CHAPTER 2 Mathematical Model of Spherical Helical Gears
2.4 Tooth generation of the concave spherical helical gear
2.4.2 Equation of meshing for concave spherical helical gears
Figure 2.7 shows a schematic relationship of coordinate systems Sw(Xw,Yw,Zw), concave spherical helical gears. Coordinate systems Sw and Sg are attached to the hob cutter and concave spherical helical gear, respectively. The rotation motion of the hob cutter is expressed by considering an auxiliary coordinate system Sc with a rotational angle . Coordinate system Sq is the fixed coordinate system attached to the machine housing. Moreover, symbol g,cave is the rotational angle of the generated gear (e.g.
concave spherical helical gear). Therefore, the homogenous coordinate transformation matrices Mcw, Mqc and Mgq can be expressed as follows:
Fig. 2.7 Coordinate systems between the hob cutter and concave spherical helical gear
(2.35), respectively.
In Fig. 2.7, point P is a common contact point of the hob cutter and work piece.
Therefore, the surface coordinates R( caveqg, ) of the work piece can be determined by transforming the hob’s surface from coordinate system Sw into the fixed coordinate system Sq as follows:
The velocity at point P of the work piece can be obtained by:
T spherical helical gear) expressed in the fixed coordinate system Sq. Differentiating Eq.
(2.37) with respect to time, the relationship among angular velocities g,cave, w and
s,cave can be obtained as follows:
cave
where symbols
dt
indicate the angular velocities of hob
cutter and hob cutter’s generating motion along the hobbing locus (see Fig. 2.6), respectively. Equation (2.44) indicates the rotation angle g ,cave of work piece
(concave spherical helical gear) in terms of angular velocities w and s,cave.
Similarly, the velocity at point P that attached to the hob cutter can be obtained as follows:
z
where V and x Vz express the linear velocities of axial and radial feeding motion, and ω(wq ) indicates the angular velocity of the hob cutter expressed in the fixed coordinate system Sq, as follows:
T
where symbol d denote the shortest vector measured from the center of the work piece to that of the hob cutter, and it can be expressed by
T
cutter can be determined as follows:
Differentiating Eqs. (2.34) and (2.35) with respect to time, the linear velocities of axial and radial feeding motions Vz and Vx can be determined by parameters of spherical angle s,cave, cutting radius Rc,cave and angular velocity s,cave as follows: determined and simplified as follows:
Therefore, the relative velocity Vq(wg) of the hob cutter and work piece at their common contact point P can be represented in the fixed coordinate system Sq as follows: The equation of meshing of hob cutter and convex spherical helical gear can be
obtained by
where the surface normal Nq can be obtained by using the following homogenous coordinate transformation matrix equation:
w
Substituting Eqs. (2.6), (2.55) and (2.53) into Eq. (2.54) yields:
0,
vector Nq, respectively. Rearranging Eq. (2.56) in terms of the independent of variables, w and s,cave, yields the following equation: Since w and s,cave are independent variables, two equations of meshing that relative
the hob cutter’s surface parameters and the cutting motion parameters can be obtained
0,
2.4.3 Mathematical model of the concave spherical helical gear
According to Fig. 2.7, the surface locus of the hob cutter, expressed in the generated concave spherical helical gear’s coordinate system Sg, can be obtained by applying the homogenous coordinate transformation matrix equation:
w
where the homogenous coordinate transformation matrices Mcw, Mqc and Mgq are expressed in Eqs. (2.39)-(2.41), respectively.
Therefore, the mathematical model of the generated gear is the combination of equation of meshing and the surface locus of hob cutter. Therefore, the mathematical model of the concave spherical helical gear can be obtained by considering Eqs.
(2.58)~(2.60), simultaneously.
2.5 Computer graphs of convex and concave spherical helical gears
The mathematical model of the convex spherical helical gear is expressed in Eqs.
(2.31)~(2.33), while the mathematical model for the concave spherical helical gear is represented in Eqs. (2.58)~(2.60). Table 2.1 summarizes some major design parameters of the hob cutter, convex pinion and concave gear. According to the developed mathematical models of the convex and concave spherical helical gears,
the tooth surfaces of the generated spherical helical gear can be plotted by using the developed computer programs. Therefore, a 3-D computer graph of the spherical helical gear set with convex pinion and concave gear can be plotted as shown in Fig.
2.8.
Table 2.1 Major design parameters of the hob cutter, convex spherical helical pinion and concave spherical helical gear
Hob cutter Convex pinion Concave gear
Normal module, mn(mm/tooth) 4 4 4
Number of teeth, Tg 1 33 47
Normal pressure angle, n(deg.) 20 20 20
Lead angle w, g(deg.) 3.823 RH 75 RH 75 LH
Face width, W (mm) - 20 20
Pitch radius, rg (mm) 30 68.328 97.316
Spherical radius, Rs (mm) - 68.328 97.316
Cutting radius, Rc (mm) - 98.328 67.316
Fig. 2.8 Computer graph of the spherical helical gear set with convex pinion and
2.6 Transverse pitch chord thicknesses of convex, concave and conventional helical gears
Since the convex and concave spherical helical gears are considered as hobbing a conventional helical gears with continuous positive and negative profile shiftings from both end sides of face width of the gears to their middle sections, respectively, the working pitch circles of the gears are different under every Z-axis cross-section of face width of the gears. Thus the pitch chord thicknesses of the convex and concave spherical helical gears are different at every Z-axis cross-section of face width of the gears. According to the gears’ design parameters of Table 2.2, Fig. 2.9 illustrates the transverse pitch chord thicknesses of the convex, concave and conventional helical gears under different Z-axis cross-sections of face width of the gears. It is found that the transverse pitch chord thickness of both ends of face width of the convex spherical helical gear is smaller than its central Z-axis cross-section of face width. Whereas, the inverse situation exists for that of the concave spherical helical gear. Moreover, the transverse pitch chord thicknesses of face width of the convex, concave and conventional helical gears under their central Z-axis cross-section are the same.
Table 2.2 Major design parameters of the hob cutter, convex, concave and conventional helical gears
Gear type Hob
cutter Convex Concave Conventional helical
Normal module, mn(mm/tooth) 4 4 4 4
Number of teeth, Tg 1 33 33 33
Normal pressure angle, n(deg.) 20 20 20 20
Lead angle w, g(deg.) 3.823 RH 75 RH 75 RH 75RH
Face width, W (mm) - 20 20 20
Pitch radius, rg (mm) 30 68.328 68.328 68.328
Spherical radius, Rs (mm) - 68.328 68.328 -
Cutting radius, Rc (mm) - 98.328 98.328 -
Fig. 2.9 Transverse pitch chord thicknesses of the convex, concave and conventional helical gears
2.7 Remarks
The mathematical models of spherical helical gears with convex and concave teeth have been developed on the basis of the CNC hobbing machine and the theory of gearing. The mathematical models can be derived as function of design parameters and motion parameters of a ZN-type hob cutter. Therefore, the design and motion parameters can provide us an efficient way to design and manufacture spherical helical gears. Moreover, the developed mathematical models of spherical helical gears with convex and concave teeth also help us to explore the possibility for further studies, such as sensitivity, kinematic errors, contact ratios and contact ellipses.
CHAPTER 3
Tooth Undercutting and Tooth Pointing Analyses
3.1. Introduction
Tooth undercutting is an important issue for gear design and manufacturing.
When tooth undercutting occurs, the tooth thickness near the gear fillets will be decreased as shown in Fig. 3.1. It is well known that gears with tooth undercutting may result in a lower load capacity of a mating gear pair. Mathematically, the phenomenon of tooth undercutting is the appearance of singular points on an active tooth surface. Therefore, the concept for checking of the tooth undercutting of the active tooth surface is to verify the appearance of singular points on the generated tooth surface. If the active tooth surface is a regular surface, it means that there is no tooth undercutting on the active tooth surface. Moreover, the tooth undercutting usually occurs near the tooth root.
Different from the location of tooth undercutting occurrence, the tooth pointing of a gear occurs near the tooth top as shown in Fig. 3.2. If the phenomenon of tooth pointing occurs, the tooth thickness of the gear on tooth topland becomes zero. When the contact location of a mating gear pair with tooth pointing locates near the tooth top, the load capacity of the mating gear pair is weak in the contact period. Therefore, the tooth pointing is also an important issue for gear design and manufacturing.
Since the spherical helical gear is hobbed by a hob cutter, the convex and concave spherical helical gears can be considered as hobbing a helical gear with its hobbing path of continuous positive-direction or negative-direction profile-shiftings in a quadric form, beginning from both sides of the tooth face width to its middle section, respectively. Therefore, the occurrence of tooth undercuttings on both ends of face width is easier than that at the middle section for a convex spherical helical gear,
Fig. 3.1 The phenomenon of tooth undercutting
Fig. 3.2 The phenomenon of tooth pointing
whereas the inverse situation exists for the concave spherical helical gear tooth surfaces. Moreover, the occurrence of tooth pointings on both ends of face width is easier than that at the middle section for a concave spherical helical gear.
Based on the developed mathematical model and theory of gearing, the tooth undercutting of the convex spherical helical gear and the tooth pointing of the concave spherical helical gear are investigated and demonstrated by seven numerical examples in this chapter. Moreover, the limit curves and the beginning points of tooth undercutting of the convex spherical helical gear and the occurrence of tooth pointing of the concave spherical helical gear at the Z cross-section are also studied.
3.2. Tooth undercutting of convex spherical helical gear
A method proposed by Litvin [3-5], which considers the relative velocity and equation of meshing between the generating tool and generated gear, is applied in this section to determine the limit curve of tooth undercutting of the convex and concave spherical helical gears. Singularities of the generated surface occur when the relative velocity V of the contact point over the generated surface equals zero. The motion r( g) of the hob cutter surface that generates the envelope surface is considered as the two-parameter motion of a rigid body. In the case of two-parameter enveloping, the condition for the appearance of a singular point on the generated tooth surface can be described as follows [3-5]:
0 V
R V
R
( ) ( ) w (wg, ) (wg, s)
w w b
b w
dt d dt
dl l
. (3.1)
where symbol R(w) represents the surface equation of the hob cutter, while symbols lb and w indicate the surface parameters of the hob cutter (see Eq. (2.3) of section 2.2). Superscripts w and g of Eq. (3.1) denote the generating tool and generated gear.
Symbol V(wg,) considers that the rotational motion parameter of the hob cutter
is a varied parameter and the moving motion parameter of spherical angle s is fixed.
Consequently, symbol V(wg,s) has to be interpreted.
Differentiating Eqs. (2.29) and (2.30), two equations of meshing for the hob cutter and gear tooth surfaces, with respect to time yield that:
) 0
Equations (3.1)-(3.3) represent a system of five equations in four unknowns:
dt
. The system of equations exists and provides a nontrivial solution if and only if the rank of the coefficient matrix for these five equations is three. Therefore, five determinants of order four for the coefficient matrix are equal to zero simultaneously. It can be proven that two of five determinants are equal to zero simultaneously, and the additional requirement is
0
and 0
To avoid the occurrence of tooth undercutting of the generated gear tooth surfaces, the generating hob cutter surface must be limited with the curve SL. Considering Eqs. (3.4)-(3.6) and two equations of meshing, simultaneously, one can solve the limited curve SL on the hob cutter surface that generates the singular points on the generated tooth surfaces. The limited curve SL on the hob cutter surface can be determined by applying the following expressions:
0
Equations (3.7)-(3.9) form a system of three equations with four unknowns, l , b
w, and s, one of these unknowns may be considered as an input variable, then solving three independent equations with three unknowns. Moreover, the differentiated equations of meshing, Eqs. (3.2) and (3.3), for the convex spherical helical gear can be rewritten respectively by
0
where the subscript q denotes the fixed coordinate system Sq (see section 2.3).
In order to derive the differentiated equations of meshing (3.10) and (3.11), let’s consider a coordinate relationship between the hob cutter and the generated gear as shown in Fig. 3.3. Axis Z represents the rotational axis of the generated gear. The g motion of the hob cutter can be represented by two parameters, rotational angle and spherical angle s. Axis Z is the rotational axis of the hob cutter, and symbol c
O is the initial position of the hob cutter center. The point P is a common point to c
both rotating bodies. Moreover, R(w) is the position vector drawn from point O to c point P, while R( g) represents a position vector drawn from an arbitrary point on the axis Z , e.g. g O , to point P. Symbol d is the relative-position vector drawn from g
point O to point g O . The locations of original points c O and c O are specified g by the position vectors ρ and (w) ρ( g), which are measured from the fixed coordinate system S . q
According to Fig. 3.3, the velocity of point P attached to the body i (iw,g) can be obtained by:
) ( ) , ) ( , ( )
( ) ( )
( ) ,
(i j i ( i i ) O j i j i
i
dt dj j
dt dj
j ρ R V ω R
V P
, (3.12)
where symbol V( ji, ) indicates the velocity of point P attached to the body i (iw,g) when parameter j (or s) is varied and another parameter s(or ) is fixed. V(Oi,j) is the velocity of point Oi (iw,g) when parameter j (or s)
is varied and parameter s (or ) is fixed. Similarly, ω( ji, ) depicts the angular velocity of body i when parameter j (or s) is varied and parameter s(or ) is fixed. Therefore, the relative velocity of point P between the hob cutter (i=w) and
Fig. 3.3 Simulation of a generation mechanism with two-parameter motion
,
The differentiation of Eq. (3.13) gives:
.
Differentiating the relative-position vector d and R(w) with respect to time yields:
) The absolute velocity of contact point P can be represented as:
).
Substituting Eqs. (3.15), (3.16) and (3.17) into Eq. (3.14), the differentiated relative velocity of point P can be represented in the coordinate system Sq as follows:
.
Similarly, the differentiation of normal vector at point P can be obtained as follows:
)
Substituting Eqs. (3.18) and (3.19) into Eqs. (3.10) and (3.11), the differentiated equations of meshing of the spherical helical gear can be rewritten as follows:
,
It is noted that the schematic mechanism of a CNC hobbing machine for the spherical helical gear generation can be referred to Fig. 2.5. Coordinate system
)
system attached to the machine housing. Symbols and g are rotational angles of the hob cutter and gear blank, respectively. s indicates the spherical angle. By compared Fig. 2.5 with Fig. 3.3, the schematic gear generation mechanism, the following position vectors can be found as:
T
s
Differentiate Eq. (3.25) with respect to time by considering that dt constants. It yields:
)
Similarly, the differentiated form of Eq. (3.26) with respect to time can be represented by
According to the hobbing mechanism of the spherical helical gear, mentioned in section 2.4, the angular velocity of the hob cutter ω(w) and the generated gear ω( g) can be represented as follows:
dt
Similarly, by considering
as constants, the differentiation of angular velocity of the hob cutter ω(wq ) and generated gear ω( gq ) can be obtained by:
According to the hobbing mechanism of the spherical helical gear, the relative velocity Vq(wg,) and Vq(wg,s) can be obtained by
3.3. Tooth pointing of concave spherical helical gear
Tooth pointing of a gear means that the tooth thickness of the tooth topland becomes zero. In other words, the tooth pointing can also be considered as the left-side and right-side tooth profiles of a gear intersect as a point at its tooth topland in a cross-section of face width. Since the concave spherical helical gear can be
considered as hobbing a helical gear with its hobbing path of continuous negative-direction profile-shiftings in a quadric form, beginning from both sides of the tooth face width to its middle section. Therefore, the occurrence of tooth pointing on both ends of face width is easier than that at the middle section for a concave spherical helical gear. Figure 3.4 illustrates the tooth pointing occurs on the tooth topland of a concave spherical helical gear. Symbol dt of Fig. 3.4 denotes the tooth thickness of the tooth topland at any Zg cross-section of face width of the concave spherical helical gear. According to the concept of tooth pointing, the condition equations of tooth pointing at the Zg cross-section of face width of the concave spherical helical gear can be considered as follows:
right
where, subscripts “left” and “right” of Eqs. (3.35)-(3.40) denote the left-side and right-side tooth profiles of the concave spherical helical gear, respectively, while symbol rtl indicates the radius of the tooth top circle. Symbols Xg and Yg denote the X and Y components of position vector Rg, respectively. Equation (3.35) explains
Fig. 3.4 Tooth pointing of concave spherical helical gear
gear intersected as a point at the Zg cross-section of the gear fact width. Equation (3.36) denotes the crossing point formed by the left-side and right-side tooth profiles of the concave spherical helical gear locates on the tooth topland. Moreover, Eqs.(3.37) and (3.38) are the equations of meshing of left-side tooth profile of the concave spherical helical gear, while Eqs.(3.39) and (3.40) are the equations of meshing of right-side tooth profile of the gear. Since Eq. (3.35) includes three independent nonlinear equations, Eqs. (3.35)-(3.40) yields a system of eight independent equations with eight variables lb,left, w,left, left, s,left, lb,right, w,right,
right
and s,right.
3.4. Numerical examples
According to the developed tooth undercutting condition equations of the convex spherical helical gear (Eqs. (3.7)-(3.9)), the tooth undercutting analysis of the proposed convex spherical helical gear is investigated. Moreover, the tooth pointing analysis of the concave spherical helical gear is also discussed based on the development tooth pointing condition equations (Eqs. (3.35)-(3.40)) of the gear. All analysis results of the tooth undercutting and tooth pointing are illustrated by the following numerical examples. Furthermore, some major design parameters of the proposed convex and concave spherical helical gears for the numerical examples are given in Table 3.1.
Example 3.1: Tooth profiles of the convex spherical helical gear with tooth undercutting under different Zg cross-sections.
This example investigates the tooth profiles of the convex spherical helical gear with tooth undercutting under different Z cross-sections. Where the Z cross-section
Table 3.1 Major design parameters of the hob cutter, convex spherical helical gear Hob cutter Convex tooth Concave tooth
Normal module, mn(mm/tooth) 4 4 4
Number of teeth, T 1 22 22
Normal pressure angle, n(deg.) 20 20 20
Lead angle (deg.) 3.823 RH 75 RH 75 RH
Face width, W (mm) - 20 20
Pitch radius, rj (mm) 30 45.552 45.552
Spherical radius, Rs (mm) - 45.552 45.552
Cutting radius, Rc (mm) - 75.552 15.552
helical gear. The analysis results are obtained based on the convex spherical helical gear data given in Table 3.1.
Figure 3.5 illustrates the tooth profiles of the convex spherical helical gear with tooth undercutting under different Zg cross-sections of the gear. It can be found that the convex spherical helical gear have tooth undercuttings on the left-side tooth profiles of Zg=10mm and Zg=5mm of the face width and on the right-side tooth profiles of Zg= 10mm and Zg= 5mm of the face width. Therefore, the tooth undercutting curves on the left- and right-side tooth surfaces of the convex spherical gear are not symmetric. Table 3.2 lists the coordinate positions of the left- and right-side tooth profiles of the convex spherical helical gear at Zg=0mm cross-section of face width of the gear, while each coordinate position of the tooth profiles of the gear can be generated by the corresponding hob cutter’s parameter lb under its working interval lb,min lb lb,max. According to Table 3.2, the left- and right-side tooth profiles of the convex spherical helical gear are geometrical symmetry at central cross-section (Zg=0mm) of face width of the gear.
Example 3.2: Limit curves of tooth undercutting of convex spherical helical gears
Example 3.2: Limit curves of tooth undercutting of convex spherical helical gears