.3 1160.7 1152.1 1143.6 1135.2 1126.9 Translational mode
1500.7 1540.7 1567.7 1585.2 1596 1601.9 1604.2 1603.9
1949.2 1941.3 1933.6 1926 1918.5 1911.3 1904.3 1897.5
2183.1 2204.5 2225 2243.6 2260.1 2274.2 2286 2295.7
3167.9 3168.4 3169 3169.9 3170.9 3172.1 3173.2 3174.4
3948.1 3921.7 3900 3883 3870.6 3862 3856.9 3854.6
4427.6 4444.5 4456.2 4464.1 4469.3 4472.8 4475.1 4476.6
5006.5 5028.4 5066.7 5114.2 5166.2 5219.9 5273.4 5325.6
5651.2 5651.7 5652.1 5652.7 5653.4 5654.4 5655.6 5657.3
6141.8 6188.6 6224.7 6248.1 6261.6 6269.6 6274.7 6278.2
6323.8 6341.9 6371.8 6416.7 6474 6539.5 6610.6 6685.7
7630.3 7934.8 8219.4 8488.3 8744.4 8989.7 9225.9 9454.1
m = N2-3 — 4025.6 4025.6 4025.6 4025.6 4025.6 4025.6 4025.6 Planet mode
5639.5 5639.5 5639.5 5639.5 5639.5 5639.5 5639.5
6263.4 6263.4 6263.4 6263.4 6263.4 6263.4 6263.4
(a) Stage 1 (b) Stage 2
Fig. 6. Typical rotational mode of the example system with N1 = 3, N2 = 4 ω = 942.0 Hz.
19L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

12. Natural frequencies and vibration modes are compared when kru
(1,2)
varies. Related mode shapes are not listed here. It is concluded
that kru
(1,2)
does not change the natural frequencies and mode shapes of translational modes (m = 2) and the second stage planet
modes (m = 3), and it only changes those of the rotational modes (m = 1). The influence of krn
(1,2)
(n = x, y) on modal characteristics
is also shown in Table 5. The results show that krx
(1,2)
(krx
(1,2)
= kry
(1,2)
) only has an effect on the values of the natural frequencies with
multiplicity m = 2 rather than the vibration types. Comparing the data in Tables 4 and 5, there is an increasing tendency of the natural
frequencies on the whole as kru
(1,2)
and krx
(1,2)
increases. Furthermore, we also investigate the impact of kcsu
(1,2)
and kcsx
(1,2)
on modal
characteristics. Similarly, a further analysis indicates that kcsu
(1,2)
only influence the natural frequencies and mode shapes of rotational
modes (m = 1) and kcsx
(1,2)
only affect the translational modes (m = 2).
Consequently, we can see that values of the coupling stiffness affect the natural frequencies rather than mode types. It can be
concluded that the coupling-twist stiffness, kru
(1,2)
and kcsu
(1,2)
, can change the natural frequencies and mode shapes of translational modes
while rotational modes are greatly dependent on the coupling-translational stiffness, krx
(1,2)
and kcsx
(1,2)
. The first and second stage planet
mode is not influenced by the coupling stiffness. Natural frequencies display an increasing trend as any coupling stiffness increases.
Therefore, we can adjust the values of coupling stiffness in reasonable ways to evade undesirable vibration modes.
(a) Stage 1 (b) Stage 2
Fig. 7. Typical translational mode of the example system with N1 = 3, N2 = 4 ω = 1540.7 Hz.
(a) Stage 1 (b) Stage 2
Fig. 8. Typical second-stage planet modes of the example system with N1 = 3, N2 = 4 ω = 4025.6 Hz.
20 L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

13. 5. Finite element mechanics model
The vibration mode structures analyzed by the lumped-parameter models are generally verified by the finite element method
or experiment. Ambarisha and Parker [21] established the lumped-parameter mathematical model and the finite element model
of a single stage planetary gear set, and the excellent agreement between those two models’ responses verified the validity of the
lumped-parameter model. Qian et al. [35] proposed a lumped-parameter model in which rotations of all components and translations of
the sun gear were taken into account to solve the natural characteristics of a planetary gear used in coal-mining applications and
examined the model by comparing it with the finite element model. In this paper, we also used the finite element mechanics model
to confirm the analytical model.
The 3D model shown in Fig. 5 is meshed with 3D tetrahedral element type. The finite element mesh for the four-planetary gear system
is shown in Fig. 12. Natural frequencies predicted by the analytical model and finite element model are shown in Fig. 13. Differences of
natural frequencies predicted by those two methods have excellent agreement with a maximum difference of 3.8 percent as shown in
Fig. 14 (a positive number indicates natural frequency derived by the analytical method is smaller than that by finite element method
and vice versa). The difference can be explained by the reason that the boundary and loading conditions are set ideally while the material
is always defective and the gears are treated as rigid bodies in the lumped parameter model. The results also prove that the first stage
planet mode is not shown in the example finite element simulation, which is matched with the analytical model.
(a) Stage 1 (b) Stage 2
Fig. 9. Typical first-stage planet mode of the example system with N1 = 6, N2 = 5 ω = 3265.7 Hz.
(a) Stage 1 (b) Stage 2
Fig. 10. Typical translational mode of the example system with N1 = 6, N2 = 5 ω = 1162.6 Hz.
21L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

14. 6. Conclusions
As the purely rotational model cannot accurately predict the natural frequencies of planetary gear systems when bearing stiffness is
an order of magnitude lower than the mesh stiffness, a translational–rotational dynamic model of a two-stage closed-form planetary gear
(a) The 1st order natural frequency (b) The 2nd order natural frequency
(c) The 3rd order natural frequency (d) The 4th order natural frequency
Fig. 11. Natural frequencies of the first four orders with N1 and N2 ranging from 3 to 6.
Table 4
Natural frequencies and their multiplicity m for a given kcsu
(1,2)
.
kru
(1,2)
/(N/m) Natural frequencies (Hz)
m = 1 m = 2 m = 3
1.5 × 106
253 751 2118 4811.8 4464.1 5114.2 6416.7 8488.3 4025.6 5639.5 6263.4
1.5 × 107
307 804 2514 5326.2 4464.1 5114.2 6416.7 8488.3 4025.6 5639.5 6263.4
1.5 × 108
384 856.2 2703 6081.3 4464.1 5114.2 6416.7 8488.3 4025.6 5639.5 6263.4
1.5 × 109
493 905.1 3085 6958.6 4464.1 5114.2 6416.7 8488.3 4025.6 5639.5 6263.4
1.5 × 1010
513 933 3815 7532 4464.1 5114.2 6416.7 8488.3 4025.6 5639.5 6263.4
Table 5
Natural frequencies and their multiplicity m for a given krx
(1,2)
.
krx
(1,2)
/(N/m) Natural frequencies (Hz)
m = 1 m = 2 m = 3
2 × 106
513 933 3815 7532 2579 3539.3 3039.2 5018.6 4025.6 5639.5 6263.4
2 × 107
513 933 3815 7532 3280 4032.7 4297.2 5318.7 4025.6 5639.5 6263.4
2 × 108
513 933 3815 7532 3758 4592.8 5093.5 6619.6 4025.6 5639.5 6263.4
2 × 109
513 933 3815 7532 4464.1 5114.2 6416.7 8488.3 4025.6 5639.5 6263.4
2 × 1010
513 933 3815 7532 5471 7012 7569 9530 4025.6 5639.5 6263.4
22 L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

15. Fig. 12. Finite element mesh for the four-planetary gear system in stage 2.
Fig. 13. Natural frequencies predicted by analytical method and finite element method.
Fig. 14. Differences of natural frequencies predicted by analytical method and finite element method.
23L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

16. set which could explicitly describe the whole system was developed. The Lagrange method was employed to get the dynamic equations.
Then, natural frequencies and vibration modes were obtained by solving the eigenvalue problem governed by corresponding equations.
Gear sets with different numbers of planets in each stage were considered here. Additionally, the influence of coupling-twist and
coupling-translational stiffness on modal characteristics were presented. Finally, the lumped-parameter model was compared with
the finite element simulation. The main results are:
1. Five vibration modes were observed: a rigid body mode, rotational modes, translational modes, the first stage and the second
stage planet modes.
2. Rotational modes: there are 12 rotational modes with natural frequencies multiplicity m = 1 for various N2. The values of natural
frequencies vary monotonically as N2 increases. All central members have pure rotation and planets in the same stage move in
phase and have the identical motion.
3. Translational modes: the central members in two stages have pure translations. There are 12 pairs of orthonormal translational
modes with natural frequencies multiplicity m = 2 for any N2. The associated natural frequencies also vary monotonically as
additional planets in the second stage are introduced.
4. The first stage planet modes: three natural frequencies with multiplicity m = N1-3 exist only if N1 N 3. Associated modes are the
first stage planet modes in which only planets in the first stage have modal deflection while other members do not move. The
modal deflection of each planet is a scalar multiple of the modal deflection of any other planet. The values of associated natural
frequencies are not changed as additional planets in the first stage are introduced.
5. The second stage planet modes: analogous to the modal characteristics of the first stage planet modes, only planets in the second stage
have modal deflections in the second stage planet modes. Three natural frequencies with multiplicity m = N2-3 exist for systems with
four or more planets in stage 2. The values of associated natural frequencies are independent of the number of planets in stage 2.
6. It is found that the coupling-twist stiffness only has an impact on translational modes while coupling-translational stiffness
only affects rotational modes.
The analytical model is verified by comparing with the finite element model, and good agreements between these two models
enhance the confidence of modeling and further analyses. Compared to prior analyses of planetary gears, the model can be used
to investigate the effects of more essential factors on gear system. The methods and results of this study have potential for
avoiding resonant response, reducing vibration of particular mode types, making vibration characteristic analysis and predicting
structural dynamic characteristics. The model established can be easily applied to multi-stage systems. Our ongoing research
will focus on the investigations of the analytical solutions of multi-stage planetary gear set when considering both translational
and torsional degrees of freedom, and the influence of coupling stiffness on planetary gear dynamic response.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 51175299).
Appendix A
M ¼ M
1ð Þ
0
0 M
2ð Þ
M
1ð Þ
¼ diag M
1ð Þ
c ; M
1ð Þ
r ; M
1ð Þ
s ; M
1ð Þ
P ; ⋯; M
1ð Þ
P
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
N1
0
B
@
1
C
A;
M
2ð Þ
¼ diag M
2ð Þ
c ; M
2ð Þ
r ; M
2ð Þ
s ; M
2ð Þ
P ; ⋯; M
2ð Þ
P
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
N2
0
B
@
1
C
A
M
ið Þ
j ¼ diag m
ið Þ
j ; m
ið Þ
j ; I
ið Þ
j =r
ið Þ2
j
; j ¼ c; r; s; i ¼ 1; 2;
G¼diag G
1ð Þ
c ; G
1ð Þ
r ; G
1ð Þ
s ; G
1ð Þ
1 ; ⋯; G
1ð Þ
N1
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
stage 1
G
2ð Þ
c ; G
2ð Þ
r ; G
2ð Þ
s ; G
2ð Þ
1 ; ⋯; G
2ð Þ
N2
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
stage 2

24. 0
B
B
@
1
C
C
A
24 L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

25. G
ið Þ
j ¼
0 −2m
ið Þ
j 0
2m
ið Þ
j 0 0
0 0 0
2
6
4
3
7
5; j ¼ c; r; s; 1; ⋯; Ni; i ¼ 1; 2
ωc ¼ diag ω
1ð Þ
c ; ⋯; ω
1ð Þ
c
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
3þN1
ω
2ð Þ
c ; ⋯; ω
2ð Þ
c
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
3þN2

32. 0
B
@
1
C
A
KΩ ¼ diag m
1ð Þ
c ; m
1ð Þ
c ; 0
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
c
; m
1ð Þ
r ; m
1ð Þ
r ; 0
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
r
; m
1ð Þ
s ; m
1ð Þ
s ; 0
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
s
; m
1ð Þ
p ; m
1ð Þ
p ; 0; ⋯; m
1ð Þ
p ; m
1ð Þ
p ; 0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
3N1
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
stage 1
0
B
B
@ ;
m
2ð Þ
c ; m
2ð Þ
c ; 0
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
c
; m
2ð Þ
r ; m
2ð Þ
r ; 0
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
r
; m
2ð Þ
s ; m
2ð Þ
s ; 0
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
s
; m
2ð Þ
p ; m
2ð Þ
p ; 0; ⋯; m
2ð Þ
p ; m
2ð Þ
p ; 0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
3N2
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
stage 2
1
C
C
A
Kb ¼ diag K
1ð Þ
b ; K
2ð Þ
b
K
1ð Þ
b ¼ K
1ð Þ
b−indep þ K
1ð Þ
cb−dep; K
2ð Þ
b ¼ K
2ð Þ
b−indep þ K
2ð Þ
cb−dep
K
1ð Þ
b−indep¼diag K
1ð Þ
cb ; K
1ð Þ
rb ; K
1ð Þ
sb ; 0; ⋯; 0
|fflfflfflffl{zfflfflfflffl}
N1
0
B
@
1
C
A
K
1ð Þ
cb−dep ¼ diag K
1ð Þc
cs ; K
1ð Þr
r ; 0 ; 0; ⋯; 0
|fflfflfflffl{zfflfflfflffl}
N1
0
B
@
1
C
A
K
2ð Þ
b−indep ¼ diag K
2ð Þ
cb ; K
2ð Þ
rb ; K
2ð Þ
sb ; 0; ⋯; 0
|fflfflfflffl{zfflfflfflffl}
N2
0
B
@
1
C
A
K
2ð Þ
cb−dep ¼ diag 0; K
2ð Þr
r ; K
2ð Þs
cs ; 0; ⋯; 0
|fflfflfflffl{zfflfflfflffl}
N2
0
B
@
1
C
A
K
ið Þ
jb ¼ diag k
ið Þ
jx ; k
ið Þ
jy ; k
ið Þ
ju
; j ¼ c; r; s; i ¼ 1; 2;
K
1ð Þr
r ¼ diag k
1;2ð Þ
rx ; k
1;2ð Þ
ry ; k
1;2ð Þ
ru
; K
1ð Þc
cs ¼ diag k
1;2ð Þ
csx ; k
1;2ð Þ
csy ; k
1;2ð Þ
csu
K
2ð Þr
r ¼ diag k
1;2ð Þ
rx ; k
1;2ð Þ
ry ; k
1;2ð Þ
ru
; K
2ð Þs
cs ¼ diag k
1;2ð Þ
csx ; k
1;2ð Þ
csy ; k
1;2ð Þ
csu
T tð Þ ¼ 0; 0;
T
1ð Þ
c tð Þ
r
1ð Þ
c
; 0; 0;
T
1ð Þ
r tð Þ
r
1ð Þ
r
; 0; 0;
T
1ð Þ
s tð Þ
r
1ð Þ
s
; 0; ⋯; 0
zfflfflffl}|fflfflffl{
3N1
0; 0;
T
2ð Þ
c tð Þ
r
2ð Þ
c
; 0; 0;
T
2ð Þ
r tð Þ
r
2ð Þ
r
; 0; 0;
T
2ð Þ
s tð Þ
r
2ð Þ
s
; 0; ⋯; 0
zfflfflfflffl}|fflfflfflffl{
3N2

39. 2
6
4
3
7
5
T
25L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

40. K
1ð Þ
m ¼
XN1
n¼1
K
1ð Þn
c1 0 0 K
1ð Þ1
c2 K
1ð Þ2
c2 K
1ð Þ3
c2
XN1
n¼1
K
1ð Þn
r1 0 K
1ð Þ1
r2 K
1ð Þ2
r2 K
1ð Þ3
r2
XN1
n¼1
K
1ð Þn
s1 K
1ð Þ1
s2 K
1ð Þ2
s2 K
1ð Þ3
s2
K
1ð Þ1
pp 0 0
symm K
1ð Þ2
pp 0
K
1ð Þ3
pp
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
K
2ð Þ
m ¼
XN1
n¼1
K
2ð Þn
c1 0 0 K
2ð Þ1
c2 K
2ð Þ2
c2 K
2ð Þ3
c2
XN1
n¼1
K
2ð Þn
r1 0 K
2ð Þ1
r2 K
2ð Þ2
r2 K
2ð Þ3
r2
XN1
n¼1
K
2ð Þn
s1 K
2ð Þ1
s2 K
2ð Þ2
s2 K
2ð Þ3
s2
K
2ð Þ1
pp 0 0
symm K
2ð Þ2
pp 0
K
2ð Þ3
pp
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
k
1ð Þn
c1 ¼ k
1ð Þ
p
1 0 − sinψ
1ð Þ
n
0 1 cosψ
1ð Þ
n
− sinψ
1ð Þ
n cosψ
1ð Þ
n 1
2
6
4
3
7
5
k
1ð Þn
c2 ¼ k
1ð Þ
p
− cosψ
1ð Þ
n sinψ
1ð Þ
n 0
− sinψ
1ð Þ
n − cosψ
1ð Þ
n 0
0 −1 0
2
6
4
3
7
5
k
1ð Þn
r1 ¼ k
1ð Þ
rn
sin
2
ψ
1ð Þ
rn − cosψ
1ð Þ
rn sinψ
1ð Þ
rn − sinψ
1ð Þ
rn
− cosψ
1ð Þ
rn sinψ
1ð Þ
rn cos
2
ψ
1ð Þ
rn cosψ
1ð Þ
rn
− sinψ
1ð Þ
rn cosψ
1ð Þ
rn 1
2
6
4
3
7
5
k
1ð Þn
r2 ¼ k
1ð Þ
rn
− sinα sinψ
1ð Þ
rn cosα sinψ
1ð Þ
rn sinψ
1ð Þ
rn
sinα cosψ
1ð Þ
rn − cosα cosψ
1ð Þ
rn − cosψ
1ð Þ
rn
sinα − cosα −1
2
6
4
3
7
5
26 L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

41. k
1ð Þn
s1 ¼ k
1ð Þ
sn
sin
2
ψ
1ð Þ
sn − cosψ
1ð Þ
sn sinψ
1ð Þ
sn − sinψ
1ð Þ
sn
− cosψ
1ð Þ
sn sinψ
1ð Þ
sn cos
2
ψ
1ð Þ
sn cosψ
1ð Þ
sn
− sinψ
1ð Þ
sn cosψ
1ð Þ
sn 1
2
6
4
3
7
5
k
1ð Þn
s2 ¼ k
1ð Þ
sn
sinα sinψ
1ð Þ
sn cosα sinψ
1ð Þ
sn − sinψ
1ð Þ
sn
− sinα cosψ
1ð Þ
sn − cosα cosψ
1ð Þ
sn cosψ
1ð Þ
sn
− sinα − cosα 1
2
6
4
3
7
5
k
1ð Þn
s3 ¼ k
1ð Þ
sn
sin
2
α cosα sinα − sinα
cosα sinα cos
2
α − cosα
− sinα − cosα 1
2
4
3
5
k
1ð Þn
r3 ¼ k
1ð Þ
rn
sin
2
α − cosα sinα − sinα
− cosα sinα cos
2
α cosα
− sinα cosα 1
2
4
3
5
k
2ð Þn
c1 ¼ k
2ð Þ
p
1 0 − sinψ
2ð Þ
n
0 1 cosψ
2ð Þ
n
− sinψ
2ð Þ
n cosψ
2ð Þ
n 1
2
6
4
3
7
5
k
2ð Þn
c2 ¼ k
1ð Þ
p
− cosψ
2ð Þ
n sinψ
2ð Þ
n 0
− sinψ
2ð Þ
n − cosψ
2ð Þ
n 0
0 −1 0
2
6
4
3
7
5
k
2ð Þn
r1 ¼ k
2ð Þ
rn
sin
2
ψ
2ð Þ
rn − cosψ
2ð Þ
rn sinψ
2ð Þ
rn − sinψ
2ð Þ
rn
− cosψ
2ð Þ
rn sinψ
2ð Þ
rn cos
2
ψ
2ð Þ
rn cosψ
2ð Þ
rn
− sinψ
2ð Þ
rn cosψ
2ð Þ
rn 1
2
6
4
3
7
5
k
2ð Þn
r2 ¼ k
2ð Þ
rn
− sinα sinψ
2ð Þ
rn cosα sinψ
2ð Þ
rn sinψ
2ð Þ
rn
sinα cosψ
2ð Þ
rn − cosα cosψ
2ð Þ
rn − cosψ
2ð Þ
rn
sinα − cosα −1
2
6
4
3
7
5
k
2ð Þn
s1 ¼ k
2ð Þ
sn
sin
2
ψ
2ð Þ
sn − cosψ
2ð Þ
sn sinψ
2ð Þ
sn − sinψ
2ð Þ
sn
− cosψ
2ð Þ
sn sinψ
2ð Þ
sn cos
2
ψ
2ð Þ
sn cosψ
2ð Þ
sn
− sinψ
2ð Þ
sn cosψ
2ð Þ
sn 1
2
6
4
3
7
5
k
2ð Þn
s2 ¼ k
2ð Þ
sn
sinα sinψ
2ð Þ
sn cosα sinψ
2ð Þ
sn − sinψ
2ð Þ
sn
− sinα cosψ
2ð Þ
sn − cosα cosψ
2ð Þ
sn cosψ
2ð Þ
sn
− sinα − cosα 1
2
6
4
3
7
5
k
2ð Þn
s3 ¼ k
2ð Þ
sn
sin
2
α cosα sinα − sinα
cosα sinα cos
2
α − cosα
− sinα − cosα 1
2
4
3
5
k
2ð Þn
r3 ¼ k
2ð Þ
rn
sin
2
α − cosα sinα − sinα
− cosα sinα cos
2
α cosα
− sinα cosα 1
2
4
3
5
27L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

42. K
1ð Þn
pp ¼ K
1ð Þn
c3 þ K
1ð Þn
r3 þ K
1ð Þn
s3
K
2ð Þn
pp ¼ K
2ð Þn
c3 þ K
2ð Þn
r3 þ K
2ð Þn
s3
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