A Study of Geometry and Deformable-body Characteristics of Non-right Angle Worm Gear Pairs
THESIS
Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University
By Sriram Madhavan, B.E. Graduate Program in Mechanical Engineering
The Ohio State University 2012
Master's Examination Committee: Dr. Ahmet Kahraman, Advisor Dr. Donald Houser
Copyright by
Sriram Madhavan
2012
Abstract
In this study, a formulation to define the three-dimensional geometry of worm gear drives having non-right angle shafts is developed. The geometry of the worm is determined by defining the geometry of the cutter and solving the corresponding equation of meshing between the worm and the cutter. The geometry of the worm gear is then defined by using a cutter which has the exact shape of the worm and solving the corresponding equation of meshing between the worm and the worm gear. Both right- and left-hand, single enveloping worm drives of ZK type with any number of worm threads are included in this formulation. With the tooth surface geometries defined, a commercial finite element gear analysis package with specific worm mesh generators is used to develop a deformable-body model of a non-right angle worm gear pair A parametric design sensitivity study is performed by using this deformable-body model to quantify the effects of basic geometric parameters including the shaft cross angle, lead angle, pressure angle, and addendum and dedendum coefficients on the maximum contact stress and mechanical efficiency of the gear pairs. In addition, variations to shaft center distance and cross angle are introduced to investigate their influence on gear pair performance.
iii
Dedication
This thesis is dedicated to my family and friends for their love and support.
iv
Acknowledgments
I would like to express my sincere gratitude to my advisor Professor Dr. Ahmet
Kahraman for providing the research opportunity, guidance throughout my research at
OSU and his effort in reviewing this thesis. I am grateful to Dr. Donald Houser for accepting to be part of my Master’s examination committee. I am also thankful to Jonny
Harianto for the support through the graduate program and while being a GRA at
Gearlab.
I would like to thank Honda R&D Americas, Ohio for providing financial support throughout my study. I would like to thank Dr. Sandeep Vijayakar for providing the
CALYX package and support throughout my research. I am also thankful to Karthikeyan
Marambedu of Advanced Numerical Solutions, Inc. for helping me out with computer programming and software troubleshooting required for the research.
Finally, I deeply appreciate the support and love of my parents, brother and friends. I would like to specially thank all my friends in Columbus for making my stay memorable and all my lab mates for their help and friendship throughout my masters program at OSU.
v
Vita
May, 2010 ...... B.E. Mechanical Engineering College of Engineering Guindy Anna University Chennai, India Oct, 2010- Present ...... Graduate Research Associate, Gear and
Power Transmission Research Laboratory,
Department of Mechanical Engineering
Ohio State University
Fields of Study
Major Field: Mechanical Engineering
vi
Table of Contents
Abstract ...... iii
Dedication ...... iv
Acknowledgments...... v
Vita ...... vi
List of Tables ...... x
List of Figures ...... xi
Nomenclature……………………………………………………………………………xiv
Chapter 1 Introduction……………………………………………………………………………… 1
1.1 Background and Motivation……………………………………………………. 1
1.2 Literature Survey……………………………………………………………….. 4
1.3 Scope and Objective……………………………………………………………. 6
1.4 Thesis Outline ………………………………………………………………….. 7
vii
Chapter 2 Definition of the Geometry of a Non-Right Angle Worm Gear Pair…………………. 9
2.1 Introduction…………………………………………………………………… 9
2.2 Definition of Cutter Geometry ……………………………...... 10
2.3 Definition of Worm Geometry……………………………………………….. 15
2.3.1 Cutter Installment and relative Motion between Cutter and Worm…….. 15
2.3.2 Equation of Meshing……………………………………………………. 21
2.3.3 Worm Surface Equations……………………………………………….. 23
2.3.4 Geometric Model of the Worm…………………………………………. 26
2.4 Geometric Model of the Worm Gear…………………………………………. 29
Chapter 3 Computational Model and Parametric Study…………………………………………. 34
3.1 Introduction…………………………………………………………………… 34
3.2 Deformable Body Finite Element Analysis of Worm Gear Pair……………… 35
3.2.1 Mesh Generation………………………………………………………... 35
3.2.2 Pre-processing in Hypoid K program…………………………………… 38
3.2.3 Post-processing in Hypoid K program………………………………….. 38
viii
3.3 Design Parameter Sensitivity Study………………………………………….. 44
3.3.1 Effect of Cross Angle and Lead Angle on max and …………. 50
3.3.2 Effect of Pressure Angle c on and
Effect of Addendum a and Dedendum d Coefficients on
and ……………………………………………………………………. 68
3.4 Manufacturing Variability Study……………………………………………… 68
Chapter 4 Conclusions…………………………………………………………………………… 76
4.1 Summary……………………………………………………………………… 76
4.2 Conclusions…………………………………………………………………… 77
4.3 Recommendations for Future Work………………………………………….. 79
References……………………………………………………………………………. 81
ix
List of Tables
Table Page
2.1 An Example set of user-defined and calculated cutter parameters.
Parameters with an asterisk next to it are user-defined ones………………… 16
3.1 Input design parameters to define a worm drive……………………………... 39
3.2 Design constraints imposed in the parametric study...... 40
3.3 Material and grease parameters………………………………………………. 49
3.4 Geometric parameters used for the vs. study…………………………… 51
3.5 Geometric parameters used for the c study………………………………… 63
3.6 Geometric parameter ranges suitable for addendum and dedendum coefficient
study…………………………………………………………………………... 70
3.7 Addendum and dedendum coefficients for worm and worm gear…………..... 71
3.8 Parameters considered in the manufacturing variability study……………….. 73
x
List of Figures
Figure Page
1.1 Worm gear drive consisting of worm and worm gear (worm wheel)……...... 2
2.1 (a) Axial section of the cutter and (b) generating cone surface………………… 11
2.2 Definition of cutter geometry parameters……………………………………… 14 2.3 Installment of the cutter on the worm surface…………………………………. 17 2.4 (a) Coordinate systems applied for generation of K type worms and
(b) definition of angle c ……………………………………………………… 18
2.5 Relative motion between the cutter and the worm………………...... 20 3.1 Definition of gear surfaces in CALYX………………………………...... 37
3.2 (a) Right angle 90 and (b, c) non-right angle design configurations
with (b) 90 and (c) 90 ...... 41
3.3 FE mesh model of a worm gear pair with m21=31, n =1, =110 , = 7.5
and c =15 …………………………………………………………………… 42
3.4 FE meshes and cross-sectional views of (a) a worm and (b) its worm gear mate…………………………………………………………. 43 3.5 Contact Patterns on (a) the worm gear and (b) the worm for a pair
For parameters m21 31, n=1, =100 , = and = ……………….. 45
3.6 Instantaneous load intensities on the worm gear of a pair for parameters
xi
m21=31, n =1, =110 , = 7.5 and c =15 at 5 different mesh steps……… 46
3.7 Contact pattern and load distribution on worm gear for parameters m21=23,
=2, =19.5 and =15 ; (a) = 70 (b) 110 ………………………. 47
3.8 Design configurations considered to study of effect of and on
max and with =1……………… ……………………………………….. 52
3.9 (a) Effect of on max at different values, and (b) effect of on
at different values for = 15 and = 1……………………………. 53
3.10 (a) Effect of on at different values, and (b) effect of on
at different values for = and = 1…………………………...... 54
3.11 Design configurations considered to study of effect of and on
max and with =2………………………………………………………….. 57
3.12 (a) Effect of on at different values, and (b) effect of on
at different values for = and = 2…………………………….. 58
3.13 (a) Effect of on at different values, and (b) effect of on
at different values for = and = 2………...... 59
3.14 Comparison between single thread (n=1) and double thread designs
(n=2) (a) Effect of on max , and (b) effect of on for =110 ………... 60
3.15 Tooth size comparison of (a) single thread and (b) double thread
worm gear design options……………………………………………………… 61
3.16 Design configurations considered to study the effect of c and on
and at =105 ………………………………………………………. 64
xii
3.17 (a) Effect of on at different values, and (b) effect of on c max
at different values for =105 ………………………………………. 65
3.18 (a) Effect of on at different values, and (b) effect of
on at different values for 110 …………………………………… 66
3.19 (a) Effect of on at different values for = , and
(b) effect of on at different values for =110 ……………………... 67
3.20 Effect of a and d on (a) max and (b) for =110 , 9
and c 20 …………………………………………………………………... 72
3.21 (a) Effect of on max , and (b) effect of on for = ,
=9 and = 20 ……………………………………………………….. …. 74 c
3.22 (a) Effect of E on , and (b) effect of E on for = ,
= and = ………………………………………………………….. 75
xiii
Nomenclature
Symbol Description
a Distance between center and apex of conical cutter
E Shortest distance between the rotational axes of the worm and worm gear surfaces when mating. E Center distance error
Ec Distance between origin of the worm and origin of the cutter
max Axial module
Transformation matrix M
Normal vector n
n Number of threads on worm
p Screw parameter
p Axial pitch x
Position vector during worm cutting process r
r Cutter radius c
Outer radius of the worm ro r Pitch radius of the worm p
Root radius of the worm rr
xiv
(R,Z) Coordinate system used in CALYX
Cutter width sc
S Worm and worm gear coordinate system
Input torque Tin
Output torque Tout
Curvilinear surface parameter u
Velocity vector during worm gear cutting process v Velocity vector during worm cutting process V Axial tooth space wax
Cartesian coordinate system (,,)x y z
Position vector during worm gear cutting process X
Pressure angle c
Axial pressure angle ax
Cross angle
Shaft misalignment
Poisson ratio
Mechanical efficiency
Curvilinear surface parameter
xv
Lead angle
Friction coefficient
Density
Tensile Stress t
Helix rotation angle
Worm rotation angle during worm gear cutting process 1
Addendum coefficient a
Dedendum coefficient d
Worm rotation angle during worm cutting process
Angular velocity vector 1
Angular velocity of the worm during worm cutting process 1
1 Angular velocity of the worm during worm gear cutting 1 process Input speed at the worm in
Output speed at the worm gear out
Subscripts
c Cutter coordinate system
1 Worm reference coordinate system
xvi
Subscripts
2 Worm gear reference coordinate system
f 1 Worm fixed coordinate system
f 2 Worm gear fixed coordinate system
max maximum
min Minimum
Superscripts c Cutter
1 Worm
2 Worm gear
xvii
CHAPTER 1
INTRODUCTION
1.1 Background and Motivation
The worm gear drive arrangement is one in which a worm (gear in the shape of a screw) meshes with a worm gear (worm wheel) to transmit motion and torque through two non-intersecting, crossed axes as shown in Figure 1.1 [1]. They are used in cross- axis applications that require higher gear ratios. Perhaps the most significant advantage of the worm gear drive system is that it provides high reduction ratios in smaller and compact spaces, unlike other gearing systems where more number of stages of reduction would be required hence increasing their sizes. The gear reduction ratio in a worm gear drive depends on the number of starts or threads (n =1, 2, 3...etc.) on the worm and the number of teeth on the worm wheel.
1
Figure 1.1 Worm gear drive consisting of the worm and worm gear (worm wheel) [1].
2
Worm gear drives are commonly used in industrial applications such as mechanical presses, rolling mills, conveyor belts in mining industries, hoist crane operations and in precision application such as dividing heads and indexing tables.
Mechanical efficiency of worm gear systems range from 20% to 98% [2]. The power losses are due to the friction between the mating gear surfaces and they are mostly caused by the relative sliding. They can provide low-noise and low-vibration performance and if properly maintained and lubricated.
Conventional worm gear drive systems have the worm and the worm wheel shafts with non-intersecting axes crossed at right angle (cross axis 90 ). Right-angle worm drives have been the preferred choice in various automotive auxiliary drives such ass electric power steering (EPS) systems, power windows, windshield wiper drives, seat adjuster gearboxes, sun roof units and power doors. In these applications, a steel worm is meshed with plastic gear molded on a steel frame. A small electric motor drives the worm at a high-speed (low-torque) to achieve much higher torques at lower speeds on the output (worm gear) side.
Most of the past focus has been on geometry and performance of worm drives having . Worm gear drives with axes inclined at non-right angles ( 90 ) might provide their own set of advantages. Such drives can provide higher contact ratios, hence a higher load carrying capacity. More importantly, they have potential to provide higher design flexibility in terms of packaging in automotive systems. For instance, non-right angle worm drive solutions with [65 ,115 ] are very desirable in EPS systems, as 3
they provide many advantages in chassis design. Accordingly, this thesis focuses on geometric design and deformable-body analysis of non-right angle worm gear drives formed by a plastic worm gear and a steel worm.
1.2 Literature Survey
Some of the earliest works on worm gear was done by Thomas [3] where he discusses the basic principles of worm gear design. Buckingham [4, 5] categorized worms on the basis of the cutting process, described the type of contact occurring in each of these worm gear drives and also described the design procedure for worm gear drives with axes crossed at right and non-right angles.
Litvin [6, 7] described the geometry of cylindrical worms such as the ZA, ZN, ZI,
ZK and Flender worms. Fang and Tsay [8] provided a mathematical model for the ZK- type worm generated through a grinding process and a mathematical model for a worm gear cut by an oversize hob cutter.
Octrue [9] determined the line of contact along worm surfaces by simulating the cutting process and used a simple extension of a two dimensional beam model to determine the stiffness along the line of contact. Assuming equal deformation at all points along the line of contact and using the Hertzian theory of cylindrical contact, he determined the load and contact pressure distributions without taking into account the frictional effects. However conventional finite element (FE) techniques required a very
4
highly refined mesh, due to fact that the contact zone is extremely small as compared to the dimensions of the gear. Vijayakar [10] and Vijayakar et al [11] overcame this problem by developing a special type of finite element called finite quasi prismatic element which combined the effects of surface integrals and boundary elements with those of finite elements making it possible to accurately define the worm and gear geometries with fewer elements.
Gopikrishnan [12] generated the geometry of the worm and the gear by simulating the cutting process and used the finite quasi prismatic elements of Vijayakar [10] to determine contact and bending stresses, and transmission error. The effect various worm gear parameters such as number of threads, number of teeth and profile angle on the root stresses of worm drives were studied by Simon [13] using a conventional FE approach with axes at right angles. The contact analysis of a worm gear set with non-right crossing angle consisting of a ZK type worm and a worm gear cut by a straight edged flyblade was studied by Liu et al [14]. The tooth contact analysis program (TCA) was used to analyze the transmission error and contact ratio of the gear set.
A computerized design and generation of ZK-type cylindrical worm gear drive was proposed by Litvin [15] who performed a loaded tooth contact analysis (TCA) to right angle and non-right shaft angle worm gear drive configurations. Houser and Su
[16] and Su [17] put forth an alternative simplified equation meshing for of worm and worm gear surfaces with axes inclined at non-right angle configurations. A ZK type worm was taken into consideration. The generalized meshing parameter considered
5
during worm gear cutting process was explicitly expressed by the two surface parameters of the worm. This expression applies to all kinds of single enveloping worm gear drives as long as the worm has a constant lead.
1.3 Scope and Objective
The above review of the literature points to in need for deformable-body finite element analysis tool for worm gear drives with non-right cross angle configurations. In addition a comprehensive study on the effect of geometric parameters on stresses and efficiency of non-right angle worm drives does not exist. In this research, the FE approach proposed by Vijayakar [10, 11] is used to perform contact analysis of worm gears with non-right angle configurations using a commercial Face Milled Hypoid Gear
Analysis program Hypoid K.
The specific objectives of this thesis are as follows:
Develop a formulation to define the geometry of ZK type worm and worm
gear surfaces with axes inclined at non-right angles.
Develop a deformable-body FE model for non-right angle worm gear drives to
predict contact stresses and mechanical efficiencies with the friction
coefficients reported by Wilson [22] for typical plastic-steel contacts with
typical greases.
6
Conduct parameter design sensitivity studies on the effect of numerous
geometric parameters (shaft cross angle, lead angle, pressure angle, number of
threads and, addendum and dedendum coefficients) and manufacturing
variations (center distance changes and shaft cross angle changes) on contact
stresses and mechanical efficiency of worm gear drives.
With a set of requirements from an automotive auxiliary drive application,
make recommendations about design of non-right angle (plastic-steel) worm
gear drives in terms of their efficiency and durability metrics.
1.4 Thesis Outline
Chapter 2 presents a complete formulation to defined the geometry of a non-right angle worm gear pair. It describes the cutter geometry and subsequently the worm surface geometry of a ZK type worm through the equation of meshing between the worm and the cutter and mathematically simulating the cutting process. The worm gear surface is generated by assuming a cutter as the exact shape of the worm in concurrence with the alternative simplified equation of meshing between the worm and worm gear surfaces [16]. Chapter 3 provides an overview of the Hypoid K and its contact solver
CALYX. Application of Hypoid K to the problem in hand (a non-right angle worm drive) is described in Chapter 3 including its FE mesh generation process and its modification to analyze the worm gear pair. A parametric design sensitivity study is performed using this model to quantify the effect of key geometric parameters and certain 7
manufacturing variability parameters on the contact stresses and mechanical efficiency of non-right worm gear drive systems. Chapter 4 summarizes this work, lists the major conclusions and provides a list of recommendation for future work.
8
CHAPTER 2
DEFINITION OF THE GEOMETRY OF A NON-RIGHT ANGLE WORM GEAR PAIR
2.1 Introduction
As stated in Chapter 1 and illustrated in Figure 1.1, a worm gear set is composed of a worm and a worm gear that are brought to mesh at a certain center distance and a certain right or near-right shaft angle. There are two common types of worm gear drives:
(i) single enveloping worm-gear drives with a cylindrical worm and (ii) double- enveloping worm gear drives with an hourglass shaped worm. This chapter focuses only on the geometry of single enveloping worm gear drives with a cylindrical worm.
Depending on the method of generation in manufacturing, there are different types of cylindrical worm surfaces such as ZA, ZN, ZI, ZK and Flender worms [6-8]. In this study, only ZK type worm drives will be considered as they are widely used due to the ease of manufacturing. While most worm gear applications are designed for a right angle
9
(90 angle between the gear axes) configuration, this study focuses on non-right angle arrangements [15, 16] for the reasons stated in Chapter 1.
This chapter provides the formulations regarding the definition of the geometry for the worm and the worm gear surfaces for non-right axes. The procedure outlined in this chapter consists of three major steps: (i) definition of the geometry of the cutter that cuts the worm, (ii) definition of geometry of the worm by applying the equation of meshing to the cutter-worm pair, and (iii) with the worm defined, definition of the geometry of the worm gear by using a cutter at the exact shape of the worm.
2.2 Definition of Cutter Geometry
In this study, the single enveloping, ZK type worm will be generated by a conical cutter surface (grinding wheel or a milling cutter). Figure 2.1(a) defines two cutting edges of a conical cutter as sides 1 and 2, which generate the two flanks of the worm tooth surface in a ZK type. In Figures 2.1(a) and (b), a cutter coordinate system
Sc(,,)x c y c z c is defined with its origin Oc located at the geometric center of the cutter and the zc axis defined along the rotation axis of the cutter. The position vector of any arbitrary point M on the cutter surface, as shown in Figure 2.1(b), can be represented in
Sc as follows
xuc ccos c cos c rcyu c ccos c sin c . (2.1) zc ( a u c sin c ) 10
xc xc
Cutting edge Cutting edge Worm side 1 Worm side 2 Any cutter surface point M
uc c
P rc zc zc P Oc
c
11 yc
a a
(a) (b) Figure 2.1 (a) Axial section of the cutter and (b) generating cone surface.
Here, and are the curvilinear surface parameters. defines the location of point uc c uc
M on the cone generatrix, a is the distance between the cone apex and the center line of the cutter, c is the pressure angle (cone angle as shown in Figure 2.2(a)) and c is the angular position of projection of point M on the mid-plane of the cutter as measured form the xc axis.
The unit normal vector from the cutter surface at the same point M is determined as
rr cc ucc nc . (2.2a) rr cc ucc
Substituting Eq. (2.1) in (2.2a), the normal vector is obtained as
sin cc cos nc sin c sin c (2.2b) cosc
In equations (2.1) and (2.2a), the sign convention ‘ ’ indicates which side of the cutter is taken into consideration. The upper sign refers to side 2 of the cutter and lower sign refers to side 1. The range for angle c is 02 c and uc is bounded by the addendum and dedendum values of the worm.
12
A typical conical cutter is shown in Figure 2.2(a). The key parameters of the cutter are the lead angle , cutter width at the mean line sc , radius of the cutter rc , the distance between the apex of the cone and center of the cutter a, and the worm axial tooth space measure w at the worm pitch cylinder. The lead angle is defined by the relation ax
1 npx tan (2.3) max
where n is the number of threads on the worm ( n 1 or 2), px is the axial pitch, and max is the axial module of the worm. From Figure 2.2(a), parameter a is defined as
s artan c (2.4) cc2
The cutter radius is given as
rc E c r p (2.5)
where Ec is distance between the origin of the worm and the origin of the cutter and rp is the pitch radius of the worm as defined in Figure 2.2(b). From the same figure, one finds
swc ax cos (2.6)
13
worm axial tooth space
wax P p waxcos p
rc c worm pitch
cylinder 14
cutter mean rp line Sc pitch radius a
(a) (b)
Figure 2.2 Definition of cutter geometry parameters
1 Here c tan (tan ax cos ) where ax is the profile angle of the worm in the axial section.
With the designer defining parameters such as the axial module ( max ), the axial tooth space ( wax ), the axial profile angle ( ax ), number of threads (n), cutter radius and pitch radius of the worm ( rc and rp , respectively) along with the parameters that are defined by using the above geometric relations to define the cutter geometry fully. In
Table 2.1, an example set of user-defined parameters as well as the other calculated parameters are listed to illustrate the process.
2.3 Definition of the Worm Geometry
2.3.1 Cutter Installment and Relative Motion between Cutter and Worm
In addition to the cutter (tool) coordinate system Sc that is rigidly connected to the cutter, define a coordinate system S1 that is rigidly connected to the worm (its origin at Oo in Figure 2.3 and rotates with worm). In addition, fixed (inertial) coordinate frame
So with origin is defined in Figure 2.3 to describe of the tool settings and the worm motion. The axes of the tool and the worm are form an c as shown in Figure 2.4. The angle between axes yo and yc is the lead angle at the worm pitch cylinder as shown in Figure 2.3. Here, axes xo and xc are collinear, Ec is the distance between origins and Oc illustrated in Figures 2.3 and 2.4.
15
Table 2.1 An example set of user-defined and calculated cutter parameters. Parameters
with an asterisk next to it are user-defined ones.
Parameter Numerical Value
Axial Tooth Space (mm)* 5.9
Axial Pressure Angle (deg)* 15.2
Axial Module (mm)* 2.598
Cutter Radius (mm)* 30
Number of Threads* 1
Pitch Radius of Worm (mm) 12.46
Cutter Width (mm) 2.046
Lead Angle (deg) 7.5
16
xx, zc co zc zo
y Oc c yc rc
pc Oo yo E tangent to c helix on the pitch Oo worm cylinder pitch yo
17 helix on r cylinder p rp the pitch cylinder pitch radius
(a) (b)
Figure 2.3 Installment of the cutter on the worm surface.
xxco, zc zc zzo, 1 1
p c
Oc
yc yc Ec Oo
yo
18
(a) (b)
Figure 2.4 (a) Coordinate systems applied for generation of K type worms and (b) definition of angle c
In the worm cutting process, the cutter with systems Sc and So are considered to be fixed while the worm performs a screw motion through the body axis coordinate frame
S1 of the worm. The worm is rotated by angle to correspond to a translation of the worm by p where p is the screw parameter of the worm as shown in Figure 2.5. The cutting process shown in Figure 2.5 is for a right hand worm with a positive p while p is negative for a left hand worm
c1 The relative velocity Vc between the cutter and the worm is given by the
c 1 difference between the velocity Vc of the cutter and the velocity Vc of the worm as
cc11 VVVc c c (2.7)
where subscript c indicates that the relative velocity is defined in . Since the cutter is
c stationary during the cutter process, Here Vc 0 and
1 Vrc11 ()O o O c c p . (2.8)
Considering
0 Ec 11 sin c , OOoc 0 , cos c 0
19
x zz, 1 o 1
xo 1
O1
20 p Oo y1 yo
Figure 2.5 Relative motion between the cutter and the worm.
where 1 is the angular velocity of the worm and using rc expression from Eq. (2.1),
Eq. (2.8) is written as
zycsin c c cos c 1 Vc 1 cos c (x c E c ) p sin c . (2.9) sin c (x c E c ) p cos c
Thus, from Eq. (2.7), the relative velocity during the cutting process is given as
zycsin c c cos c c1 Vc 1 cos c (x c E c ) p sin c . (2.10) sinc (x c E c ) p cos c
2.3.2 Equation of Meshing
The derivation of the equation of meshing between the cutter and the worm is based on the theory of gearing [6, 7]. Equation of meshing is the mathematical
c1 representation of the fact that, at a given surface point, the relative velocity vector Vc and the surface normal vector nc (common to both surfaces) are perpendicular to each other:
c1 nVcc0 (2.11)
c1 Here, subscripts c again indicates that both vectors nc and Vc are represented in the same coordinate system Sc .
21
Since there are two cutting sides on the cutter, two separate equations of meshing are both sides as the position vector normal vectors differ for both sides. The equation of meshing between the side 1 cutting surface and the worm surface for a right hand worm is obtained as
(uc a sin c )cos c ( E c sin c cot c p sin c )sin c (2.12a) (Epc cot c )cos c 0.
The equation of meshing between the side 2 cutting surface and the worm surface for a right hand worm has the form
uc asin c cos c E c sin c cot c p sin c sin c (2.12b) Epc cot c cos c 0.
It is noted that both of these expressions are only a function of the curvilinear surface parameters uc and c . Defining
A ucc asin ,
B Ecsin c cot c p sin c
C( Ec p cot c )cos c
Eq. (2.12a) is written as
ABCcoscc sin 0 . (2.13)
The solution to this quadratic problem is
22
BC B2 C 2 ( A 2 B 2 )( C 2 A 2 ) sin c . (2.14a) AB22
Here uc 0 . For any value of uc , Eq. (2.14) provides two solutions for c . The desired solution for c is near 180 (often slightly larger than ). By defining
C E pcot cos A ucc asin , B ( Ec sin c cot c p sin c ) and c c c , the same equations (2.13) and 2.14) can be used to solve Eq. (2.12b) for side 2 as well. The solution to this quadratic problem is
BC B2 C 2 ( A 2 B 2 )( C 2 A 2 ) sin c . (2.14b) AB22
2.3.3 Worm Surface Equations
With surface coordinates uc and c for every cutter contact point computed in the previous section, the worm surface generated by the cutter is represented as the family of lines of contact between the cutter and the worm. From Figure 2.5, position of a worm tooth surface point in S1 is defined as
r11(,,)(,)uuc c M o M oc r c c c (2.15)
23
The coordinate transformation from system Sc to S1 is expressed in terms of the product of coordinate transformation matrices Moc (from to So ) and M1o (from So to S1 ), defined respectively as
1 0 0 Ec 0 coscc sin 0 Moc , (2.16a) 0 sincc cos 0 0 0 0 1
cos sin 0 0 sin cos 0 0 M . (2.16b) 1o 0 0 1 p 0 0 0 1
With this, Eq. (2.15) transforms position vector in to its equivalent in such that
xx1 c yy11 MMo oc c . (2.17) zz1 c
With this, the tooth surface coordinates and the normal of any point of a right hand worm surface generated side 1 of the conical cutter are written explicitly as
24
xu1 ccos c cos c cos cos c cos c sin c sin sin c sin c sin aEsin cc sin cos yu1 c( cos c cos c sin cos c cos c sin c cos sin c sin c cos ) (2.18a) aEsin cc cos sin z1 uc(sin c cos c cos c sin c sin c ) p a cos c
nxcos sin c cos c sin (cos c sin c sin c sin c cos c ) 1 n sin sin cos cos (cos sin sin sin cos ) (2.18b) y1 c c c c c c c n sin sin sin cos cos z1 c c c c c
Similarly, the tooth surface coordinates and the normal of any point of a right hand worm surface generated side 2 of the conical cutter are
xu1 c(cos c cos c cos cos c cos c sin c sin sin c sin c sin ) aEsin cc sin cos yu1 c( cos c cos c sin cos c cos c sin c cos sin c sin c cos ) aEsin cc cos sin z1 uc( sin c cos c cos c sin c sin c ) p a cos c
(2.19a)
nxcos sin c cos c sin (cos c sin c sin c sin c cos c ) 1 n sin sin cos cos (cos sin sin sin cos ) (2.19b) y1 c c c c c c c n sin sin sin cos cos z1 c c c c c 25
Similar equations can also be derived for a left hand worm considering a negative screw parameter p.
2.3.4 Geometric Model of the Worm
Once the tooth surface coordinates of any point on the worm surface are generated due to the cutter that has been obtained, a geometric model of the worm can be put forth. Following expression can be obtained from Eq. (2.18a) by squaring and adding the values of x1 and y1.
2 2 2 2 2 2 2 2 x11 y xcc ycos ccc 2 x E 2cos ccccc sin y z z sin cc E . (2.20)
At the outside and root radii of the worm, respectively, Eq. (2.20) reduces to
2 2 2 x11 y fo(,) u c c r o , (2.21a)
2 2 2 x11 y fr(,) u c c r r . (2.21b)
Here, ro and rr are the outer and root radii of the worm.
Combining Eq. (2.14a) with Eq. (2.21a), following pair of equations is obtained:
f(,) u r2 o c c o 2 2 2 2 2 2 (2.22) BC B C ( A B )( C A ) sin c AB22
26
Similarly combining Eq. (2.21b) with Eq. (2.14b) yields
f(,) u r2 r c c r 2 2 2 2 2 2 (2.23) BC B C ( A B )( C A ) sin c AB22
The solutions to Eq. (2.22) and Eq. (2.23) define the ranges of for both uc and c as umin uc u max and min c max .
Cross-section of the worm on xy11 plane can be defined by numerically assigning coordinate z1 the value of zero (i.e. z1 0) in Eq. (2.18a). It results in the following explicit relation between parameter and surface parameters uc and c
1 ua(sin cos cos sin sin ) cos . (2.24) p c c c c c c c
Since umin uc u max varies from the addendum to the dedendum of the worm tooth, uc values according to the selected step size of ()/umax u min m determines the discrete points on the cross section of the worm, where m is the number of points along the worm tooth surface cross-section. Hence, solving equations (2.14) and (2.24) for and c , and then substituting them back into Eq. (2.18a) results in x1 and y1 values for these discrete points on the addendum to dedendum.
27
With the coordinates along the cross-section are known, the worm surface in three-dimensions can be generated through screw motion of the points on cross section.
This corresponds to the following coordinate transformation:
(1) x cos sin 0 0 x 1 1 (1) y sin cos 0 0 y1 1 . (2.25) (1) 0 0 1 pz 1 z1 0 0 0 1 1 1
Here (,,)x y z are the coordinates of a surface point on at along the cross-section ( 1 1 1
(1) (1) (1) 0) and (,,)x1 y 1 z 1 are the coordinates corresponding the same reference point on a helix after a rotation along the worm axis through an angle of (06 ).
Superscript 1 indicates that the worm is a single-thread one.
For generating a double threaded worm, the following index transformation can be used to generate the second thread,
(2) x 1 0 0 0 x 1 1 (2) y 0 1 0 0 y1 1 (2.26) (2) 0 0 1 0 z1 z1 0 0 0 1 1 1
(2) (2) (2) Here, (,,)x1 y 1 z 1 represents the points corresponding to each on a helix of the second thread. Similarly coordinate transformations need to be done for equations
28
modeled for side 2 of the cutting edge of the cutter to obtain the complete geometry of the worm surface.
2.4 Geometric Model of the Worm Gear
The worm gear is usually cut by a cutter whose geometry is identical to the worm defined in the previous section. The equation of meshing between the worm and worm gear surfaces is given as [6, 7]
z1cos 1 E cot sin 1 nxy 1 z 1 sin 1 E cot cos 1 n 1 (2.27) pm(121 cos ) (x1 cos 1 y 1 sin 1 E ) nz 1 0. m21 sin
T Here x1,, y 1 z 1 is the position vector of a point on the surface of a worm tooth and
T nx1, n y 1, n z 1 is the corresponding normal vector. 1 is the worm rotation angle, E is the shortest distance between the rotational axes of the worm and worm gear surfaces
m when mating with each other, 21 is the gear ratio and is the crossing shaft angle (i.e. the angle between axes of the worm and worm gear).
An alternative equation of meshing can be arrived at for worm gear drives by replacing the cutter rotation motion about its axis by an equivalent translation along its
axis. The advantage of this equation is that meshing parameter 1 can be expressed as an explicit function of the two surface parameters u and . Thus the generated tooth can be clearly described by the independent surface parameters.
29
If the worm has a constant lead, the rotation of the worm about its axis can be 1 kinematically replaced by a translation tp 1 in the direction of its rotational axis
[16]. The equation of meshing with translating meshing motion can be derived using the general equation of meshing of gears
12 nvii.0 (2.28) where i represents a particular coordinate system. The coordinate systems taken into consideration here are S and S that are attached to the worm and worm gear, 1 2 respectively (i.e. rotates with it), and S and S that are fixed worm and worm gear f 1 f 2 coordinate systems. Here S and coincide when the worm has zero translation. The 1 position and unit surface normal vectors of any arbitrary point on the worm are given as
x1 nx1 X11 y , n11 ny . (2.29a,b) z 1 nz1
The same point and its normal at can be expressed in as 1
x n f 1 x1 xf 1 nx1 X ff1yy 1 1 , n f1nn yf 1 y 1 . (2.30a,b) zp z f 1 11 nnzf11 z
30
The relative velocity at the meshing point in S is given by f 1
v12 v 1 v 2 (2.31) f1 f 1 f 1 where v1 is the velocity of the meshing point of the worm surface and v2 is the f 1 f 1 velocity of the meshing point of the gear surface. Here
T v1 = 0,0,p 1 (2.32a) f 1 1 where p 1 is the translational velocity of S and 1 is the angular velocity of the 1 1 1
2 rotating worm. Similarly, v f 1 is defined by the determinant
if1 j f 1 k f 1 2 1 1 v f 10mm 21 1 sin 21 1 cos (2.32b)
x1 E y 1 z 1 p 1
T 11 where 0mm21 1 sin 21 1 cos 0 is the angular velocity vector of the gear in
T S and x E y z p represents a vector from the gear center O to the f 1 1 1 1 1 2 meshing point X1. With these, relative velocity vector is found to be
31
(z1 p 1 )sin y 1 cos 12 1 v f 1 m 21 1( x 1 E )cos . (2.33) p (xE1 )sin m21
From Eq. (2.28), Eq. (2.30b) and Eq. (2.33), an alternative equation of meshing is derived as
(z1 p 1 )sin y 1 cos nxy 1 ( x 1 E )cos n 1
p (2.34) (x11 E )sin nz 0. m21
Here, 1 can be expressed explicitly by the two worm surface parameters uc and c as
p 1(uc , c ) p ( x 1 E )cot n y 1 ( x 1 E ) n z 1 n x 1 m21 sin (2.35) 2 p z1 y 1cot ( nx 1 ) .
Hence, through a series of transformations, any point X1 can be expressed in S2 as
x21( x E )cos( m 2111 ) ( y cos ( z 11 p )sin )sin( m 211 ),
y21( x E )sin( m 2111 ) ( y cos ( z 11 p )sin )cos( m 211 ), (2.36)
z2 y 1sin ( z 1 p 1 )cos .
T Here X2 x 2,, y 2 z 2 . Above equations map any point X1 on the worm thread surface to the corresponding point X2 on the generated worm gear tooth surface, 32
completing the process of defining the worm and worm gear surfaces. In the next
Chapter, these surface coordinates will be utilized to form FE models of the worm and worn gear to perform deformable-body contact analyses for determining the deflections, stresses and mechanical efficiency of candidate worm gear pairs.
33
CHAPTER 3
COMPUTATIONAL MODEL AND PARAMETRIC STUDY
3.1 Introduction
The previous chapter dealt with formulation of equations required to generate the worm and worm gear surfaces at any given cross angle . The finite element modeling of the worm gear pair is performed using Hypoid Face Milling program (Hypoid K) of
Advanced Numerical Solutions, Inc. Since the study involves contact of two bodies, the contact zones are typically two orders of magnitude smaller than the working depths of the gear teeth and require a large degree of freedom concentrated within for the contact equations to be well conditioned. Also, the contact zone moves along the surface of the gear tooth. In order to accurately capture the contact between the gear teeth, one would need to use a highly refined conventional finite element mesh over the entire contact zone and remesh the localized contact elements for each position of the mesh cycle [18].
Hypoid K program resolves this issue by using the contact analysis solver CALYX which utilizes a hybrid algorithm of finite elements to predict far field displacements and an elastic half space model to predict relative displacements local to the contact region [19]. 34
CALYX does not require having a highly refined finite element mesh and can more efficiently locate which points are actually in contact.
The approach CALYX uses to solve the contact problem assumes that (i) the finite element solution predicts deflection well for points far away from the contact region, and (ii) the contact region is sufficiently smaller than the dimensions of the gears themselves and employs a semi-analytical technique. It does this effectively by overlaying a contact grid separate from the finite element mesh. After a search routine of tooth separation is employed, this grid is only placed on the points that have the potential to be in contact. The finite element solution and the semi-analytical solution are then matched at a subsurface [19].
This chapter first discusses the details of deformable body finite element analysis of a worm gear pair using Hypoid K program. Then it presents the results if a design parameter sensitivity study to quantify the effect of geometric parameters and manufacturing variability on the contact stresses and the mechanical efficiency of the worm gear pair.
3.2 Deformable Body Finite Element Analysis of Worm Gear Pair
3.2.1 Mesh Generation
The first step in development of the deformable-body model is the FE mesh generation of the worm and worm gear surfaces. Mesh generator code developed by
Advanced Numerical Solutions, Inc. is used to generate the FE mesh. Specific input 35
modules are available for spur and helical gears, but for worm gears, there is a need for the development of an intermediate input routine using C++ compiler to define the worm and worm gear surfaces [20].
The main parameters required for the definition of the gear in CALYX are the cones that define the gear surfaces (Figure 3.1). The cones are represented by a line in the RZ plane in the form of ArR AzZ B 0 . The inputs for the intermediate routine of the mesh generator program are as follows
(i) Geometric parameters of the worm gear pair from Table 3.1.
(ii) The equation of meshing between the cutter and the worm as represented by
Eq. (2.12a) and (2.12b).
(iii) The tooth surface coordinates and normal of any point on the generated
worm surface as expressed by Eq. (2.18a,b) and Eq. (2.19a,b).
(iv) The solution to the surface parameters uc and c from Eq. (2.22) and (2.23).
(v) Explicit relation for the worm rotation angle from Eq. (2.23) and
coordinate transformation equations Eq. (2.25) and (2.26) to generate the
final worm surface.
(vi) Worm gear surface generation equations defined by Eq. (2.34) (2.36).
These equations can be used to generate FE mesh for worm gear drives with right hand worms, similar FE mesh can be generated for worm gear drives with left hand worms by incorporating the appropriate input parameters and surface equations in the routine. 36
Figure 3.1 Definition of gear surfaces in CALYX [20]
37
3.2.2 Pre-processing in Hypoid K Program
Here, Hypoid Face Milling Program (Hypoid K) is customized to analyze a worm gear pair. The input parameters are the worm gear pair design parameters (Table 3.1), basic material properties (Table 3.2), friction coefficient, input torque, number of time steps and the FE mesh files of the gear surfaces generated in the mesh generation stage
[21]. A shaft cross angle range of 90 25 is considered in this design study. Hence,
values between 65 and 115 are considered depending upon the hand of the worm.
For a 90 (right-angle) worm gear pair, the worm can be left-handed or right-handed as shown Figure 3.2(a). For a right-handed worm gear pair, cross angles that are greater than are possible while 90 is preferred for left-handed drives as illustrated in
Figures 3.2(b) and 3.2(c).
Hypoid-K program meshes the worm and the worm gear at certain center distance
E and . With these input parameters, a 3D FE model can be developed as seen in
Figure 3.3. A cross-sectional views of the worm and worm gear in Figure 3.4 depicts the tooth shapes clearly.
3.2.3 Post Processing in Hypoid K program
In the post processing stage, CALYX solver searches for contact zones and plots the contact forces and load distributions on the worm and worm gear surfaces. Contact stresses are expressed as cumulative maximum contact stress patterns on the worm and
38
Table 3.1 Input design parameters to define a worm drive
Design Parameters Symbol Unit
Number of Threads on Worm n -
Axial Module max mm
Outer Diameter of the Worm do mm
Root Diameter of the Worm dr mm
Lead Angle degrees
Pressure Angle c degrees
m Gear Ratio 21 -
d Outer Diameter of the Worm Gear wgo mm
Root Diameter of the Worm Gear dwgr mm
mm Throat Diameter of the Worm dwgt mm Face Width of the Worm Gear b
Cross Angle degrees
Center Distance E mm
and Addendum and Dedendum Coefficients a d -
39
Table 3.2 Design constraints imposed in the parametric study
Parameter Value
Gear ratio ( m21) 31:1
Worm gear pitch diameter ( dwgp ) [mm] 90
Input torque (T ) [Nm] 3 in 175 Tensile stress of worm gear ( t ) [MPa]
40
RH and LH, = 90 LH, = 65 - 80
(a) (b)
41
RH, = 100 - 115
(c)
Figure 3.2 (a) Right angle 90 and (b, c) non-right angle design configurations with (b) 90 and (c) 90 .
42
Figure 3.3 FE mesh model of a worm gear pair with m21=31, n =1, =110 , = 7.5 and c =15 .
43
Figure 3.4 FE meshes and cross-sectional views of (a) a worm and (b) its worm gear mate.
worm gear surfaces. Figure 3.5(a) represents a contact stress pattern on the worm gear tooth. The contact pattern on the mating worm surface is as represented in Figure 3.5(b).
The corresponding load distribution patterns on the worm gear tooth surfaces are shown in Figure 3.6. They represent the load intensities along the gear teeth at five different mesh steps. This explains qualitatively how the contact moves along the worm gear teeth.
Here it is noted that a left-hand worm drive with 90 is equivalent to right-hand worm drive with 90 in terms of contact stresses and load distributions. In Figure 3.7, maximum contact stress contours and load distribution patterns for a left-hand, 70 design and a right-hand, 110 counterpart are shown to be exact mirror images of each other. As such, this study is performed for only right- hand worm drives with 90 .
3.3 Design Parameter Sensitivity Study
With the FE model of the worm gear pair established, a series of analyses are performed here to quantify the effect of the geometric parameters and certain manufacturing variability on the two primary objective functions: (i) maximum contact stress max , and (ii) mechanical efficiency .
In this study, the user-defined gear ratio is specified as 31:1. As such, the number of teeth of the worn gear-worm pair must be either 31T-1T for single-thread designs ( 44
(a)
45
(b)
Figure 3.5 Contact Patterns on (a) the worm gear and (b) the worm for a pair for parameters m21 31, n=1, =100 ,
= 7.5 and c =15
Mesh step 1 of 11 Mesh step 3 of 11
Mesh step 5 of 11 Mesh step 7 of 11
46
Mesh step 9 of 11
Figure 3.6 Instantaneous load intensities on the worm gear of a pair for parameters m21=31, n =1, =110 , =7.5 and c =15
at 5 different mesh steps.
(a)
47
(b)
Figure 3.7 Contact pattern and load distribution on worm gear for parameters m21=23, n =2, =19.5 and c =15 ; (a) = 70
(b) = 110 .
n 1) or 62T-2T for double-thread designs ( n 2 ). Worm gear pitch diameter is kept at dwgp 90 mm. In addition, the tensile stress limit of the worm gear material ( t 175
MPa) is imposed as a maximum contact stress limit. An input (pinion) torque of Tin 3
Nm is used in all analyses. This design limits are listed in Table 3.2.
A plastic worm gear is meshed with a steel worm in this study. Basic material parameters for this combination are listed in Table 3.3. These material properties are extracted from Wilson [22] who performed basic durability and friction coefficient tests of a number of plastic-grease combinations. The friction coefficient listed in Table 3.3 (
0.015) was also reported by Wilson based to twin-disk traction experiments.
The mechanical efficiency of the worm gear pair is defined here as
TT out out out (3.1) inT in m12 T in
where m12 in out . Hypoid K computes the torque loss caused by the friction force on each contact grid cell with a user defined .
The geometric parameters taken into consideration are (i) cross angle , (ii) lead angle
, (iii) pressure angle c , and iv) addendum and dedendum coefficients. Rest of this section, present the results of this parametric study. Influence of these parameters on the primary objective functions in both single-thread and a double-thread worm is quantified.
48
Table 3.3 Material and grease parameters
Material and Grease Worm Worm Gear
Parameters
Young’s Modulus [GPa] 200 11.6
Poisson Ratio ( ) 0.3 0.38
Density ( ) [kg/m3] 7800 1600
Friction Coefficient ( ) 0.015
49
3.3.1 Effect of Cross Angle and Lead Angle on and max
Table 3.4 specified the ranges of values considered for the combined cross angle
and lead angle study. With four discrete values considered for each parameter, a total of 16 different candidate single-thread ( n 1) designs are defined and analyzed as shown in Figure 3.8. Pressure angle and addendum and dedendum coefficients are kept constant at c 15 , a =1/1 and d =1.2/1.2.
Figures 3.9(a) and 3.10(a) present the variation of max and with , respectively, for each of the discrete values. Same data is presented in Figures 3.9(b) and 3.10(b) as a function of for each discrete value. It is seen from Figure 3.9(a) that max increases with increased With an increase in , the worm outside diameter decreases as a function of the tan, causing both and values to increase due to decrease in relative sliding between the worm and worm gear surfaces. A simplified relation between and is given by Shigley [23] for a right-angle worm drive as
cos tan c . (3.2) cosc cot
The values predicted by Hypoid K in a more detailed manner (Figure 3.10) are indeed quite close to values obtained from this equation. Therefore, it can be stated that for a given c and an increase in results in increased when 45 . Hence the trend
50
Table 3.4 Geometric parameters used for the vs. study
Parameter Range of Values
Cross Angle ( ) [deg] 100, 105, 110, 115
Lead Angle ( ) [deg] 5, 7.5, 10, 12.5
Pressure Angle ( c ) [deg] 15
Addendum Coefficients ( a ) 1/1
Dedendum Coefficients ( d ) 1.2/1.2
51
100
Cross 105 Angle ( ) deg. 110
115
5 7.5 10 12.5 52
Lead Angle ( ), deg.
Figure 3.8 Design configurations considered to study of effect of and on max and with n =1.
255 (a) 235
215
195 λ-5 175 λ-7.5 [MPa] 155 λ-10 135 λ-12.5 115
95
75 100 105 110 115 [degrees]
255 (b) 235
215
195 γ-100 175 γ-105 [MPa] 155 γ-110
135 γ-115
115
95
75 5 7.5 10 12.5 [degrees]
Figure 3.9 (a) Effect of on max at different values, and (b) effect of on at
different values for c = 15 and n = 1.
53
94 (a)
92
λ-5 90 [%] λ-7.5
λ-10 88 λ-12.5
86
84 100 105 110 115 [degrees]
94 (b)
92
γ-100 90 γ-105
γ-110 [%] 88 γ-115
86
84 5 7.5 10 12.5 [degrees]
Figure 3.10 (a) Effect of on at different values, and (b) effect of on at
different values for c = 15 and n = 1.
54
observed in the results is in compliance with the governing equation. It is also seen from
Figures 3.9 and 10 that increases in the cross angle result in increases in max and
Due to an increase in the relative sliding between the worm and worm gear surfaces decreases, leading to an increase in max . The same reason can be attributed to the increase in as well, since the worm diameter does not vary with the increase is marginal.
The same study is repeated next for double-thread equivalents ( n 2 ). The 16 different candidate designs analyzed are as shown in Figure 3.11. Figures 3.12 and 3.13 show trends similar to Figures 3.8 and 3.9 with max and increasing with increasing
and . It is, however, noted that the value at a given pair of and values is significantly higher for in comparison to that of n 1. In order to better quantify this difference, the n 1 and results are compared in Figure 3.14 within the entire range of with 110 . In Figure 3.14(a), it is observed that the values for
are nearly 25% to 37% higher than those for . This significant increase in
m can be attributed to reduced tooth thickness since the axial module ax of the gear reduces to half as the pitch diameter of the worm gear is kept constant as shown in
Figures 3.15. Meanwhile, there is only a modest increase in values in Figure 3.14(b), which shows about 1% higher for .
Due to increased values in the double-thread designs, design choices are limited since one must stay below established stress limits for the plastic material (
max t ). From Figure 3.13, it is desirable to have a large to maximize . Yet,
55
Figure 3.12 shows that should be very low (say 7.5 ) for max t (t 175
MPa in Table 3.2). This limits the efficiency to values below 86%, which is not acceptable. Hence, this study indicates that worm gear pairs with n 1 provide a broader wider ranges of for selection of candidate worm gear designs for this high gear ratio application with limited gear diameter. For this reason, only single-thread designs will be considered in the following sections.
Since the influence of is more pronounced on both max and values when compared to the effect of , an optimal trade-off is required to arrive upon suitable values for both and . Cross angle values of 105 and 110 along with lead angle values of 7.5 , 9 and 10.5 are chosen in this regard for further analysis.
56
100
Cross 105 Angle ( ) 110
deg.
57
115 5 7.5 10 12.5
Lead Angle ( ), deg.
Figure 3.11 Design configurations considered to study of effect of and on max and with n =2.
350 (a)
300
λ-5 250 λ-7.5 [MPa] λ-10 200 λ-12.5
150
100 100 105 110 115 [degrees]
350 (b)
300
250 γ-100 γ-105 [MPa] γ-110 200 γ-115
150
100 5 7.5 10 12.5 [degrees] Figure 3.12 (a) Effect of on max at different values, and (b) effect of on at
different values for c = 15 and n = 2.
58
96 (a)
94
92 λ-5 λ-7.5 [%] 90 λ-10
88 λ-12.5
86
84 100 105 110 115 [degrees]
96 (b)
94
92 γ-100 γ-105
[%] 90 γ-110 γ-115 88
86
84 5 7.5 10 12.5 [degrees]
Figure 3.13 (a) Effect of on at different values, and (b) effect of on at
different values for c = 15 and n = 2.
59
300 (a)
250 Single Thread
200 Doubl e [MPa] Thread
150
100 5 7.5 10 12.5 [degrees] 96 (b)
94
92 Single [%] Thread 90 Double Thread 88
86
84 5 7.5 10 12.5
[degrees]
Figure 3.14 Comparison between single thread (n=1) and double thread designs (n=2) (a)
Effect of on max , and (b) effect of on for =110 .
60
(a)
61
(b)
Figure 3.15 Tooth size comparison of (a) single thread and (b) double thread worm gear design options.
3.3.2 Effect of Pressure Angle on and c max
Table 3.5 specifies the ranges of the cross angle , lead angle and the pressure angle considered in this parametric study. A total of 9 different candidate designs are obtained and analyzed to study the effect of c in conjunction with and on both max and . Figure 3.16 illustrates the design options for 105 .
Figures 3.17(a) for and 3.18(a) for 110 shows the variation of with at three discrete values of The same data is presented in Figures 3.17(b) and
3.18(b) for as as the variable at discrete values. It can be observed from Figures
3.17 and 3.18 that decreases by approximately 10% with increase in c from
c 10 to c 20 a given and value. Increase in c strengthens the worm gear teeth thereby reducing the stress levels.
The corresponding effects on are shown in Figures 3.19 (a) and (b) for and , respectively, indicates that changes in do not affect much. For a
0.015 very low value ( in Table 3.3), Eq. (3.2) can be used to confirm that has a secondary effect on regardless of . Based on these results, parameter values of
105 and 110 , 9 and 10.5 and c 20 were identified for further analyses as the suitable values to obtain high values and values within allowable stress levels (i.e. max t ). Within these ranges, a candidate design #1 having ,
62
Table 3.5 Geometric parameters used for the c study
Parameter Range of Values
Cross Angle ( ) [deg] 105, 110
Lead Angle ( ) [deg] 7.5, 9, 10.5
Pressure Angle ( c ) [deg] 10, 15, 20
Addendum Coefficients ( a ) 1/1
Dedendum Coefficients ( d ) 1.2/1.2
63
10
Pressure Angle 15 64 ( c )
deg. 20 7.5 9 10.5
Lead Angle ( ), deg.
Figure 3.16 Design configurations considered to study the effect of c and on max and at =105
200 (a) 190 180 170 160 λ-7.5 [MPa] 150 λ-9 140 λ-10.5 130 120 110 100 10 15 20
[degrees]
200 (b) 190 180 170 160 PA-10
[MPa] 150 PA-15 140 PA-20 130 120 110 100 7.5 9 10.5
[degrees]
Figure 3.17 (a) Effect of c on max at different values, and (b) effect of on at different values for =105 .
65
220 (a)
200
180 λ-7.5 [MPa] 160 λ-9 λ-10.5 140
120
100 10 15 20 [degrees]
220 (b)
200
180 PA-10 [MPa] 160 PA-15 PA-20 140
120
100 7.5 9 10.5
[degrees]
Figure 3.18 (a) Effect of c on max at different values, and (b) effect of on at different values for = 110
66
91.5 (a)
91
90.5 λ-7.5 [%] 90 λ-9 89.5 λ-10.5
89
88.5
88 10 15 20
[degrees]
91.5 (b)
91
90.5
90 λ-7.5 [%] λ-9 89.5 λ-10.5
89
88.5
88 10 15 20
[degrees]
Figure 3.19 (a) Effect of c on at different values for =105 , and (b) effect of on at different values for =110 . 67
10.5 and c 20 results in max 165 MPa and 90.95%. Another candidate design #2 with 110 , 9 and results in max 168 MPa and
90.55%.
3.3.3 Effect of Addendum a and Dedendum d Coefficients on max and
In order to study the effect of addendum and dedendum coefficients, a and d , on max and , a baseline design with , and selected from the ranges specified in Table 3.6. A total of six and combinations are selected for analysis as listed in Table 3.7. They are denoted by letters A to F. These combinations cover different values of and for both the worm and worm gear. It can be
observed from Figures 3.20(a) and (b) that combinations B to E yield the same and
values while combinations A and F show marginal increases in and values.
This suggests values within the range of 1 to 1.2 can be chosen for and d based primarily on the stress considerations.
3.4 Manufacturing Variability Study
Two typical manufacturing variations are introduced: (i) shaft misalignments
(cross angle error) and (ii) center distance error. This is done by defining the gear geometries and FE meshes under nominal conditions and introducing variations to E and
during the contact analysis stage in the form of EE and . Table 3.8 lists the
E and values used here on a baseline design having , , and
68
a and d values defined by combination B in Table 3.7. It can be observed from
Figures 3.21 and 3.22 that both max and are quite insensitive to and E within the ranges considered. A very slight increase in is for positive and negative
while the values were shown to remain unchanged with these ranges of errors
69
Table 3.6 Geometric parameter ranges suitable for addendum and dedendum coefficient study
Parameter Range of values
Cross Angle ( ) [deg] 105 and 110
Lead Angle ( ) [deg] 9 and 10.5
20 Pressure Angle ( c ) [deg]
70
Table 3.7 Addendum and dedendum coefficients for worm and worm gear
Worm Worm Gear Combination a d a d
A 0.6 0.8 0.6 0.8
B 1 1.2 1 1.2
C 1.2 1 1.2 1
D 1 1.1 1 1.1
E 1.1 1 1.1 1
F 1.6 1.4 1.6 1.4
71
180 (a)
175
170
165 [MPa]
160
155
150 A B C D E F and
91 (b) 90.9 90.8 90.7 90.6 [%] 90.5 90.4 90.3 90.2 90.1 90 A B C D E F and
Figure 3.20 Effect of a and d on (a) max and (b) , for =110 , =9 and c = 20 .
72
Table 3.8 Parameters considered in the manufacturing variability study
Parameter Range of Values
Center Distance Error ( E ) [mm] 0.3, 0.15, -0.15, -0.3
Shaft Misalignment ( ) [deg] 0.2, 0.1, -0.1, -0.2
73
180 (a)
175
170
165
[MPa] 160
155
150 -0.2 -0.1 0 0.1 0.2 [degrees] 91 (b) 90.9 90.8
90.7
90.6
90.5 [%] 90.4
90.3
90.2
90.1
90 -0.2 -0.1 0 0.1 0.2 [degrees]
Figure 3.21 (a) Effect on max , and (b) effect of on for =110 , =9 and
c = 20 .
74
180 (a)
175
170
165 [MPa]
160
155
150 -0.3 -0.15 0 0.15 0.3 [mm] 91 (b) 90.9 90.8 90.7 90.6 90.5 [%] 90.4 90.3 90.2 90.1 90 -0.3 -0.15 0 0.15 0.3 [mm]
Figure 3.22 (a) Effect of E on max , and (b) E on for =110 , =9 and c = 20 .
75
CHAPTER 4
CONCLUSIONS
4.1 Summary
In this study, first, a formulation regarding definition of the geometry of worm and worm gear surfaces inclined at non-right cross angles were developed. The geometry of the worm was defined by defining the geometry of the cutter first and solving the corresponding equation of meshing between the worm and the cutter. The geometry of the worm gear was defined next by using a cutter which has the exact shape of the worm and solving the corresponding equation of meshing between the worm and the worm gear. Both right- and left hand drives with any number of worm threads were included in this formulation.
With the tooth surface geometries defined by above formulation, a deformable- body finite element analysis of a non-right angle worm gear pair was developed by using a commercial gear contact analysis package Hypoid K. The three-dimensional finite
76
element mesh of the worm and worm gear surfaces were generated through a mesh generator routine. Hypoid K was customized to analyze non-right angle worm gear pairs
A parametric design sensitivity study was performed by using this model the effect of geometric parameters (cross angle ( ), lead angle ( ), pressure angle ( c ) and addendum and dedendum coefficients) and manufacturing variability parameters (shaft misalignments and center distance variations) on the maximum contact stress max and mechanical efficiency of the worm gear pair, with friction parameters obtained from measurements of a companion study.
4.2 Conclusions
Based on the results presented in Chapter 3, several conclusions can be made on the effect of the geometric parameters ( , , and addendum and dedendum coefficients, a and d ) and the manufacturing variability parameters (shaft misalignments and center distance variations) on the maximum contact stress and mechanical efficiency of the worm gear drive system:
Both lead angle and cross angle impact both max and with having a
bigger effect. Values of and should be chosen suitably to obtain the
candidate designs with max below the allowable stress levels along with high
values.
77
Increases in both and increase max values, with they are more significant
with increases in . Meanwhile, higher ( 90 ) and lower values result in
reduced values. This indicates that the and values should be determined
with both and considered.
Maximum contact stresses were found to be as much as 25% higher in double-
thread worm gear drive designs when compared to single-thread equivalents
having the same worm gear diameter and gear ratio. As such, single-thread
designs might be more suitable for applications requiring high gear ratios within
small spaces. However, the single-thread designs were found to have values
that are about 1% lower than those equivalent double-thread designs.
Increased pressure angle c results in a significant decrease of max while it has
a negligible effect on . Thus, a higher value of is desired for reduced stress
levels.
Addendum and dedendum coefficients, a and d , within the range 1 and 1.2
are observed to be best for both and .
Shaft misalignments and center distance variations within the reasonable ranges
were shown to have secondary effects on and .
78
4.3 Recommendations for Future Work
The following topics can be listed as potential future work in an attempt to improve upon the work that was conducted:
In this study, only the geometry of ZK-type, single enveloping cylindrical worm
gears was considered. This study can be extended to other type of cylindrical
worms like the ZA, ZI, ZN and Flender worms. Geometry of double enveloping
worm gears with hourglass type worms can also be considered in this regard.
The worm wheel designed in this study was a single throated worm gear drive.
The two other types of worm gear drives are (i) non-throated worm gears in which
neither the worm or the worm gear has a throat or a groove and (ii) double-
throated worm gears in which both the worm and worm gear are throated.
Geometry of such type of worm gear drives can also be explored.
Different hob geometry profiles can be defined for the cutter of the worm wheel.
Design parameters such as hob swivel angle, hob profile modifications etc. can
also be studied to understand their impact on the cutting process and resultant
geometries.
Modifications on the worm and worm gear surfaces could be incorporated in the
mesh generator routine (e.g. crowning of gears). The effect of these
modifications on the stresses would be a useful addition to the parametric design
study.
79
The parametric design sensitivity study can be extended to other material-grease
combinations. Metal-metal, plastic-metal and many other interactions between
the materials along with lubricant combinations can be studied.
Other objective functions such as root stresses and transmission error can be
added on to the current parametric design study.
Experimental verification of contact patterns on worm and worm gear surfaces
and efficiency studies in test rigs would help in improving the current
computational model.
80
References
[1] “Worm Gear” Retrieved from http://science.howstuffworks.com/transport/engines-
equipment/gear5.htm
[2] Maitra, G.M., 1994, “Handbook of Gear Design”, Tata Mc-Graw-Hill Education
Private Ltd.
[3] Thomas, H.G., 1913, “Worm Gearing”, McGraw-Hill Book Company, Inc.
[4] Buckingham, E., 1949, “Analytical Mechanics of Gears”, McGraw-Hill Book
Company, Inc.
[5] Buckingham, E. and Ryffel, H., 1960, “Design of Worm and Spiral Gears”,
Industrial Press.
[6] Litvin, F. L., 1997, “Development of Gear Technology and Theory of Gearing”,
NASA Reference Publication 1406.
[7] Litvin, F. L. and Fuentes, A., 2004, “Gear Geometry and Applied Theory”, 2nd
ed., Cambridge University Press, Cambridge, England.
[8] Fang, H.S. and Tsay, C.B, 1996, “Mathematical Model and Bearing Contacts
of the ZK-Type Worm Gear Set by Oversize Hob Cutters,” Mech. Mach.
Theory, 31, pp. 271-282.
[9] Octrue, M., 1988, “A New Method for Designing Worm Gear”, American Gear
Manufacturers Association, Technical Paper 88 FTM 6.
81
[10] Vijayakar, S.M., 1987, “Finite Element Methods for Quasi-Prismatic Bodies with
Application to Gears”, Ph.D. Dissertation, The Ohio State University, Columbus,
Ohio.
[11] Vijayakar, S.M., Busby, H., Houser, D.R., 1987, “Finite Element Analysis of
Three-Dimensional Conformal Contact with Friction”, Computers and Structures,
33, pp. 49-62.
[12] Gopikrishnan, M., “Development of Finite Element Mesh Based on
Manufacturing Setup for Worm Gear Contact Analysis”, MS Thesis, The Ohio
State University, Columbus, Ohio.
[13] Simon, V., 1996, “Stress Analysis in Worm Gears with Ground Concave Worm
Profile”, Mech. Mach. Theory, 31, pp. 1121-1130.
[14] Liu, C.C., Chen, H.J., Tsay, B.C., Ariga, Y., 2006, “Meshing Simulations of the
Worm Gear Cut by a Straight-edged Flyblade and the ZK-type Worm with a Non-
90 Crossing Angle”, Mech. Mach. Theory, 41, pp. 987-1002.
[15] Litvin, F. L., Fuentes, A., and Yukishima, K., 2007, “Geometry and Investigation
of Klinberg-Type Worm Gear Drive”, Journal of Mechanical Design, 129,
pp.17-22.
[16] Houser, D.R. and Su, X., 2000, “Alternative Equation of Meshing for Worm-Gear
Drives and its Application to Determining Undercutting and Reverse
Engineering”, Journal of Mechanical Design,122, pp. 207-212
82
[17] Su, X., 1999, “Generation, Measurement and Reverse Engineering of three-
Dimensional Gears”, Ph.D. Dissertation, The Ohio State University, Columbus,
Ohio.
[18] Vijayakar, S., 1991, “A Combined Surface Integral and Finite Element Solution
for a Three-Dimensional Contact Problem,” International Journal for Numerical
Methods in Engineering, 31, pp. 525-545.
[19] Advanced Numerical Solutions, LLC, Helical 3D Tour, Retrieved from
http://ansol.us/Products/Helical3D/GalleryPage8/.
[20] “Terminology used by Mesh Generator”, 2008, Advanced Numerical Solutions
LLC.
[21] “Hypoid Face Milled User’s Manual”, 2005, Advanced Numerical Solutions LLC.
[22] Wilson, S., 2012, “Traction and Wear Evaluation Of a Number of Plastic Materials
and Greases under Combined Rolling and Sliding Contact Conditions,” MS
Thesis, The Ohio State University, Columbus, Ohio.
[23] Shigley, J., Mischke, C., Budynas, R.G., 2003, “Mechanical Engineering Design”,
McGraw-Hill Company.
83