kinematic and power flow analysis of bev

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Kinematic and power-flow analysis of bevel gears planetarygear trains with gyroscopic complexity

Germano Del Pio, Ettore Pennestrì⁎, Pier Paolo ValentiniDipartimento di Ingegneria dell'Impresa, University of Rome Tor Vergata, via del Politecnico, 1 00133 Roma, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 February 2013Received in revised form 12 July 2013Accepted 21 August 2013Available online 21 September 2013

In this paper the authors propose a method for the kinematic and power-flow analysis of bevelepicyclic gear trains with gyroscopic complexity. By gyroscopic complexity, we mean thepossibility of the gear carrier to be a floating link as, for instance, in robotic gear wrists.Thanks to the new formulas herein deduced, the methods based on the graph representationof planetary spur gear trains have been extended to bevel gear trains. In particular, the wellknownWillis equation has been modified to maintain its validity for bevel gears. The modifiedWillis equation was then embodied in new power ratio expressions. Under our simplifyinghypotheses of absence of friction and constant angular speeds, it is shown that gyroscopictorques do not enter into power flow analysis. Two numerical examples are discussed.

© 2013 Elsevier Ltd. All rights reserved.

Keywords:Planetary bevel gear trainsPower flowKinematicsWillis formulaMechanical efficiency

1. Introduction

In recent times different papers on the mechanics of multi degrees-of-freedom planetary gear trains have been published. Thefocus of this effort was the development of systematic methods for mechanical efficiency analysis. This renewed interest is alsodue to the application of epicyclic gear trains as one of the main powertrain component of hybrid vehicles. The capability tohandle torques from different power sources is an almost unique feature of this type of transmissions.

A fundamental step in mechanical efficiency analysis is the ascertainment of the amount of power flow through the meshinggears. Although not self evident, due to power circulation, some meshing gears may sustain a power higher than the input one.Power circulation, that usually occurs with very low transmission ratios, must be detected at the early design stages in order todimension properly meshing gears and lubricating methods.

Most of the contributions are related to spur gear trains. In this case the kinematics can be studied with the classic scalar Willisequation.1 The relationship between the absolute angular speeds of bevel gear trains is not scalar and this complicates theanalysis.

This paper focuses on kinematic and power flow analysis of planetary trains with bevel gears. It can be considered as anattempt to extend the modus operandi of the analysis methods devised for spur gear trains to bevel gear trains.

A complete review of all the scientific contributions on the topic is outside the purpose of this paper. Thus the following reviewcannot be considered exhaustive.

Belfiore, Pennestrì and Sinatra [1] presented a Maple procedure for kinematic and power-flow analysis of spur planetary geartrains based on the graph based method of Pennestrì and Freudenstein [2,3].

Mechanism and Machine Theory 70 (2013) 523–537

⁎ Corresponding author. Tel.: +39 0672597138.E-mail address: [email protected] (E. Pennestrì).

1 Following the tradition of many textbooks, the authors called “Willis equation” the expression which relates the absolute angular velocities of a differentialgear train. However, it should be acknowledged that this equation was well known before Rev. Robert Willis (1800–1875), Jacksonian Professor of AppliedMechanics in the University of Cambridge.

0094-114X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.mechmachtheory.2013.08.016

Contents lists available at ScienceDirect

Mechanism and Machine Theory

j ourna l homepage: www.e lsev ie r .com/ locate /mechmt


Kaharaman et al. [4] proposed a modular general formulation valid for 1-dof automatic transmission composed of complex–compound planetary gear sets.

Chen and Teh [5] applied the concept of virtual-power [6] to find ready-to-use formulas for the mechanical efficiency analysisof 2-dof gear trains. For a Simpson differential gear train, Chen [7] analyzed the sensitivity of the mechanical efficiency.

Mathis and Remond [8] proposed a unified model for the kinematic, torque and efficiency analysis of epicyclic gear trains. Anapplication of the method to a Ravigneaux type gear train is discussed.

Galvagno [9] discussed the influence of dynamics on the mechanical efficiency analysis of a 2-dof differential spur gear train.Pennestrì et al. [10] presented a systematic approach for the modeling and analysis of power split transmissions which include

an epicyclic gear train. The method is a refinement of the one proposed by Pennestrì and Freudenstein [2,3,11].However, less common are the power flow and efficiency analyses of gear trains with bevel gears.Day et al. [12] proposed a matrix method for the kinematic analysis of planetary bevel gear trains using the concept of the

fundamental circuit. However, the method is limited to gear carriers rotation about a fixed axis.Freudenstein et al. [13] extended the concept of fundamental circuit to the analysis of geared robotic wrists. This category of

wrists, enumerated by Belfiore [14], has gyroscopic complexity when the gear carrier is not adjacent to the frame link.Further contributions on planetary geared robotic wrists are due to Tsai [15] who hinted that the motion of a bevel-gear-type

end-effector can be described by an equivalent open-loop chain. The analysis equations follow by considering relative rotationbetween every two adjacent links in the equivalent open-loop chain and coaxial conditions.

Litvin and Zheng proposed a matrix method for the kinematic analysis of differential trains with bevel gears [16].Gupta and Ma [17], extended the tabular superposition method to derive the relations among the three wrist joint variables

and three coaxial actuation variables.Nelson and Cipra [18] proposed a systematic graph-based matrix method for kinematic, power-flow andmechanical efficiency

suited to the complete solution of bevel gear sets as well as planar epicyclic sets.Uyguroglu and Demirel [19,20] applied oriented linear graph techniques toward the kinematic and static moment analysis of

robotic bevel-gear trains.Staicu proposed recursive matrix relations for kinematics and dynamics analysis of different orienting bevel gear train [21–23].

The relations are particularly useful for inverse dynamics.Laus et al. [24] combined graph and screw theory for the analysis of mechanical efficiency of bevel gear trains with complex

architecture. In particular Davies' equations [25] have been applied to gear trains. The analysis equations are deduced in a waysimilar to the one followed for electrical networks. The mechanical analog of the electrical resistance is introduced to take intoaccount power losses.

Chen [26] introduced the constraint equations for kinematics and power flow analysis. The method is very systematic and canbe potentially implemented in a general purpose code. The use of constraint equations for kinematic and dynamic analysis ofplanetary gear trains was also explored by Mantriota and Pennestrì [27] by means of multibody dynamics approach.

The classical tabular method, based on Willis equation, cannot be applied to complex planetary bevel gear trains.In this paper, we propose a scalar equation between the absolute angular velocities of the simplest bevel gear train. By means

of this equation, the method of fundamental circuits [28,29] can be extended to the kinematic analysis of planetary bevel geartrains.

Moreover, the power ratios through the bevel gears and gear carrier forming a fundamental circuit are herein deduced. Theresult is also novel. In fact, such ratios include, as a particular case, those deduced by Pennestrì and Freudenstein [2,3] forplanetary spur gear trains.

By means of these ratios, graph based methods of power flow analysis of planetary gear trains can include the presence ofbevel gears.

The paper is divided into three parts. The first two parts are dedicated to methods of kinematic and power flow analysis,respectively, and the third one to applications.

k

Gj i

R(a) R(b)

Fig. 1. Labeled graph representation of a fundamental circuit. G: Gear pair; R(a), R(b): revolute pairs with different axes; i and j, gear wheels; k gear carrier(transfer vertex).

524 G. Del Pio et al. / Mechanism and Machine Theory 70 (2013) 523–537


2. Kinematic analysis

The mechanism is considered in a right-hand Cartesian coordinate system fixed to its framemember. Stationary conditions areassumed.

The topology of a gear train can be meaningfully identified bymeans of a labeled graphwhere links are represented by verticesand kinematic pairs by edges, respectively. Edges are labeled according to the type of kinematic pair, i.e. R: revolute pairs and G:gear pair. A letter within parentheses specifies the axis level of the revolute pair.

This approach allows us to locate fundamental circuits in the planetary gear train (Fig. 1). The fundamental circuit identifies thesimplest gear unit, that is two meshing gears connected through a gear carrier (see Fig. 1).

A planetary gear train can be considered as an assembly of elementary gear units each of them associated with a fundamentalcircuit. By identifying the fundamental circuits one recognizes such units and the application of the analysis equations can bemade in a very systematic manner (e.g. [28,29,10]).

The gears of a generic fundamental circuit are denoted with the letters i and j and the gear carrier with k. The frame member ofa mechanism is denoted with F.

The vectors will be interpreted as a multiplication of an algebraic scalar with a generic unit vector baba . It is imposed thatbaba ¼ baab.With reference to the mth fundamental circuit (see Fig. 2), the following angles are introduced:

• θm angle between the axis of relative rotation of the two bevel gears and the X axis (see Fig. 6);• ϕqF angle between the absolute angular velocity vector of a generic link q and the X axis;• αm and βm, angles between X axis and the axes of bevel gears i, j, respectively. These are also the anomalies of the torques

T!

im and T!

jm acting on the gear wheels i and j of the mth fundamental circuit (see Fig. 7);• γm, angle between the X axis and the gear torque T

!km acting on the gear carrier of themth fundamental circuit (see Fig. 7).

The angles are measured positively counterclockwise.Moreover, the following nomenclature is introduced:

• ωab is the module of the relative angular velocity of link a with respect to link b; it is imposed that ω!ab ¼ −ω!ba;• ωqF is the module of absolute angular velocity of a generic link q.

In particular, denoted by Nj and Ni, respectively, the number of gear teeth of the bevel gears i and j:

• the bevel gear ratio Rm ¼ Nj

Ni± is always negative unless the semi-vertex angle of one bevel gear is greater than 90°;

• the unit vectorbaik associatedwith a gear rotation axis is directed outward from the center of spherical motion (see Figs. 2 and 3).

In a graph representing a bevel gear train, we may distinguish two types of fundamental circuits. With reference to the geararrangements shown in Fig. 4, for the sake of brevity, we shall define them as Type A and Type B [30].

Type A fundamental circuits are those in which one of the gears and the gear carrier both rotate about a fixed axis (see Fig. 4).Type B fundamental circuits are those in which the bevel gears do not share revolute joints with the frame link.When type B circuit is present in the graph, the bevel gear train will have gyroscopic complexity. The analysis of

planetary gear trains with type B circuits requires further conditions [15] on angular velocities of links connected byrevolute pairs.

For the purpose of kinematic analysis, for each fundamental circuit, the following equations can be written:

ωijbaij ¼ ωikbaik−ωjkbajk ; ð1aÞ

X

Y i

j

kaik

ajk

aij

m m

mθ

α β

Fig. 2. Externally meshing bevel gears. Rm b 0.

525G. Del Pio et al. / Mechanism and Machine Theory 70 (2013) 523–537


aik

ajk

aij

X

Y

i

j

k

mm

m

α βθ

Fig. 3. Internally meshing bevel gears. Rm N 0.

Type A

Type B

ij

jk

ik iF

kF

ij jkik

kF

j

k

i

j

ik

F

F

1

Π

Π Π≡

→

→→

→

→

→→ →

ω

ω

ωω

ω

ω

ωω ω

→

Fig. 4. Type A (top) and Type B (bottom) fundamental circuits. The orientation and modules of vectors are represented qualitatively.



ωik

ωjk¼ Rm ; ð1bÞ

ωiFbaiF ¼ ωijbaij þωjFbajF : ð1cÞ

Eq. (1c) is required for fundamental circuits that do not contain the framemember, while Eqs. (1a) and (1b) have to be writtenfor every fundamental circuit.

The unit vectors of the rotation axes of the relative motion between the fundamental circuit links can be expressed asfollows

baij ¼ cos θmsin θm

� �; baik ¼ cos αm

sin αm

� �; bajk ¼ cos βm

sin βm

� �; ð2Þ

baiF ¼ cos ϕiFsin ϕiF

� �; bajF ¼ cos ϕjF

sin ϕjF

� �; bakF ¼ cos ϕkF

sin ϕkF

� �: ð3Þ

Manipulating Eqs. (1a)–(1c) and taking into account Eqs. (2) and (3), the angle θm is computed by means of the followingexpression

tan θm ¼ Rm sin αm−sin βm

Rm cos αm−cos βm: ð4Þ

One can test the correctness of Eq. (4) for the simple case shown in Fig. 5, where Rm = −1, αm = 180° and βm = 90°. FormEq. (4) one obtains tan θm = −1, or θm = 135°.

A visual inspection is required to determine whether ϕiF, ϕjF, and ϕkF need to be computed by means of Eqs. (1a), (1b), and(4) or whether they can be directly obtained by inspection from the mechanism drawing.

The solution of the system of equations formed by Eqs. (1a), (1b), (1c) and (4), written for each fundamental circuit of theplanetary gear train, gives the magnitudes and directions of the unknown angular velocities.

Although geared wrists are spatial mechanisms, our algebraic treatment is planar. In fact, for a given fundamental circuit, in aunit of two bevel gears i, j and a gear carrier k, the relative angular velocities ω!ij, ω

!ik, and ω!jk, must lie all on the same plane (sayП

in Fig. 4), as stated by Eq. (1a).Also ω!ij, ω

!iF , and ω!jF must lie on the same plane (say П1), in fact, adding ω!kF at left and right sides we obtain

ω!ik þ ω!kF ¼ ω!ij þ ω!jk þ ω!kF ; ð5Þ

which is equivalent to Eq. (1c).For type A fundamental circuits, ω!ik, ω

!ij, ω!

jF , ω!

jk and ω!kF lie on the same plane, therefore П and П1 necessarily coincide. Fortype B fundamental circuits, the two planes may coincide or may form a constant angle. However, in current industrial solutions

mm

m

=180° =90°

=135°

Rm= -1

α β

θ

Fig. 5. Simple test case.



(e.g. [31–33]) of robotic geared wrists, all Type B fundamental circuits have revolute axes on the same plane (see Fig. 4).Therefore, in the present treatment the planes П and П1 are always considered coincident.

In any case, the relationships between angular velocities are not altered by the positions of these planes in space or by the poseof the end-effector. This justifies the analytic treatment in a plane.

In other words, the relationships between the modules of angular velocities do not change if the plane П changes its positionin space.

Our conclusions are consistent with the findings of other investigators (e.g. [33,19]).For these reasons a planar description of angular speeds of the links belonging to a fundamental circuit has been herein

adopted.

3. A generalized form of Willis equation

The previous equations can be manipulated in order to obtain a scalar generalized Willis equation valid for bevel gears (seeFig. 6):

ωik ¼ Rmωjk ; ð6Þ

ωikbaik ¼ ωiFbaiF−ωkFbakF ; ð7Þ

ωjkbajk ¼ ωjFbajF−ωkFbakF : ð8Þ

From these last two equations follow

ωik ¼ ωiFbaiF � bxbaik � bx−ωkF

bakF � bxbaik � bx ; ð9aÞ

ωjk ¼ ωjF

bajF � bxbajk � bx−ωkFbakF � bxbajk � bx : ð9bÞ

Combining Eqs. (6), (9a) and (9b) one obtains

cimωiF−cjmRmωjF þ ckm Rm−1ð ÞωkF ¼ 0; ð10Þwhere

cim ¼ baiF � bxbaik � bx ¼ cos ϕiF

cos αm; ð11aÞ

cjm ¼bajF � bxbajk � bx ¼ cos ϕjF

cos βm; ð11bÞ

ckm ¼Rm

bakF � bxbajk � bx−bakF � bxbaik � bx

Rm−1¼

Rm cos ϕkF1

cosβm− 1

cosαm

� �Rm−1

: ð11cÞ

Eq. (10) is a generalized form of the Willis equation. This equation can be applied for the kinematic analysis of planetary bevelgear trains, including those with gyroscopic complexity. The deduced expression allows the extension of the graph based method

ij

iF

jF

jF-

jF

iF

m

→

→

→ω

ω

ω

ω→

Fig. 6. Nomenclature.



of Freudenstein and Yang [28,29] to bevel gear trains. For each fundamental circuit, Eq. (10) must be particularized. Solving thesystem of the obtained equations, the unknown angular velocities are systematically computed.

4. Power-flow analysis

The gear train is analyzed under steady state conditions, neglecting power losses and inertia forces. The convention formeasuring torque vectors is the same as outlined for angular velocities. Let us denote by T

!im , T

!jm and T

!km , respectively, the

torques applied on the gear wheels i and j and the gear carrier k of the mth fundamental circuit.Taking into account only the torque components in the П or П1 plane, the equilibrium condition of the mth gear unit

yields

T!

im þ T!

jm þ T!

km ¼ 0 : ð12Þ

Since the torques on the gears are applied through revolute joints, we can state that the direction of a torque vector applied ona gear coincides with the axis of its revolute joint.

For power flow analysis we are interested in the generalized forces (torques) that do work. Under the hypotheses of negligiblefriction, gyroscopic torques in geared wrists do not do work.

Let us consider a type A fundamental circuit (see Fig. 8). The gyroscopic torque is acting on gear j (satellite) since gear i rotatesabout a fixed axis.

The absolute angular velocity ω!kF of the gear carrier can be interpreted as the precession velocity and ω!jk as the spin velocity.The absolute angular speed of gear j is

ω!jF ¼ ω!kF þ ω!jk

and its local components, in a Cartesian frame system attached to the gear, are

ωxjF ¼ ωjk þωkF cos η ;

ωyjF ¼ ωkF sin η ;

ωzjF ¼ 0 ;

where η is the angle between the axes of the revolute pairs.

jk

j

k

iFkmT

kF

x

y

Gyroscopic torqueis orthogonal to

the plane

Π

→

→ω

→ω

η

Fig. 8. Nomenclature.



Assuming constant angular speeds, the application of Newton–Euler equations (e.g. [34]) gives

Mx ¼ 0;My ¼ 0

Mz ¼ Jy− Jx� �

ωxjFω

yjF ;

where Jy and Jx are moment of inertia of the gear j about axes x and y, respectively.Hence, the gyroscopic torque Mg

� � ¼ 0 0 Mzf gT is always orthogonal to the planeП containing the angular speedsω!kF andω!jk . Consequently, assuming absence of friction in the kinematic pairs, the gyroscopic torque will not produce any work andshould not be taken into account in any power balance condition.

The externally applied torque vectors in the equilibrium Eq. (12) lie all in theП plane. Since the gyroscopic torque vectorM!

g isorthogonal to this plane, it does not appear in the Eq. (12).

A similar reasoning can be applied to the case of Type B fundamental circuits.For the purpose of power flow analysis each fundamental circuit is represented by means of a box with three nodes, one for

each link. Boxes are connected through nodes (shared links). Powers Pim, Pjm and Pkm are positive when entering the mth box,negative when leaving it.

The following torque equilibrium and power balance equations hold for every fundamental circuit (see Fig. 7)

Tim cos αm þ Tjm cos βm þ Tk;m cos γm ¼ 0; ð13Þ

Tim sin αm þ Tjm sin βm þ Tk;m sin γm ¼ 0; ð14Þ

Pim ¼ T!

im � ω!iF ¼ TimωiF cos ϕiF−αmð Þ; ð15Þ

Pjm ¼ T!

jm � ω!jF ¼ TjmωjF cos ϕjF−βm

� �; ð16Þ

Pkm ¼ T!

km � ω!jF ¼ TkmωkF cos ϕkF−γmð Þ; ð17Þ

Pim þ Pjm þ Pkm ¼ 0; ð18Þ

cimωiF−cjmRmωjF þ ckm Rm−1ð ÞωkF ¼ 0; ð19Þ

where γm is the angle formed by T!

km with the abscissa axis, and Pim is the share of power on gear i flowing through the mthfundamental circuit.

In a graph representation of a gear train, a link corresponds to a vertex. By analogy with the Kirchhof current law, the sum ofpowers through a node must be zero. Hence, for the generic qth moving member the following power balance condition can bewritten,

XNk¼1

Pqk ¼ 0; ð20Þ

where N is the number of adjacent members.

Fig. 7. Torque equilibrium: Nomenclature.



Manipulating these equations, one obtains the power ratios2

Pjm

Pim¼

cos αm RmcjmωjF

ωiF−cim

� �cos ϕkF−γmð Þcos ϕiF−αmð Þ− Rm−1ð Þckmcos γm

cos βm cimωiF

ωjF−Rmcjm

� �cos ϕkF−γmð Þcos ϕjF−βm

� �þ Rm−1ð Þckmcos γm

ð21aÞ

Pkm

Pjm¼

cos βm Rm−1ð ÞckmωkF

ωjF−Rmcjm

� cos ϕiF−αmð Þcos ϕjF−βm

� �þ cimcos αm

cos γm RmcjmωjF

ωkF− Rm−1ð Þckm

� cos ϕiF−αmð Þcos ϕkF−γmð Þ−cimcos αm

: ð21bÞ

For each fundamental circuit, Eq. (18) and either Eq. (21a) or (21b) need to be included in the power-flow analysis system ofequations, together with link power balance Eq. (20).

Eqs. (21a) and (21b) represent an extension to bevel gears of the power ratios deduced by Pennestrì and Freudenstein [2,3] forthe case of spur gear trains.

The expressions (Eqs. (21a) and (21b)) may look cumbersome, however, once the kinematic analysis is completed, the anglesαm, βm can be obtained by visual inspection.

The angle γm follows from the equation

tan γm ¼ sin αm−Rm sin βm

cos αm−Rm cos βmð22Þ

deduced by combining Eqs. (13), (14) and

Tjm

Tim¼ −Rm: ð23Þ

5. Numerical examples

5.1. Kinematic analysis

The proposed method is applied first to planetary gear trains with type A and then to those with type B circuits. In particular,the Humpage reduction gear (see Fig. 9) and the Cincinnati Milacron gear wrist (see Fig. 10) will be considered.

For brevity, the units of measurement have been omitted from the text.

Example 1. Humpage reduction gear

The Humpage reduction gear has only type A circuitsAs summarized in Table 1, the graph of the gear train contains three fundamental circuits.For each fundamental circuit, Eqs. (1a), (1b) and (1c) are particularized:

• Fundamental circuit 1

ω23ba23 ¼ ω24ba24−ω34ba34;ω24

ω34¼ −56

20;

ω2Fba2 F ¼ ω23ba23 þω3Fba3F :• Fundamental circuit 2

ω31ba31 ¼ ω34ba34−ω14ba14;ω34

ω14¼ −76

56:


ω35ba35 ¼ ω34ba34−ω54ba54;ω34

ω54¼ −35

24;

ω3Fba3 F ¼ ω35ba35 þω5Fba5F :2 In our derivations Eq. (13) is used instead of Eq. (14).



The value ω2F = 1 rad/s is prescribed. The angles summarized in Table 2 are preliminary obtained. Moreover, also byinspection ϕ1F = β2 = 180°, ϕ2F = α1 = 0°, α4 = β2 = 180°, ϕ5F = β3 = 180°. The angles θ1, θ2, and θ3 are unknowns andneed to be computed by means of Eq. (4):

tan θ1 ¼ R1sin α1−sin β1

R1cos α1−cos β1¼ 14:71B;


R2cos α2−cos β2¼ 105:30B;


R3cos α3−cos β3¼ 102:11B:

Therefore, the unit vectors relevant for our analysis are:

ba2F ¼ cos ϕ2Fsin ϕ2 F

� �; ba3F ¼ ba31 ¼ cos θ2

sin θ2

� �; ba4F ¼ ba14 ¼ cos β2

sin β2

� �;

ba5F ¼ cos ϕ5Fsin ϕ5 F

� �; ba23 ¼ cos θ1

sin θ1

� �; ba24 ¼ cos α1

sin α1

� �;

ba34 ¼ cos β1sin β1

� �; ba35 ¼ cos θ3

sin θ3

� �; ba54 ¼ cos β3

sin β3

� �:

One can solve the system of eight equations in the eight unknowns formed by particularizing Eqs. (1a), (1b) and (1c) for eachfundamental circuit. The solution is herein reported:

ω2F ¼ 1; ba2F ¼ 10

� �: ∴ω!2 F ¼ 1

0

� �

ω3F ¼ −0:254; ba3F ¼ −0:2640:965

� �: ∴ω!3F ¼ 0:067

−0:245

� �

ω4F ¼ ω41 ¼ −0:208; ba4F ¼ ba14 ¼ −10

� �: ∴ω!4F ¼ 0:208

0

� �

ω5F ¼ −0:014; ba5F ¼ −10

� �: ∴ω!5F ¼ 0:014

0

� �:

The solutions are in agreement with those given by Nelson and Cipra [18] for the same mechanism. Their system of equationswas

ω!3Fω!4Fω!5F

8<:

9=; ¼

0:067 0:9330:208 0:7920:014 0:986

24

35 ω2F

ω1F

�bxþ −0:245 0:2450 00 0

24

35 ω2F

ω1F

�by ð24Þ

with ω2F = 1 and ω1F = 0.Alternatively, one could apply the proposed modified Willis Eq. (10). In this case the coefficients (Eqs. (11a)–(11c)) must be

computed in advance

c21 ¼ cos ϕ2F

cos α1¼ 1; c31 ¼ cos ϕ3F

cos β1¼ −0:528;

c41 ¼R1

cos ϕ4 F

cos β1−cos ϕ4F

cos α1

R1−1¼ −1:737;

c32 ¼ cos ϕ3F

cos α2¼ −0:528; c12 ¼ cos ϕ1 F

cos β2¼ 1;

c42 ¼R2

cos ϕ4F

cosβ2−cos ϕ4 F

cos α2

R2−1¼ −0:273;

c33 ¼ cos ϕ3F

cos α3¼ −0:528; c53 ¼ cos ϕ3 F

cos β3¼ 1;

c43 ¼R3

cos ϕ4 F

cos β3−cos ϕ4F

cos α3

R3−1¼ −0:221:

Since there are three fundamental circuits, the modified Willis equation must be written three times:

I) ω2F − (−0.258)(−2.8)ω3F + (−1.737)(−2.8 − 1)ω4F = 0.II) (−0.528)ω3F − (1)(−1.357)ω1F + (−0.273)(−1.357 − 1)ω4F = 0III) (−0.528)ω3F − (1)(−1.458)ω5F + (−0.221)(−1.458 − 1)ω4F = 0.

The solution of the system is ω3F = −0.254, ω4F = −0.208, and ω5F = −0.014.



Example 2. Cincinnati Milacron gear wrist (Table 3)

In this analysis, due to the presence of type B circuits, for the computation of the cim, cjm and ckm coefficients, the followingkinematic conditions must be taken into account

ω41f g ¼ ω4310

� �þω32

cosεsinε

� �þω21

10

� �; ð25Þ

ω51f g ¼ ω52cosεsinε

� �þω21

10

� �; ð26Þ

ω31f g ¼ ω32cosεsinε

� �þω21

10

� �: ð27Þ

Writing the Willis equation for each fundamental circuit yields:


c41ω41−R1c51ω51 þ c31ω31 R1−1ð Þ ¼ 0; ð28Þ

where

c41ω41 ¼ ω43 þω32cosε þω21;

c51ω51 ¼ ω52 þω21

cosε;

c31ω31 R1−1ð Þ ¼ R1 ω32 þω21

cosε

� �− ω32cosε þω21ð Þ:


c52ω51−R2c62ω61 þ c22 R2−1ð Þω21 ¼ 0; ð29Þ

where

c52ω51 ¼ ω52 þω21

cosε;

c62 ¼ 1;

c22 R2−1ð Þ ¼ R2−1

cosε

� �:


c33ω31−R3c73ω71 þ c23 R3−1ð Þω21 ¼ 0; ð30Þ

where

c33 ¼ ω32

ω31þ ω21

ω31cosε;

c73 ¼ 1;

c23 R3−1ð Þ ¼ R3−1

cosε:

Solving the system of equations formed by Eqs. (28), (29) and (30) one obtains:

ω32 ¼ R3 ω71−ω21ð Þ; ð31Þ

ω43 ¼ R1 R3−R2ð Þω21 þ R2ω61−R3ω71½ �; ð32Þ

ω52 ¼ R2 ω61−ω21ð Þ: ð33Þ

5.2. Power-flow analysis

In this section the power-flow analyses of the Humpage reduction gear and of the Cincinnati Milacron geared wrist by meansof the proposed method are discussed.



Example 3. Humpage reduction gear

For each fundamental circuit, Eqs. (18) and (21a) are written:

I) Since the link 1 is the driving link, let P21 = 1.

P21 þ P31 þ P41 ¼ 0;P31

P21¼ −0:5

II) Since link 1 is the frame, P12 = 0 and only Eq. (18) is required

P32 þ P12 þ P42 ¼ 0:

III)

P33 þ P53 þ P43 ¼ 0P53

P33¼ 0:114:

The Eq. (20) is written for each member, except the driving, driven and frame links:

Gear 3) P31 + P32 + P33 = 0.Gear 4) P41 + P42 + P43 = 0.

1 N=76

N=244

3 N=56

5 N=352 N=20

F=1

1

2

3

2

3

5

4

P53

P41

P31

P21

P32

P33

P43

P42

1

Fig. 9. Kinematic structure of the Humpage reduction gear and power flow graph.



The solution of the system of previous equations is

P21 ¼ 1:000; P31 ¼ −0:500; P41 ¼ −0:500;

P32 ¼ 9:272; P42 ¼ −9:272; P33 ¼ −8:772;

P43 ¼ þ9:772; P53 ¼ −1:000:

Example 4. Cincinnati Milacron gear wrist

The following numerical data are prescribed or known from kinematic analysis:

• gear ratios: R1 = −1, R2 = −1, R3 = 0.4;• driving angular velocities: ω21 = −1, ω61 = 1, ω71 = −0.1;• known angles: α1 = 0°, β1 = 45°, α2 = 225°, β2 = 180°, α3 = 225°, β3 = 180°;• computed angles: θ1 = −157.5°, θ2 = 22.5°, θ3 = −21.524°, γ1 = 22.5°, γ2 = −157.5°, γ3 = −113.476°;• angular velocities of link 3: ω37 = 0.694, ω32 = 0.360, ω31 = 0.788, with ϕ31 = −18.854°;• angular velocities of link 4: ω43 = −2.360, ω45 = 4.361, ω41, with ϕ41 = −171.040°;• angular velocities of link 5: ω56 = 3.696, ω52 = −2, ω51 with ϕ51 = 30.361°.

The coefficients of the Willis equation are c41 = −0.988, c51 = 1.220, c31 = 1.142, c52 = −1.220, c62 = 1.000, c22 = 1.207,c33 = −1.338, c73 = 1, and c23 = 1.690.

45

3

2

7

6

2 1

3

4

5

2

6

3

7

P62P52 P51

P41

P2i

P23

P22P31

P33

P73

P6i

P7i

Pout

ε

Fig. 10. Kinematic structure and flow graph of the Cincinnati Milacron gear wrist.

Table 1Fundamental circuits (F.C.) of the Humpage gear train.

m Gear i Gear j Gear carrier k Rm

1 2 3 4 − 5620

2 3 1 4 − 7656

3 3 5 4 − 3524



For each fundamental circuit, Eqs. (18) and (21a) are written:

I)

P41 þ P51 þ P31 ¼ 0P51

P41¼ −1:677

II)

P52 þ P62 þ P22 ¼ 0P22

P62¼ 1:707

III)

P33 þ P73 þ P23 ¼ 0P23

P73¼ 7:678:

When the power balance condition is applied at each node link, the following equations are obtained:

P73 þ P7i ¼ 0;P62 þ P6i ¼ 0;

P22 þ P23 þ P2i ¼ 0;P41 þ Pout ¼ 0;P52 þ P51 ¼ 0;P33 þ P31 ¼ 0

Imposing Pout = 1, then the solution of the system formed by the previous equations gives: P41 = −1, P51 = 1.677,P52 = −1.677, P31 = −0.677, P33 = 0.677, P22 = 1.057, P23 = −0.599, P62 = 0.619, P73 = −0.078, P2i = −0.459, P6i = −0.619,P7i = 0.078.

It should be observed that in the power-flow analysis of multi degrees-of-freedom gear trains, the output power is imposedand the input powers are computed.

According to our results, under the prescribed kinematic conditions, link 7 is driven, whereas links 6 and 2 are driving.

6. Conclusions

In the paper novel procedures of kinematic and power flow analysis of planetary bevel gear trains with gyroscopic complexityare presented. All the methods are based on the graph representation of the gear train. This feature allows a systematic approachto the analysis.

Innovative features of this work are:

• a scalar equation that relates the absolute angular velocities of a basic bevel gear train;• the equations that express the power ratios in a basic bevel gear train.

These equations have, as particular case, the corresponding equations valid for spur gears.

Table 2Rotation axes angles within each circuit.

m αm βm

1 0° 60°2 60° 180°3 60° 180°

Table 3Fundamental circuits (F.C.) of the Cincinnati Milacron geared wrist.

m Gear i Gear j Gear carrier k Gear ratio

1 4 5 3 R12 5 6 2 R23 3 7 2 R3



Thanks to the availability of these equations, graph based methods of kinematic and power flow analysis of planetary geartrains have been extended to a new class of mechanical devices.

Although geared robotic wrists provide good load and dexterity capabilities, their application in the automation field has beensomewhat limited. To the best of the authors' knowledge, this is the first time that a power flow-analysis for this class ofmechanical devices has been discussed.

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