Gear Geometry and Applied Theory Episode 2 Part 6 ppsx

Gear tooth surfacesare in line contact for involute helical gear drives with parallel axes and in point contactfor involute helical gear drives with crossed axes.. The conditions of mesh

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CB672-15 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:44

The derivation of line L is based on the following considerations:

(i) Equation (15.8.1) yields

Here,∂rc/∂uc,∂rc /∂θc, and v(c c σ)are three-dimensional vectors represented in

sys-tem S cof the pinion rack-cutter

(ii) Equation (15.8.2) yields

(iii) Equations (15.8.3) and (15.8.4) represent a system of four linear equations in two

unknowns: du c/dt and dθc/dt This system has a certain solution for the unknowns

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NOTE. In most cases, it is sufﬁcient for derivation of function F = 0 to use instead

of (15.8.10) only one of the three following equations:

An exceptional case, when application of (15.8.10) is required, is discussed inSection 6.3

Singularities of the pinion may be avoided by limitation by line L of the rack-cutter

surfacec that generates the pinion The determination of L [Fig 15.8.1(a)] is based

on the following procedure:

(1) Using equation of meshing f (u c, θc , ψσ)= 0, we may obtain in the plane of

pa-rameters (u c, θc) the family of contact lines of the rack-cutter and the pinion Eachcontact line is determined for a ﬁxed parameter of motionψ σ

(2) The sought-for limiting line L is determined in the space of parameters (u c, θc) by

simultaneous consideration of equations f = 0 and F = 0 [Fig 15.8.1(a)] Then,

we can obtain the limiting line L on the surface of the rack-cutter [Fig 15.8.1(b)] The limiting line L on the rack-cutter surface is formed by regular points of the rack- cutter, but these points will generate singular points on the pinion tooth surface Limitations of the rack-cutter surface by L enable us to avoid singular points on the

pinion tooth surface Singular points on the pinion tooth surface can be obtained by

coordinate transformation of line L on rack-cutter surface c to surfaceσ

Pointing

Pointing of the pinion means that the width of the topland becomes equal to zero.Figure 15.8.2(a) shows the cross sections of the pinion and the pinion rack-cutter Point

Ac of the rack-cutter generates the limiting point A σ of the pinion when singularity of

the pinion is still avoided Point B c of the rack-cutter generates point B σ of the pinion

proﬁle Parameter s a indicates the chosen width of the pinion topland Parameter αt

indicates the pressure angle at point Q Parameters h and h indicate the limitation of

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Figure 15.8.1: Contact lines L c σ and limiting line L: (a) in plane (u c,θ c); (b) on surface c.

location of limiting points A c and B cof the rack-cutter proﬁles Figure 15.8.2(b) shows

functions h1(N1) and h2(N1) (N1is the pinion tooth number) obtained for the followingdata:αd= 25◦,β = 30◦, parabola coefﬁcient of pinion rack-cutter a c = 0.002 mm−1,

sa = 0.3 m, parameter s12= 1.0 [see Eq (15.2.3)], and module m = 1 mm.

15.9 STRESS ANALYSIS

This section covers stress analysis and investigation of formation of bearing contact

of contacting surfaces The performed stress analysis is based on the ﬁnite elementmethod [Zienkiewicz & Taylor, 2000] and application of a general computer program[Hibbit, Karlsson & Sirensen, Inc., 1998] An enhanced approach for application ofﬁnite element analysis is presented in Section 9.5

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-Figure 15.8.2: Permissible dimensions h1and h2 of cutter: (a) cross sections of pinion and

rack-cutter; (b) functions h1(N1) and h2(N1 ).

Using the developed approach for stress analysis, the following advantages can beobtained:

• Finite element models of the gear drive can be automatically obtained for any position

of pinion and gear obtained from TCA Stress convergence is assured because there

is at least one point of contact between the contacting surfaces

• Assumption of load distribution in the contact area is not required because the contactalgorithm of the general computer program [Hibbit, Karlsson & Sirensen, Inc., 1998]

is used to get the contact area and stresses by application of torque to the pinion whilethe gear is considered at rest

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1 2

3

Figure 15.9.1: Whole gear drive ﬁnite element model.

1 2

3

Figure 15.9.2: Contacting model of ﬁve pairs of teeth derived for stress analysis.

437

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1 2

of the grinding worm.

Contact Stresses (MPa)

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+4.956e-04 +1.500e+02 +3.000e+02 +4.500e+02 +6.000e+02 +7.500e+02 +9.000e+02

1 2

The use of several teeth in the models has the following advantages:

(i) Boundary conditions are far enough from the loaded areas of the teeth

(ii) Simultaneous meshing of two pairs of teeth can occur due to the elasticity of faces Therefore, the load transition at the beginning and at the end of the path ofcontact can be studied

sur-Numerical Example

Stress analysis has been performed for the gear drive with the design parameters shown

in Table 15.7.1 A ﬁnite element model of three pairs of contacting teeth has beenapplied for each chosen point of the path of contact Elements C3D8I [Hibbit, Karlsson

& Sirensen, Inc., 1998] of ﬁrst order (enhanced by incompatible modes to improve their

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(Ave Crit.: 75%)

S, Mises

+1.598e-03 +1.843e+02 +3.686e+02 +5.529e+02 +7.372e+02 +9.215e+02 +1.106e+03

1 2

edge contact is avoided.

bending behavior) have been used to form the ﬁnite element mesh The total number

of elements is 45,600 with 55,818 nodes The material is steel with the properties

of Young’s Modulus E = 2.068 × 105 MPa and Poisson’s ratio of 0.29 A torque of

500 Nm has been applied to the pinion Figure 15.9.3 shows the contact and bendingstresses obtained at the mean contact point for the pinion

The variation of contact and bending stresses along the path of contact has been alsostudied Figure 15.9.4 illustrates the variation of contact and bending stresses for thepinion Stress analysis has also been performed for a conventional helical involute drivewith an error of the shaft angle ofγ = 3 arcmin (Fig 15.9.5) Recall that the tooth

surfaces of an aligned conventional helical gear drive are in line contact, but they are inpoint contact with errorγ The results of computation show that error γ causes an

edge contact and an area of severe contact stresses

Figure 15.9.6 shows the results of ﬁnite element analysis for the pinion of a ﬁed involute helical gear drive wherein an errorγ = 3 arcmin occurs As shown in

modi-Fig 15.9.6, a helical gear drive with modiﬁed geometry is indeed free of edge contactand areas of severe contact stresses

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16 Involute Helical Gears with Crossed Axes

16.1 INTRODUCTION

Involute helical gears are widely applied in the industry for transformation of rotationbetween parallel and crossed axes Figure 16.1.1 shows an involute helical gear drivewith crossed axes in 3D-space A gear drive formed by a helical gear and a worm gear is

a particular case of a gear drive with crossed axes (Figure 16.1.2) Gear tooth surfacesare in line contact for involute helical gear drives with parallel axes and in point contactfor involute helical gear drives with crossed axes

The theory of involute gears and research in this area have been presented by Litvin

[1968], Colbourne [1987], Townsend [1991], and Litvin et al [1999, 2001a, 2001c,

2001d] and the theory of shaving and honing technological processes are discussed

in the works of Townsend [1991] and Litvin et al [2001a] Despite the broad

in-vestigation that has been accomplished in this area, the quality of misaligned lute helical gear drives is still a concern of manufacturers and designers The maindefects of such misaligned gear drives are (i) appearance of edge contact, (ii) highlevels of vibration, and (iii) the shift of the bearing contact far from the centrallocation

invo-To overcome the defects mentioned above, some corrections of gear geometry havebeen applied in the past: (i) correction of the lead angle of the pinion (requires regrind-ing), and (ii) crowning in the areas of the tip of the proﬁle and the edge of the teeth(based on the experience of manufacturers) A more general approach for localization

of bearing contact has been proposed in Litvin et al [2001c].

The conditions of meshing of crossed involute gears are represented in this chapter

(i) representation of an edge contact as the result of the shift of lines of action in

a misaligned gear drive(ii) relation of the sensitivity to an edge contact with the nominal value of thecrossing angle

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Figure 16.1.1: Involute helical gears with crossed axes in 3D-space.

Worm

Helical gear

Figure 16.1.2: Gear drive formed by a worm and a helical gear.

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Figure 16.2.1: Illustration of generation of a

screw involute surface.

Algorithms for simulation of meshing (including simulation of edge contact) arerepresented The theory is supported by numerical examples

16.2 ANALYSIS AND SIMULATION OF MESHING OF HELICAL GEARS

Conceptual Considerations

It is well known [Litvin, 1968, 1989] that a screw involute surface can be generated

by a screw motion of a straight line MD (Fig 16.2.1), while in the process of motion

the generating line keeps its orientation as the tangent to the helix on the base

cylin-der The generated surface is a developed one [Litvin, 1968, 1989; Zalgaller, 1975], and the normals to the surface along MD are collinear Figure 16.2.2(a) shows that

the generated surface is formed as a family of straight lines that are tangent to thehelix on the base cylinder Tooth surfaces of mating helical gears (in an aligned gear

drive with parallel axes) contact each other along the straight lines MD mentioned

above

Assume now that, using coordinate transformation, the lines of contact are sented in plane that is tangent to both base cylinders of the helical gears with parallel

repre-axes Figure 16.2.3 shows plane that is tangent to the base cylinder of the pinion.

Angleαot1 is the pressure angle in the transverse section Points O1and O2representthe centers of base circles of mating helical gears

Using coordinate transformation, we represent in plane lines of contact of helical

gears with parallel axes The lines of contact L represented in plane (Fig 16.2.3) are parallel straight lines Plane of a gear drive with parallel axes is the plane of action

in the ﬁxed coordinate system rigidly connected to the housing of the gear drive

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Figure 16.2.2: Types of contact of helical gears: (a) line contact in a gear drive with parallel axes, (b) point contact of crossed helical gears.

We emphasize the special features of the contact lines L:

(1) We recall that Fig 16.2.1 shows that the screw involute surface is generated by

the screw motion of line MD The normals to the screw involute surface (at the instant position of MD) are collinear and their orientation does not depend on the parameter of location of the point on MD We may consider that locally MD is a

small strip of the screw involute surface with collinear normals

Figure 16.2.3: Illustration of contact lines on plane of action

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Figure 16.2.4: Line A of action of involute planar

gearing.

(2) The normal to the screw involute surface at any contact point is orthogonal to the

contact line Therefore, the normals to the tooth surface are also orthogonal tocontact lines represented on plane

(3) The normals to the screw involute surface have the same constant orientation inthe plane of action

Figure 16.2.4 illustrates a planar involute gearing Line T1–T2is tangent to the base

circles with radii r b1 and r b2 Simultaneously, line T1–T2is the common normal to themeshing involute proﬁles

We may extend the conceptual considerations of meshing of a planar involute gearing

to the case of crossed involute helical gears The gear tooth surfaces of crossed helicalgears contact each other at a point [Fig 16.2.2(b)], but not at a line as in the case ofhelical gears with parallel axes [Fig 16.2.2(a)] Figure 16.2.5 shows the base cylinders

of crossed helical gears It is obvious that the crossed base cylinders of the gears may

have a common tangent line but not a common tangent plane There are two lines A1

and A2that are tangent to the base cylinders and to the base helices at the same time

We may call A1 and A2the lines of action of crossed helical gears Lines A1 and A2correspond to the meshing of the respective sides of the tooth surface

It is shown below (see Appendices 16.A and 16.B) that the design of crossed helicalgears may be accomplished by observation of a special relation between the shortest

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Figure 16.2.5: Lines of action of aligned crossed helical gears.

center distance of the gears and the crossing angle Such a design provides that the lines

of action intersect each other at a point that belongs to the shortest center distance This

design is called canonical.

Figure 16.2.6 shows the lines of action for which the rules of canonical design are

not observed, and the lines of action A1and A2 are crossed but do not intersect each

other Each line A iis still a tangent to both base cylinders and base helices of the crossed

helical gears The crossing of lines of action A1and A2is the result of an errorγ of the nominal value of the shaft angle γoof the gears or the result of an errorE of center

distance (see below)

Analytical Determination of Line of Action

of Crossed Helical Gears

Analytical determination of the line of action of misaligned crossed helical gears isimportant for detection of edge contact Edge contact occurs when the line of action isshifted from the theoretical position and is located outside of the face width (Figs 16.2.7and 16.2.8) The derivations that are represented as follows open the way for analyticaldetermination of the appearance of edge contact We have represented intersected andcrossed lines of action of crossed helical gears in Figs 16.2.5 and 16.2.6, respectively

In addition to this presentation, it is important to represent the lines of action in the

plane that is tangent to the base cylinder of one of the crossed helical gears, say the

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Figure 16.2.6: Lines of action in gear drive with errorγ

of shaft angle.

Figure 16.2.7: Illustration of (a) plane 1 and

contact lines on ... 16< /span>

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CB6 72- 16 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:51

Figure 16 .2. 8:... JTH

Figure 16 .2. 5: Lines of action of aligned crossed helical gears.... JTH

Figure 16 .2. 4: Line A of action of involute planar

gearing.

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