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Nonlinear dynamics of a planetary gear system with multiple clearances
Mechanism and Machine Theory 38 (2003) 1371–1390 www.elsevier.com/locate/mechmt
Nonlinear dynamics of a planetary gear system with multiple clearances Tao Sun
a,*
, HaiYan Hu
b
a
b
Department of Precision Mechanical Engineering, Shanghai University, Shanghai 200072, PR China Institute of Vibration Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China Received 19 April 2002; received in revised form 12 March 2003; accepted 16 May 2003
Abstract Presented in this paper is on the nonlinear dynamics of a planetary gear system with multiple clearances taken into account. A lateral–torsional coupled model is established with multiple backlashes, time-varying mesh stiffness, error excitation and sun-gear shaft compliance considered. The solutions are determined by using harmonic balance method from the equations in matrix form. The theoretical results from HBM are verified by using the numerical integration. Finally, effects of parameters are discussed. 2003 Elsevier Ltd. All rights reserved. Keywords: Planetary gear transmission; Nonlinear vibration; Fourier transform; Harmonic balance method; Dynamic
1. Introduction Planetary gear systems have been widely used in engineering owing to their advantages such as little space required, large ratio of transmission and high efficiency. One of the most popular applications is to the automatic transmissions in automobiles. Because of machinery complexity, most of earlier studies on the planetary gear systems were confined to their static behaviors and sharing characteristics. Over the past two decades, the dynamics of planetary transmissions has drawn much attention. However, almost all the published studies on the planetary transmissions focused only on their linear vibration [1,2]. In the advanced mechanical systems running at high speed, such as exact antennas and automatic weapon systems, which usually contain a number of planetary gear sets, the gear systems *
Corresponding author. Fax: +86-21-56334458. E-mail address: [email protected] (T. Sun).
0094-114X/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0094-114X(03)00093-4
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Nomenclature b C, c e f I K, k M, m N P, p q r t T X, x n, g U a s X, x u
backlash damping static transmission error nonlinear displacement function rotary inertia stiffness mass describing function force displacement radius time torque displacement transverse displacement angle pressure angle dimensionless time frequency phase angle
Subscripts a alternating components b base circle c carrier d relative to h high-sped part i ordinal number l low-speed part m mean component p planet gear r ring gear s sun gear n horizontal g vertical
often undergo startup and brake interactively or run at high speed and under light load. The current studies have shown that a gear pair would likely lose contact and the tooth separations
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occur due to the unavoidable backlash. Accordingly, the backlash, namely the clearance, tends to bring gear systems to exhibit typical nonlinear dynamical behaviors. A gear pair is bound to have some backlash, which may be either designed to provide better lubrication and to eliminate interference or due to manufacturing errors and wear. Backlashinduced nonlinear vibrations may cause tooth separation and impacts in unload or lightly loaded geared drives. Such impacts result in intense vibration and noise problems and large dynamic loads, which may affect reliability and life of the gear drive. Experimental studies on the dynamic behavior of a spur gear pair with backlash started almost 40 years ago and still continue [3–5]. For instance, Kubo et al. [4] observed a jump in the frequency response of a gear pair with backlash even though the test set-up was heavily damped. Such experimental studies, albeit limited in scope, have clearly shown that the dynamics of a gear pair can hardly be predicted on the basis of a linear model. Consequently, the nonlinear dynamics and mathematical models of a gear pair with backlash have been intensively studied in the past decade. Although most of the nonlinear mathematical models used to describe the dynamic behavior of a gear pair are somewhat similar to each other, they differ in terms of the excitation mechanisms considered and especially the solution technique used. The nonlinearity of a gear backlash has to be modeled by a discontinuous and nondifferentiable function, which represents a strong nonlinear interaction in the dynamic equation of whole system. Comparin and Singh addressed this problem in [6] and pointed out that most techniques available in the literatures cannot be directly applied to solving this problem. Many researchers have recognized this problem implicitly and therefore employed either digital or analog simulation techniques in their studies. Kahraman and Singh [7] made a detailed review of nonlinear gear dynamics available in current publications. Their theoretical study also made contributions to the nonlinear dynamics of a spur gear pair with backlash subject to the static transmission error. Although there is a vast body of literature concerned with nonlinear dynamics of a general gear pair with clearance, the studies on nonlinear dynamics of a planetary transmission system with multiple clearances are still very limited. Kahraman took the possibility of tooth separation in a planetary gear system into account [2]. In his study, however, the model was not considered as a nonlinear dynamic system. Instead, a step function in linear model was used to distinguish the tooth contact and the tooth separation roughly. As a result, the back collisions of teeth were not included in his study. Compared with above-mentioned works, the nonlinear dynamics of a planetary gear system is much more complicated. Such a system is inherently nonlinear owing to the multiple clearances, and includes the temporally and spatially varying system parameters. Moreover, the request for prediction and examination of dynamic behaviors is progressively urgent in the design of more quiet and reliable planetary transmissions. The published studies on the nonlinear dynamics of geared drive with backlash, however, focus on only a single gear pair, rather than any planetary gear systems. Accordingly, the focus of this paper is on the nonlinear dynamics of planetary gear systems. In this paper, some contributions will be made to a number of key issues such as nonlinear dynamics modeling, solution techniques to the nonlinear differential equations and dynamic behaviors of planetary gear systems.
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2. Dynamic model of system 2.1. Model and assumptions The planetary gear system of concern in this study is a single-stage 2K-H type planetary gear set as shown in Fig. 1. The system consists of a high-speed part h, a low-speed part l, a sun gear s, a ring gear r, n planet gears p and a carrier c. Here, n, the number of planet gears, is taken as 3 throughout the paper. All the gears are mounted on their flexible shafts supported by rolling element bearings. To establish the mathematical model of system, a few assumptions are introduced in the case of speed reduction in the planetary gear shown in Fig. 1 as following. (1) The inertial effects of prime mover (high-speed part) and load inertia (low-speed part) are taken as those of lumped mass. Hence, the planetary gear system has 4 þ n rotational degrees of freedom (DOF), including the rotational displacements of low-speed part, high-speed part, sun gear, carrier and n planetary gears, respectively. (2) Considering the bending stiffness of the shaft of sun gear, the compliance of bearings and potential displacement as a rigid body caused by floating, the horizontal and vertical transverse DOF of the sun gear are included. (3) As the bending stiffness of the planet gear shaft is very large, the deflection of this shaft can be neglected. Thus, the transverse displacement of planet gear is not considered. (4) Because the ring gear itself is a part of gearbox, the displacement of ring gear as a rigid body is insignificant and the center of ring gear is assumed not to move. Based on the above assumptions, the dynamic model of the planetary gear system is established as shown in Fig. 2. All symbols in Fig. 2 can be found in the nomenclature presented and further explanations are given in subsequent sections. The total number of DOF in the model is n þ 4 þ 2,
p c h s
r
Fig. 1. A planetary gear set.
l
T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371–1390
(a)
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(b)
Fig. 2. The dynamic model of a planetary gear set.
Ring gear
Planet gear
line of action
Sun gear (a)
(b)
Fig. 3. Mesh relation in a planetary gear set.
including n þ 4 rotational DOF and 2 transverse DOF, respectively. Obviously, this is a complicated lateral–torsional coupled nonlinear system with multiple clearances and time-varying parameters. 2.2. Equivalent displacements In the case of speed reduction, the meshing relation of the planetary gears is shown in Fig. 3. Regarding the angular displacement of each gear in driving, the direction of the revolution caused by the driving torque is assumed to be positive. Namely, the angular displacements hs and hc of the sun gear and the carrier are in the same direction. And the angular displacements hr and hp of the ring gear and the planet gear are reversed in the directions of hs and hc . In order to establish the equations of motion easily, both torsional and transverse displacements are unified on the pressure line in terms of equivalent displacements.
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The equivalent transverse displacements in the pressure line direction caused by rotational displacements are written as follows: xh ¼ rbs hh ;
xs ¼ rbs hs ;
xr ¼ rbr hr ;
xpi ¼ rbp hpi ;
xc ¼ rbc hc ;
xl ¼ rbc hl ;
i ¼ 1; 2; 3;
ð1Þ where rb (subscripts ÔsÕ, ÔrÕ, ÔpÕ) are the base circle radius of gears, while rbc is the nominal base circle radius of the carrier defined as following: rbc ¼ rbs þ rbp ¼ rbr rbp :
ð2Þ
With regard to the direction of the equivalent displacements, the direction of deflection caused by driving torque is assumed to be positive. It should be pointed out that in the case of speed reduction, the rotational displacement of the ring gear r is set to be zero since the ring gear is fixed on gearbox in the ‘‘2K-H’’ type planetary gear set shown in Fig. 1. That is, hr ¼ 0. However, the tooth deflection of the ring gear is included when calculating the mesh stiffness. By using X to represent the relative displacements in the direction of press line, the relative displacements are obtained as follows according to the meshing relation shown in Fig. 3 and the equivalent displacements 9 Xhs ¼ xh xs ; > > = Xspi ¼ xs xpi xc ; ð3Þ Xrpi ¼ xpi xc xr ¼ xpi xc ; > > ; Xcl ¼ xc xl i ¼ 1; 2; 3: To describe the plane motion of the sun gear, a kinetic Cartesian coordinate system attached to the carrier, instead of general fixed coordinate system, is introduced. The origin-o of this coordinate system coincides with the center of carrier. And the coordinate axes marked as ðn; gÞ rotate along with the carrier. Therefore, ns and gs represent the horizontal and vertical transverse displacements of sun gear with regard to the movable reference frame respectively. As the variation of pressure angle caused by the tiny translation of sun gear can be neglected, the equivalent displacement in the pressure line direction derived from the transverse displacements ns and gs is represented as xsdi ¼ ns sinð/i aÞ þ gs cosð/i aÞ:
ð4Þ
Here a is the pressure angle of gear pair, and /i means that the ith planet gear is mounted at a theoretical position angle /i on the carrier with respect to positive direction of axis-n. If the center of the first planet gear is assigned to positioned at angle 0, /i will be /i ¼ 2pði 1Þ=n;
i ¼ 1; 2; . . . ; n;
ð5Þ
where n represents the numbers of planet gears. 3. Equations of motion In the dynamical model shown in Fig. 2, the gear mesh is described by a nonlinear displacement function f . Here, f is defined as a nonlinear function in the relative gear mesh displacement q and with backlash 2 b as parameter
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8 q b; < f ð qÞ ¼ 0; : qþ b;
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q> b; b 6 q6 b; q < b:
ð6Þ
The planetary gear system includes two different kinds of gear pairs, the external gear pair, namely, the sun gear/planet gear-i pair (subscripts ÔsÕ and pi), and the internal gear pair, namely, the ring gear/planet gear-i pair (subscripts ÔrÕ and pi). Describing each pair of gear mesh by the backlash 2b, the time-varying mesh stiffness k, the viscous damping C and the static transmission error e, the dynamic tooth load of sun gear/planet gear-i pair and ring gear/planet gear-i pair can be given as following: Wspi ¼ kspi f ðXspi þ xsdi espi Þ;
ð7aÞ
Wrpi ¼ krpi f ðXrpi erpi Þ:
ð7bÞ
The mesh stiffness of internal and the external gear pairs are obtained by using finite element method (FEM). A finite element model on tooth contact is developed by using 3-dimensional FEM, and the nonlinear time varying mesh stiffness can be determined by finite element analysis. Mesh stiffness function of the internal and external gear pairs are then formed by cubic spline interpolation and approximation of the discrete mesh points in a mesh period. They are both periodic functions as shown in Fig. 4. To get the frequency behavior of mesh stiffness, the fast Fourier transform (FFT) algorithm is introduced. The damping force of gear pair is represented as Dspi ¼ Cspi ðX_ spi þ x_ sdi e_ spi Þ;
ð8aÞ
Drpi ¼ Crpi ðX_ rpi e_ rpi Þ:
ð8bÞ
Thus, the equations of lateral–torsional coupled motion of the nonlinear planetary gear system can be established by using the Lagrange principle as following: 3.2
4.0 Mesh stiffness (10 8 N/m)
Mesh stiffness (10 8 N/m)
3.0 2.8 2.6 2.4 2.2 2.0 1.8
3.6 3.2 2.8 2.4
1.6 1.4
0.0
0.2
0.4 0.6 Mesh position (a)
0.8
1.0
2.0 0.0
0.2
0.4
0.6
0.8
Mesh position (b)
Fig. 4. Mesh stiffness of gears in a cycle: (a) internal gear pair, (b) external gear pair.
1.0
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9 > > > > > > > > > > i¼1 i¼1 > > > 3 3 > P P € _ > ms ns þ Cn ns Dspi sinð/i aÞ þ kn ns Wspi sinð/i aÞ ¼ 0; > > > = i¼1 i¼1 3 3 P P ms g€s þ Cg g_ s þ Dspi cosð/i aÞ þ kg gs þ Wspi cosð/i aÞ ¼ 0; > > > > i¼1 i¼1 > > > Mpi€xpi Dspi þ Drpi Wspi þ Wrpi ¼ 0; > > > 3 3 3 3 > P P P P > > Mc€xc Dspi Drpi þ Ccl X_ cl Wspi Wrpi þ kcl Xcl ¼ 0; > > > i¼1 i¼1 i¼1 i¼1 > > ; _ Ml€xl Ccl Xcl kcl Xcl ¼ Pl ði ¼ 1; 2; 3Þ; Mh€xh þ Chs X_ hs þ khs Xhs ¼ Ph ; 3 3 P P Ms€xs Chs X_ hs þ Dspi khs Xhs þ Wspi ¼ 0;
ð9Þ
where Ih ; 2 rbs Kcl kcl ¼ 2 ; rbc
Mh ¼
Is Ipi ; Mpi ¼ 2 ; 2 rbs rbp Th Tl Ph ¼ ; Pl ¼ : rbs rbc Ms ¼
Ml ¼
Il ; 2 rbc
Mc ¼
Ic mp ; þ3 2 cos2 a rbc
khs ¼
Khs ; 2 rbs
Here, the subscripts ÔhÕ, ÔsÕ, pi, ÔcÕ and ÔlÕ represent the high-speed part, the sun gear, the planet gear, the carrier and the low-speed part, respectively. I represents the inertia, m the actual mass, M the equivalent mass, C the damping coefficient, Th and Tl the input and output torque respectively, Ph and Pl the equivalent force, Khs and Kcl the torsional stiffness, khs and kcl the equivalent stiffness in the direction of line of action, kn and kg the bearing stiffness. Eq. (9) features considerable difficulties in its solving process as follows. (1) It is a semi-definite system, which predicates prospective trivial solutions corresponding to rigid body motions. (2) The function f is nonlinear multivariate, and the number of variables is even different according to the external and internal gear pairs. (3) As both linear and nonlinear restoring forces exist in the equation, it is not possible to write out the governing equation in matrix form, while a general solution technique applicable to the systems of multiple degrees of freedom must be based on matrix form. Therefore, Eq. (9) is simplified further by using a set of new variables 9 Xhs ¼ xh xs ; > > > > Xsdi ¼ xs xpi xc ns sinð/i aÞ þ gs cosð/i aÞ espi ; > > = Xn ¼ ns ; ð10Þ Xg ¼ gs ; > > > > Xrdi ¼ xpi xc erpi ; > > ; Xcl ¼ xc xl ði ¼ 1; 2; 3Þ: The new coordinate variables defined in above not only have intuitional physical meaning, but also eliminate the rigid body motions. Furthermore, f can be written as a set of functions in single variable in terms of variables given in Eq. (10). Hence, the set of simplified governing equations is obtained by combining Eqs. (9) and (10)
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3 Mhs X Mhs Cspi X_ sdi þ khs Xhs kspi f ðXsdi Þ ¼ Ph ; Ms i¼1 Mh i¼1 3
X M M M M M Msdi sdi sdi sdi sdi sdi Msdi X€sdi þ Cspi X_ sdi Chs X_ hs þ Cspi X_ sdi þ þ Ccl X_ cl Ms Mp Ms Mc ms Mc i¼1
Mhs Mhs X€hs þ Chs X_ hs Ms
3 X
3 sinð/i aÞMsdi _ cosð/i aÞMsdi Msdi Msdi X Cn n s þ Cg g_ s Crpi X_ rdi þ Crpi X_ rdi ms ms Mp Mc i¼1 3
X Msdi Msdi Msdi Msdi Msdi khs Xhs þ kspi f ðXsdi Þ þ þ þ kspi f ðXsdi Þ Ms Mp Ms Mc ms i¼1
Msdi sinð/i aÞMsdi _ cosð/i aÞMsdi Msdi kcl Xcl kn ns þ k g gs krpi f ðXrdi Þ Mc ms ms Mp
þ
3 Msdi X €espi krpi f ðXrdi Þ ¼ Msdi Mc i¼1
ms X€n þ Cn X_ n
3 X
ði ¼ 1; 2; 3Þ;
Cspi X_ sdi sinð/i aÞ þ kn Xn
i¼1
ms X€g þ Cg X_ g þ
3 X
3 X
kspi f ðXsdi Þ sinð/i aÞ ¼ 0;
i¼1
Cspi X_ sdi cosð/i aÞ þ kg Xg þ
i¼1
3 X
kspi f ðXsdi Þ cosð/i aÞ ¼ 0;
i¼1
Mrdi Mrdi X€rdi Cspi X_ sdi þ Mp
3 3 X X Mrdi Mrdi Mrdi Mrdi _ _ Cspi Xsdi þ Crpi Xrdi þ Crpi X_ rdi Ccl X_ cl M M M M c p c c i¼1 i¼1
3 X Mrdi Mrdi Mrdi Mrdi kspi f ðXsdi Þ þ kspi f ðXsdi Þ þ krpi f ðXrdi Þ kcl Xcl Mp Mc Mp Mc i¼1
þ
3 X Mrdi erpi krpi f ðXrdi Þ ¼ Mrdi € M c i¼1
Mcl X€cl
ði ¼ 1; 2; 3Þ;
3 3 3 Mcl X Mcl X Mcl X Cspi X_ sdi þ Ccl X_ cl Crpi X_ rdi kspi f ðXsdi Þ þ kcl Xcl Mc i¼1 Mc i¼1 Mc i¼1
3 Mcl X Mcl krpi f ðXrdi Þ ¼ P; Mc i¼1 Ml
where Mp Mc Mc Ml ; Mcl ¼ ; Mp þ Mc Mc þ Ml Ms Mpi Mc ms ¼ : Mpi Mc ms þ Ms Mc ms þ Ms Mpi ms þ Mpi Mc Ms
Mhs ¼ Msdi
Ms Mh ; Ms þ Mh
Mrdi ¼
ð11Þ
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If a displacement vector q is introduced as following: q ¼ f Xhs ;
Xsd1 ;
Xsd2 ;
ns ;
Xsd3 ;
gs ;
Xrd1 ;
Xrd2 ;
Xrd3 ;
T
ð12Þ
Xcl g :
The equation of motion is given in the matrix form as M€ q þ C q_ þ K f ð qÞ ¼ p;
ð13Þ
where the mass matrix M reads M ¼ diag½ Mhs ;
Msd1 ;
Msd2 ;
Msd3 ;
ms ;
ms ;
Mrd1 ;
Mrd2 ;
Mrd3 ;
Mcl ;
ð14Þ
f ð qÞ is the vector of nonlinear functions in single variable with uniform shape in terms of the coordinates transformed in Eq. (10) 8 qi bi ; qi > bi ; < qi Þ ¼ 0; bi 6 fi ¼ f ð ð15Þ qi 6 bi ; : q i < bi ; qi þ bi ; where bi ¼ 0; for i ¼ 1; 5; 6; 10; iþ1 ¼ b bspi ; biþ6 ¼ brpi ; for i ¼ 1; 2; 3:
ð16aÞ ð16bÞ
The stiffness matrix K, the damping matrix C and the force vector p are given in Appendix A. Eq. (13) can bepfurther simplified by using a number of dimensionless parameters. For this ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi purpose, let xn ¼ ksp1 =Msd1 , where ksp1 is the mean value of mesh stiffness between the sun gear and planet gear-1. Other dimensionless parameters are defined as ii xn Þ; kij ¼ kij =ðm ii x2 Þ; q¼ q=bc ; M ¼ I; X ¼ x=xn ; cij ¼ cij =ðm n
pi ðtÞ ¼
ii bc x2n Þ; pi ðtÞ=ðm
bi =bc ; bi ¼
i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; n; 8 < qi bi ; qi > bi ; fi ¼ f ðqi Þ ¼ 0; bi 6 qi 6 bi ; : qi þ bi ; qi < bi ;
ð17Þ
where bc is a character length and x is the excitation frequency. Thus, we have the dimensionless dynamic equations M€ qðsÞ þ C q_ ðsÞ þ Kf ðqðsÞÞ ¼ pðsÞ;
ð18Þ
where s ¼ t xn is dimensionless time. 4. Solutions by harmonic balancing Kahraman and Singh [5] and Comparin and Singh [6] located the periodic vibration of a gear pair with backlash excited by external torque and internal excitations by employing the harmonic balance method (HBM). Successful examples of HBM for solving nonlinear systems with multiple clearances have not been available in archival publications. In this section, the HBM is presented to reveal the nonlinear dynamics of multi-degree-of-freedom planetary gear systems excited by transmission errors.
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4.1. Harmonic balance method Attention is paid to the periodic vibrations of system under the harmonic excitation. The procedure of solving Eq. (18) by HBM includes four aspects as following. (1) Form of excitation: According to the assumption of harmonic excitations, excitations given in Eq. (17) can be represented approximately as pi ¼ pmi þ pai cosðXs þ upi Þ;
ð19aÞ
where subscripts ÔmÕ and ÔaÕ represent the mean and alternating components of force, and upi is phase angle of force. (2) Form of response: For the harmonic excitations given by Eq. (19a), the entries in approximate solution vector q are assumed in the form qi ¼ qmi þ qai cosðXs þ ui Þ;
ð19bÞ
where qmi and qai are the mean and alternating components of the steady state response, and ui is phase angle. (3) Nonlinear function: The steady state solution qi ðsÞ ¼ qi ðs þ T Þ in Eq. (19b) is assumed to be periodic with period T ¼ 2p=X. Accordingly, f ðqi ðsÞÞ ¼ f ðqi ðs þ T ÞÞ must also be periodic. A key step of HBM is to represent the nonlinear function f in Eq. (18) in the following form: f ðqi Þ ¼ Nmi qmi þ Nai qai cosðXs þ ui Þ; where Nmi and Nai are the describing functions defined as Z 2p 1 f ðqi Þdhi ; Nmi ðqmi ; qai Þ ¼ 2pqmi 0 Z 2p 1 Nai ðqmi ; qai Þ ¼ f ðqi Þ cos hi dhi ; hi ¼ Xs þ ui : pqai 0 The substitution of Eqs. (17) and (19b) into Eqs. (19d) and (19e) gives qai ½Gðlþ Þ Gðl Þ ; Nmi ¼ 1 þ 2qmi Nai ¼ 1 12½H ðlþ Þ ðl Þ ;
ð19cÞ
ð19dÞ ð19eÞ
ð20aÞ ð20bÞ
where
pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 l þ 1 l2 Þ; jlj 6 1; ð2=pÞðl sin GðlÞ ¼ jlj; jlj > 1; 8 l < 1; < 1; pffiffiffiffiffiffiffiffiffiffiffiffiffi H ðlÞ ¼ ð2=pÞðl sin1 l þ l 1 l2 Þ; jlj 6 1; : 1; l > 1; l ¼ ð bi qmi Þ=qai :
ð20cÞ
ð20dÞ ð20eÞ
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For i ¼ 1, 5, 6, 10, we have bi ¼ 0:
ð20fÞ
Substituting Eq. (20f) into Eqs. (20a)–(20e) yields Nmi ¼ 1; Nai ¼ 1:
ð20gÞ
This result agrees with the physical fact that describing function of linear function is really 1. (4) Algebraic equations: Considering the mean value and the fundamental harmonics of timevarying periodic mesh stiffness in Fig. 4, the entries in stiffness matrix K in Eq. (18) can be written as kij ¼ kmij þ kaij cosðXs þ /kij Þ:
ð21aÞ
To get the mean value and the harmonics of the time-varying mesh stiffness, the FFT algorithm is introduced to the periodic function of mesh stiffness. Hence, K is written in terms of two separate matrices for mean stiffness and alternating stiffness K ¼ K m þ DK;
ð21bÞ
where K m ¼ ½kmij n n ;
DK ¼ ½kaij cosðXs þ /ij Þ n n :
ð21cÞ
Eqs. (19a)–(19c) can be written in the following matrix form p ¼ fpmi gn 1 þ fpai cosðXs þ upi Þgn 1 ;
ð22aÞ
q ¼ fqmi gn 1 þ fqai cosðXs þ ui Þgn 1 ;
ð22bÞ
f ðqÞ ¼ fNmi qmi gn 1 þ fNai qai cosðXs þ ui Þgn 1 :
ð22cÞ
By substituting Eqs. (19a)–(19c) into Eq. (18) and balancing the same harmonics, one obtains the algebraic equations of system 8 < K m ym þ 12 ðK 1 y3 þ K 2 y4 Þ pm ¼ f0gn 1 ; ð23Þ K y þ K 1 ym X2 My1 XCy2 p1 ¼ f0gn 1 ; : m 3 2 K m y4 þ K 2 ym X My2 XCy1 p2 ¼ f0gn 1 ; where y1 ¼ fqai cos ui gn 1 ; y4 ¼ fNai qai sin ui gn 1 ; p1 ¼ fpai cos upi gn 1 ; K 1 ¼ fkaij cos /ij gn n ;
y2 ¼ fqai sin ui gn 1 ;
y3 ¼ fNai qai cos ui gn 1 ;
ym ¼ fNmi qmi gn 1 ; p2 ¼ fpai sin upi gn 1 ;
ð24Þ
K 2 ¼ ½kaij sin /ij n n :
Eq. (23) includes following 3n unknown variables to be solved qai ; qmi ; ui ; i ¼ 1; 2; . . . ; n:
ð25Þ
As it is impossible to solve the nonlinear algebraic equation (23) by any analytical methods, a numerical routine (DNEQBJ of IMSL [8]) is used to determine the solution. The routine
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DNEQBJ uses a secant method with BroydenÕs update to find the zeros of a set of nonlinear algebraic equations. 4.2. Solutions and validation As the first example, a 2K-H type planetary gear system shown in Fig. 1 is studied. The geometric parameters of system are listed in Table 1 and other parameters are given in Table 2. Furthermore, the characteristic length is set to bc ¼ 0:01 mm. The excitations are associated with the mesh stiffness fluctuation and errors. The frequency responses of system are determined by using the HBM in the previous subsection. As a comparison, the frequency responses are shown in Fig. 5(a)–(e), together with the responses of corresponding linear systems without any backlash. These figures demonstrate that a planetary gear system with clearances taken into account exhibits particular behaviors as following. (1) For all the response results shown in Fig. 5(a)–(e), except for rigid and reduplicate frequencies, the system undergoes resonance at five different frequencies, all of which are near the natural frequencies of corresponding linear system. (2) Similar to the studies on single gear pair by Kahraman and Singh [5], the dynamics of the planetary gear system with clearances exhibits strong nonlinearity. As shown in Fig. 5(a) and (b), both regimes of no impacting (no tooth separation) and single-sided impacting (tooth
Table 1 Parameters of a planetary gear set Number of teeth Module (mm) Pressure angle Amount of crowning (mm) Face width (mm) Load (N m)
Sun gear
Planet gear
Ring gear
15 0.25 20 0.3835 14 50
24 0.25 20 0.161 14 50
63 0.25 20 0.7056 14 50
Table 2 Parameters in case study Parameter ms mp Ih Ip Ic esp1 ¼ esp2 ¼ esp3 erp1 ¼ erp2 ¼ erp3 bsp1 ¼ bsp2 ¼ bsp3 brp1 ¼ brp2 ¼ brp3
Unit Kg Kg Kg mm2 Kg mm2 Kg mm2 lm lm lm lm
Value 0.417 0.254 202 140 6000 20 20 50 50
Parameter Il ks Kcl Khs ksp1m ¼ ksp2m ¼ ksp3m ksp1a ¼ ksp2a ¼ ksp3a krp1m ¼ krp2m ¼ krp3m ksp1m ¼ ksp2m ¼ ksp3m Damping ratio
Unit
Value 2
Kg mm N/m N mm/Rad N mm/Rad N/m N/m N/m N/m –
1980 4.24e8 4.95e8 4.58e8 2.42e8 0.43e8 2.91e8 0.47e8 0.02
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4
0.05
3 Amplitude
Amplitude
0.04 0.03 0.02
2
1
0.01 0.00 0.0
0 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.6
0.8
1.0
1.2
1.4
Frequency
Frequency
(a)
(b)
4.0
1.6
1.8
2.0
0.12
3.5 0.10 Amplitude
Amplitude
3.0 2.5 2.0
0.08 0.06
1.5 0.04 1.0 0.02
0.5
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.00 0.0
2.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Frequency
Frequency
(d)
(c)
0.08
Amplitude
0.06
0.04
0.02
0.00 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Frequency (e)
Fig. 5. Frequency response of a planetary gear set ( ) linear system; (––) nonlinear system: (a) frequency response of Xhs , (b) frequency response of Xsd1 , (c) frequency response of Xrd1 , (d) frequency response of Xs and (e) frequency response of Xcl .
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Alternating amplitude
3
2
1
0 0.8
1.0
1.2
1.4 Frequency
1.6
1.8
2.0
Fig. 6. Comparison of frequency response Xsd1 obtained by different methods: (·) numerical integration; (––) harmonic balance method.
separation, but no back collision) exist in the frequency response. One can detect the dual-valued solutions and jump discontinuities near resonant frequencies. These phenomena predict inherent characters of nonlinearity. (3) As the motions of a MDOF system at different degrees of freedom are coupled each other, sudden discontinuities are also observed in Fig. 5(c) and (d), although there are no clearances for these parts. This indicates the interactions between gear pairs and other components. (4) Not all the resonant frequencies of system are dangerous for the nonlinear vibration. The jump discontinuities of frequency response occur only near the resonance sensitive to vibration of gear pairs. Next, we verify the solutions of HBM through the numerical simulations, where a variable-step Runge–Kutta algorithm of fifth–sixth order in [9] is used. The frequency responses obtained by using HBM and Runge–Kutta algorithm are compared in Fig. 6. The two methods get an agreement although the response amplitudes at resonant frequencies are slightly different. Discrepancies may come from the assumption of single harmonics in HBM. Though the validity of HBM is confirmed, one should be aware of the following as also mentioned by Kahraman in [5]. (1) Several problems may appear in the implementation of numerical integration to the nonlinear dynamics due to clearances and caution must be exercised [6] even though it is more precise. (2) The HBM is incapable of predicting any quasi-periodic or chaotic vibrations.
5. Parametric studies The dynamic behaviors of a planetary gear set, such as the frequency response, the transition frequency and the existence of various impacting regimes, depend on the parameters of system to some extent. Therefore, parametric studies are presented here.
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We first examine the effect of mesh stiffness. As well known, the mesh stiffness of a gear pair is a time-variant periodic function, which can be represented as kðtÞ ¼ km þ ka sinðXs þ uÞ:
ð26Þ
To describe the extent of variation, a stiffness ratio is introduced as ka k^ ¼ : km
ð27Þ
Three different results of Xrd1 are obtained by varying k^ only for krp1 , the mesh stiffness between the ring gear and planet gear-1, while other parameters remain unchanged as shown in Table 2. Fig. 7 shows the jump discontinuities in all three cases no matter how large k^ is taken. When k^ is increased, the transition frequencies of jump become lower, the transition frequencies of jumping-up and jumping-down leave each other. This gives rise to the dual-valued solution region. The comparison indicates that although the existence of typical nonlinear dynamic behaviors of a planetary gear system with clearances does not depend on the variation of stiffness, the extent of nonlinearity is affected really by the stiffness ratio. An increase of variation of stiffness ratio enhances the nonlinearity of response. Next, we examine the effect of static transmission error e, which is the excitation on the planetary gear system in this study. Three case studies are shown in Fig. 8 for dimensionless e being 0.2, 2 and 4, respectively. When e is very small, say, e ¼ 0:2, the system dynamics is linear and the tooth pairs do not lose contact all the time although the system has clearances. When e is increased to 2, the dynamics of system becomes nonlinear. The single-sided impact vibration and the jump discontinuities are observed. When e is increased further to 4, the nonlinear behaviors get more complicated. The double-sided impact (back collision) vibration are also observed near the resonant frequency, X ¼ 1:58. Moreover, as e is increased, the amplitudes of response near the resonance become larger. This is reasonable since e plays a role of alternating components in the excitation on planetary gear system.
3.5
Alternating amplitude
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Frequency
Fig. 7. Frequency response of Xrd1 for different mesh stiffness: ( ) k^ ¼ 0; (- - -) k^ ¼ 0:2; (––) k^ ¼ 0:4.
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5.0 4.5 4.0
Amplitude
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Frequency
Fig. 8. Frequency response of Xsd1 for different errors: ( ) e ¼ 0:2; (- - -) e ¼ 2:0; (––) e ¼ 4:0.
6. Conclusions In this study, some advances are made in the nonlinear dynamics of a general planetary gear system with backlash and time-varying mesh stiffness, excited by the static transmission errors. First, a planetary gear system with multiple clearances taken into consideration is studied from the viewpoint of nonlinear dynamics. Second, a lateral–torsional coupled nonlinear dynamic model for the planetary gear set is developed and the governing equation is formulated in matrix form. Third, the frequency response of the planetary gear set under a harmonic excitation is determined by using the HBM and verified by using numerical simulation. Finally, the effects of some important parameters, such as the variation of mesh stiffness and static transmission errors, on the nonlinear dynamics are discussed. The study enriches the current literature in the nonlinear dynamics of mechanical systems with multiple clearances and the dynamics of planetary gear systems. The results of this study yield some guidelines to be instrumental to improvement of the planetary transmission performance. The ongoing research work focuses on the load sharing characteristics of planetary transmissions with multiple clearances since the advantages of a planetary gear system greatly rely on the sharing characteristics. Future work will include the analysis of higher order harmonics of planetary gear systems with multiple clearances, meshing phase difference and unequal backlash.
Acknowledgement We thank National Natural Science Foundation of China for supporting this study.
Appendix A. Entries in matrices K, C, p Both K and C are 10 · 10 square matrices, while p is a 10 · 1 vector.
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(1) The entries in the stiffness matrix K are given as follows: k1;i ¼ khs ;
k1;i ¼ Mhs kspi ; Ms
i ¼ 1; 2; 3;
k2;i ¼ Msd1 khs ; k2;2 ¼ ksp1 ; k2;iþ1 ¼ Msd1 þ Msd1 þ Msd1 kspi ; i ¼ 2; 3; Ms Ms Mc ms k2;5 ¼ sinð/1 aÞ Msd1 kn ; k2;6 ¼ cosð/1 aÞ Msd1 kg ; ms ms
k2;7 ¼ Msd1 Msd1 krp1 ; k2;iþ6 ¼ Msd1 krpi ; i ¼ 2; 3; k2;10 ¼ Msd1 kcl ; Mc Mp Mc Mc
k3;1 ¼ Msd2 khs ; k3;iþ1 ¼ Msd2 þ Msd2 þ Msd2 kspi ; i ¼ 1; 3; Ms Ms Mc ms k3;3 ¼ ksp2 ; k3;5 ¼ sinð/2 aÞ Msd2 kn ; k3;6 ¼ cosð/2 aÞ Msd2 kg ; ms ms
k3;iþ6 ¼ Msd2 krpi ; i ¼ 1; 3; k3;8 ¼ Msd2 Msd2 krp2 ; k3;10 ¼ Msd2 kcl ; Mc Mc Mp Mc
k4;1 ¼ Msd3 khs ; k4;iþ1 ¼ Msd3 þ Msd3 þ Msd3 kspi ; i ¼ 1; 2; k4;4 ¼ ksp3 ; Ms Ms Mc ms k4;5 ¼ sinð/3 aÞ Msd3 kn ; k4;6 ¼ cosð/3 aÞ Msd3 kg ; ms ms
k4;iþ6 ¼ Msd3 krpi ; i ¼ 1; 2; k4;9 ¼ Msd3 Msd3 krp3 ; k4;10 ¼ Msd3 kcl ; Mc Mc Mp Mc k5;iþ1 ¼ sinð/i aÞkspi ; i ¼ 1; 2; 3; k5;5 ¼ kn ; k6;iþ1 ¼ cosð/i aÞkspi ; i ¼ 1; 2; 3; k6;6 ¼ kg ;
k7;2 ¼ Mrd1 Mrd1 ksp1 ; k7;iþ1 ¼ Mrd1 kspi ; i ¼ 2; 3; Mc Mp Mc Mrd1 i ¼ 2; 3; k7;10 ¼ kcl ; Mc
k8;iþ1 ¼ Mrd2 kspi ; i ¼ 1; 3; k8;3 ¼ Mrd2 Mrd2 ksp2 ; Mc Mc Mp k8;8 ¼ krp2 ; k8;iþ6 ¼ Mrd2 krpi ; i ¼ 1; 3; k8;10 ¼ Mrd1 kcl ; Mc Mc
k9;iþ1 ¼ Mrd3 kspi ; i ¼ 1; 2; k9;4 ¼ Mrd3 Mrd3 ksp3 ; Mc Mc Mp k9;9 ¼ krp3 ; k9;iþ6 ¼ Mrd3 krpi ; i ¼ 1; 2; k9;10 ¼ Mrd3 kcl ; Mc Mc
k7;7 ¼ krp1 ;
k7;iþ6 ¼ Mrd1 krpi ; Mc
T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371–1390
k10;iþ1 ¼ Mcl kspi ; Mc k10;10 ¼ kcl :
i ¼ 1; 2; 3;
k10;iþ6 ¼ Mcl krpi ; Mc
i ¼ 1; 2; 3;
The entries not listed above are taken as zeros. (2) The entries in the damping matrix C are listed below c1;i ¼ Chs ;
c1;i ¼
Mhs Cspi ; Ms
i ¼ 1; 2; 3;
Msd1 Msd1 Msd1 Msd1 c2;1 ¼ Chs ; c2;2 ¼ Csp1 ; c2;iþ1 ¼ þ þ Cspi ; i ¼ 2; 3; Ms Ms Mc ms Msd1 Msd1 c2;5 ¼ sinð/1 aÞ Cn ; c2;6 ¼ cosð/1 aÞ Cg ; ms ms
Msd1 Msd1 Msd1 Msd1 c2;7 ¼ Crpi ; i ¼ 2; 3; c2;10 ¼ Ccl ; Crp1 ; c2;iþ6 ¼ Mc Mp Mc Mc
Msd2 Msd2 Msd2 Msd2 c3;1 ¼ Chs ; c3;iþ1 ¼ þ þ Cspi ; i ¼ 1; 3; Ms Ms Mc ms Msd2 Msd2 c3;3 ¼ Csp2 ; c3;5 ¼ sinð/2 aÞ Cn ; c3;6 ¼ cosð/2 aÞ Cg ; ms ms
Msd2 Msd2 Msd2 Msd2 c3;iþ6 ¼ þ Crp2 ; c3;10 ¼ Crpi ; i ¼ 1; 3; c3;8 ¼ Ccl ; Mc Mc Mp Mc
Msd3 Msd3 Msd3 Msd3 c4;1 ¼ Chs ; c4;iþ1 ¼ þ þ Cspi ; i ¼ 1; 2; c4;4 ¼ Csp3 ; Ms Ms Mc ms Msd3 Msd3 c4;5 ¼ sinð/3 aÞ Cn ; c4;6 ¼ cosð/3 aÞ Cg ; ms ms
Msd3 Msd3 Msd3 Msd3 c4;iþ6 ¼ Crpi ; i ¼ 1; 2; c4;9 ¼ Ccl ; Crp3 ; c4;10 ¼ Mc Mc Mp Mc c5;iþ1 ¼ sinð/i aÞCspi ;
i ¼ 1; 2; 3;
c5;5 ¼ Cn ;
c6;iþ1 ¼ cosð/i aÞCspi ; i ¼ 1; 2; 3; c6;6 ¼ Cg ;
Mrd1 Mrd1 Mrd1 c7;2 ¼ Cspi ; i ¼ 2; 3; Csp1 ; c7;iþ1 ¼ Mc Mp Mc Mrd1 Mrd1 c7;7 ¼ Crp1 ; c7;iþ6 ¼ Crpi ; i ¼ 2; 3; c7;10 ¼ Ccl ; Mc Mc
Mrd2 Mrd2 Mrd2 c8;iþ1 ¼ Cspi ; i ¼ 1; 3; c8;3 ¼ Csp2 ; Mc Mc Mp Mrd2 Mrd1 c8;8 ¼ Crp2 ; c8;iþ6 ¼ Crpi ; i ¼ 1; 3; c8;10 ¼ Ccl ; Mc Mc
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Mrd3 Mrd3 c9;iþ1 Csp3 ; i ¼ 1; 2; c9;4 ¼ Mc Mp Mrd3 Mrd3 c9;9 ¼ Crp3 ; c9;iþ6 ¼ Crpi ; i ¼ 1; 2; c9;10 ¼ Ccl ; Mc Mc Mcl Mcl c10;iþ1 ¼ Cspi ; i ¼ 1; 2; 3; c10;iþ6 ¼ Crpi ; i ¼ 1; 2; 3; Mc Mc c10;10 ¼ Ccl : Mrd3 ¼ Cspi ; Mc
Again, the entries not listed above are set to be zeros. (3) The entries in the force vector p are given as following Mhs Mcl Ph ; p5 ¼ 0; p6 ¼ 0; p10 ¼ Pl ; p1 ¼ Mh Ml piþ6 ¼ Mrdi €erpi ; i ¼ 1; 2; 3: piþ1 ¼ Msdi €espi ; i ¼ 1; 2; 3;
References [1] T. Hidaka, Analysis of dynamic tooth load on planetary gear, Bulletin of the Japanese Society of Mechanical Engineers 23 (1980) 315–322. [2] A. Kahraman, Load sharing characteristics of planetary transmissions, Mechanism and Machine Theory 29 (1994) 1151–1165. [3] R.G. Munro, The dynamic behavior of spur gears. PhD Dissertation, Cambridge University, 1962. [4] A. Kubo, K. Yamada, T. Aida, S. Sato, Research on ultra high speed gear devices (reports 1–3). Transactions of the Japan Society of Mechanical Engineers 38 (1972) 2692-2715. [5] A. Kahraman, R. Singh, Nonlinear dynamics of a spur gear pair, Journal of sound and vibration 142 (1990) 49–75. [6] R.J. Comparin, R. Singh, Nonlinear frequency response characteristics of an impact pair, Journal of sound and vibration 134 (1989) 259–290. [7] A. Kahraman, R. Singh, Nonlinear dynamics of a geared rotor-bearing system with multiple clearances, Journal of Sound and Vibration 144 (1991) 469–506. [8] Microsoft INC. IMSL Libraries Reference (Pro Edition) 1994–95. [9] H.N. Ozguven, D.R. Houser, Dynamic analysis of high speed gears by using loaded static transmission error, Journal of Sound and Vibration 125 (1988) 71–83.
Nonlinear dynamics of a planetary gear system with multiple clearances Tao Sun
a,*
, HaiYan Hu
b
a
b
Department of Precision Mechanical Engineering, Shanghai University, Shanghai 200072, PR China Institute of Vibration Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China Received 19 April 2002; received in revised form 12 March 2003; accepted 16 May 2003
Abstract Presented in this paper is on the nonlinear dynamics of a planetary gear system with multiple clearances taken into account. A lateral–torsional coupled model is established with multiple backlashes, time-varying mesh stiffness, error excitation and sun-gear shaft compliance considered. The solutions are determined by using harmonic balance method from the equations in matrix form. The theoretical results from HBM are verified by using the numerical integration. Finally, effects of parameters are discussed. 2003 Elsevier Ltd. All rights reserved. Keywords: Planetary gear transmission; Nonlinear vibration; Fourier transform; Harmonic balance method; Dynamic
1. Introduction Planetary gear systems have been widely used in engineering owing to their advantages such as little space required, large ratio of transmission and high efficiency. One of the most popular applications is to the automatic transmissions in automobiles. Because of machinery complexity, most of earlier studies on the planetary gear systems were confined to their static behaviors and sharing characteristics. Over the past two decades, the dynamics of planetary transmissions has drawn much attention. However, almost all the published studies on the planetary transmissions focused only on their linear vibration [1,2]. In the advanced mechanical systems running at high speed, such as exact antennas and automatic weapon systems, which usually contain a number of planetary gear sets, the gear systems *
Corresponding author. Fax: +86-21-56334458. E-mail address: [email protected] (T. Sun).
0094-114X/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0094-114X(03)00093-4
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Nomenclature b C, c e f I K, k M, m N P, p q r t T X, x n, g U a s X, x u
backlash damping static transmission error nonlinear displacement function rotary inertia stiffness mass describing function force displacement radius time torque displacement transverse displacement angle pressure angle dimensionless time frequency phase angle
Subscripts a alternating components b base circle c carrier d relative to h high-sped part i ordinal number l low-speed part m mean component p planet gear r ring gear s sun gear n horizontal g vertical
often undergo startup and brake interactively or run at high speed and under light load. The current studies have shown that a gear pair would likely lose contact and the tooth separations
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occur due to the unavoidable backlash. Accordingly, the backlash, namely the clearance, tends to bring gear systems to exhibit typical nonlinear dynamical behaviors. A gear pair is bound to have some backlash, which may be either designed to provide better lubrication and to eliminate interference or due to manufacturing errors and wear. Backlashinduced nonlinear vibrations may cause tooth separation and impacts in unload or lightly loaded geared drives. Such impacts result in intense vibration and noise problems and large dynamic loads, which may affect reliability and life of the gear drive. Experimental studies on the dynamic behavior of a spur gear pair with backlash started almost 40 years ago and still continue [3–5]. For instance, Kubo et al. [4] observed a jump in the frequency response of a gear pair with backlash even though the test set-up was heavily damped. Such experimental studies, albeit limited in scope, have clearly shown that the dynamics of a gear pair can hardly be predicted on the basis of a linear model. Consequently, the nonlinear dynamics and mathematical models of a gear pair with backlash have been intensively studied in the past decade. Although most of the nonlinear mathematical models used to describe the dynamic behavior of a gear pair are somewhat similar to each other, they differ in terms of the excitation mechanisms considered and especially the solution technique used. The nonlinearity of a gear backlash has to be modeled by a discontinuous and nondifferentiable function, which represents a strong nonlinear interaction in the dynamic equation of whole system. Comparin and Singh addressed this problem in [6] and pointed out that most techniques available in the literatures cannot be directly applied to solving this problem. Many researchers have recognized this problem implicitly and therefore employed either digital or analog simulation techniques in their studies. Kahraman and Singh [7] made a detailed review of nonlinear gear dynamics available in current publications. Their theoretical study also made contributions to the nonlinear dynamics of a spur gear pair with backlash subject to the static transmission error. Although there is a vast body of literature concerned with nonlinear dynamics of a general gear pair with clearance, the studies on nonlinear dynamics of a planetary transmission system with multiple clearances are still very limited. Kahraman took the possibility of tooth separation in a planetary gear system into account [2]. In his study, however, the model was not considered as a nonlinear dynamic system. Instead, a step function in linear model was used to distinguish the tooth contact and the tooth separation roughly. As a result, the back collisions of teeth were not included in his study. Compared with above-mentioned works, the nonlinear dynamics of a planetary gear system is much more complicated. Such a system is inherently nonlinear owing to the multiple clearances, and includes the temporally and spatially varying system parameters. Moreover, the request for prediction and examination of dynamic behaviors is progressively urgent in the design of more quiet and reliable planetary transmissions. The published studies on the nonlinear dynamics of geared drive with backlash, however, focus on only a single gear pair, rather than any planetary gear systems. Accordingly, the focus of this paper is on the nonlinear dynamics of planetary gear systems. In this paper, some contributions will be made to a number of key issues such as nonlinear dynamics modeling, solution techniques to the nonlinear differential equations and dynamic behaviors of planetary gear systems.
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2. Dynamic model of system 2.1. Model and assumptions The planetary gear system of concern in this study is a single-stage 2K-H type planetary gear set as shown in Fig. 1. The system consists of a high-speed part h, a low-speed part l, a sun gear s, a ring gear r, n planet gears p and a carrier c. Here, n, the number of planet gears, is taken as 3 throughout the paper. All the gears are mounted on their flexible shafts supported by rolling element bearings. To establish the mathematical model of system, a few assumptions are introduced in the case of speed reduction in the planetary gear shown in Fig. 1 as following. (1) The inertial effects of prime mover (high-speed part) and load inertia (low-speed part) are taken as those of lumped mass. Hence, the planetary gear system has 4 þ n rotational degrees of freedom (DOF), including the rotational displacements of low-speed part, high-speed part, sun gear, carrier and n planetary gears, respectively. (2) Considering the bending stiffness of the shaft of sun gear, the compliance of bearings and potential displacement as a rigid body caused by floating, the horizontal and vertical transverse DOF of the sun gear are included. (3) As the bending stiffness of the planet gear shaft is very large, the deflection of this shaft can be neglected. Thus, the transverse displacement of planet gear is not considered. (4) Because the ring gear itself is a part of gearbox, the displacement of ring gear as a rigid body is insignificant and the center of ring gear is assumed not to move. Based on the above assumptions, the dynamic model of the planetary gear system is established as shown in Fig. 2. All symbols in Fig. 2 can be found in the nomenclature presented and further explanations are given in subsequent sections. The total number of DOF in the model is n þ 4 þ 2,
p c h s
r
Fig. 1. A planetary gear set.
l
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(a)
1375
(b)
Fig. 2. The dynamic model of a planetary gear set.
Ring gear
Planet gear
line of action
Sun gear (a)
(b)
Fig. 3. Mesh relation in a planetary gear set.
including n þ 4 rotational DOF and 2 transverse DOF, respectively. Obviously, this is a complicated lateral–torsional coupled nonlinear system with multiple clearances and time-varying parameters. 2.2. Equivalent displacements In the case of speed reduction, the meshing relation of the planetary gears is shown in Fig. 3. Regarding the angular displacement of each gear in driving, the direction of the revolution caused by the driving torque is assumed to be positive. Namely, the angular displacements hs and hc of the sun gear and the carrier are in the same direction. And the angular displacements hr and hp of the ring gear and the planet gear are reversed in the directions of hs and hc . In order to establish the equations of motion easily, both torsional and transverse displacements are unified on the pressure line in terms of equivalent displacements.
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The equivalent transverse displacements in the pressure line direction caused by rotational displacements are written as follows: xh ¼ rbs hh ;
xs ¼ rbs hs ;
xr ¼ rbr hr ;
xpi ¼ rbp hpi ;
xc ¼ rbc hc ;
xl ¼ rbc hl ;
i ¼ 1; 2; 3;
ð1Þ where rb (subscripts ÔsÕ, ÔrÕ, ÔpÕ) are the base circle radius of gears, while rbc is the nominal base circle radius of the carrier defined as following: rbc ¼ rbs þ rbp ¼ rbr rbp :
ð2Þ
With regard to the direction of the equivalent displacements, the direction of deflection caused by driving torque is assumed to be positive. It should be pointed out that in the case of speed reduction, the rotational displacement of the ring gear r is set to be zero since the ring gear is fixed on gearbox in the ‘‘2K-H’’ type planetary gear set shown in Fig. 1. That is, hr ¼ 0. However, the tooth deflection of the ring gear is included when calculating the mesh stiffness. By using X to represent the relative displacements in the direction of press line, the relative displacements are obtained as follows according to the meshing relation shown in Fig. 3 and the equivalent displacements 9 Xhs ¼ xh xs ; > > = Xspi ¼ xs xpi xc ; ð3Þ Xrpi ¼ xpi xc xr ¼ xpi xc ; > > ; Xcl ¼ xc xl i ¼ 1; 2; 3: To describe the plane motion of the sun gear, a kinetic Cartesian coordinate system attached to the carrier, instead of general fixed coordinate system, is introduced. The origin-o of this coordinate system coincides with the center of carrier. And the coordinate axes marked as ðn; gÞ rotate along with the carrier. Therefore, ns and gs represent the horizontal and vertical transverse displacements of sun gear with regard to the movable reference frame respectively. As the variation of pressure angle caused by the tiny translation of sun gear can be neglected, the equivalent displacement in the pressure line direction derived from the transverse displacements ns and gs is represented as xsdi ¼ ns sinð/i aÞ þ gs cosð/i aÞ:
ð4Þ
Here a is the pressure angle of gear pair, and /i means that the ith planet gear is mounted at a theoretical position angle /i on the carrier with respect to positive direction of axis-n. If the center of the first planet gear is assigned to positioned at angle 0, /i will be /i ¼ 2pði 1Þ=n;
i ¼ 1; 2; . . . ; n;
ð5Þ
where n represents the numbers of planet gears. 3. Equations of motion In the dynamical model shown in Fig. 2, the gear mesh is described by a nonlinear displacement function f . Here, f is defined as a nonlinear function in the relative gear mesh displacement q and with backlash 2 b as parameter
T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371–1390
8 q b; < f ð qÞ ¼ 0; : qþ b;
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q> b; b 6 q6 b; q < b:
ð6Þ
The planetary gear system includes two different kinds of gear pairs, the external gear pair, namely, the sun gear/planet gear-i pair (subscripts ÔsÕ and pi), and the internal gear pair, namely, the ring gear/planet gear-i pair (subscripts ÔrÕ and pi). Describing each pair of gear mesh by the backlash 2b, the time-varying mesh stiffness k, the viscous damping C and the static transmission error e, the dynamic tooth load of sun gear/planet gear-i pair and ring gear/planet gear-i pair can be given as following: Wspi ¼ kspi f ðXspi þ xsdi espi Þ;
ð7aÞ
Wrpi ¼ krpi f ðXrpi erpi Þ:
ð7bÞ
The mesh stiffness of internal and the external gear pairs are obtained by using finite element method (FEM). A finite element model on tooth contact is developed by using 3-dimensional FEM, and the nonlinear time varying mesh stiffness can be determined by finite element analysis. Mesh stiffness function of the internal and external gear pairs are then formed by cubic spline interpolation and approximation of the discrete mesh points in a mesh period. They are both periodic functions as shown in Fig. 4. To get the frequency behavior of mesh stiffness, the fast Fourier transform (FFT) algorithm is introduced. The damping force of gear pair is represented as Dspi ¼ Cspi ðX_ spi þ x_ sdi e_ spi Þ;
ð8aÞ
Drpi ¼ Crpi ðX_ rpi e_ rpi Þ:
ð8bÞ
Thus, the equations of lateral–torsional coupled motion of the nonlinear planetary gear system can be established by using the Lagrange principle as following: 3.2
4.0 Mesh stiffness (10 8 N/m)
Mesh stiffness (10 8 N/m)
3.0 2.8 2.6 2.4 2.2 2.0 1.8
3.6 3.2 2.8 2.4
1.6 1.4
0.0
0.2
0.4 0.6 Mesh position (a)
0.8
1.0
2.0 0.0
0.2
0.4
0.6
0.8
Mesh position (b)
Fig. 4. Mesh stiffness of gears in a cycle: (a) internal gear pair, (b) external gear pair.
1.0
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9 > > > > > > > > > > i¼1 i¼1 > > > 3 3 > P P € _ > ms ns þ Cn ns Dspi sinð/i aÞ þ kn ns Wspi sinð/i aÞ ¼ 0; > > > = i¼1 i¼1 3 3 P P ms g€s þ Cg g_ s þ Dspi cosð/i aÞ þ kg gs þ Wspi cosð/i aÞ ¼ 0; > > > > i¼1 i¼1 > > > Mpi€xpi Dspi þ Drpi Wspi þ Wrpi ¼ 0; > > > 3 3 3 3 > P P P P > > Mc€xc Dspi Drpi þ Ccl X_ cl Wspi Wrpi þ kcl Xcl ¼ 0; > > > i¼1 i¼1 i¼1 i¼1 > > ; _ Ml€xl Ccl Xcl kcl Xcl ¼ Pl ði ¼ 1; 2; 3Þ; Mh€xh þ Chs X_ hs þ khs Xhs ¼ Ph ; 3 3 P P Ms€xs Chs X_ hs þ Dspi khs Xhs þ Wspi ¼ 0;
ð9Þ
where Ih ; 2 rbs Kcl kcl ¼ 2 ; rbc
Mh ¼
Is Ipi ; Mpi ¼ 2 ; 2 rbs rbp Th Tl Ph ¼ ; Pl ¼ : rbs rbc Ms ¼
Ml ¼
Il ; 2 rbc
Mc ¼
Ic mp ; þ3 2 cos2 a rbc
khs ¼
Khs ; 2 rbs
Here, the subscripts ÔhÕ, ÔsÕ, pi, ÔcÕ and ÔlÕ represent the high-speed part, the sun gear, the planet gear, the carrier and the low-speed part, respectively. I represents the inertia, m the actual mass, M the equivalent mass, C the damping coefficient, Th and Tl the input and output torque respectively, Ph and Pl the equivalent force, Khs and Kcl the torsional stiffness, khs and kcl the equivalent stiffness in the direction of line of action, kn and kg the bearing stiffness. Eq. (9) features considerable difficulties in its solving process as follows. (1) It is a semi-definite system, which predicates prospective trivial solutions corresponding to rigid body motions. (2) The function f is nonlinear multivariate, and the number of variables is even different according to the external and internal gear pairs. (3) As both linear and nonlinear restoring forces exist in the equation, it is not possible to write out the governing equation in matrix form, while a general solution technique applicable to the systems of multiple degrees of freedom must be based on matrix form. Therefore, Eq. (9) is simplified further by using a set of new variables 9 Xhs ¼ xh xs ; > > > > Xsdi ¼ xs xpi xc ns sinð/i aÞ þ gs cosð/i aÞ espi ; > > = Xn ¼ ns ; ð10Þ Xg ¼ gs ; > > > > Xrdi ¼ xpi xc erpi ; > > ; Xcl ¼ xc xl ði ¼ 1; 2; 3Þ: The new coordinate variables defined in above not only have intuitional physical meaning, but also eliminate the rigid body motions. Furthermore, f can be written as a set of functions in single variable in terms of variables given in Eq. (10). Hence, the set of simplified governing equations is obtained by combining Eqs. (9) and (10)
T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371–1390
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3 Mhs X Mhs Cspi X_ sdi þ khs Xhs kspi f ðXsdi Þ ¼ Ph ; Ms i¼1 Mh i¼1 3
X M M M M M Msdi sdi sdi sdi sdi sdi Msdi X€sdi þ Cspi X_ sdi Chs X_ hs þ Cspi X_ sdi þ þ Ccl X_ cl Ms Mp Ms Mc ms Mc i¼1
Mhs Mhs X€hs þ Chs X_ hs Ms
3 X
3 sinð/i aÞMsdi _ cosð/i aÞMsdi Msdi Msdi X Cn n s þ Cg g_ s Crpi X_ rdi þ Crpi X_ rdi ms ms Mp Mc i¼1 3
X Msdi Msdi Msdi Msdi Msdi khs Xhs þ kspi f ðXsdi Þ þ þ þ kspi f ðXsdi Þ Ms Mp Ms Mc ms i¼1
Msdi sinð/i aÞMsdi _ cosð/i aÞMsdi Msdi kcl Xcl kn ns þ k g gs krpi f ðXrdi Þ Mc ms ms Mp
þ
3 Msdi X €espi krpi f ðXrdi Þ ¼ Msdi Mc i¼1
ms X€n þ Cn X_ n
3 X
ði ¼ 1; 2; 3Þ;
Cspi X_ sdi sinð/i aÞ þ kn Xn
i¼1
ms X€g þ Cg X_ g þ
3 X
3 X
kspi f ðXsdi Þ sinð/i aÞ ¼ 0;
i¼1
Cspi X_ sdi cosð/i aÞ þ kg Xg þ
i¼1
3 X
kspi f ðXsdi Þ cosð/i aÞ ¼ 0;
i¼1
Mrdi Mrdi X€rdi Cspi X_ sdi þ Mp
3 3 X X Mrdi Mrdi Mrdi Mrdi _ _ Cspi Xsdi þ Crpi Xrdi þ Crpi X_ rdi Ccl X_ cl M M M M c p c c i¼1 i¼1
3 X Mrdi Mrdi Mrdi Mrdi kspi f ðXsdi Þ þ kspi f ðXsdi Þ þ krpi f ðXrdi Þ kcl Xcl Mp Mc Mp Mc i¼1
þ
3 X Mrdi erpi krpi f ðXrdi Þ ¼ Mrdi € M c i¼1
Mcl X€cl
ði ¼ 1; 2; 3Þ;
3 3 3 Mcl X Mcl X Mcl X Cspi X_ sdi þ Ccl X_ cl Crpi X_ rdi kspi f ðXsdi Þ þ kcl Xcl Mc i¼1 Mc i¼1 Mc i¼1
3 Mcl X Mcl krpi f ðXrdi Þ ¼ P; Mc i¼1 Ml
where Mp Mc Mc Ml ; Mcl ¼ ; Mp þ Mc Mc þ Ml Ms Mpi Mc ms ¼ : Mpi Mc ms þ Ms Mc ms þ Ms Mpi ms þ Mpi Mc Ms
Mhs ¼ Msdi
Ms Mh ; Ms þ Mh
Mrdi ¼
ð11Þ
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T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371–1390
If a displacement vector q is introduced as following: q ¼ f Xhs ;
Xsd1 ;
Xsd2 ;
ns ;
Xsd3 ;
gs ;
Xrd1 ;
Xrd2 ;
Xrd3 ;
T
ð12Þ
Xcl g :
The equation of motion is given in the matrix form as M€ q þ C q_ þ K f ð qÞ ¼ p;
ð13Þ
where the mass matrix M reads M ¼ diag½ Mhs ;
Msd1 ;
Msd2 ;
Msd3 ;
ms ;
ms ;
Mrd1 ;
Mrd2 ;
Mrd3 ;
Mcl ;
ð14Þ
f ð qÞ is the vector of nonlinear functions in single variable with uniform shape in terms of the coordinates transformed in Eq. (10) 8 qi bi ; qi > bi ; < qi Þ ¼ 0; bi 6 fi ¼ f ð ð15Þ qi 6 bi ; : q i < bi ; qi þ bi ; where bi ¼ 0; for i ¼ 1; 5; 6; 10; iþ1 ¼ b bspi ; biþ6 ¼ brpi ; for i ¼ 1; 2; 3:
ð16aÞ ð16bÞ
The stiffness matrix K, the damping matrix C and the force vector p are given in Appendix A. Eq. (13) can bepfurther simplified by using a number of dimensionless parameters. For this ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi purpose, let xn ¼ ksp1 =Msd1 , where ksp1 is the mean value of mesh stiffness between the sun gear and planet gear-1. Other dimensionless parameters are defined as ii xn Þ; kij ¼ kij =ðm ii x2 Þ; q¼ q=bc ; M ¼ I; X ¼ x=xn ; cij ¼ cij =ðm n
pi ðtÞ ¼
ii bc x2n Þ; pi ðtÞ=ðm
bi =bc ; bi ¼
i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; n; 8 < qi bi ; qi > bi ; fi ¼ f ðqi Þ ¼ 0; bi 6 qi 6 bi ; : qi þ bi ; qi < bi ;
ð17Þ
where bc is a character length and x is the excitation frequency. Thus, we have the dimensionless dynamic equations M€ qðsÞ þ C q_ ðsÞ þ Kf ðqðsÞÞ ¼ pðsÞ;
ð18Þ
where s ¼ t xn is dimensionless time. 4. Solutions by harmonic balancing Kahraman and Singh [5] and Comparin and Singh [6] located the periodic vibration of a gear pair with backlash excited by external torque and internal excitations by employing the harmonic balance method (HBM). Successful examples of HBM for solving nonlinear systems with multiple clearances have not been available in archival publications. In this section, the HBM is presented to reveal the nonlinear dynamics of multi-degree-of-freedom planetary gear systems excited by transmission errors.
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4.1. Harmonic balance method Attention is paid to the periodic vibrations of system under the harmonic excitation. The procedure of solving Eq. (18) by HBM includes four aspects as following. (1) Form of excitation: According to the assumption of harmonic excitations, excitations given in Eq. (17) can be represented approximately as pi ¼ pmi þ pai cosðXs þ upi Þ;
ð19aÞ
where subscripts ÔmÕ and ÔaÕ represent the mean and alternating components of force, and upi is phase angle of force. (2) Form of response: For the harmonic excitations given by Eq. (19a), the entries in approximate solution vector q are assumed in the form qi ¼ qmi þ qai cosðXs þ ui Þ;
ð19bÞ
where qmi and qai are the mean and alternating components of the steady state response, and ui is phase angle. (3) Nonlinear function: The steady state solution qi ðsÞ ¼ qi ðs þ T Þ in Eq. (19b) is assumed to be periodic with period T ¼ 2p=X. Accordingly, f ðqi ðsÞÞ ¼ f ðqi ðs þ T ÞÞ must also be periodic. A key step of HBM is to represent the nonlinear function f in Eq. (18) in the following form: f ðqi Þ ¼ Nmi qmi þ Nai qai cosðXs þ ui Þ; where Nmi and Nai are the describing functions defined as Z 2p 1 f ðqi Þdhi ; Nmi ðqmi ; qai Þ ¼ 2pqmi 0 Z 2p 1 Nai ðqmi ; qai Þ ¼ f ðqi Þ cos hi dhi ; hi ¼ Xs þ ui : pqai 0 The substitution of Eqs. (17) and (19b) into Eqs. (19d) and (19e) gives qai ½Gðlþ Þ Gðl Þ ; Nmi ¼ 1 þ 2qmi Nai ¼ 1 12½H ðlþ Þ ðl Þ ;
ð19cÞ
ð19dÞ ð19eÞ
ð20aÞ ð20bÞ
where
pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 l þ 1 l2 Þ; jlj 6 1; ð2=pÞðl sin GðlÞ ¼ jlj; jlj > 1; 8 l < 1; < 1; pffiffiffiffiffiffiffiffiffiffiffiffiffi H ðlÞ ¼ ð2=pÞðl sin1 l þ l 1 l2 Þ; jlj 6 1; : 1; l > 1; l ¼ ð bi qmi Þ=qai :
ð20cÞ
ð20dÞ ð20eÞ
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T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371–1390
For i ¼ 1, 5, 6, 10, we have bi ¼ 0:
ð20fÞ
Substituting Eq. (20f) into Eqs. (20a)–(20e) yields Nmi ¼ 1; Nai ¼ 1:
ð20gÞ
This result agrees with the physical fact that describing function of linear function is really 1. (4) Algebraic equations: Considering the mean value and the fundamental harmonics of timevarying periodic mesh stiffness in Fig. 4, the entries in stiffness matrix K in Eq. (18) can be written as kij ¼ kmij þ kaij cosðXs þ /kij Þ:
ð21aÞ
To get the mean value and the harmonics of the time-varying mesh stiffness, the FFT algorithm is introduced to the periodic function of mesh stiffness. Hence, K is written in terms of two separate matrices for mean stiffness and alternating stiffness K ¼ K m þ DK;
ð21bÞ
where K m ¼ ½kmij n n ;
DK ¼ ½kaij cosðXs þ /ij Þ n n :
ð21cÞ
Eqs. (19a)–(19c) can be written in the following matrix form p ¼ fpmi gn 1 þ fpai cosðXs þ upi Þgn 1 ;
ð22aÞ
q ¼ fqmi gn 1 þ fqai cosðXs þ ui Þgn 1 ;
ð22bÞ
f ðqÞ ¼ fNmi qmi gn 1 þ fNai qai cosðXs þ ui Þgn 1 :
ð22cÞ
By substituting Eqs. (19a)–(19c) into Eq. (18) and balancing the same harmonics, one obtains the algebraic equations of system 8 < K m ym þ 12 ðK 1 y3 þ K 2 y4 Þ pm ¼ f0gn 1 ; ð23Þ K y þ K 1 ym X2 My1 XCy2 p1 ¼ f0gn 1 ; : m 3 2 K m y4 þ K 2 ym X My2 XCy1 p2 ¼ f0gn 1 ; where y1 ¼ fqai cos ui gn 1 ; y4 ¼ fNai qai sin ui gn 1 ; p1 ¼ fpai cos upi gn 1 ; K 1 ¼ fkaij cos /ij gn n ;
y2 ¼ fqai sin ui gn 1 ;
y3 ¼ fNai qai cos ui gn 1 ;
ym ¼ fNmi qmi gn 1 ; p2 ¼ fpai sin upi gn 1 ;
ð24Þ
K 2 ¼ ½kaij sin /ij n n :
Eq. (23) includes following 3n unknown variables to be solved qai ; qmi ; ui ; i ¼ 1; 2; . . . ; n:
ð25Þ
As it is impossible to solve the nonlinear algebraic equation (23) by any analytical methods, a numerical routine (DNEQBJ of IMSL [8]) is used to determine the solution. The routine
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DNEQBJ uses a secant method with BroydenÕs update to find the zeros of a set of nonlinear algebraic equations. 4.2. Solutions and validation As the first example, a 2K-H type planetary gear system shown in Fig. 1 is studied. The geometric parameters of system are listed in Table 1 and other parameters are given in Table 2. Furthermore, the characteristic length is set to bc ¼ 0:01 mm. The excitations are associated with the mesh stiffness fluctuation and errors. The frequency responses of system are determined by using the HBM in the previous subsection. As a comparison, the frequency responses are shown in Fig. 5(a)–(e), together with the responses of corresponding linear systems without any backlash. These figures demonstrate that a planetary gear system with clearances taken into account exhibits particular behaviors as following. (1) For all the response results shown in Fig. 5(a)–(e), except for rigid and reduplicate frequencies, the system undergoes resonance at five different frequencies, all of which are near the natural frequencies of corresponding linear system. (2) Similar to the studies on single gear pair by Kahraman and Singh [5], the dynamics of the planetary gear system with clearances exhibits strong nonlinearity. As shown in Fig. 5(a) and (b), both regimes of no impacting (no tooth separation) and single-sided impacting (tooth
Table 1 Parameters of a planetary gear set Number of teeth Module (mm) Pressure angle Amount of crowning (mm) Face width (mm) Load (N m)
Sun gear
Planet gear
Ring gear
15 0.25 20 0.3835 14 50
24 0.25 20 0.161 14 50
63 0.25 20 0.7056 14 50
Table 2 Parameters in case study Parameter ms mp Ih Ip Ic esp1 ¼ esp2 ¼ esp3 erp1 ¼ erp2 ¼ erp3 bsp1 ¼ bsp2 ¼ bsp3 brp1 ¼ brp2 ¼ brp3
Unit Kg Kg Kg mm2 Kg mm2 Kg mm2 lm lm lm lm
Value 0.417 0.254 202 140 6000 20 20 50 50
Parameter Il ks Kcl Khs ksp1m ¼ ksp2m ¼ ksp3m ksp1a ¼ ksp2a ¼ ksp3a krp1m ¼ krp2m ¼ krp3m ksp1m ¼ ksp2m ¼ ksp3m Damping ratio
Unit
Value 2
Kg mm N/m N mm/Rad N mm/Rad N/m N/m N/m N/m –
1980 4.24e8 4.95e8 4.58e8 2.42e8 0.43e8 2.91e8 0.47e8 0.02
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T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371–1390
4
0.05
3 Amplitude
Amplitude
0.04 0.03 0.02
2
1
0.01 0.00 0.0
0 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.6
0.8
1.0
1.2
1.4
Frequency
Frequency
(a)
(b)
4.0
1.6
1.8
2.0
0.12
3.5 0.10 Amplitude
Amplitude
3.0 2.5 2.0
0.08 0.06
1.5 0.04 1.0 0.02
0.5
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.00 0.0
2.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Frequency
Frequency
(d)
(c)
0.08
Amplitude
0.06
0.04
0.02
0.00 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Frequency (e)
Fig. 5. Frequency response of a planetary gear set ( ) linear system; (––) nonlinear system: (a) frequency response of Xhs , (b) frequency response of Xsd1 , (c) frequency response of Xrd1 , (d) frequency response of Xs and (e) frequency response of Xcl .
T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371–1390
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Alternating amplitude
3
2
1
0 0.8
1.0
1.2
1.4 Frequency
1.6
1.8
2.0
Fig. 6. Comparison of frequency response Xsd1 obtained by different methods: (·) numerical integration; (––) harmonic balance method.
separation, but no back collision) exist in the frequency response. One can detect the dual-valued solutions and jump discontinuities near resonant frequencies. These phenomena predict inherent characters of nonlinearity. (3) As the motions of a MDOF system at different degrees of freedom are coupled each other, sudden discontinuities are also observed in Fig. 5(c) and (d), although there are no clearances for these parts. This indicates the interactions between gear pairs and other components. (4) Not all the resonant frequencies of system are dangerous for the nonlinear vibration. The jump discontinuities of frequency response occur only near the resonance sensitive to vibration of gear pairs. Next, we verify the solutions of HBM through the numerical simulations, where a variable-step Runge–Kutta algorithm of fifth–sixth order in [9] is used. The frequency responses obtained by using HBM and Runge–Kutta algorithm are compared in Fig. 6. The two methods get an agreement although the response amplitudes at resonant frequencies are slightly different. Discrepancies may come from the assumption of single harmonics in HBM. Though the validity of HBM is confirmed, one should be aware of the following as also mentioned by Kahraman in [5]. (1) Several problems may appear in the implementation of numerical integration to the nonlinear dynamics due to clearances and caution must be exercised [6] even though it is more precise. (2) The HBM is incapable of predicting any quasi-periodic or chaotic vibrations.
5. Parametric studies The dynamic behaviors of a planetary gear set, such as the frequency response, the transition frequency and the existence of various impacting regimes, depend on the parameters of system to some extent. Therefore, parametric studies are presented here.
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T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371–1390
We first examine the effect of mesh stiffness. As well known, the mesh stiffness of a gear pair is a time-variant periodic function, which can be represented as kðtÞ ¼ km þ ka sinðXs þ uÞ:
ð26Þ
To describe the extent of variation, a stiffness ratio is introduced as ka k^ ¼ : km
ð27Þ
Three different results of Xrd1 are obtained by varying k^ only for krp1 , the mesh stiffness between the ring gear and planet gear-1, while other parameters remain unchanged as shown in Table 2. Fig. 7 shows the jump discontinuities in all three cases no matter how large k^ is taken. When k^ is increased, the transition frequencies of jump become lower, the transition frequencies of jumping-up and jumping-down leave each other. This gives rise to the dual-valued solution region. The comparison indicates that although the existence of typical nonlinear dynamic behaviors of a planetary gear system with clearances does not depend on the variation of stiffness, the extent of nonlinearity is affected really by the stiffness ratio. An increase of variation of stiffness ratio enhances the nonlinearity of response. Next, we examine the effect of static transmission error e, which is the excitation on the planetary gear system in this study. Three case studies are shown in Fig. 8 for dimensionless e being 0.2, 2 and 4, respectively. When e is very small, say, e ¼ 0:2, the system dynamics is linear and the tooth pairs do not lose contact all the time although the system has clearances. When e is increased to 2, the dynamics of system becomes nonlinear. The single-sided impact vibration and the jump discontinuities are observed. When e is increased further to 4, the nonlinear behaviors get more complicated. The double-sided impact (back collision) vibration are also observed near the resonant frequency, X ¼ 1:58. Moreover, as e is increased, the amplitudes of response near the resonance become larger. This is reasonable since e plays a role of alternating components in the excitation on planetary gear system.
3.5
Alternating amplitude
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Frequency
Fig. 7. Frequency response of Xrd1 for different mesh stiffness: ( ) k^ ¼ 0; (- - -) k^ ¼ 0:2; (––) k^ ¼ 0:4.
T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371–1390
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5.0 4.5 4.0
Amplitude
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Frequency
Fig. 8. Frequency response of Xsd1 for different errors: ( ) e ¼ 0:2; (- - -) e ¼ 2:0; (––) e ¼ 4:0.
6. Conclusions In this study, some advances are made in the nonlinear dynamics of a general planetary gear system with backlash and time-varying mesh stiffness, excited by the static transmission errors. First, a planetary gear system with multiple clearances taken into consideration is studied from the viewpoint of nonlinear dynamics. Second, a lateral–torsional coupled nonlinear dynamic model for the planetary gear set is developed and the governing equation is formulated in matrix form. Third, the frequency response of the planetary gear set under a harmonic excitation is determined by using the HBM and verified by using numerical simulation. Finally, the effects of some important parameters, such as the variation of mesh stiffness and static transmission errors, on the nonlinear dynamics are discussed. The study enriches the current literature in the nonlinear dynamics of mechanical systems with multiple clearances and the dynamics of planetary gear systems. The results of this study yield some guidelines to be instrumental to improvement of the planetary transmission performance. The ongoing research work focuses on the load sharing characteristics of planetary transmissions with multiple clearances since the advantages of a planetary gear system greatly rely on the sharing characteristics. Future work will include the analysis of higher order harmonics of planetary gear systems with multiple clearances, meshing phase difference and unequal backlash.
Acknowledgement We thank National Natural Science Foundation of China for supporting this study.
Appendix A. Entries in matrices K, C, p Both K and C are 10 · 10 square matrices, while p is a 10 · 1 vector.
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(1) The entries in the stiffness matrix K are given as follows: k1;i ¼ khs ;
k1;i ¼ Mhs kspi ; Ms
i ¼ 1; 2; 3;
k2;i ¼ Msd1 khs ; k2;2 ¼ ksp1 ; k2;iþ1 ¼ Msd1 þ Msd1 þ Msd1 kspi ; i ¼ 2; 3; Ms Ms Mc ms k2;5 ¼ sinð/1 aÞ Msd1 kn ; k2;6 ¼ cosð/1 aÞ Msd1 kg ; ms ms
k2;7 ¼ Msd1 Msd1 krp1 ; k2;iþ6 ¼ Msd1 krpi ; i ¼ 2; 3; k2;10 ¼ Msd1 kcl ; Mc Mp Mc Mc
k3;1 ¼ Msd2 khs ; k3;iþ1 ¼ Msd2 þ Msd2 þ Msd2 kspi ; i ¼ 1; 3; Ms Ms Mc ms k3;3 ¼ ksp2 ; k3;5 ¼ sinð/2 aÞ Msd2 kn ; k3;6 ¼ cosð/2 aÞ Msd2 kg ; ms ms
k3;iþ6 ¼ Msd2 krpi ; i ¼ 1; 3; k3;8 ¼ Msd2 Msd2 krp2 ; k3;10 ¼ Msd2 kcl ; Mc Mc Mp Mc
k4;1 ¼ Msd3 khs ; k4;iþ1 ¼ Msd3 þ Msd3 þ Msd3 kspi ; i ¼ 1; 2; k4;4 ¼ ksp3 ; Ms Ms Mc ms k4;5 ¼ sinð/3 aÞ Msd3 kn ; k4;6 ¼ cosð/3 aÞ Msd3 kg ; ms ms
k4;iþ6 ¼ Msd3 krpi ; i ¼ 1; 2; k4;9 ¼ Msd3 Msd3 krp3 ; k4;10 ¼ Msd3 kcl ; Mc Mc Mp Mc k5;iþ1 ¼ sinð/i aÞkspi ; i ¼ 1; 2; 3; k5;5 ¼ kn ; k6;iþ1 ¼ cosð/i aÞkspi ; i ¼ 1; 2; 3; k6;6 ¼ kg ;
k7;2 ¼ Mrd1 Mrd1 ksp1 ; k7;iþ1 ¼ Mrd1 kspi ; i ¼ 2; 3; Mc Mp Mc Mrd1 i ¼ 2; 3; k7;10 ¼ kcl ; Mc
k8;iþ1 ¼ Mrd2 kspi ; i ¼ 1; 3; k8;3 ¼ Mrd2 Mrd2 ksp2 ; Mc Mc Mp k8;8 ¼ krp2 ; k8;iþ6 ¼ Mrd2 krpi ; i ¼ 1; 3; k8;10 ¼ Mrd1 kcl ; Mc Mc
k9;iþ1 ¼ Mrd3 kspi ; i ¼ 1; 2; k9;4 ¼ Mrd3 Mrd3 ksp3 ; Mc Mc Mp k9;9 ¼ krp3 ; k9;iþ6 ¼ Mrd3 krpi ; i ¼ 1; 2; k9;10 ¼ Mrd3 kcl ; Mc Mc
k7;7 ¼ krp1 ;
k7;iþ6 ¼ Mrd1 krpi ; Mc
T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371–1390
k10;iþ1 ¼ Mcl kspi ; Mc k10;10 ¼ kcl :
i ¼ 1; 2; 3;
k10;iþ6 ¼ Mcl krpi ; Mc
i ¼ 1; 2; 3;
The entries not listed above are taken as zeros. (2) The entries in the damping matrix C are listed below c1;i ¼ Chs ;
c1;i ¼
Mhs Cspi ; Ms
i ¼ 1; 2; 3;
Msd1 Msd1 Msd1 Msd1 c2;1 ¼ Chs ; c2;2 ¼ Csp1 ; c2;iþ1 ¼ þ þ Cspi ; i ¼ 2; 3; Ms Ms Mc ms Msd1 Msd1 c2;5 ¼ sinð/1 aÞ Cn ; c2;6 ¼ cosð/1 aÞ Cg ; ms ms
Msd1 Msd1 Msd1 Msd1 c2;7 ¼ Crpi ; i ¼ 2; 3; c2;10 ¼ Ccl ; Crp1 ; c2;iþ6 ¼ Mc Mp Mc Mc
Msd2 Msd2 Msd2 Msd2 c3;1 ¼ Chs ; c3;iþ1 ¼ þ þ Cspi ; i ¼ 1; 3; Ms Ms Mc ms Msd2 Msd2 c3;3 ¼ Csp2 ; c3;5 ¼ sinð/2 aÞ Cn ; c3;6 ¼ cosð/2 aÞ Cg ; ms ms
Msd2 Msd2 Msd2 Msd2 c3;iþ6 ¼ þ Crp2 ; c3;10 ¼ Crpi ; i ¼ 1; 3; c3;8 ¼ Ccl ; Mc Mc Mp Mc
Msd3 Msd3 Msd3 Msd3 c4;1 ¼ Chs ; c4;iþ1 ¼ þ þ Cspi ; i ¼ 1; 2; c4;4 ¼ Csp3 ; Ms Ms Mc ms Msd3 Msd3 c4;5 ¼ sinð/3 aÞ Cn ; c4;6 ¼ cosð/3 aÞ Cg ; ms ms
Msd3 Msd3 Msd3 Msd3 c4;iþ6 ¼ Crpi ; i ¼ 1; 2; c4;9 ¼ Ccl ; Crp3 ; c4;10 ¼ Mc Mc Mp Mc c5;iþ1 ¼ sinð/i aÞCspi ;
i ¼ 1; 2; 3;
c5;5 ¼ Cn ;
c6;iþ1 ¼ cosð/i aÞCspi ; i ¼ 1; 2; 3; c6;6 ¼ Cg ;
Mrd1 Mrd1 Mrd1 c7;2 ¼ Cspi ; i ¼ 2; 3; Csp1 ; c7;iþ1 ¼ Mc Mp Mc Mrd1 Mrd1 c7;7 ¼ Crp1 ; c7;iþ6 ¼ Crpi ; i ¼ 2; 3; c7;10 ¼ Ccl ; Mc Mc
Mrd2 Mrd2 Mrd2 c8;iþ1 ¼ Cspi ; i ¼ 1; 3; c8;3 ¼ Csp2 ; Mc Mc Mp Mrd2 Mrd1 c8;8 ¼ Crp2 ; c8;iþ6 ¼ Crpi ; i ¼ 1; 3; c8;10 ¼ Ccl ; Mc Mc
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T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371–1390
Mrd3 Mrd3 c9;iþ1 Csp3 ; i ¼ 1; 2; c9;4 ¼ Mc Mp Mrd3 Mrd3 c9;9 ¼ Crp3 ; c9;iþ6 ¼ Crpi ; i ¼ 1; 2; c9;10 ¼ Ccl ; Mc Mc Mcl Mcl c10;iþ1 ¼ Cspi ; i ¼ 1; 2; 3; c10;iþ6 ¼ Crpi ; i ¼ 1; 2; 3; Mc Mc c10;10 ¼ Ccl : Mrd3 ¼ Cspi ; Mc
Again, the entries not listed above are set to be zeros. (3) The entries in the force vector p are given as following Mhs Mcl Ph ; p5 ¼ 0; p6 ¼ 0; p10 ¼ Pl ; p1 ¼ Mh Ml piþ6 ¼ Mrdi €erpi ; i ¼ 1; 2; 3: piþ1 ¼ Msdi €espi ; i ¼ 1; 2; 3;
References [1] T. Hidaka, Analysis of dynamic tooth load on planetary gear, Bulletin of the Japanese Society of Mechanical Engineers 23 (1980) 315–322. [2] A. Kahraman, Load sharing characteristics of planetary transmissions, Mechanism and Machine Theory 29 (1994) 1151–1165. [3] R.G. Munro, The dynamic behavior of spur gears. PhD Dissertation, Cambridge University, 1962. [4] A. Kubo, K. Yamada, T. Aida, S. Sato, Research on ultra high speed gear devices (reports 1–3). Transactions of the Japan Society of Mechanical Engineers 38 (1972) 2692-2715. [5] A. Kahraman, R. Singh, Nonlinear dynamics of a spur gear pair, Journal of sound and vibration 142 (1990) 49–75. [6] R.J. Comparin, R. Singh, Nonlinear frequency response characteristics of an impact pair, Journal of sound and vibration 134 (1989) 259–290. [7] A. Kahraman, R. Singh, Nonlinear dynamics of a geared rotor-bearing system with multiple clearances, Journal of Sound and Vibration 144 (1991) 469–506. [8] Microsoft INC. IMSL Libraries Reference (Pro Edition) 1994–95. [9] H.N. Ozguven, D.R. Houser, Dynamic analysis of high speed gears by using loaded static transmission error, Journal of Sound and Vibration 125 (1988) 71–83.