Dynamic modeling and vibration characteristics of a two-stage closed-form planetary gear train

Mechanism and Machine Theory 97 (2016) 12–28

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Dynamic modeling and vibration characteristics of a two-stage closed-form planetary gear train Lina Zhang a,b, Yong Wang a,b,⁎, Kai Wu c, Ruoyu Sheng a, Qilin Huang a a b c

School of Mechanical Engineering, Shandong University, Jinan, China Key Laboratory of High-efficiency and Clean Mechanical Manufacture (Shandong University), Ministry of Education, China Weichai Power Co., Ltd., Weifang, China

a r t i c l e

i n f o

Article history: Received 27 January 2015 Received in revised form 17 October 2015 Accepted 21 October 2015 Available online xxxx Keywords: Planetary gear Multi-stage Vibration characteristics Dynamics Coupling stiffness

a b s t r a c t 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 couplingtranslational stiffness only affects rotational modes. © 2015 Elsevier Ltd. All rights reserved.

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 ⁎ Corresponding author at: School of Mechanical Engineering, Shandong University, Jin-Shi Road 17923, Jinan 250061, China. E-mail addresses: [email protected] (L. Zhang), [email protected] (Y. Wang).

http://dx.doi.org/10.1016/j.mechmachtheory.2015.10.006 0094-114X/© 2015 Elsevier Ltd. All rights reserved.

L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

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[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 dynamic models of compound gear sets and predicted the free vibration characteristics for each power flow configuration separately. Finally, 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 dynamic model of closed-form planetary gear set to investigate its natural frequency and free vibration modes. Lin and Parker [2,8,31] developed 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 element 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 singlestage 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.

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L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

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 0

rP ¼ rp þ rOn

ð1Þ 0

rOn ¼ R cosðΩc t þ θÞi þ R sinðΩc t þ θÞj

0

0

ð2Þ 0

rp ¼ ðx cosΩc t−y sinΩc t Þi þ ðy cosΩc t þ x sinΩc t Þj

ð3Þ

Substituting Eqs. (2) and (3) into Eq. (1) yields 0

0

0

0

0

rP ¼ rOn þ rP ¼ ½R cosðΩc t þ θÞ þ x cosðΩc t Þ−y sinðΩc t Þi þ ½R sinðΩc t þ θÞ þ y cosðΩc t Þ þ x sinðΩc t Þ j ¼ r x0 i þ r y0 j

Fig. 2. The lumped-parameter model of the single stage.

ð4Þ

L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

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Fig. 3. Schematic drawing of coordinate systems.

The absolute acceleration of P is →′ 0 0 0 0 a ¼ ax0 i þ ay0 j ¼ €r x0 i þ €r y0 j

ð5Þ

The component of absolute acceleration of P in the dynamic coordinate system is

 ! a ¼ ax0 cosΩc t þ ay0 sinΩc t i þ ay0 cosΩc t−ax0 sinΩc t j

 2 2 €−2Ωc x −Ω2c y−RΩ2c sinθ j ¼ €x−2Ωc y −Ωc x−RΩc cosθ i þ y 



ð6Þ

where 2Ωcẋ and 2Ωcẏ denotes the Coriolis accelerations of P. Ω2c x and Ω2c y are centripetal accelerations with respect to the dynamic coordinate system. RΩ2c cos θ and RΩ2c sin θ are centripetal accelerations with respect to the absolute coordinate system. Considering that R = 0 when the component is a center member, RΩ2c cos θ and RΩ2c 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, (i) (i) (i) (i) (i) and ξ(i) n , ηn (n = 1,…, Ni) are planet translations. The rotational displacements are uj = rj θj ( j = c,s,r,1,2,…, Ni), where θj 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(i) (i) ness. ψ(i) n = 2π(n − 1)/Ni is the circumferential angle of the planet n which is defined positive when it is counterclockwise. α s , αr 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

 ðiÞ ði Þ ði Þ ði Þ ðiÞ ðiÞ ðiÞ ðiÞ ði Þ ði Þ ðiÞ δsn ¼ us −xs sinψsn þ ys cosψsn − −un þ ξn sinα þ ηn cosα

ð6Þ

(2) The relative displacement of ring-planet mesh

 ðiÞ ði Þ ði Þ ði Þ ðiÞ ðiÞ ðiÞ ði Þ ðiÞ ðiÞ ði Þ δrn ¼ ur −xr sinψrn þ yr cosψrn − un −ξn sinα þ ηn cosα

ð7Þ

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L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

Fig. 4. The relative displacement between members of the single stage.

(3) Radial compression between planet and carrier ðiÞ

ði Þ

ði Þ

ðiÞ

ðiÞ

ðiÞ

δnrad ¼ xc cosψn þ yc sinψn −ξn

ð8Þ

(4) Tangential compression between planet and carrier ðiÞ

ðiÞ

ðiÞ

ðiÞ

ði Þ

ði Þ

ðiÞ

δn tan ¼ yc cosψn −xc sinψn þ uc −ηn

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: ð1;2Þ

¼ ur −ur

ð1Þ

ð2Þ

ð10aÞ

ð1;2Þ

¼ xr −xr

ð1Þ

ð2Þ

ð10bÞ

ð1;2Þ

¼ yr −yr

ð1Þ

ð2Þ

ð10cÞ

δr

Xr

Yr

The compressions between carrier in the first stage and sun gear in the second stage are ð1;2Þ

¼ uc −us

ð1Þ

ð2Þ

ð11aÞ

ð1;2Þ

¼ xc −xs

ð1Þ

ð2Þ

ð11bÞ

ð1;2Þ

¼ yc −ys

ð1Þ

ð2Þ

ð11cÞ

δcs

X cs

Y cs

L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

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2.4. Dynamic equations of motion and 3D modeling According to the analyses above, the displacement vector is 8 stage 1 > > |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} > |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} : c r s planet 1 planet N1 9T stage 2 > zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ > = ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ xc ; yc ; uc ; xr ; yr ; ur ; xs ; ys ; us ; ξ1 ; η1 ; u1 ; ⋯; ξnp2 ; ηnp2 ; unp2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} > |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} > ; c r s planet 1

ð12Þ

planet N 2

(i) Mass and mass moment of inertia are denoted as m(i) g , Ig (g = c, r, s, p). Equations of motion of the model shown in Fig. 1 are assembled in matrix form

h i € þ ωc G q þ Kb þ Km −ω2c KΩ q ¼ Tðt Þ Mq

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 frequencies 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 € þ ðKb þ Km Þq ¼ 0 Mq

ð14Þ


 2 Kb þ Km −ωi M ϕi ¼ 0

ð15Þ

Natural frequencies ωi and modal vectors ϕi are calculated by solving Eq. (15). The modal vector is expressed as  h i ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ  ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ϕi ¼ pc ; pr ; ps ; p1 ; ⋯; pN1 ; pc ; pr ; ps ; p1 ; ⋯; pN2 ; (i) (i) (i) T 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 p(i) n = [ξn , ηn , un ] (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 Sun

Mass (kg) I (kg · mm2) Base diameter (mm) Mesh stiffness (N/m) Bearing stiffness (N/m) Torsional stiffness (N/m) Coupling stiffness (N/m)

Ring Planet

Carrier

0.72 0.77 2.04 43.6 800.85 6815.3 11 32.25 57.8 8 (1) k(1) sn = krn = 4.5 × 10 9 (i) (i) (i) (i) kjx = kjy = 1.2 × 10 ( j = c, s) krx = kry = 1.3 × 109k(i) px = 9 (2) kju(i) = 0( j = s, r); k(1) cu = 0, kcu = 1 × 10 9 10 (1,2) k(1,2) = k(1,2) csx csy = 2 × 10 , kcsu = 1.2 × 10

Low-speed stage Sun

14 0.63 179097 291.2 120 21.5 8 (2) (2) ksn = krn = 5 × 10 9 (i) kpy = 1 × 10

Planet 0.48 424.83 30

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L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

Fig. 5. 3D model of the planetary gear set.

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 (i) (i) T mode, planets in the same stage all move in phase and have identical motions [ξ(i) 1 , η1 , u1 ] . 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 deflection 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.

Table 2 Geometric parameters of the example gear set. Parameter

Number of teeth Module (mm) Pressure angle Modification Face width (mm)

High-speed stage

Low-speed stage

Sun

Planet

Ring

Sun

Planet

Ring

16 3 20o 0.3 22

38

92

24

62

0.12 20

0.54 20

16 3 20o 0.475 34.5

0.471 30

0.309 32

L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

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Table 3 Natural frequencies (Hz) and their multiplicities m for the example systems. N2

3

4

5

6

7

8

9

10

Vibration mode

0 485 946 2061 3115.7 3703 3840.7 4655.8 5599.6 6638.4 8309 10211 1186.8 1500.7 1949.2 2183.1 3167.9 3948.1 4427.6 5006.5 5651.2 6141.8 6323.8 7630.3 —

0 500 942 1957 3120.9 3794 3879.8 4764.6 5599.7 6934.4 8935 10211 1178.1 1540.7 1941.3 2204.5 3168.4 3921.7 4444.5 5028.4 5651.7 6188.6 6341.9 7934.8 4025.6 5639.5 6263.4

0 508 937 1869 3122.1 3811.5 3957.1 4861.1 5599.8 7233.6 9523.3 10211 1169.3 1567.7 1933.6 2225 3169 3900 4456.2 5066.7 5652.1 6224.7 6371.8 8219.4 4025.6 5639.5 6263.4

0 513 933 1795 3122.1 3815 4023.7 4946.9 5599.8 7532 10079 10212 1160.7 1585.2 1926 2243.6 3169.9 3883 4464.1 5114.2 5652.7 6248.1 6416.7 8488.3 4025.6 5639.5 6263.4

0 516 928 1731 3121.7 3816.2 4076.2 5024 5599.9 7828.1 10211 10608 1152.1 1596 1918.5 2260.1 3170.9 3870.6 4469.3 5166.2 5653.4 6261.6 6474 8744.4 4025.6 5639.5 6263.4

0 518 923 1675 3121.2 3816.7 4118.4 5094.1 5600 8118.5 10211 11112 1143.6 1601.9 1911.3 2274.2 3172.1 3862 4472.8 5219.9 5654.4 6269.6 6539.5 8989.7 4025.6 5639.5 6263.4

0 519.21 918.48 1626.9 3120.7 3817 4152.9 5158.7 5600.2 8402.6 10211 11594 1135.2 1604.2 1904.3 2286 3173.2 3856.9 4475.1 5273.4 5655.6 6274.7 6610.6 9225.9 4025.6 5639.5 6263.4

0 520 913.71 1583.5 3120.3 3817.1 4181.9 5219 5600.5 8680 10211 12058 1126.9 1603.9 1897.5 2295.7 3174.4 3854.6 4476.6 5325.6 5657.3 6278.2 6685.7 9454.1 4025.6 5639.5 6263.4

Rotational mode

Multiplicity m=1

m=2

m = N2-3

Translational mode

Planet mode

(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 k(1,2) ru .

(a) Stage 1

(b) Stage 2

Fig. 6. Typical rotational mode of the example system with N1 = 3, N2 = 4 ω = 942.0 Hz.

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L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

(a) Stage 1

(b) Stage 2

Fig. 7. Typical translational mode of the example system with N1 = 3, N2 = 4 ω = 1540.7 Hz.

Natural frequencies and vibration modes are compared when k(1,2) varies. Related mode shapes are not listed here. It is concluded ru does not change the natural frequencies and mode shapes of translational modes (m = 2) and the second stage planet that k(1,2) ru modes (m = 3), and it only changes those of the rotational modes (m = 1). The influence of k(1,2) rn (n = x, y) on modal characteristics (k(1,2) = k(1,2) is also shown in Table 5. The results show that k(1,2) rx rx ry ) 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 (1,2) and k(1,2) increases. Furthermore, we also investigate the impact of k(1,2) frequencies on the whole as k(1,2) ru rx csu and kcsx on modal (1,2) characteristics. Similarly, a further analysis indicates that kcsu only influence the natural frequencies and mode shapes of rotational modes (m = 1) and k(1,2) csx 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 and k(1,2) concluded that the coupling-twist stiffness, k(1,2) ru csu , can change the natural frequencies and mode shapes of translational modes and k(1,2) while rotational modes are greatly dependent on the coupling-translational stiffness, k(1,2) rx csx . 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. 8. Typical second-stage planet modes of the example system with N1 = 3, N2 = 4 ω = 4025.6 Hz.

L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

(a) Stage 1

21

(b) Stage 2

Fig. 9. Typical first-stage planet mode of the example system with N1 = 6, N2 = 5 ω = 3265.7 Hz.

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. 10. Typical translational mode of the example system with N1 = 6, N2 = 5 ω = 1162.6 Hz.

22

L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

(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 k(1,2) csu . k(1,2) ru /(N/m)

Natural frequencies (Hz) m=1

1.5 1.5 1.5 1.5 1.5

× × × × ×

106 107 108 109 1010

253 307 384 493 513

m=2 751 804 856.2 905.1 933

2118 2514 2703 3085 3815

4811.8 5326.2 6081.3 6958.6 7532

4464.1 4464.1 4464.1 4464.1 4464.1

m=3 5114.2 5114.2 5114.2 5114.2 5114.2

6416.7 6416.7 6416.7 6416.7 6416.7

8488.3 8488.3 8488.3 8488.3 8488.3

4025.6 4025.6 4025.6 4025.6 4025.6

5639.5 5639.5 5639.5 5639.5 5639.5

6263.4 6263.4 6263.4 6263.4 6263.4

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

Table 5 Natural frequencies and their multiplicity m for a given k(1,2) rx . k(1,2) /(N/m) rx

Natural frequencies (Hz) m=1

2 2 2 2 2

× × × × ×

106 107 108 109 1010

513 513 513 513 513

m=2 933 933 933 933 933

3815 3815 3815 3815 3815

7532 7532 7532 7532 7532

2579 3280 3758 4464.1 5471

m=3 3539.3 4032.7 4592.8 5114.2 7012

3039.2 4297.2 5093.5 6416.7 7569

5018.6 5318.7 6619.6 8488.3 9530

4025.6 4025.6 4025.6 4025.6 4025.6

5639.5 5639.5 5639.5 5639.5 5639.5

6263.4 6263.4 6263.4 6263.4 6263.4

L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

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.

23

24

L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

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

 ð1Þ M¼ M 0

0 ð2 Þ M



0 ð1Þ

M

1

ð1Þ ð1Þ C B ð1Þ ð1Þ ð1Þ ¼ diag@Mc ; Mr ; Ms ; MP ; ⋯; MP A; |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} N1

0 ð2Þ

M

1

ð2Þ ð2Þ C B ð2Þ ð2Þ ð2Þ ¼ diag@Mc ; Mr ; Ms ; MP ; ⋯; MP A |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} N2


 ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ2 M j ¼ diag m j ; m j ; I j =r j ; j ¼ c; r; s; i ¼ 1; 2;  1   B ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ  ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ C C G¼diagB @Gc ; Gr ; Gs ; G1 ; ⋯; GN1 Gc ; Gr ; Gs ; G1 ; ⋯; GN2 A |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}  stage 1 stage 2 0

L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

2 ði Þ Gj

ði Þ

0

6 ¼ 4 2mðiÞ 0

−2m j 0 0

j

0

3

7 0 5; j ¼ c; r; s; 1; ⋯; Ni ; i ¼ 1; 2 0

 1   ð1Þ  ð2Þ ð2Þ C B ð1Þ ωc ¼ diag @ωc ; ⋯; ωc ωc ; ⋯; ωc A |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}  3þN 3þN 0

1

2

0 stage 1 zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ B ð1Þ ð1Þ ð 1 Þ ð 1 Þ ð 1 Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ KΩ ¼ diag B mc ; mc ; 0 ; mr ; mr ; 0 ; ms ; ms ; 0 ; mp ; mp ; 0; ⋯; mp ; mp ; 0 ; @|fflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} c r s 3N 1 1 stage 2 zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ C ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ mc ; mc ; 0 ; mr ; mr ; 0 ; ms ; ms ; 0 ; mp ; mp ; 0; ⋯; mp ; mp ; 0 C |fflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} A c

r

s

3N2


 ð1Þ ð2Þ Kb ¼ diag Kb ; Kb

ð1Þ

ð1Þ

ð1 Þ

ð2Þ

ð2Þ

ð2 Þ

Kb ¼ Kb−indep þ Kcb−dep ; Kb ¼ Kb−indep þ Kcb−dep 0

1

ð1Þ B ð1Þ ð1Þ ð1Þ C Kb−indep ¼diag@Kcb ; Krb ; Ksb ; 0; ⋯; 0 A |fflfflfflffl{zfflfflfflffl} N1

0 ð1Þ Kcb−dep

1

B ð1Þc ð1Þr C ¼ diag @Kcs ; Kr ; 0 ; 0; ⋯; 0 A |fflfflfflffl{zfflfflfflffl} N1

0

1

ð2Þ B ð2Þ ð2Þ ð2Þ C Kb−indep ¼ diag @Kcb ; Krb ; Ksb ; 0; ⋯; 0 A |fflfflfflffl{zfflfflfflffl} N2

0

1

ð2Þ ð2Þr ð2Þs B C Kcb−dep ¼ diag @0; Kr ; Kcs ; 0; ⋯; 0 A |fflfflfflffl{zfflfflfflffl} N2


 ðiÞ ðiÞ ðiÞ ðiÞ K jb ¼ diag k jx ; k jy ; k ju ; j ¼ c; r; s; i ¼ 1; 2;

ð1Þr



 ð1;2Þ ð1;2Þ ð1;2Þ ð1Þc ð1;2Þ ð1;2Þ ð1;2Þ ¼ diag krx ; kry ; kru ; Kcs ¼ diag kcsx ; kcsy ; kcsu

ð2Þr



 ð1;2Þ ð1;2Þ ð1;2Þ ð2Þs ð1;2Þ ð1;2Þ ð1;2Þ ¼ diag krx ; kry ; kru ; Kcs ¼ diag kcsx ; kcsy ; kcsu

Kr

Kr

2 6 Tðt Þ ¼ 40; 0;

ð1Þ T c ðt Þ ð1Þ

rc

; 0; 0;

ð1Þ T r ðt Þ ð1Þ

rr

 3T 3N1  3N 2 ð1Þ ð2Þ ð2Þ ð2Þ T s ðt Þ zfflfflffl}|fflfflffl{ T c ðt Þ T r ðt Þ T s ðt Þ zfflfflfflffl}|fflfflfflffl{7 ; 0; 0; ð1Þ ; 0; ⋯; 0 0; 0; ð2Þ ; 0; 0; ð2Þ ; 0; 0; ð2Þ ; 0; ⋯; 05  rs rc rr rs 

25

26

L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

2

ð1Þ

Km

N1 X

6 6 6 n¼1 6 6 6 6 6 6 ¼6 6 6 6 6 6 6 6 6 4

3 ð1Þn Kc1

0 N1 X

ð1Þn

Kr1

n¼1

N1 X

0

ð1Þ1 Kc2

0

Kr2

ð1Þ1

ð1Þn

ð1Þ1

Ks1

Ks2

n¼1

ð1Þ2 Kc2 ð1Þ2

Kr2

ð1Þ2

Ks2

ð1Þ1

Kpp symm

0 ð1Þ2

Kpp

ð1Þ3 Kc2

7 7 7 7 7 ð1Þ3 7 Kr2 7 7 7 7 ð1Þ3 7 Ks2 7 7 7 7 0 7 7 0 7 5 ð1Þ3 Kpp

3

2

ð2Þ

Km

N1 X ð2Þn 6 Kc1 6 6 n¼1 6 6 6 6 6 6 ¼6 6 6 6 6 6 6 6 6 4

2 ð1Þn kc1

ð1Þn

kr2

N1 X

ð2Þ1

0

Kc2

0

Kr2

ð2Þ1

ð2Þn

ð2Þ1

Ks1

Ks2

n¼1

ð2Þ2

Kc2

ð2Þ2

Kr2

ð2Þ2

Ks2

ð2Þ1

Kpp symm

1 0 ð1Þ − sinψn

¼

ð1Þ 6 kp 4

¼

ð1Þ 6 krn 4 − cosψðrn1Þ

2

ð2Þn

Kr1

n¼1

¼

2 ð1Þn kr1

N1 X

ð1Þ 6 kp 4

2 ð1Þn kc2

0

ð1Þ

− cosψn ð1Þ − sinψn 0

2

3 ð1Þ − sinψn ð1Þ 7 cosψn 5 1

0 1 ð1Þ cosψn

ð1Þ

sinψn ð1Þ − cosψn −1

ð1Þ

sin ψrn ð1Þ sinψrn ð1Þ − sinψrn

ð1Þ

− sinα sinψrn ð1Þ 6 ¼ krn 4 sinα cosψð1Þ rn sinα

0 ð2Þ2

Kpp

ð2Þ3 Kc2 7 7 7 7 7 ð2Þ3 7 Kr2 7 7 7 7 ð2Þ3 7 Ks2 7 7 7 7 0 7 7 0 7 5 ð2Þ3 Kpp

3 0 7 05 0

ð1Þ

ð1Þ

− cosψrn sinψrn 2 ð1Þ cos ψrn ð1Þ cosψrn

ð1Þ

cosα sinψrn ð1Þ − cosα cosψrn − cosα

3 ð1Þ − sinψrn ð1Þ 7 cosψrn 5 1

3 ð1Þ sinψrn ð1Þ 7 − cosψrn 5 −1

L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

2 ð1Þn ks1

¼

2 ð1Þn ks2

ð1Þ

ð1Þ

¼

¼

ð1Þ ksn 4

¼

2 sin α ð1Þ krn 4 − cosα sinα

2

sin α cosα sinα − sinα

2 ð2Þn kc1

¼

ð2Þ 6 kp 4

1 0 ð2Þ − sinψn

2

ð2Þn

kc2

2 ð2Þn kr1

ð2Þ

− cosψn ð1Þ 6 ¼ kp 4 − sinψð2Þ n 0

¼

2

2

ð2Þn

kr2

2

¼

¼

ð2Þ ksn 4

¼

2 sin α ð2Þ 4 krn − cosα sinα

ð2Þ

sinα sinψsn ð2Þ cosψsn − sinα

2

sin α cosα sinα − sinα

2

ð2Þn kr3

ð2Þ

ð2Þ 6 ksn 4 − sinα

− sinα

3 − sinα cosα 5 1

3 0 7 05 0

ð2Þ

ð2Þ

− cosψrn sinψrn 2 ð2Þ cos ψrn ð2Þ cosψrn ð2Þ

cosα sinψrn ð2Þ − cosα cosψrn − cosα

sin ψsn ð2Þ sinψsn ð2Þ − sinψsn

¼

2 ð2Þn ks3

ð2Þ

ð2Þ 6 ksn 4 − cosψðsn2Þ

2 ð2Þn ks2

ð2Þ

− sinα sinψrn ð2Þ 6 ¼ krn 4 sinα cosψð2Þ rn sinα 2

ð2Þn ks1

ð2Þ

sinψn ð2Þ − cosψn −1

sin ψrn ð2Þ sinψrn ð2Þ − sinψrn

3 ð1Þ − sinψsn ð1Þ 7 cosψsn 5 1

3 ð2Þ − sinψn ð2Þ 7 cosψn 5 1

0 1 ð2Þ cosψn

ð2Þ 6 krn 4 − cosψðrn2Þ

1

3 − sinα − cosα 5 1

− cosα sinα 2 cos α cosα

− sinα

ð1Þ

cosα sinψsn ð1Þ − cosα cosψsn − cosα

cosα sinα 2 cos α − cosα

2

ð1Þn kr3

ð1Þ

− cosψsn sinψsn 2 ð1Þ cos ψsn ð1Þ cosψsn

ð1Þ

sinα sinψsn ð1Þ cosψsn − sinα

ð1Þ 6 ksn 4 − sinα

2 ð1Þn ks3

2

sin ψsn ð1Þ sinψsn ð1Þ − sinψsn

ð1Þ 6 ksn 4 − cosψðsn1Þ

3 ð1Þ − sinψsn ð1Þ 7 cosψsn 5

ð2Þ

3 ð2Þ sinψrn ð2Þ 7 − cosψrn 5 −1 ð2Þ

− cosψsn sinψsn 2 ð2Þ cos ψsn ð2Þ cosψsn ð2Þ

cosα sinψsn ð2Þ − cosα cosψsn − cosα

cosα sinα 2 cos α − cosα

3 ð2Þ − sinψrn ð2Þ 7 cosψrn 5 1

3 ð2Þ − sinψsn ð2Þ 7 cosψsn 5 1

3 ð2Þ − sinψsn ð2Þ 7 cosψsn 5 1

3 − sinα − cosα 5 1

− cosα sinα 2 cos α cosα

3 − sinα cosα 5 1

27

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L. Zhang et al. / Mechanism and Machine Theory 97 (2016) 12–28

ð1Þn

ð1Þn

ð1Þn

ð1Þn

ð2Þn

ð2Þn

ð2Þn

ð2Þn

Kpp ¼ Kc3 þ Kr3 þ Ks3

Kpp ¼ Kc3 þ Kr3 þ Ks3

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

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