Thesis Overviews - 以盤狀圓弧刀具創成之曲線齒的數學模式推導與接觸分析研究

CHAPTER 1 Introduction

1.4 Thesis Overviews

In this thesis, a gear with curvilinear teeth generated by a disk-type circular-arc cutter is

briefly proposed. In all, this thesis includes seven chapters, and the contents of each chapter are described as follows:

1. CHAPTER 1 is the introduction of the background, motivations, literature reviews and a brief overview of this thesis.

2. CHAPTER 2 introduces the curvilinear-tooth gear and the circular-arc tooth gear.

Advantages and disadvantages of these two gears are briefly discussed.

3. The mathematical model of the circular-arc curvilinear-tooth gear set has been developed in CHAPTER 3. The circular-arc profile on its curvilinear-tooth gear is generated by a disk-type rack cutter with circular-arc normal section geometry.

However, the generation method of the circular-arc curvilinear-tooth gear is simulated by a circular-arc profile rack cutter. Equations of the circular-arc curvilinear-tooth gear can be obtained by applying the theory of gearing and the equations of the cutter. Computer graph illustrates the shape of the gear pair, and it also proves the correctness of the developed mathematical model. In addition, the equations of the gear set are the bases for further investigations, such as tooth contact analysis, contact ratio and contact ellipses.

4. CHAPTER 4 includes the study of tooth undercutting of the proposed gear pair. The undercutting points are the singular points on the generated tooth surface.

Consequently, it can be derived according the theory of gearing. Some tables and graphs are built up by applying the developed computer programs to give suggestions for the manufactures and designers to prevent the tooth undercutting.

5. The TCA technique is applied to find the KE of the circular-arc curvilinear-tooth gear in CHAPTER 5. Tooth contact simulation model includes the center distance

variation, horizontal axial misalignment and vertical axial misalignment of the gear set. The KE curve are tried to pre-design into a parabolic form in order to prevent the jump of the junction. Several numerical examples are presented to demonstrate the influences of gear design parameters and assembly errors on the KE and contact analysis of the mating gear pair.

6. Contact pattern is an important symbol of gear loading capability. The contact pattern can be simulated based on the developed mathematical model of the gear set.

Some numerical examples express the relationships between the design parameters and contact patterns. In this thesis, the contact patterns are evaluated by applying the surface topology method.

7. CHAPTER 7 concludes this thesis by summarizing the major findings of the accomplished work, and also discusses some potential topics for future study.

CHAPTER 2 The Curvilinear-Tooth Gear and the Circular-Arc Tooth Gear

2.1 Introduction

The shape of a gear can be described into two ways: one mentioned the profile of the cross section of gear. Fig. 2-1(a) is the involute-shape gear profile of a spur gear shown in Fig.

2-1(b). As everyone knows that most applications of the gear profiles are in the involute shape.

Fig. 2-1 Schema of gear profile and gear tooth trace of a spur gear

The other description for a gear shape is the trace of its tooth, e.g. gears with trace parallel to the rotational axis as marked with shadow zones in Fig. 2-1(b) is called a spur gear.

Different profiles and traces of gears have their own characteristics, the usage of gear is according to there characteristics. In this chapter, two kinds of gear types will be introduced.

One is the curvilinear-tooth trace gear and another one is the circular-arc tooth profile gear.

2.2 The Curvilinear-Tooth Gear

Curvilinear-tooth gear is one kind of cylindrical gears as shown in Fig. 2-2, and has circular-arc tooth traces. Since the circular arc tooth trace is symmetrical, thus gear axial thrust force may be eliminated. Besides, the curvilinear-tooth gear has the characteristics as follows [2]:

1. Curvilinear-tooth gears endure a higher bending and contacting strength; thus, the size of transmission gear boxes may be reduced while transmitting the power or torque.

2. The teeth of curvilinear-tooth gear pair engaged simultaneously smoothly with low noise.

3. Lubrication oil is retained within the concave tooth surfaces, therefore there is always an oil film between the two engaged surfaces, resulting in good lubrication qualities.

The spur, helical, and herringbone gear pairs with parallel axes are in line contact.

However, the line contact of gear tooth surfaces can be realized only for an ideal contact of the gear drive. In reality, the gear assembly errors of axial misalignments and errors of lead angles result in the so-called edge contact, a specific instantaneous contact caused by curve-to-surface tangency as shown in Fig. 2-3. Therefore, the stress is concentrated at the contact edge. Gear manufacturers always cut gears with tooth crowning to make the line contact become a point contact, and to avoid the edge contact of gears as shown in Fig. 2-4.

has an axial misalignment without crowning, tooth edge contact will occur and this will induce a serious stress concentration. In this circumstance, the noise and vibration of the gear pair may occur during the gear meshing. The curvilinear-tooth gear also has a characteristic similar to a crowned tooth, as shown in Fig. 2-5. The contact type of the curvilinear-tooth gear is in a point contact situation no matter there is an axial misalignment or not.

Fig. 2-2 Schema of curvilinear-tooth gear [3]

2.3 The Circular-Arc Tooth Gear

The theory of circular-arc tooth gears is first proposed by Wildhaber [7] in U.S. patent, 1926. Later, Novikov [8], also proposed another patent for helical gears with circular-arc teeth.

Therefore, the circular-arc tooth gear is also called the Wildhaber-Novikov (W-N) gear. The profiles of the Wildhaber’s and Novikov’s are quite similar to each other. It is noted that the

Top land Contact stress

Contact region Front end

Line contact

Top land Contact stress

Front end

(b) Helical gear pair with axial misalignment Edge contact

(a) Helical gear pair without axial misalignment

x y

Rear end

Fillet

Rear end Contact region

Fig. 2-3 Contact region of a helical gear pair

Top land Contact stress

Front end

(a) Helical gear with crowning teeth pair without axial misalignment

Top land Contact stress

(b) Helical gear with crowning teeth pair with axial misalignment Contact region Front end

Contact point

x y

Fillet

Rear end Fillet

Rear end

Contact point

Contact region

Fig. 2-4 Contact region of a crowned teeth gear pair

Top land Contact stress

Contact region

(a) Curvilinear gear pair without axial misalignment

(b) Curvilinear gear pair with axial misalignment Contact point

Front end

x y

Fillet

Rear end

Top land Contact stress

Contact region Front end

Fillet

Rear end Contact point

Fig. 2-5 Contact region of a curvilinear-tooth gear pair

circular-arc curves as shown in Fig. 2-6, where the R^{( )}^p and R^{( )}^g are the radii of the two cutters, where O_R^{( )}^p and O_R^{( )}^g are the center of these circular-arc. In order to generate a pair of conjugate gear surfaces, the rack cutters can be considers as two separate rack cutters which may be imagine as a mold and its corresponding cast express in Fig. 2-7. One of these rack cutters generates the pinion and the other generates gear.

Rack cutter Σ^p O^R Rack cutter Σ^g

O^R

(p) (g)

Fig. 2-6 Schema of the normal section of rack cutters

Rack cutter centrode Cutter Σ

Cutter Σ

Fig. 2-7 Imaginary of the separating rack cutters

Since the normal section of each rack cutter has circular-arc edges, thus the tool needs a higher precision requirement. However, the technology is brilliant enough to produce high precision circular-arc cutter now.

It is known that a circular-arc gear has the following two major advantages:

1. Reducing the contacting stress. Theoretically, the gear surfaces are contact when two gears are meshed with each other. However, the contact area spread over a contact ellipse due to elastic deformation under loading. After operating for a period of time, the two meshing surfaces fit each other into a suitable shape for meshing. It is said that the curvatures on the contact point of these two gears tend to equal gradually.

The contact surfaces enlarge quickly with the operation loading. Hence, the loading limit is three to five times higher than an involute gear pair [18].

2. Better conditions of lubrication. The sliding velocity of every point on the gear surface are equal in a slight value, it means the wear problem of the circular-arc tooth gear is not serious. Rotation speed of the circular-arc tooth gear is high on the vertical direction of the gear surface, so the oil- membrane can be formed easily.

Thus the lubrication of the circular-arc gear pair is ten times better than an involute gear pair [19]. With the better lubrication, the life time of this type of gears becomes longer and abrasion becomes less serious.

The circular-arc gear can provide power transmission with a high loading, and it is widely used in the transmission system of hoists, elevators, mechanisms for mining industry, and many transmission systems with a high load/weight ratio. Although the circular-arc gear tooth surfaces are in point contact at every instant, the contact area is also larger than that of involute gear pair under load with elastic deformation.

Nevertheless, there are also three major disadvantages of the circular-arc gear as below:

1. The profile of the rack cutter is in a circular-arc shape, and it is manufactured by several processes requiring high precision, and the heat treatment process may somewhat cause distortion of the cutter shape. Therefore, the menufacturing cost of this cutter is higher.

2. There are no international standards for the circular-arc gears, e.g.. the radius of circular-arc edge. Hence, the circular-arc gear cannot be exchanged facilely. It means the comparability with the circular-arc gear is not good.

3. The kinematical error occurs if there has error of center distance. However, the variation of center distance takes no effects for involute gear pair.

2.4 Remarks

Gears are used in power transmissions. Therefore, good gears always include two essential factors: high loading capacity and low sensitivity to assembly errors. The goal can be achieved by designing the profile and tooth-trace for the gear pair. Two types of gears are introduced in this chapter, one is the curvilinear-tooth gear which has a curvilinear tooth-trace, and the other one is the circular-arc tooth gear which is generated by two conjugate rack cutters. Both of these two gears benefit to developing the performance of the gear pair.

CHAPTER 3 Mathematical Model of Gears with Curvilinear-Teeth Generated by a Disk-Type Circular-Arc Cutter

3.1 Introduction

In this chapter, the mathematical model of curvilinear-teeth generated by the disk-type circular-arc cutter will first be developed. Based on the theory of gear [10][20], the surface equation of circular-arc gears with curvilinear-teeth is derived as the envelope of the locus of circular-arc cutter. Hence, an imaginary rack cutter surface with circular-arc normal section will be first established to simulate the generation of the gear tooth.

3.2 Mathematical Model of Circular-Arc Gears with Curvilinear-Teeth 3.2.1 Generation Method

The curvilinear-tooth gear is generated by the cutting machine as shown in Fig. 3-1.

Gear teeth are produced by a rotating disk-type cutter. The spindle of the disk-type cutter with radius R rotates on the axis B_ab − with an angular velocity B ω_t and translating velocity

1 1r

ω to the right, where r is the pitch radius of the gear blank and ₁ ω₁ is the angular velocity of it. The cutting process of a curvilinear-tooth gear was developed by Liu [2] as steps below:

1. The gear blank rotates with an angular velocity ω₁ in clockwise. At this moment, the disk-type cutter rotates with an angular velocity ω_t in counterclockwise and translates with a velocity r_{1 1}ω to right. A curvilinear-tooth space may be generated

ω

r¹

r

ω

R^ab

W

x

y z

y

Disk-type cutter

Circular-arc

shape cutter Work piece

Fig. 3-1 The cutting process of a curvilinear-tooth gear with a disk-type tool [2]

2. After generating the tooth space of the gear, the gear blank stop rotating and then spin to the next working position with considering adjustment for the backlash δ_bl.

3. In order to cut the other tooth space, the disk-type cutter rotates with an angular velocity ω_t in clockwise and translates with a velocity r_{1 1}ω to left, the gear blank rotates in the counterclockwise with angular velocity ω₁ in the mean time (contrary to the Fig. 3-1), then the other tooth space is generated by this process. In order to prevent the interfering, the radius of the disk-type cutter used to generate the different side of the tooth surface should be modified. In this thesis, the difference between the different side of the gear tooth is defined as S . ^G

4. Generating cycle is repeated and this sequence is continued until all the spaces and teeth are formed.

Obviously, the structure of cutting machine to produce the curvilinear-tooth gear is complex, and the assembling of the cutter is so inconvenient. Adjusting and grinding is also arduous for the curvilinear-tooth gear because of its special trace. Therefore, the manufacturing cost of curvilinear-tooth gears is higher than that of spur gears or helical gears.

In this thesis, the cutting machine is adopted and the straight line cutter shape is replaced by the circular-arc cutter shape. Gears made by this process are called the curvilinear-tooth gears generated by a disk-type circular-arc cutter.

Although the gear is generated by a disk-type cutter, however it can be considered that the gear is generated by an imaginary rack cutter as shown in Fig. 3-2. The surfaces of the two cutters are labeled as ∑ and _g ∑ , and we may imagine that the two surfaces are rigidly _p connected to each other and are in tangency along the curve a−a as shown in Fig. 3-3. The normal section of each rack cutter is a circular arc. Fig. 3-4 shows the relationships among the

pitch plan of rack cutter and the axode of the two gears. The locus of an instantaneous axis of rotation represented in a coordinate system that is attached to a movable body is known as the body axode.

ω

r

ω

Fig. 3-2 The relationship between imaginary rack cutter and blank

a

Σ

a

Σ

Π

Fig. 3-3 Surfaces of the imaginary cutters ∑_g and ∑_p

ω

Pitch plane of the rack cutter ω

I V I

r r

1 2

O

1 1

O

Pinion axode

Gear axode

Fig. 3-4 The axodes of the gears

The two gears are rotating with an angular velocity ω₁ and ω₂ in opposite directions about their respective rotation axes. The radii of the two axodes are ₁

1 tangent line I I− of the two axodes is called the instantaneous axis of rotation. The tangent plane to the axodes is also the pitch plane of the rack cutters. The gear is generated by the cutter ∑ and the pinion is generated by the cutter _g ∑ . Namely, the mathematical model of _p

the gear is generated by the surface ∑ , and another one is generated by _g ∑ . Finally, the _p mathematical model of the generated gear tooth surface is the combination of the meshing equation and the locus equation of imaginary rack cutter surface. Points on gear surface can be calculated by solving the developed gear mathematical model by using numerical methods.

3.2.2 Equation of the Disk-Type Circular-Arc Cutter

The teeth surfaces of a pair of conjugate circular-arc with curvilinear-teeth gear can be generated by two imaginary circular-arc rack cutters with curvilinear trace. These two rack cutters of ∑ and _g ∑ are shown in Fig. 3-5(a). The normal sections of the cutters are also _p shown in Fig. 3-5(b). Parameters A and B determine the initial and end points of the circular-arc curve, respectively, as shown in Fig. 3-6. O_R^{( )}^g is the center of the circular-arc

MN with a radius of R^{( )}^g ; S is the tooth thickness measured along the pitch line of the ^G rack cutter; θ^{( )}^g is the design parameter of the rack cutter which determines the point on the circular arc MN . The normal section of the circular-arc rack cutter is rigidly attached to coordinate system S_r^{( )}^g (X_r^{( )}^g ,Y_r^{( )}^g ,Z_r^{( )}^g ) with its origin O_r^{( )}^g , as shown in Fig. 3-6. The

( )

X

Fig. 3-6 Normal section of the circular-arc rack cutter ∑ _g

where θ^{( )}^g is a design parameter of the circular-arc rack cutter, ranging from θ_min^{( )}^g to θ_max^{( )}^g , and α denotes the normal pressure angle defined in Fig. 3-6, and R^{( )}^g represents the radius of the circular-arc MN . The symbol “± ” represents the different side of the cutter ∑ , _g where “−“ indicates the right side circular-arc rack cutter ∑ and “_gR +“ indicates ∑ . _gL

To form a circular-arc rack cutter with curvilinear-trace, the normal section of the circular-arc (Fig. 3-6) should attach to coordinate system ^S^r^{( )}^g

(

^X^r^{( )}^g ^,^Y^r^{( )}^g ^,^Z^r^{( )}^g

)

, as shown in Fig. 3-7. It is noted that the circular-arc ab is the cutting path of the disk-type rack cutter.

The cutting path of the cutter consequently causes a crowning effect on the generated tooth flank. Coordinate system ^S^c^{( )}^g

(

^X^c^{( )}^g ^,^Y^c^{( )}^g ^,^Z^c^{( )}^g

)

is rigidly attached to the middle of transverse section of the imaginary rack cutter. Coordinate system ^S^r^{( )}^g

(

^X^r^{( )}^g ^,^Y^r^{( )}^g ^,^Z^r^{( )}^g

)

is attached to

coordinate system ^S^c^{( )}^g

(

^X^c^{( )}^g ^,^Y^c^{( )}^g ^,^Z^c^{( )}^g

)

with a variable angle γ^{( )}^g . The center of the curvilinear-trace is located at point C^{( )}^g with a radius of R and _ab W represents the width of the gear pair.

The imaginary rack cutter surface ∑ represented in coordinate system _g

( )

( )^g ( )^g , ( )^g , ( )^g

c c c c

S X Y Z can be obtained by applying the following homogeneous coordinate transformation matrix equation as follows:

( )g ( )g

Matrix M is a homogeneous coordinate transformation matrix transforming from _cr coordinate system S_r^{( )}^g (X_r^{( )}^g ,Y_r^{( )}^g ,Z_r^{( )}^g ) to ^S^c^{( )}^g

(

^X^c^{( )}^g ^,^Y^c^{( )}^g ^,^Z^c^{( )}^g

)

. Using the Eq.(3.1), Eq.(3.3), and Eq.(3.4) the mathematical model of the disk-type circular-arc cutter represented in coordinate system ^S^c^{( )}^g

(

^X^c^{( )}^g ^,^Y^c^{( )}^g ^,^Z^c^{( )}^g

)

can be obtained as following:

Substituting Eq.(3.1) into Eq.(3.5) yields the imaginary rack cutter surface represented

The normal cross section of rack cutter Σ is shown in Fig. 3-8. Similarly, the _p

mathematical model of the rack cutter surface Σ can be established by following the _p above-mentioned steps:

X r

Fig. 3-8 Normal section of the circular-arc rack cutter ∑ _p

( )

Eqs.(3.7), (3.9) and (3.10) result in the unit normal vector of the generating surface represented in the coordinate system ^S^c^{( )}^g

(

^X^c^{( )}^g ^,^Y^c^{( )}^g ^,^Z^c^{( )}^g

)

as follow:

where the upper sign of symbol “± ” indicates the left-side of the rack cutter surfaces.

Similarly, the unit normal vector of the rack cutter surface ∑ can be also obtained by _p following the similar process:

( ) ( )

The relative velocity of the gear with respect to the rack cutter ∑ can be obtained by _g considering the gear generation mechanism as shown in Fig. 3-9. The rack cutter translates to the left tangent to the axode of the gear. Therefore, the velocity of the point on the rack cutter represented in coordinate system ^S^f

(

^X^f^,^{Y Z}^f^, ^f

)

^is: during its generation.

Fig. 3-9 Kinematical relationship between the imaginary rack cutter Σ and the gear_g

The velocity of the point on the rack cutter ∑ can be expressed in the coordinate _g

The relative velocity of the gear with respect to the left and right side of the cutter ∑ _g can be attained by subtracting Eq.(3.13) and Eq.(3.14):

(

1 1 ^{( )}

)

Similarly, the relative velocity of the pinion with respect to the left and right sides of the rack cutter ∑ can also be attained by applying the same steps according to the kinematical _p relationships shown in Fig. 3-10:

(

^{2 2} ^{( )}

)

( ) ( )

Fig. 3-11 illustrates the geometrical relationship between the contact surfaces and tangent point. The contact points on gear surfaces ∑ , ₁ ∑ are ₂ M and ₁ M , respectively. ₂ T is the tangent plane of these two surfaces. Relative velocity V⁽¹²⁾ is defined in physical terms as the velocity of point M of ₁ ∑ as seen by an observer at point ₁ M of ₂ ∑ . ₂ Theoretically, in the generation process, gear and cutter are in pure rolling or sliding on the contact surfaces. It means that these two contact surfaces never embed into each other. Thus, the relative velocity of the gear with respect to the cutter along their common normal direction is equal to zero. Then, it can be said that the relative velocity V⁽¹²⁾ lies on the common tangent surface and perpendicular to the unit normal vector n at the common tangent point.

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