On grinding manufacture technique and tooth contact and stress analysis of ring-involute spherical gears

On grinding manufacture technique and tooth contact and stressanalysis of ring-involute spherical gears Li Ting*, Pan Cunyun School of Mechatronics and Automation, National University of

On grinding manufacture technique and tooth contact and stress

analysis of ring-involute spherical gears

Li Ting*, Pan Cunyun

School of Mechatronics and Automation, National University of Defense Technology, Changsha 410073, China

a r t i c l e i n f o

Article history:

Received 29 April 2008

Received in revised form 3 March 2009

Accepted 17 March 2009

Available online 21 May 2009

Keywords:

Spherical gear

Generating method

Plate grinding wheel

Contact analysis

Contact point path

a b s t r a c t

The spherical gear is a new gear-driven mechanism with two degrees of freedom (DOF), which can transfer spatial motion and power A grinding machine is designed for manufac-turing spherical gear with a plate grinding wheel by using generating method Based on the mathematical models of spherical gears, the tooth contact analysis (TCA) of spherical gear pairs is performed, and the contact point paths on tooth surfaces of spherical gears are investigated Finally, the tooth contact and bending stress analysis are simulated using Finite element method (FEM)

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1 Introductions

The spherical gears with continuous ring-involute teeth inFig 1were invented by Pan[1]in 1990 As a new 2 DOF gear-driven mechanism, it overcomes the two defects of Trallfa spherical gear[2]with discrete cone tooth: error of transmission principle and difficulty in machining The spherical gear can be used for robot arms and wrists, the seeker mechanism of missile, the flexible wrist mechanism of spatial robot in Fig 2, the control system of antenna’s attitude on satellite in

Fig 3, the lunar rover with complex function of wheel and leg, the attitude control system of solar array on spacecraft, and so on

Although the manufacturing method of spherical gears has already been developed, yet few investigations are on spher-ical gear, until Yang et al introduced a spherspher-ical gear with discrete arc teeth[3]and a spherical gear with discrete ring-invo-lute teeth[4] Tsai et al discussed the application of the rapid prototyping and manufacturing technology to form a spherical gear with skew axes[5]

This new type of spherical gears is different from spur involute gears and spherical involute conical gears, so its manu-facture methods are also different from that of traditional gears InFig 1, the spherical gears are machined by milling cutter and can satisfy the low-precision requirements However, in some fields, the spherical gears cannot be applied because of its low-precision Therefore, in this paper, a grinding manufacture method is put forward to improve the machining precision of spherical gears, and the machining theory comes from the mesh of a spherical gear and a teeth turner[6,7] The generation of the teeth turner is similar with that of a rack formed by a gear: when the teeth number of the spherical gear is infinite, the radius of its reference sphere will also be infinite, and the spherical gear becomes a teeth turner.Fig 4a shows the meshing of

a spherical gear and a teeth turner, andFig 4b is its section drawing Since the spherical gear can move in three-dimensional

0094-114X/$ - see front matter Ó 2009 Elsevier Ltd All rights reserved.

* Corresponding author Tel.: +86 0731 4574932.

E-mail address: bee.lt@163.com (L Ting).

Contents lists available atScienceDirect

Mechanism and Machine Theory

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / m e c h m t

space and the teeth turner allows plane movement, the spherical gear and teeth turner mechanism can achieve a shift in movement between spatial and plane movement The teeth turner can be machined plate grinding wheel, and then the plate grinding wheel can grind the spherical gear according to the meshing movement relationship between a spherical gear and a teeth turner

The tooth contact analysis (TCA) is an important factor which can evaluate gear transmission characteristics In Refs

[8–10], the TCA of spur gears, bevel gears and worm gears are investigated, but the meshing theory and tooth surface characteristics of the spherical gear are different from those of any other gears Up to now, few or no investigation about the TCA analysis and stress analysis of spherical gears can be found So, the efforts of this paper are to build the mathematical models of spherical gears with continuous convex ring-involute tooth and concave ring-involute tooth With these mathematical models, the meshing model and tooth contact analysis are performed, and the instantaneous contact points can be obtained In addition, the contact and bending stress of spherical gear drive is performed These play the important

Fig 2 Flexible transmission shaft with spherical gear pairs.

Fig 1 Spherical gears.

roles in studying the transmission characteristic of spherical gear mechanism, and also are useful to the practical applica-tions of spherical gear pairs

2 Design of grinding machine

A grinding machine for the generation of spherical gears must have 5 DOFs Four of these DOFs are necessary for the con-trol of the related motions of the plate grinding wheel and a spherical gear The fifth DOF is required to provide the desired velocity of grinding and is not related to the process of surface generation

2.1 Coordinate systems applied for grinding machine

According to meshing theory[6]of spherical gear pairs and the freedom requirement, the grinding machine (Fig 5) was designed for generating spherical gears The machine is provided with five DOF for three rotational motions, and two trans-lational motions The transtrans-lational motions are performed in two perpendicular directions Two rotational motions are pro-vided as rotation of the spherical gear and the rotation that enables the machine to change the angle between the axes of spherical gear and the plate grinding wheel The third rotational motion is provided as rotation of plate grinding wheel

Fig 3 Installed spherical gear mechanism.

Fig 4 Spherical gear and teeth turner mechanism.

and generally is not related to the process of generation The motions of the other 4 DOF are provided as related motions in the process of surface generation

As shown inFig 6, the coordinate systems S1ðx1;y1;z1Þ and Sgðxg;yg;zgÞ are rigidly connected to the spherical gear and the plate grinding wheel, respectively For further discussion, we distinguish three reference frames designated inFig 5as ,a

and b The reference frame / is the fixed one that is the housing of the machine, which is rigidly connected to coordinate system Sfðxf;yf;zfÞ Reference frames // and /// perform translations in two perpendicular directions Coordinate system

Seðxe;ye;zeÞ performs rotational motion with respect to Sf about the yf axis with an angle h Coordinate system

Sdðxd;yd;zdÞ and Seare parallel to each other and the location of Sdwith respect to Seis represented by ð0; 0; zðOd Þ

e Þ And then coordinate system S1performs rotational motion with respect to Sdabout the zdaxis with an angle b Coordinate system

Smðxm;ym;zmÞ and Sfare parallel to each other and the location of Smwith respect to Sfis represented by ðxðO m Þ

f ;0; zðO m Þ

f Þ Coor-dinate system Snðxn;yn;znÞ performs rotational motion with respect to Smabout the xmaxis with 180° And then, coordinate system Sgperforms rotational motion with respect to Snabout the znaxis with an angle /

According to the relationship of coordinate transformation inFig 6, the transformation matrix from coordinate system Sg

to S is as follows:

Fig 5 Schematic of grinding machine.

O f

x f

z f

y f

θ

x e

O e

y e

z e

O d

O1

z d

x d

y d

x1

y1

z1

O m

x m

z m

y m

O n

z n

x n

y n

y g

f

x

(O m)

f z

e

z

z g

Fig 6 Coordinate systems applied for grinding machine.

M1g¼ M1dMdeMefMfmMmnMng

¼

cos b cos h cosu sin b sinu  cosusin b  cos b cos h sinu cos b sin h xðO m Þ

f cos b cos h  zðO m Þ

f cos b sin h

 cos h cosusin b  cos b sinu  cos b cosuþ cos h sin b sinu  sin b sin h xðOm Þ

f cos h sin b þ zðO m Þ

f sin b sin h

f sin h þ zðO m Þ

f cos h  zðO d Þ

e

2

6

6

6

4

3 7 7 7 5

ð1Þ

Here, M1d;Mde;Mef;Mfm;Mmnand Mngare the transformation matrixes from coordinate system Sdto S1, from coordinate sys-tem Seto Sd, from coordinate system Sf to Se, from coordinate system Smto Sf, from coordinate system Snto Sm, and from coordinate system Sgto Sn, respectively

2.2 Kinematic meshing relationship between spherical gear and plate grinding wheel

According to the motion relationship of the machine (inFig 5), we can simplify the coordinate systems in the xfzfplane as shown inFig 7, because the motion for generation of ring-direction tooth shape of spherical gear only requires the rotational motion of spherical gear with respect to its polar axis In the xfzfplane, if the spherical gear rotates about its spherical center

O1, the meshing motion between the spherical gear and the plate grinding wheel is the same as that between a spur involute gear and a rack However, inFig 5, the spherical gear is installed on the rotating floor When the rotating floor rotates about the yf axis (inFig 6), the spherical gear is driven and sways There is a distance D (equal to zðO d Þ

e inFig 6) between the spher-ical center O1and the swaying center Of The advantage of the machine design is that it is easy to install spherical gear since its spherical center O1need not be fixed with rotational center, and the DOF of the machine does not increase

Consider now that the initial position is the position in which the polar axis z1of spherical gear coincides with the zgaxis

of plate grinding wheel r (inFig 7) is the radius of reference sphere of the spherical gear When the rotating floor rotates about the yf axis with an angle h, spherical center O1also rotates about the origin Of with the equal angle h Accordingly, the plate grinding wheel must move not only about the xf axis, but also about the zf axis Based on the meshing movement rule of a gear and a rack, the following equations can be obtained:

Dxf ¼ D sin h þ rh

Dzf ¼ Dð1  cos hÞ



ð2Þ

3 Mathematical models of plate grinding wheels

The spherical gear pair for the meshing simulation comprises a convex-tooth spherical gear and a concave-tooth spherical gear Correspondingly, the grinding tools are a concave-tooth plate grinding wheel and a convex-tooth plate grinding wheel Assume that the concave-tooth plate grinding wheel surfacesP

vand convex-tooth plate grinding wheel surfacesP

c gen-erate the convex-tooth spherical gear surfacesP

and concave-tooth spherical gear surfacesP

, respectively The rotation

α

r

x f

Reference line

x1

1

O

1

O

Convex tooth in position 1

Reference line

Plate grinding wheel in position 1

D

O f

z f

x g

z g

O g

Plate grinding wheel in position 2 Convex tooth in position 2

θ

Fig 7 Kinematic meshing relationship between spherical gear and plate grinding wheel.

of the plate grinding wheel is used to provide the desired velocity of grinding and is not related to the process of surface generation, so its surface equation can be represented by its generatrix equation

Assume that the pressure angle and the module of rack section of plate grinding wheel areaand m, respectively Accord-ing toFig 8, the section of plate grinding wheel consists of two straight edges Therefore, the mathematical models of two straight edges of the concave-tooth plate grinding wheel can be represented in coordinate system Sf by

RðvÞ

g ¼

kvpm  ð0:25pm þ m tana lvsinaÞ

0

m  lvcosa

1

2

6

6

3 7

where the design parameter lv¼ A 0A

  represents the distance between the initial point A0and the moving point A The upper and lower signs of Eq.(3)represent the cutter surface equation of the inner and outer side on the concave-tooth plate grinding wheel kvðkv¼ 0; 1; 2 and 3Þ represents the tooth number of the concave-tooth plate grinding wheel

The unit normal to the concave-tooth plate grinding wheel surface can be gotten by

nðvÞ

Similarly, the mathematical models of these two straight edges of the convex-tooth plate grinding wheel can be repre-sented in coordinate system Sf by

RðcÞ

g ¼

kcpm  ð0:25pm  m tanaþ lcsinaÞ

0

m  lccosa

1

2

6

6

3 7

where kcðkc¼ 0; 1 and 2Þ represents the tooth number of convex-tooth plate grinding wheel

The unit normal to the convex-tooth plate grinding wheel surface can be gotten by

nðcÞ

4 Mathematical models of spherical gears

As shown inFig 5, when spherical gear is grinded, its tooth shape along the involute-direction is generated by its rotating motion about the yf axis and the translational motions of the plate grinding wheel according to Eq.(2) Except that they are meshing in linear contact in the initial position, in any other position they are meshing in point contact In the generation process, the plate grinding wheel performs translational motion with a velocity V, while the spherical gear rotates with an angular velocityx According to the theory of gearing[8], the plane axode of the plate grinding wheel and the pitch sphere of spherical gear roll over each other without sliding on the pitch plate Therefore, with Eqs.(3) and (4)and concave-tooth plate grinding wheel surfacesP

v, the mathematical model of the generated convex-tooth spherical gear surface can be repre-sented in coordinate system Sf as follows:

z g

x g

O g

A0

A

α

m

y g

Fig 8 Section of plate grinding wheel.

The unit normal vector of the generated convex-tooth spherical gear can be gained by

n1¼ L1gnðvÞ

Similarly, with Eqs.(5) and (6)and the same generation mechanism, the mathematical model of concave-tooth spherical gear surface generated by convex-tooth plate grinding wheel surfaces P

c can be expressed in coordinate system Sf as follows:

R2¼ M1gRðcÞ

The unit normal vector of the generated concave-tooth spherical gear also can be obtained by

n2¼ L1gnðcÞ

Here, L1gis the rotating transformation matrix from coordinate system Sgto S1

5 Meshing model of spherical gear pairs

Fig 9 shows the simplified model of the spherical gear mechanism The origin O0 of the fixed coordinate system

S0ðx0;y0;z0Þ coincides with the origin O1of coordinate system S1(fixed with convex-tooth spherical gear) The coordinate systems S1ðx1;y1;z1Þ and S2ðx2;y2;z2Þ are fixed with the convex-tooth spherical gear and the concave-tooth spherical gear, respectively Since the tooth contact form of meshing spherical gears is point contact between a convexity and a saddle sur-face, according to the tooth contact analysis method and the tooth surface characteristic of spherical gear, the position vec-tors and unit normal vecvec-tors of both convex-tooth spherical gear and concave-tooth spherical gear should be represented in the same coordinate system S0 The instantaneous common contact point of convex-tooth spherical gear and concave-tooth spherical gear is the same point in coordinate system S0 Furthermore, the unit normal vectors of convex-tooth spherical gear and concave-tooth spherical gear should be collinear to each other Therefore, the following equations should hold at the point of tangency of the meshing spherical gear pair:

Here, Rð1Þ

0 and Rð2Þ

0 are the position vectors of the convex-tooth spherical gear and concave-tooth spherical gear, while nð1Þ

0 and

nð2Þ0 represent the unit normal vectors, respectively, in coordinate system S0 Since jnð1Þ0 j ¼ jnð2Þ0 j ¼ 1, Eqs.(11) and (12)yield five independent nonlinear equations with six independent parameters b1;lv;h0

1;b2;lcand h0

2 The vector n, the swaying axis,

is perpendicular to the swaying plane in which two spherical gears mesh Any meshing position of the spherical gear pair can

be gotten by making the convex-tooth spherical gear rotate about the vector n with an angle h0

1from the initial position in which two spherical gears’ polar axes z1and z2coincide with to each other, and the concave-tooth spherical gear rotate about the vector p going through the center O2with an angle h0

2 If the input rotation angle h0

1of the convex-tooth spherical gear is known, other five independent parameters can be solved by nonlinear solver Therefore, a common contact point of two tooth surfaces can be obtained

O2

O1

θ1

2

0

x0

x1

y1

y0

z0

y2

x2

n

z1

p

θ2

x20

z20

y20

O20

6 Transmission error model of spherical gear mechanism

According to the movement characteristic of spherical gear mechanism[7], its transmission error can be described by er-ror of the swaying angle and the azimuth angle of the mechanism’s output axis (polar axis z2of the concave-tooth spherical gear inFig 9) InFig 10, because of the assembly errors, the z2axis does not coincide with the ideal z20axis (no assembly error) of the concave-tooth spherical gear hz 20

z 0 and hz 2

z 0represent the swaying angles of the z20axis and z2axis, respectively

nz20

z 0 and nz2

z 0represent the unit normal vectors of the plane / and plane //, respectively The azimuth angleaz 20of the z20axis is equal to the angle between the normal vector nz 20

z 0 and the y0axis, and the azimuth angleaz 2of the z2axis is equal to the angle between the normal vector nz 2

z 0and the y0axis Therefore, the error of swaying angle and azimuth angle can be obtained

as follows:

Dh¼ hz 20

z 0  hz 2

7 Contact point path of spherical gear pairs

The contact path of spherical gears is complex There are two reasons for it First, since every tooth of traditional gear (spur gear, bevel gear, spiral bevel gear, and so on) is the same, the contact path on every tooth is uniform whenever the teeth are in point contact or linear contact However, the teeth of spherical gear pairs are diverse Second, the spherical gear mechanism has two DOF, and the spherical gear can rotate about any alterable axis through its spherical center (for example, the vector n inFig 9) Therefore, the transmission characteristic of spherical gear pair is more complex and flexible than that

of the traditional gear pairs, and the tooth contact paths of spherical gears are also alterable and complex because of the above two reasons The following research will be focused on the path of tooth contact point of spherical gear pair and will

be divided into three cases to be investigated

7.1 Contact points path of tooth meshing in a fixed swaying plane

When convex-tooth spherical gear rotates about a fixed swaying axis vector n with an angle h0

1, the meshing movement of two spherical gears in the swaying plane (passes through the origin O1and O2and is perpendicular to the vector n) is the same as that of the a spur involute gear pair

z0

O0

y0

x0

z2

y2

x2

O2

p2

z20

α2

20 0

z z

0

z z

θ

20 0

nz z

2 0

nz z

α20

Fig 10 Transmission error model of spherical gear mechanism.

Table 1

Design parameters of spherical gear.

Example 1 The major spherical gear parameters are given inTable 1 Based on the above developed mathematical model of spherical gears, the three-dimensional tooth profiles of convex-tooth and concave-tooth spherical gears can be plotted

Fig 11 shows the convex-tooth and concave-tooth spherical gears Assume that the swaying axis n is the y0 axis, and

h01¼ 60 to 60°, the contact paths on the tooth surfaces of two spherical gears are shown inFig 11

7.2 Contact points path of tooth meshing in alterable swing planes

In the initial position, the polar axes of two spherical gears are not collinear to each other Assume that the convex-tooth spherical gear has rotated about a fixed axis vector p10with an angle hp10 From this position, the convex-tooth spherical gear rotates about another fixed axis vector p1with an angle hp1 Along with the increase of hp1, the contact paths on the tooth surfaces of two spherical gears are the special curves InFig 9, to get the transformation matrixes from coordinate system

S1and S2to coordinate system S0, it can be simulated by using the following procedures:

Step 1: In initial position, the angles between the axis vector p10and three axes (x0;y0and z0) of the coordinate system S0 area;bandc, respectively To describe the coordinate transformation from S1to S0in the initial position, we use coordinate system S10ðx10;y10;z10Þ coinciding with the convex-tooth spherical gear in this position Then the transformation matrix from the coordinate system S10to S0can be obtained by:

M010¼ M010ða;b;c;hp10Þ

¼

cos2að1  cos hp10Þ þ cos hp10 cosacos bð1  cos hp10Þ  coscsin hp10 cosacoscð1  cos hp10Þ þ cos b sin hp10 0 cosacos bð1  cos hp10Þ þ coscsin hp10 cos2bð1  cos hp10Þ þ cos hp10 cos b coscð1  cos hp10Þ  cosasin hp10 0 cosacoscð1  cos hp10Þ  cos b sin hp10 cos b coscð1  cos hp10Þ þ cosasin hp10 cos2cð1  cos hp10Þ þ cos hp10 0

2

6

6

6

3 7 7 7

ð15Þ

Step 2: If the angles between the axis vector p1and three axes (x0;y0and z0) of the coordinate system S0area0;b0andc0, the angles between p1and the three axes (x10;y10and z10Þ of the coordinate system S10area1;b1andc1, respectively.a1;b1and

c1can be calculated by:

a1¼ cos1 xð10Þ0  p1

xð10Þ0



 jp1j

0

B

1 C A; b1¼ cos1 yð10Þ0  p1

yð10Þ0



 jp1j

0 B

1 C A; c1¼ cos1 zð10Þ0  p1

zð10Þ0



 jp1j

0 B

1

Here, the axes xð10Þ

0 ;yð10Þ

0 and zð10Þ

0 are the representation of the three coordinate axes of S10in S0 And then, when the convex-tooth spherical gear rotates about p1with hp1, the transformation matrix from the coordinate system S1to S10can be ob-tained by:

Step 3: The transformation matrix from the coordinate system S1to S0is that:

Step 4: According to the transmission characteristic of spherical gear pairs, when the convex-tooth spherical gear rotates about p1 with an angle hp1, to simulate the meshing movement of the spherical gear pair, it can be regarded as the convex-tooth spherical gear rotating a fixed axis n with an angle h0

1and the concave-tooth spherical gear rotating a fixed axis p(parallel to the vector unit n in ideal condition[7]) with an angle h0

2 The unit vector n and angle h0

1can be calculated by:

Fig 11 Contact point paths of spherical gear pair with convex tooth and concave tooth.

n ¼

nx

ny

nz

6 75 ¼ z0 zð1Þ

h01¼ cos1 z0 zð1Þ0

jz0j zð1Þ0



 

0

B

1

Assume thatan;bnandcnrepresent the angles between the unit vector n and three coordinate axes x0;y0and z0of S0, we can obtain the following calculations:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2

xþ n2

yþ n2 z

q

0

B

1 C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2

xþ n2

yþ n2 z

q

0 B

1 C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2

xþ n2

yþ n2 z

q

0 B

1

InFig 9, the origin O20of the fixed coordinate system S20ðx20;y20;z20Þ coincides with the origin O2of the coordinate system S2, and the coordinate axes of S20and S0are parallel to each other From the above discussion, we can get that the angles be-tween p and three coordinate axes x20;y20and z20of S20is equal to 180—an;bnandcn, respectively Therefore, the transfor-mation matrix from the coordinate system S2to S20is that:

M202¼ M010180an;bn;cn;h02

ð22Þ

InFig 9, assume that C represents the center distance between two spherical gears We obtain the transformation matrix from the coordinate system S20to S0

M020¼

2

6

6

3 7

Step 5: the transformation matrix from the coordinate system S2to S0is that:

Lastly, according to step 1 step 5, we can obtain the mathematical model and the unit normal vector of the convex-tooth spherical gear in S0

Similarly, the mathematical model and the unit normal vector of the concave-tooth spherical gear in S0are as follows:

where, L01and L02are the rotational transformation matrixes from coordinate systems S1and S2to S0, respectively With the Eqs.(25)–(28), Eqs.(11) and (12), the contact point position of two spherical gear tooth surfaces can be obtained by using nonlinear solver

Example 2 Assume that the p10is the x0axis, p1is the y0axis and hp10¼ 0; 0:2;2 and 20 Along with the change of hp1, the contact paths on the tooth surfaces of two spherical gears are shown inFig 12 The curves with the same color represent