APUNTES_GLEASSON_Basics of Spiral Bevel & Hypoid Gears

Spiral Bevel and Hypoid Gear Cutting Systems: Basics of Spiral Bevel & Hypoid Gears

MTA-Z WIR 21.03.2012 Klingelnberg AG · Training Center

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[Basics of Bevel Gears] · Slide 1

Basics of Spiral Bevel and Hypoid Gears 1.

Application and Classification of Bevel Gears

2.

Gear Cutting Systems

39

3.

Gear Geometry and Basic Definitions

67

4.

Hypoid Offset and Hand of Spiral

91

5.

Face Milling / Face Hobbing Cutting System in Comparison

113

6.

Basics of Continuous Indexing

133

7.

Particularities of Epycycloide Tooth Length Curvature

150

8.

Face Hobbing: Generating - / FORM - Cutting System

165

9.

The Ease Off

183

10. Bias / Tooth Twist

193

11. Calculation of Radial- & Axial Forces

231

12. The Influence of Cutter Diameter Klingelnberg AG · Training Center


 page

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1

1.

Application and Classification of Bevel Gears 1.1

Application of Spiral Bevel Gears

1.1.1 Type of Axle Design 1.2

Characteristics of Classification:

1.2.1

– shaft angle

1.2.2

– offset of axes

1.2.3

– tooth depth

1.2.4

– tooth length curvature

1.2.5

– indexing system

1.2.6

– generating system

1.2.7

– methods to apply length crowning

Klingelnberg AG · Training Center

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Application of Spiral Bevel Gears :

General

The main field of application for Spiral Bevel- and/or Hypoid Gears are driven rearor front axle drives in automotive vehicles.



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General

As mentioned, Spiral Bevel and Hypoid gears - in particular those manufactured in mass production - are mainly utilized in drive lines of automotive axles. Type of vehicles could be .... Trucks, buses Passenger cars

Transporters Sports Utility Vehicles Tractors

(SUV‘s)

Off-road vehicles Klingelnberg AG · Training Center

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General

Other non-automotive applications could be for example ...

Stationary gear transmission of all kind

Helicopters

motorbikes

Railways

Outboard marine drives

Azimuthing Thrusters Klingelnberg AG · Training Center


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General

Spiral Bevel- and/or Hypoid Gears are designed to transmit torque in gear drive-lines where the axial direction of power needs to be changed. With axles of automotive vehicle applications this is in general 90° Between INPUT and OUTPUT • the RPM is reduced by the ratio of tooth combination nout = nin  z1/z2

Z2

• the torque is increased by the ratio of tooth combination Mdout = Mdin  z2/z1

sense of rotation OUTPUT


Z1

sense of rotation INPUT

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General

The term of “spiral bevel gear” is frequently mistaken for “differential gear” or vice versa. In driven axles of vehicles the differential gear is generally an integral part of the spiral bevel gear drive. The differential serves to compensate the different angular velocities between right and left wheel of the axle in condition of driving the vehicle in a bend. Differential gears are generally designed as straight bevel gears cut by the REVACYCLE method.

Hypoid Gearset with Differential Klingelnberg AG · Training Center


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Type of Axle Designs

Spiral Bevel- & Hypoid gears are used in driven axles of vehicles in quite different types of design. Without being complete, the most popular types of axles are presented in the following:

- Salisbury axles - Banjo axles - Timken axles - IRDS axles - PTO units - Outer Planetary- (or Hub Reduction) axles - Single reduction -, Tandem- or Tridem axles Klingelnberg AG · Training Center

Type of Axle Design:


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for Passenger Cars, VANs & light Trucks

“Salisbury” Axles • the axle carrier is split in a plane parallel to the gear axis • gear axis is located “inboard” of the split of carrier • the setting of radial backlash and preload of gear- and differential bearings is controlled either with adjustable threaded rings or with selectable shims

.... here shown in form of an IRDS type of axle Klingelnberg AG · Training Center


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“Salisbury” Axles

.... here shown in form of rigid or solid beam type of axles




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“Salisbury” Axles • good to recognize in this view : the split of carrier parallel to the axis of gear or differential respectively • setting of pre-load of differential bearings with this design is applied by means of spread of the carrier set with adjusted shims hold with snap rings



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“Banjo” Axles

Land Rover

• the axle carrier is split in a plane parallel to the gear axis • gear axis is located “outboard” of the split of carrier • gear- / differential bearings are hold in half cups which are fastened with pedestal caps • the setting of radial backlash and pre-load of gear- and differential bearings is controlled with adjustable threaded rings

TOYOTA Pick-Up «Tundra»

• the pre-assembled unit is mounted in the axle; the “Banjo” shape of the axle crossbeam has given the name to this type of axle (see next page) Note:

Banjo type of axles are sometimes also referred to as “Hotchkiss” type



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for medium- to heavy duty trucks

“Banjo” Axles noticeably the “Banjo” shape of axle crossbeam which is typically manufactured in either steel casting or welded construction

 .  Meritor

DC „Actros“

ca. 13  16 t* / 450HP

ca. 5  15 t* ( * = load per axle)

AAM ca. 12 t* / 26‘000Nm Gear Ø 400mm



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“Banjo” Axles FORD 9“ axle with straddle mounted pinion (does not show in this view) in a heavy webbed carrier designed for the transmission of very high torque. This type of axle used to be very popular in the 50’s – 70’s but has been replaced mostly for cost issues

FORD 9“ as a standard beam axle Klingelnberg AG · Training Center


FORD 9“ as a special axle for racing application [Basics of Bevel Gears] · Slide 15

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“Timken” Axles • the axle carrier is split in a plane parallel to the pinion axis • this type of design is quite rarely applied in present times



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AdvanTEK® type of Axle Design • relatively recent DANA design • carrier is split parallel to pinion axis • features an advantageous design for automatic assembly




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Independent Rear Drive Systems • compared to rigid type of axles this design allows a better segregation of the axle carrier vibrations being transmitted to the body of the vehicle  reduction of gear noise • reduction of unsuspended masses (wheels, brakes)  improvement of driving conditions of the vehicle



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Segregated suspended Axles • the carrier of the gear-set is generally designed as „Salisbury“ type; the cover of the assembly opening serves hereby as conjunction to the suspended axle • depending on the axle design, the suspended points of mounting are not rigidly connected but dampened to the cross beam by means of absorber elements



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Power Take-Off Units • Power Take-Off Units take the power from a front wheel drive transaxle, and transfer it to the rear wheels; in general PTO’s are therefore part of a 4WD concept • PTO’s generally consist of rather low bevel gear ratios whereby typically the GEAR serves as the driving member (hence the pinion becomes the driven member)

driven front axle

take - off to rear axle Klingelnberg AG · Training Center


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for Trucks and Buses source: MAN

Axles for heavy duty application depending to required gross weight of vehicle and allowed load per axle, driven single reduction axles can be designed in either Single-, TANDEM- or TRIDEM arrangement. Axles for heavy duty application are typically designed as Banjo type.

Hypoid axle without () and with () through-drive source: MAN

„TANDEM“ axle

„TRIDEM“ axle arrangement




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for Trucks and Buses

Hub Reduction Axles • the total ratio of the axle is finally reduced by means of a set of planetary gears in the hubs • therefore torque and stresses are reduced for the set of bevel gears which allows to reduce gear diameter • smaller diameter of ring gear results in higher clearance of the axle to the ground



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Hub Reduction Axle with Through Drive • application of a power divider in case of several driven axles

power divider

sectional drawing

overall view Klingelnberg AG · Training Center

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Hub Reduction Axle with Through Drive

TATRA Pendulum Axle  1 rim 2 hub of rear wheels 3 outer wheel 4 cover 5 oil fill bolt 6 sun gear 7 planetary gear 8 outer wheel carrier 9 bell hub 10 brake drum 11 brake carrier 12 axle beam 13 bevel pinion 14 axle carrier 15 gear 16 cover 17 shift fork 18 pressure switch 19 coupling flange 20 drive shaft



21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Schaltmuffe transm. shaft gear differential carrier differential gear shaft of through drv. locking plate bevel ring gear differential carrier center bolt thread ring rear axle shaft brake pad wheel bolt brake camshaft roller tension spring brake pad oil drain bolt Gestängesteller oil fill bolt fender guard [T1e_v10_WIR]

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Classification of Spiral Bevel Gears :

General

With the knowledge of the main criteria of classification of Spiral Bevel gears it becomes a lot easier to understand the various gear manufacturing - and tooling systems presently existing and their particular cross relations. Some of the major characteristics are correlated and make it therefore even easier to classify the different systems correctly .


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Bevel Gears can be classified to: Shaft Angle = 90°

shaft angle shaft angle

 

= 90° > 90°

shaft angle



< 90°  90°

Note: • different size of Generating Plane Gear for same diameter of gear • i.e. different machine capacity (range of machine axes) is required for same size of gears • gear sets with shaft angles  90° are also referred as “angular bevel gear drives” Klingelnberg AG · Training Center


 90° [T1e_v10_WIR]

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… additional information to: Shaft Angle Beveloid® • Beveloid® gears are helical gears with variable profile displacement along the face width. Applying gears of this kind, crossing angles of gear axes of up to approx. 10° to 15° can be realized with either intersecting or non intersecting (skew) axes. • bevel gears with small shaft angles can be replaced with Beveloid® gears • using Beveloid® gears rather than spiral bevel gears can be advantageous as the manufacturing of bevel gears with very small shaft angles requires a disproportionately large generating crown gear which calls for the application of large cutting machines even for relatively small gear dimensions. Klingelnberg AG · Training Center


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… additional information to: Shaft Angle Beveloid® in combination with Angular Hypoid • this combination of bevel gears is e.g. actually applied in drivelines of certain 4WD cars



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offset = 0

Bevel Gears can be classified to:

offset

av = 0 Spiral Bevel gears with intersecting axes

offset

av  Hypoid gears with non intersecting axes

Note: • the expression HYPOID is derived from the word HYPerbolOID ( >>> see next page)

offset > 0

OFFSET of AXES

• the most general case of Bevel Gears is represented with Angular Hypoid (   90° ) (this case, however, is applied quite rarely Klingelnberg AG · Training Center


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OFFSET of AXES

With Hypoid gears the pitch surfaces are theoretically represented by Hyperboloids and not by conical faces.

HRH High Ratio Hypoid

This fact becomes important only for gears with quite large Hypoid offset ( to avoid possibility of interference). For Hypoid offset as typically used with automotive applications, however, the configuration of Hypoid elements are very nearly cones. For practical reasons of manufacturing therefore, bevel or cone shaped blanks are generally used for Hypoid ring gears and pinions.



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Bevel Gears can be classified to: TOOTH DEPTH

tooth depth

=

constant

tooth depth

=

not constant, tapered

Note: for tapered tooth depth, dependant on certain design parameters and/or the applied cutting system, the apex of – pitch angle – face angle and – root angle .... do not coincide Klingelnberg AG · Training Center


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… additional information to:

TOOTH DEPTH Depending on the cutting method or gear design, inconstant or tapered tooth depth appears either with a tapered or with a parallel width of bottom land: Standard Taper

Duplex Taper or Tilted Root Line Taper



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Bevel Gears can be classified to: TOOTH LENGTH CURVATURE ARC of CIRCLE gears cut to single indexing method

Note:

EPICYCLOIDE INVOLUTE gears cut to gears cut to continuous indexing method Klingelnberg Palloid method

the SPIRAL in terms of length curvature is not applied with any kind of cutting system



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Bevel Gears can be classified to: TOOTH LENGTH CURVATURE STRAIGHT LINE Straight Bevel Gears

STRAIGHT LINE Skew Bevel Gears or Helical Bevel Gears

Straight Bevel Gears in these days are primarily used for: – low torque low speed transmissions – applications where no gear noise criteria apply  Straight Bevel Gears will therefore not further be considered in this presentation



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Bevel Gears can be classified to: INDEXING SYSTEM indexing

=

continuous indexing is also referred to as  “Face Hobbing” or “3 - axis gears”

indexing

=

discontinuous – or single indexing is also referred to as  “Face Milling” or “2 - axis gears”

Note: the description 2- / 3-axis gears refers to the minimal number of coupled axes which are required to manufacture a respective pinion Klingelnberg AG · Training Center

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Bevel Gears can be classified to: GENERATING SYSTEM Pinion & Gear

=

generated Ring Gear Pinion

Gear

= non generated  FORM cut

Pinion

=

generated Ring Gear Pinion



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Bevel Gears can be classified to: LENGTH CROWNING SYSTEM

Fixed Settings individual cutters set with individual machine settings to cut convex and concave side in 2 operations Cutter Eccentricity (Dual Part Cutters) as used with KLINGELNBERG Zyclo Palloid System Blade Succession Angle (Single Part Cutters) as used with OERLIKON cutting systems: TC / ETC / EN / HN / FN Cutter Tilt most common and most flexible system to apply length crowning as used with all Completing methods Klingelnberg AG · Training Center

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Classification of Spiral Bevel Gears :

Summary

Position of Axes

Tooth Depth

Tooth Length Curvature

Indexing Method

Method of Length Crowning

Profile Curvature

 no vertical offset (Sp. Bevel Gear)

 constant tooth depth

 Epicycloid

 Continuous (Face Hobbing)

 cutter tilt

 pinion + gear generated

 blade succession angle N-Type (Oerlikon)  cutter eccentricity (Cyclo Palloid)  cutter tilt

 tapered tooth depth

 Involute

 (PALLOID method)

 Arc of a Circle

 Discontinuous (Face-Milling)

 with vertical offset (Hypoid Gear)



 pinion generated + gear form-cut (FORM method)  pinion + gear generated

 fixed settings for individual flanks or cutter tilt

 pinion + gear generated

 fixed settings for individual flanks or cutter tilt

 pinion generated + gear form-cut (FORM method)

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2.

Gear Cutting Systems 2.1 The Generating Rack 2.2 The Generating Plane Gear 2.3 Generated Cutting Systems 2.4 FORM Cutting Systems 2.5 Neutral Data 2.6 General Features of Cutting Systems - discontinuous systems - continuous systems



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The Generating Rack: • the generating principle is derived from a straight rack with straight tooth profile • in case of spur or helical gears, the involute tooth profiles are generated by rolling (generating) a cylinder with constant center distance along the rack

Animation: generation of tooth profiles



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The Generating Principle: • as the rack advances, with corresponding rotation of the pinion, the point of contact moves uniformly along the path of contact • the generating speed of the rack is equal to the peripheral speed of the basic circle to which the path of contact is tangential • hence the point of contact can be visualized as being a point on a cord unwound from this circle



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The Generating Plane Gear: • Cylinders rolling on a Straight Rack will form spur or helical gears

• for Hypoid gears the Generating Plane Gear is theoretically represented by a Helical Cone Face, however, this in • in similarity the Generating Plane Gear general is replaced by a Generating can be considered as a ring shaped rack Plane Gear.



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The Generating Plane Gear: The generating process for Bevel Gears is based on the common Generating Plane Gear. The Plane Gear rotates between gear and pinion, tangent to both pitch cones.

Generating Plane Gear for Spiral Bevel

The tracks of the rotating cutter blades represent one tooth of the Generating Plane Gear. Note:

in case of the continuous indexing system, z0 number of blade groups represent z0 number of successive teeth of the Generating Gear

The action is as though the gear or the pinion being cut were rolling with an imaginary gear. Klingelnberg AG · Training Center

Generating Plane Gear for Hypoid

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The Generating Plane Gear: • the tooth profiles in direction of tooth depth are generated by the generating motion • in general the flanks of the Generating Plane Gear - similar to the generating Rack - have straight profiles in direction of tooth depth • in order to introduce profile crowning or other profile modifications, the flanks of the tools (cutting edges) might be curved

Note: in order to generate tooth profiles, a generating roll motion (rolling angle) of about 25 to 35° is required





rotation of cutter

 rotation of Generating Plane Gear

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Generated Cutting Systems:

Axis of Generating Plane Gear

• Generating Plane Gear with straight tooth profiles • generating roll applied to cut D/P • generating roll applied to cut R/G • generated tooth profiles on D/P • generated tooth profiles on R/G In case of Spiral Bevel gears with 90° shaft angle the Number of Teeth of the Generating Plane Gear (Zp) calculates as:

Zp  Z12  Z22 Z1 No. of teeth Pinion Z2 No. of teeth R. Gear Klingelnberg AG · Training Center

D/P = Drive Pinion R/G = Ring Gear [Basics of Bevel Gears] · Slide 45

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Generated Cutting Systems: • for cutting systems which pinion and gear are generated, the Generating Gear is a Plane Gear • the rotating tool represents a tooth of the Plane Gear which is in mesh with either the gear or pinion to be generated • the axis of the tool is herby parallel to the axis of the Generating Gear i.e. the axis of the Plane Gear Note: a usually small amount of tilt of the axis of the tool required to create some length crowing is NOT considered in this illustration

• during the generating roll, the axis of the tool rotates around the axis of the Generating Gear i.e. around the axis of the Plane Gear Start Video Klingelnberg AG · Training Center


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FORM Cutting Systems:

Axis of Generating Gear

• Generating Gear with straight tooth profiles • generating roll applied to cut D/P • plunge process applied to cut R/G • generated tooth profiles on D/P • straight tooth profiles on R/G

Number of Teeth of the Generating Gear (Zp) is in this case:

Zp  Z2 Z1 No. of teeth Pinion Z2 No. of teeth R. Gear Klingelnberg AG · Training Center

D/P = Drive Pinion R/G = Ring Gear [Basics of Bevel Gears] · Slide 47

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FORM Cutting Systems: • for systems which the gear is cut with the FORM method, the Generating Gear for the pinion is represented by the Bevel Gear which meshes with the pinion • the rotating tool represents here again a tooth of the Generating Gear which is in mesh with the pinion to be generated • the axis of the tool is hereby inclined or tilted towards the axis of the Generating Gear • during the generating roll the axis of the tool rotates around the axis of the Generating Gear; at the same time the axis of the tool wobbles relative to the axis of the Generating Gear Start Video Klingelnberg AG · Training Center


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Comparison Form of Profile: FORM - / GENERATED Gear Profile • compared to the GENERATED gear, the profile of the FORM type gear is wider at the tip and across the root section FORM type gear


GENERATED type gear


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Comparison Form of Profile: FORM - / GENERATED Pinion Profile • both pinion profiles are generated; compared to the pinion of a FORM cut gear the profile of the GENERATED gear is wider at the tip and across the root section FORM type gear


GENERATED type gear


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Neutral Data: Task: In order to allow for all possible motions between work and tool, an effective, non restricting mathematical approach for the kinematics of machine and tool is needed. • minimum number of parameters • all 6 degrees of freedom • no restrictions of any physical machines

Approach of Neutral Data: • • • •

description of the tool description of blank geometry specification of basic machine settings additional motions by Taylor series up to 6th order


HORIZONTAL motion (  )

M/c Root angle

work rotation tool rotation

Mounting Distance

+ additional Free Form Motion

crossing point

Machine Center-to-Back

Basic Setting

pitch apex



setting value

Neutral Data

S Radial Distance a M Tilt angle  M Swivel angle  a Work Offset a M M/c Root angle a Md+ M/c Center-to-Back mccp Sli Sliding Base a RA0 Ratio of Roll b QM Mean Cradle angle m

Work Axis

-CL



Horizontal Setting

parameter  does NOT exist with BASIC SETTINGS !! generation roll

Machine Center

+

Cradle Axis

Sliding Base


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Neutral Data: Work Axis

Basic Motions defined by Plane Gear

HELICAL motion

+ [Basics of Bevel Gears] · Slide 52

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Basic Setting Tilt angle Swivel angle

Radial Distance

V

Spiral angle

mean Cradle angle

Work Axis

Neutral Data

S Radial Distance a M Tilt angle  M Swivel angle  a Work Offset a M M/c Root angle a Md+ M/c Center-to-Back mccp Sli Sliding Base a RA0 Ratio of Roll b QM Mean Cradle angle m

VERTICAL motion

HORIZONTAL motion

setting value

Basic Gear Data

setting value

V H

Vertical Horizontal

Work Offset Cradle Axis H


see aux. document [T1e_v10_WIR]


Neutral Data:

Possible additional Free Form Motions:

following motions can be added (superimposed) during generation roll: • • • • • •

superimposed rotation of work superimposed depth position superimposed offset position superimposed offset position superimposed root angle pos. superimposed radial position

  1.... m .... 2

Modified Roll Helical Motion Horizontal Motion Vertical Motion Angular Motion Radial Motion

angle of Generating - Roll 2

6

2

6

  aβ  b     m   c      m   ...  g     m   ()

  aconst. ( )b      m   c      m   ...  g     m  2

  aconst.  ()b      m   c      m   ...  g     m 

6

2

  aconst.  ()b     m   c      m   ...  g     m  2

 ( )b      m   c      m   ...  g     m    aconst. 2

6

6

const. ()   a   b      m   c      m   ...  g     m  Klingelnberg AG · Training Center

() () () () () ()

6


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Arrangement of Axes:

setting axes

The Conventional Gear Generator with Cradle, Tilt & Swivel Altogether there are 10 axes required: • 6 setting axes remain set and rigidly clamped during the cutting process • 4 working axes are partially coupled: with the continuous indexing method there are simultaneously 3 axes coupled at a time : respectively

A + B + WT A + B + WS

for generating for plunging

working axes


Eccentricity:

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Referring to Conventional Generators with Cradle

Depending on the design of conventional generators, the Radial Distance is either set by the rotation of an eccentric drum or by linear displacement of a cross slide. Eccentricity = 0 the centers of tool (cutter) spindle and cradle coincide  this does not correspond to any reasonably applicable position of the cutter !

Eccentricity > 0

Eccentricity > 0

the center of tool (cutter-) spindle is swiveled along an arc of a circle off the center of the cradle

the center of the tool spindle is specified by the Radial Distance and the Mean Cradle Angle

WT [ °]

EX [mm]

EX [mm]

m [ °]

EX [ °]

EX [ °]

EX = 0



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Cutter Tilt:


• with this concept tilting the tool spindle is realized by means of the rotation of a obliquely split drum. • the Neutral Point of the tilting drum is preferably located near the crossing point of the cradle axis and the plane of blade pitch point. Cutter Tilt = 0° the rotational setting of swivel DLM () is not relevant

Cutter Tilt > 0

Cutter Tilt > 0

max. cutter tilt results at a rotation of 180° of the tilt block

the orientation of the tilted cutter is realized with the drum of swivel DLM ()

 max

 max / 2 EK = 180°

EK = 0°


Radial Distance:

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Depending on the design of conventional generators, the Radial Distance is either set by the rotation of an eccentric drum or by linear displacement of a cross slide. Radial Distance = 0 the centers tool (cutter-) spindle and cradle coincide  this does not correspond to any reasonably applicable position of the cutter !

Radial Distance > 0

Radial Distance > 0

the center of the tool spindle is displaced in lateral direction off the cradle center

the center of the tool spindle is specified by the Radial Distance and the Mean Cradle Angle

EX [mm]

WT [ °]

EX [mm]

m [ °] DLM [ °] EX = 0

DLM = 0° EX [mm]



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Cutter Tilt:


• with this concept, tilting the tool spindle is realized by means of the pivoting the spindle block around the axis of the spindle worm drive • the amount of cutter tilt affects the location of the center of the plane of blade pitch points Cutter Tilt = 0° without any tilt of cutter the rotational setting of swivel DLM (s) is not relevant

Cutter Tilt > 0

Cutter Tilt > 0

the amount of cutter tilt  causes an alteration of the center of the tool

the orientation of the tilted cutter is realized with the drum of swivel DLM ()

 Klingelnberg AG · Training Center

Cutting Machines:

Oerlikon C-type Machines (horizontal concept)

3 linear axes • X axis • Y axis • Z axis

Z B

3 rotational axes • A cutter axis • B work axis • C rotational axis

A

X

C

Chip flow  subject to the sense of rotation of the cutter chips are falling either directly or indirectly via inclined surfaces of into the chip conveyor Klingelnberg AG · Training Center

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Y


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30

Cutting Machines:

Oerlikon C-type Machines (vertical concept)

3 linear axes • X axis • Y axis • Z axis

X Y

3 rotational axes • A cutter axis • B work axis • C rotational axis

A C

Chip flow  Irrelevant to the sense of rotation of the cutter chips are falling directly into the chip conveyor Klingelnberg AG · Training Center

Cutting Systems:

Z B [T1e_v10_WIR]


Single Indexing

In case both flanks of a gear are cut simultaneously with a single indexing tool we get constant width of bottom land (*). In order to control lengthwise crowning, the pinion is to be finished with independent machine- and cutter settings. (*)

cutting: PINON

cutting: GEAR

in case of single sided cutting processes the width of bottom land might vary from toe to heel

 top- & bottom land width is constant  tooth depth needs to be tapered  tooth length curvature: Arc of a Circle Applications are: • Gleason

5-cut system (Fixed Settings)



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31

Cutting Systems:

Single Indexing

In case both flanks of pinion/gear are cut with a circular tool we get - with reference to the pitch point of the tool - constant slot width  top- & root land width is constant  tooth depth needs to be tapered  face- & root angle need to be adjusted in order to achieve uniform slot width from toe to heel (Duplex-Cone)  tooth length curvature: Arc of a Circle

Applications are: • Oerlikon ARCON • Gleason Completing (Duplex Helical) Klingelnberg AG · Training Center


Cutting Systems:

[T1e_v10_WIR]

Single Indexing

In case of constant tooth depth, the slot width needs to be tapered  tooth length curvature: Arc of a Circle  tooth depth is constant  thickness of top- & root land is tapered

Applications are: • WIENER

system

• CURVEX

system (Modul)

• SARATOV

system

• ROCHAT

system



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32

Cutting Systems:

Continuous Indexing

In case of constant tooth depth, the slot width necessarily needs to be tapered. The slot width at the outer diameter is wider than that at the inner diameter.  top land thickness is tapered  tooth depth is constant  lengthwise curvature: Epicycloide Applications are: • Oerlikon Oerlikon

N1-FN, HN-FN SPIROFLEX, SPIRAC

• Oerlikon

SPIRON

• Klingelnberg CYCLO PALLOID • Gleason

TRIAC®, PENTAC-FH®


Cutting Systems:


[T1e_v10_WIR]

Continuous Indexing

In case of constant tooth depth, the slot width necessarily needs to be tapered. The slot width at the outer diameter is wider than at the inner diameter.  normal module is constant  tooth depth is constant  lengthwise curvature: Involute

Application is exclusively: • Klingelnberg PALLOID



Note: more and detailed information related to individual cutting systems see class: T8 Cutting Methods & Tooling Systems



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33

3.

Gear Geometry 3.1 General Nomenclature 3.2 Pressure Angle / Line of Action 3.3 Contact Ratio 3.4 Path of Contact 3.5 Addendum / Dedendum / Whole Depth / Clearance 3.6 Axial - / Radial Backlash 3.7 Replacement Helical Gears 3.8 Undercut / Profile Displacement 3.9 Tooth Thickness Correction 3.10 Blank Dimensions


Gear Geometry: many denominations as specified for helical gears are used likewise for spiral bevel gears.

General Nomenclature root circle whole depth

addendum

For more detailed gearing expressions and denominations see Standard AGMA, DIN or ISO23509 Specifications

[T1e_v10_WIR]


dedendum tooth fillet

bottom land

backlash top land flank

circular thickness working depth

Note: by convention the smaller member of gear is called Pinion, it is usually the driving member Klingelnberg AG · Training Center

chordal thickness

base radius [Basics of Bevel Gears] · Slide 68

tip clearance

pitch radius [T1e_v10_WIR]

34

Gear Geometry:

Pressure Angle / Line of Action

Line of Action is tangent to both base circles

Path of Action (Length of Action) is the locus of successive contact points between a pair of gears during mesh; it is limited by both of the tip circles



Pitch of Action is the tooth pitch measured along the path of action

Pressure Angle  is the angle at a pitch point between the line of pressure which is normal to the tooth surface, and the plane tangent to the pitch surface. The pressure angle gives the direction of the normal to the tooth profile. Klingelnberg AG · Training Center

>> meshing gears (animation) [T1e_v10_WIR]


Gear Geometry:

Zone of Action

Zone of Action for involute, parallel-axis gears with either spur or helical teeth, is the rectangular area in the plane of action bounded by the length of action and the face width. In practical case (i.e. for gears with length and profile crowning) the Zone of Action is restricted by the effective face width and the effective working depth.



Face Width

Zone of Action Length of Action Line of Contact

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35

Gear Geometry:

Contact Ratio for Bevel Gears

Profile Contact Ratio   =

Length of Action angle a Pitch of Action angle  Pitch of Action Length of Action

high profile contact ratio results of: 

• long path of action - large whole depth of teeth - small pressure angle

a

Pitch of Action angle Length of Action angle

• small pitch of action - large number of teeth

advantage of high profile contact ratio: • more pairs of teeth in contact • smooth meshing Klingelnberg AG · Training Center

disadvantage of high profile contact ratio: • low strength of rupture (... as teeth are generally slender and high >> small tooth root section)


Gear Geometry:

[T1e_v10_WIR]


Overlap Contac Ratio   =

Overlap angle  angular pitch 

high overlap contact ratio results of:  • large overlap angle  - large tooth width - large spiral angle - small tool diameter

• small angular pitch - large number of teeth - small normal module

advantage of high overlap contact ratio: disadvantage of high overlap contact ratio: • high axial thrust to bearings • more pairs of teeth in contact • smooth meshing if  is an integral No. Klingelnberg AG · Training Center


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36

Gear Geometry:


Total Contact Ratio  In case of conjugate gears there would theoretically be:

ε     

In case of bevel gears manufactured with length- and profile crowning there is:

Total Contact Ratio  2

ε γ  ε α  εβ

2

A more accurate determination of the contact ratio is possible by means of tooth contact analysis. Based on the effective shape, length and flattening of the contact under load the so-called effective contact ratio can be calculated. torque = 600Nm

torque = 100Nm


[T1e_v10_WIR]


Gear Geometry:

Path of Contact

The Path of Contact on either of the tooth flanks is the course along which the theoretical single point contacts develop during the meshing period.

start of mesh for DRIVE sense of rotation: FORWARD

Under load the single point contacts spread in direction of the Contact Lines

Path of Contact Contact Lines

The Path of Contact appears different depending on the Ease Off i.e. in particular on the amount of Tooth Twist and/or Profile Crowning (  see section 9.7 / 10.5 )

start of mesh for COAST sense of rotation: REVERSE



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37

Gear Geometry:

Spiral Angle

The Spiral Angle - as well as other major gear parameters - is defined in the plane of the Generating Plane Gear. It is the angle between the tangent to any point along the tooth flank and the line of the tangent point to the apex of pitch angle.

• in general the spiral angle  is specified at the Mean Point  m ( i.e. at center of tooth width)

m

• the spiral angle is - minimal at the small end of gear - maximal at the large end of gear Klingelnberg AG · Training Center

Gear Geometry:

Module C.P.

P.C.D    mt   z

C.P. 

Transverse Module

b = =

mt

P.C.D. d C.P.   z z 

Diametral Pitch D.P.  Note:

dm1

Circular Pitch

mt 

[T1e_v10_WIR]


z  25.4   P.C.D. C.P. m te small D.P. large D.P.

dm2

D.P. 1inch

P.C.D. = pitch circle diameter

 coarse pitch  fine pitch

Mean Normal Modul

d2 = P.C.D.

dm

= mean diameter

z

= number of teeth

mn

dm1 cos  m1 dm2  cos  m2 mn   z1 z2 Klingelnberg AG · Training Center


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38

Gear Geometry:

Reference Planes

Transverse Plane ... is perpendicular to the axial plane and to the pitch plane

Normal Plane ... is normal to a tooth surface at the pitch point. In a spiral bevel gear, one of the positions of a normal plane is at a mean point. This plane is normal to tangent of the tooth length curvature and to the pitch plane at the mean point


For cylindrical gears: transverse section

normal section

[T1e_v10_WIR]


Gear Geometry:

Addendum / Dedendum / Whole Depth

for spiral bevel gears - similar to spur and helical gears - addendum and dedendum of uncorrected profile depth (tooth height) are selected as a factor of normal module. ha = hf = ks =

hf1

ha2

ha1

hf2

ks ha h hf

ha*  mnm hf*  mnm (hf*- ha*)  mnm

ks

typically factors ha* and hf* of tooth depth are selected in the range of: range

default (Face Hobbing)

addendum dedendum tip clearance

ha = hf = ks =

0.9 ... 1.10  m n 1.1 ... 1.35  m n 0.2 ... 0.30  m n

whole depth

h

ha + hf


=

ha = 1.00  mn hf = 1.25  mn ks = 0.25  mn


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39

Gear Geometry:

Tip - / Root Clearance

Pinion Root Clearance: ks1 minimum distance between pinion root to the tip of gear without any influence of the root fillet radius

hf1

ha2

ha1

hf2

ks 1

ks 2

Gear Root Clearance: ks2

minimum distance between root of ring gear to the tip of pinion without any influence of the root fillet radius. corrected addendum - dedendum respectively

ha1 = (ha* + xh)  mnm hf1 = (hf*- xh)  mnm

ha2 = (ha* - xh)  mnm hf2 = (hf*+ xh)  mnm

Note: the tip- root clearances of pinion and gear are supposed to be equal: ks1= ks2 Due to manufacturing tolerances (or – errors) of whole depth and/or tooth thickness, due to insufficient clearance interference might occur either in the root of pinion or in the root of gear. Klingelnberg AG · Training Center

[T1e_v10_WIR]


Gear Geometry:

Addendum / Dedendum / Whole Depth

factors related to tooth dimensions are specified differently acc. to standards below: KLINGELNBERG tooth depth factor addendum factor (related to Mean Normal Module m mn )

AGMA 2105

GLEASON

K1

= K/4

Cnom

C1

ha*

mean addemndum factor profile shift coefficient

hhm = xh

working depth

hmw = 2 x m mn x ha*

dedendum factor

hf* = hm/m mn - ha*

tooth whole depth

he = ( ha* + hf* ) x m mn

clearance factor

c = hf * - ha*

= hm/mmn - ( K/2 )

thickness modification factor

xsm = xs

= K3/2

thickness modification factor (old OERLIKON specification)

Ds = 2 xs


= K/2 x (0.5 - C1) hmw = m mn x K1

hmw = mmn x K/2 hf * = hm /m mn - ( K/4 )

hm = 1.15 hmw + 0.05 m mn


hm = 1.15 hmw + 0.05 m mn

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40

Gear Geometry:

Axial Backlash

with Spiral Bevel or Hypoid gears backlash is usually specified as radial backlash but set in terms of axial withdrawal back from metal-to-metal contact which represents the zero-backlash condition.

1

Slow-

Rapid-feed

There is no straight forward and definite relation between axial (J) and radial (jne) backlash as parameters such as - pitch cone angle pinion 1 - pressure angles n - and actual contact position

2

3

Axial Backlash

J

would have to be considered.

jne  2  ΔJ  sin  n  cos 1 4



Gear Geometry:

[T1e_v10_WIR]

Radial Backlash

radial backlash is therefore measured and set separately. Typically the pinion spindle is locked and radial backlash (B/L) is measured with an indicator either as: transversal radial backlash jte (measured in tangential direction at heel)

or normal radial backlash jne (measured perpendicular to tangent of flank at heel)

recommendation:

jne  (0.03  mnm ) + 0.05

Note: root land and tooth fillet of gear and pinion must be free of interference even at condition of zero- (or minimal) backlash Klingelnberg AG · Training Center


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41

Gear Geometry:

Virtual Helical Gears

for reasons of simplicity, certain parameters at any point along the face width of spiral bevel gears can be calculated, checked or explained on the base of the so-called Equivalent- or Virtual Helical Gears (according to Tredgold) Virtual Helical Gears are either considered with either: > infinitely small face width > helical angle corresponding to spiral angle at respective position

center lines of Equivalent Helical gears are crossing center lines of Bevel gears

or (acc. to DIN 3991 / ISO 10300) > original face width of bevel gear > helical angle = mean spiral angle Klingelnberg AG · Training Center

Gear Geometry:

[T1e_v10_WIR]


Undercut / Profile Displacement

undesirable undercut might occur with - high ratios - low number of teeth of pinion - deep profile (high dedendum factor)

Profile Displacement is therefore applied to avoid undercut, whereby:

undercut might appear in particular at toe end of pinion in case of parallel tooth depth (Face Hobbing) due to high dedendum factor in relation to corresponding normal module at toe

- upper limit: - lower limit:

xh1  m n   xh2  m n

hf1

h hf

profile displacement:

ha

hf T = mn x 

Pinion: (+) xh x mn positive Gear: (-) xh x mn negative

h hf2 T = mn *


h ha1

ha2

ha h

pointed teeth on pinion to avoid undercut


 [T1e_v10_WIR]

42

Gear Geometry:

Undercut / Profile Displacement

The influence of profile displacement with reference to the pinion is shown with below sections of tooth profiles e.g. z1=9 z2=41

Profile displacement …: • None to Small  undercut might appear • Larger  undercut disappears  pointed topland might appear towards toe Klingelnberg AG · Training Center

[T1e_v10_WIR]


Gear Geometry:

Tooth Thickness Correction

undesirable pointed teeth might appear at the pinion toe end due to: - large profile displacement - large pressure angles - large face width of pinion (in particular F.H. design) Tooth Thickness Displacement can therefore be applied in order to: - avoid pointed teeth (particularly for F.H. on pinion toe) - to balance root stresses (i.e. to balance life to rupture) between pinion and gear Note:



tooth thickness correction: Pinion: (+) xs x mn positive Gear: (-) xs x mn negative

typically tooth thickness will be - increased on pinion ( xs1 + ) - decreased on gear ( xs2 – )


T = mn *


xs x mn T = mn x



xs x mn [T1e_v10_WIR]

43


Gear Geometry:

General

es

Back cone

b

hfe

b 2

Re Ri

Rm



dai

f

E

a

k

crossing point Kb

[T1e_v10_WIR]


Blank Dimensions (parallel tooth depth) snm1

Blank Dimensions Pinion In order to avoid interference of the tips of blades with the front end of a bearing journal, a face angle modification can be applied.

OERLIKON denominations

b1

Ra

1 Rm 1

hk

1

AZ tI

h

Face- and root angles remain parallel but may deviate from pitch angle by some degrees.

Ri1

hf 1

tE

Note:

dki 1

dz

dm 1

d1

k1

dka 1

Limit for positive angle correction: k =  – k1 = 0° … ca. 5°  pointed tip of teeth at toe of pinion

1

Limit for negative angle correction: k =  – k1 = 0° … ca. – 3°  undercut at pinion toe Klingelnberg AG · Training Center

Axis of the mating gear

de

Face Width half of Face Width Outer Cone Distance Mean Cone Distance Inner Cone Distance Mounting Distance Angle Correction Generating Cone Angle Face Angle Pitch Angle Root Angle Outer Tip Circle Diameter Pitch Diameter Inner Tip Circle Diameter Whole Depth Addendum Dedendum

hae

: : : : : : : : : : : : : : : : :

hg

b b/2 Re Rm Ri Kb k E a  f dae de dai hges hae hfe

Blank Dimensions (constant tooth depth)

d ae

Gear Geometry:

- Zt 1

AP1 A1


[T1e_v10_WIR]

44

Gear Geometry:

Blank Dimensions (parallel tooth depth) snm2

OERLIKON denominations

In order to avoid interference of the tips of blades with the front end of a bearing journal, a face angle modification can be applied. Face- and root angle remain parallel but may deviate from pitch angle by some degrees.

Rm

2 Ri

2

2 k2

ha2 h2

b2

hf2

dki 2

tI

tE

A2

On the ring gear, any positive angle modification of the pinion appears negative …& vice versa

2

AP2

Ra

+ Zt 2

Blank Dimensions Ring Gear

dm 2

Note:

D2 (P.Diam.) dka 2

This kind of tip- / root-angle modification is applicable for Face Hobbing only

Gear Geometry:


hae

Raur

Back cone

Rinr

a

d

fe

de

dai



f



a

f

Axis of the mating gear

dae

Face Width Crown to Crossing Point Front Crown to Crossing Point Mounting Distance Addendum Angle Dedendum Angle Face Angle Pitch Angle Root Angle Outer Tip Circle Diameter Pitch Diameter Inner Tip Circle Diameter Whole Depth (*) Addendum (*) Dedendum (*) Crossing Point to Face Apex Crossing Point to Pitch Apex Crossing Point to Root Apex Shaft Angle

hges

: : : : : : : : : : : : : : : : : : :

Blank Dimensions (tapered tooth depth)

b

b Raur Rinr Kb a f a  f dae de dai hges hae hfe Gza Gzt Gzf 

[T1e_v10_WIR]


hfe


Crossing point

GZa GZt GZf

tB

0

Note: Tooth depth dimensions (*) refer generally to the outer diameter of gears


[T1e_v10_WIR]

45

4.

Offset and Hand of Spiral 4.1 General Definitions 4.2 Spiral Bevel Gears (LH - and RH - pinion) 4.3 Hypoid (LH - and RH - pinion) with positive offset 4.4 Hypoid (LH - and RH - pinion) with negative offset 4.5 Example of Application 4.6 Features of Spiral Bevel - / Hypoid Gears in comparison


General Definition:


[T1e_v10_WIR]

Hand of Spiral

the Hand of Spiral of bevel gears / pinions is specified in direction of view from cone apex: left hand right hand

L.H.  c.c.w. R.H.  c.w.

direction of view for hand of spiral

The correct or appropriate hand of spiral is usually selected as such that the main application of gear set results in axial thrust of pinion towards the main bearing of the shaft.  see section 11 : calculation of axial and radial forces Klingelnberg AG · Training Center


[T1e_v10_WIR]

46

General Definition:

DRIVE - / COAST Flank

Ring Gear:

HEEL

DRIVE flank

convex flank = DRIVE flank concave flank = COAST flank

COAST flank HEEL

Pinion: concave flank = DRIVE flank convex flank = COAST flank

TOE TOE

Heel

=

outer (large) end of tooth

Toe

=

inner (small) end of tooth

DRIVE flank COAST flank

DRIVE flanks

loaded (in mesh)

in sense of rotation forward

COAST flanks

loaded (in mesh)

in sense of rotation reverse in coasting conditions forward

or: Klingelnberg AG · Training Center

[T1e_v10_WIR]


Spiral Bevel Gears:

Offset = 0

Spiral Bevel gear sets can be designed and applied without restrictions: pinion: L.H. r. gear: R.H.

• shaft direction may be either to right or to left side of ring gear • pinion may be positioned either in front or behind ring gear

m2

m1 = m2 Spiral angels are typically selected: pinion / gear


pinion: R.H. r. gear: L.H.

m1

dm1

Spiral angles are the same for pinion and ring gear:

m = 30° ... 35°


[T1e_v10_WIR]

47

ZEROL Gears:

Offset = 0

a special application of Spiral Bevel Gears is known as ZEROL gears.

m1 = m2 = 0° The main aspect of this a application is to keep axial thrust of the pinion as small as possible and thrust in all directions positive for pinion and ring gear.

m = 0°

• relatively large cutter diameter need to be considered • in particular with Face Hobbing spiral angles at the mean point should be selected as such that spiral angle at toe will not get negative  at toe: i > 0° Klingelnberg AG · Training Center

m = 3 ... 5°

[T1e_v10_WIR]


Hypoid Gears :

Positive Offset ( L.H. pinion / R.H. gear )

Hypoid gears are usually designed with positive offset, i.e. with larger spiral angle on pinion than on gear.

Hypoid pinion

av

m1 > m 2

• by indicating the offset “below center” it is customary to look at the face of the gear with the pinion shaft to the right

Spiral Bevel pinion

m2

pinion: R.H. r. gear: L.H.

av

Spiral angels are typically selected: pinion gear Klingelnberg AG · Training Center

m1 m2

m1

dm1'

This results in a larger pinion diameter than compared with the spiral bevel pinion of the same ratio

= 45° ... 50° = 27° ... 32° [Basics of Bevel Gears] · Slide 96

[T1e_v10_WIR]

48

Hypoid Gears :

Positive Offset ( R.H. pinion / L.H. gear )

Hypoid gears are usually designed with positive offset, i.e. with larger spiral angle on pinion than on gear.

m1 > m 2

av

This results in a larger pinion diameter than compared with the spiral bevel pinion of the same ratio. • by indicating the offset “above center” it is customary to look at the face of the gear with the pinion shaft to the right

pinion: R.H. r. gear: L.H. av

Spiral angels are typically selected: pinion gear

m1 m2


Hypoid Gears:

= 45° ... 50° = 27° ... 32° [T1e_v10_WIR]


Negative Offset ( L.H. pinion / R.H. gear )

Hypoid gears can exceptionally be designed with negative offset, i.e. with smaller spiral angle on pinion than on gear. m1 < m2

av

This results in a smaller pinion diameter than compared with the spiral bevel pinion of the same ratio.

• by indicating the offset “above center” it is customary to look at the face of the gear with the pinion shaft to the right For geometrical reasons (small pinion diameter) negative Hypoid offset can be only very small. This case is rarely applied in automotive drives. Klingelnberg AG · Training Center

m2

pinion: L.H. r. gear: R.H.


av m1

[T1e_v10_WIR]

49

Hypoid Gears:

Negative Offset ( R.H. pinion / L.H. gear )

Hypoid gears can exceptionally be designed with positive offset, i.e. with smaller spiral angle on pinion than on gear. m1 < m2

av

This results in a smaller pinion diameter than compared with the spiral bevel pinion of the same ratio. pinion: R.H.

• by indicating the offset “below center” r. gear: L.H. it is customary to look at the face of the gear with the pinion shaft to the right

av

For geometrical reasons (smaller pin. diameter) negative Hypoid offset can be only very small. This case is quite rarely applied in automotive drives. (Exception: PTO – type gear sets with i < 2 ) Klingelnberg AG · Training Center

Bevel Gears :


[T1e_v10_WIR]

Summary of Cases most common application

Spiral Bevel: pinion L.H. / gear R.H.

Hypoid - positive offset: pinion L.H. / gear R.H.

Hypoid - negative offset: pinion L.H. / gear R.H.

Spiral Bevel: pinion R.H. / gear L.H.

Hypoid - positive offset: pinion R.H. / gear L.H.

Hypoid - negative offset: pinion R.H. / gear L.H.



[T1e_v10_WIR]

50

Hypoid Gears :

Range of Hypoid Offset

Hypoid offset is selected by the designer who in consideration of the application of the Hypoid axle - selects a balanced choice between a number of properties of opposing characteristics.

D2

av

>> see page 107

Typically Hypoid offset is selected in the range indicated below: Application:

av in % of Gear  D2

Passenger Car Truck

av

D2

10 ... 25 % 8 ... 12 %


[T1e_v10_WIR]


Hypoid Gears :

Arrangement of Gears and Offset for 4WD

• in automotive drive-lines the Hypoid offset (the drive shaft) is usually arranged below the center line of ring gear. ( lower center of gravity, stronger pinion, higher total contact ratio, etc.) • in case of 4WD concepts it is therefore required that the selected hand of spiral of gear and pinion for the same direction of rotation of ring gear is providing either “Driving” or “Coasting” conditions for rear- and front axle. In order to apply positive offset for both drives, the pinion needs to be placed once in front and once behind the gear. The offset will therefore be above center for e.g. front axle. L.H. pinion / R.H. gear

R.H. pinion / L.H. gear

offset + offset +

pinion with equal sense of rotation: pinion in front of gear pinion behind gear forward direction rear axle Klingelnberg AG · Training Center


forward direction front axle [T1e_v10_WIR]

51

Hypoid Gears :

Arrangement of Gears and Offset for 4WD

• in case of 4WD concepts it is required that the selected hand of spiral of gear and pinion for the same direction of rotation of ring gear is providing either “Driving” or “Coasting” conditions for rear- and front axle. • in order to apply positive offset for both drives with the pinion arranged in both cases at the same side of gear, the sense of rotation must necessarily be different for rear- and front axle. • the same hand of spiral for both gear sets results as an advantage of this concept L.H. pinion / R.H. gear

L.H. pinion / R.H. gear

offset + offset +

pinion with unequal sense of rotation: pinion of both axles on same side of gear forward direction rear axle Klingelnberg AG · Training Center

forward direction front axle

Spiral Bevel Gears:

Tandem- / Drive-Through Axle

• In case of tandem- or multiple drivethrough axles there appears a similar problem concerning the arrangement of the gear-set and the selection of hand of spiral. Due to the reverse direction of drive shaft and pinion, the last axle requires a mirrored arrangement of the set and inverse hand of spiral Pinion R.H. / Gear L.H.

offset >0 non-offset HypoidBevel Spiral gears Klingelnberg AG · Training Center

[T1e_v10_WIR]


see page 19

 last AP axle  drive-throughAP axle

Pinion L.H. / Gear R.H.

forward direction of coupled (Tandem-) axles forward direction of coupled (Tandem-) axles [Basics of Bevel Gears] · Slide 104

[T1e_v10_WIR]

52

Geometry of Gear Blanks for:

ratio: face width pinion:

Z2 d2 dm2   Z1 d1 dm1 b1  b2 i

diameter pinion:

1 dm1   dm2 i

spiral angles:

m1  m2

1

b2

dm1 d1

b1

SPIRAL BEVEL Gears:

2 dm2 d2

2

Z1 Z2 (for 90° shaft angle) Z2 2  arctan Z1 1  2  90 pitch angles:

1  arctan


1

[T1e_v10_WIR]


Geometry of Gear Blanks for: SPIRAL BEVEL Gears:

pinion (1)

pressure angles are usually balanced for DRIVE / COAST flanks

the Limit Pressure Angle lim is lim = 0° for Spiral Bevel Gears Drive flanks: cv pinion + cx gear

0 v1  Nv1

Coast flanks: cx pinion + cv gear

0 x1  Nx1


convex flank concave flank

convex flank

0 x 2  Nv1

0 v 2  Nx1

concave flank

gear (2) [Basics of Bevel Gears] · Slide 106

[T1e_v10_WIR]

53

Geometry of Gear Blanks for:

b1

1

face width pinion:

b2

2

1 dm1  k   dm2 i cos m2 enlargement factor: k cos  m1 spiral angles:  m1  m2

dm2

diameter pinion:

pitch angles:

1  arctan

Z1 Z2

d2

2

1  2  90  2  arctan

Z2 Z1

1

 Z2  1,2  f  ,offset,cutter,cutting  system   Z1  Klingelnberg AG · Training Center

d1

Z2 d2 dm2  rsp.  Z1 d1 dm1 b1  b2 i

ratio:

dm1

HYPOID Gears:

[T1e_v10_WIR]


Geometry of Gear Blanks for: HYPOID Gears:

pinion (1)

pressure angles are usually not balanced for DRIVE / COAST flanks in order to result in a balanced length of the Line of Action in both directions of operation. the Limit Pressure Angle lim > 0° is usually positive for positive Hypoid offset Drive flanks: cv pinion + cx gear

0 v1  Nv1  lim

Coast flanks: cx pinion + cv gear

0 x1  Nx1  lim


convex flank concave flank

convex flank

0 x 2  Nv1  lim

0 v 2  Nx1  lim [Basics of Bevel Gears] · Slide 108

concave flank

gear (2) [T1e_v10_WIR]

54

... some properties in comparison properties

SPIRAL BEVEL Gears:

HYPOID Gears:

diameter pinion

:

smaller (  )

bigger

tangential force at dm1 resulting input torque

:

bigger

smaller (  )

()

()

strength to rupture & pitting :

smaller (  )

bigger

()

spiral angle pinion

:

smaller (  )

bigger

()

axial thrust of pinion as result of input torque

:

smaller (  )

bigger

()

length of tooth trace

:

smaller (  )

bigger

()

face contact ratio

:

smaller (  )

bigger

()

sliding velocities on flanks :

smaller (  )

bigger

()

resistance against scoring :

higher

smaller (  )

heat generated into oil

:

smaller (  )

bigger

efficiency factor

:

bigger

smaller (  )


() ()

[T1e_v10_WIR]


Flank Sliding Velocities SPIRAL BEVEL Gears:

()

direction of velocities

HYPOID Gears:

Note: sliding velocities are larger in case of Hypoid gears

• sliding velocities appear only in profile direction • no sliding velocities appear along the pitch line Klingelnberg AG · Training Center

• sliding velocities appear both in length- and profile direction • sliding velocities appear also along the pitch line [Basics of Bevel Gears] · Slide 110

[T1e_v10_WIR]

55

Flank Sliding Velocities

consequence for lapping of gear-sets

SPIRAL BEVEL Gears:

HYPOID Gears:

lapping removal with protuberance

lapping removal with protuberance

lapping removal without protuberance

lapping removal without protuberance

• no lapping effect along pitch line

• lapping effect along pitch line too

• increasing lapping removal towards tip and root starting p.line

• more regular lapping removal across profile depth

• narrow contact after long lapping

• better lapping abilities


[T1e_v10_WIR]


Flank Sliding Velocities

consequence for lapping of gear-sets

Spiral Bevel & Hypoid Gears: protuberance to avoid lapping step

no protuberance could create lapping step

• the appearance of a lapping step is avoided by applying a protuberance to the tool tips; protuberance at the tool tip is causing “root relief” in the root of teeth being cut • protuberance is usually applied only to pinions to be cut and lapped or to rough-cut gears & pinions which are to be flank-ground only (fillet radii and bottom land to remain not ground!) • ring gears (in particular gears of high ratios) don’t need protuberance due to the fact that lapping removal is much less compared to pinion Klingelnberg AG · Training Center


[T1e_v10_WIR]

56

5.

Face Milling / Face Hobbing in comparison 5.1 Cutting Method - Features 5.2 Indexing System 5.3 Tooth Depth 5.4 Tip- / Root Clearance 5.5 Tooth Gap Width 5.6 Generating Plane Gear 5.7 Contact Lines / Cutting Lines 5.8 Blade System 5.9 Machine Capacity 5.10 Deflection Characteristic



[T1e_v10_WIR]

Face Milling methods

Face Hobbing methods

Single Indexing

Continuous Indexing

in general cutting methods are either Multi-Cut or Completing

continuous indexing methods are generally Completing systems

Spread Blade (Fixed Setting) 2 cutting operations for Gear Roughing Finishing

3 cutting operations for Pinion Roughing Finishing convex flank Finishing concave flank

DUPLEX completing method for Pinion and Gear

1 cutting operation for Gear & Pinion - basically 1 m/c required only - less handling operations - less floor space - more flexibility in production Alternatively: diverse multi-cut operations are possible 1 or 2 operations for Gear 2 or 3 operations for Pinion

 there is also the possibility of a so-called Mixed Completing (i.e. pinion Completing + Gear rough- & then finish-cut) Klingelnberg AG · Training Center


[T1e_v10_WIR]

57

Face Milling

some facts in comparison

• Hard Finishing Processes can either be lapping or grinding

Face Hobbing

• Hard Finishing Process for mass production is restricted to lapping • HFP could be skiving, however, FORM method for ring gears cannot be applied

• in case of ground applications HT distortions can be eliminated

• HT distortions can be pre-compensated but never completely be eliminated

• designed EaseOff can be achieved within a few microns deviations only

• Lapped i.e. final EaseOff can only be assumed based on experience

• design / application of small cutter developments are extremely difficult

• design / application of small cutter developments easy to perform (some restrictions in lapping apply)

• cutting times generally higher than FH due to required indexing and idle times for back-roll • higher expenditure in production for rough-cutting and grinding of single components

• smaller expenditure in production for cutting and lapping in pairs [T1e_v10_WIR]


Face Milling

Face Hobbing

Arc of a Circle

RW

ro of tat cu ion tte r

Indexing Method

ro t of ation cut ter


• Cutting times generally shorter than F.M.

Epicycloide RW

ro of tatio wo n rk

in of dex wo ing rk

RBM

• each single tooth is cut in a consecutive manner

• all teeth are cut “simultaneously”

• tooth length curvature = Arc of Circle  radius of curvature is constant and generally larger than with F.Hob. Klingelnberg AG · Training Center

• tooth length curvature = Epicycloide  radius of length curvature is not constant and generally smaller than with F.Mill.


[T1e_v10_WIR]

58

Face Milling

Indexing Method

Face Hobbing rotation of work

outside blade

outside blade

inside blade

inside blade

center of cutter

cutter center

rotation of cutter

rotation of cutter

• radii of inside (IB) - and outside (OB) blades are different

• radii of IB- and OB blades are theoretically identical ( see section. 6.3) (without profile- and tooth thickness correction)



Face Milling (5-Cut)

Length Crowning

rfl concave (pinion) rfl convex (pinion)

[T1e_v10_WIR]

Face Milling

• with all single indexing – (except some Jobbing –) cutting methods, roughing as well as finishing means cutting both flanks of the ring gear with 1 single tool only • Thereby inevitably two different radii of length curvature between the concave and convex flank are created

rfl concave (gear) rfl convex (gear)

• With the 5-cut method the pinion needs therefore to be cut with 2 different tools in order to create a reasonable amount of length curvature • reason for two individual tool diameter to generate the pinion: rfl convex (pinion) >> rfl concave (pinion)



[T1e_v10_WIR]

59

Face Milling (Completing) Length Crowning Face Milling (Completing) LB = large LB = small

Completing with relatively SMALL tool diameter

Completing with relatively LARGE tool diameter

• difference of radii of length curvature concave – convex is relatively large • a lot of crowning needs to be reduced to obtain reasonable length of contact Klingelnberg AG · Training Center

Face Milling (Completing)

• difference of radii of length curvature concave – convex is relatively small • less length crowning needs to be reduced to obtain a reasonable TCP [T1e_v10_WIR]


Face Hobbing

Length Crowning

m

m

a) plunge position below b) with center virtual tilttilt (without of of tool tool)

• Plunge Cut of Gear

• Plunge Cut of Gear - with (virtual) tilt of tool  > 0° - replaced by pl. pos. below center

 a)  b)

• length crowning caused by difference of cx/cv radii is reduced Klingelnberg AG · Training Center

- without tilt of tool  = 0° - at center roll position T = m

• the cut root line is straight


[T1e_v10_WIR]

60

Face Milling (Completing) Tip – Root Clearance

Face Hobbing

root of pinion to tip of gear

root of pinion to tip of gear

root of gear to tip of pinion

root of gear to tip of pinion

• cut root line of gear is heavily curved • clearance root-gear to tip-pinion is considerably reduced at toe and heel • tip clearance at mean point must be increased in order to avoid interference at both ends of tooth Klingelnberg AG · Training Center

Face Milling


Tooth Depth

• tooth depth is tapered • slot width is constant or tapered (*) - constant - slightly tapered

for Tilted Root Line for Standard Root Line

• width of top land is even (*) • width of root land is even (* see above) Klingelnberg AG · Training Center

• cut root line of pinion is slightly curved ( similar as with Face Milling) • cut root line of gear is straight • there is no negative disturbance of the tip-root clearance • whole depth does must not be increased [T1e_v10_WIR]

Face Hobbing

• tooth depth is parallel • slot width is tapered • top land pinion might become pointed at toe in case of high ratios  2nd face angle is to be applied in order to avoid pointed teeth


[T1e_v10_WIR]

61

Face Milling

Tapering of Tooth Depth

Tool Diameter Tool  / Mean Cone Distance

rW = 4.5“ rW / Rm = 0.7

Face Milling

Tool Diameter Tool  / Mean Cone Distance

rW = 7.5“ rW / Rm = 1.1

• The tapering of tooth depth is characterized by the Sum of Dedendum Angles • In case of DUPLEX taper the sum of dedendum angles depends on the mean spiral angle, the relative size of tool diameter and face module • The sum of dedendum angles of pinion and gear is:   f = f_pinion +  f_gear  90  m et ΣΚ f _ Duplex   R  tan  n  cos m  e2

  Rm 2  sin m 2    1      rW   

The larger the Tool Diameter and the smaller the Spiral Angle the larger the Sum of Dedendum Angles; i.e. the more tapered the tooth depth Klingelnberg AG · Training Center

[T1e_v10_WIR]


Face Milling

Tooth Gapwidth

Face Hobbing he

he

hi < he hi = he hi

• in particular with all completing cutting • the width of tooth gap is tapered methods the gap width is constant • the size of fillet radii in the root are • the size of fillet radii in the root can be restricted by the width of gap at toe optimized with reference to the whole • in case of relatively large cutter tooth length diameter and/or large tooth width, the • regardless to cutter diameter and tooth blade point width OB/IB might not fully width, the point width of blades can be overlap at the heel selected for optimal overlap Klingelnberg AG · Training Center


[T1e_v10_WIR]

62

Face Milling

Tooth Depth: EXCEPTION !

• parallel tooth depth appears exceptional for gear cutting methods: - WIENER 2 - Track - CURVEX (Modul) - SARATOV - SemiCompleting (= grinding method producing circular tooth length form for gears pre-cut to Face Hob system)

• tooth depth is tapered in general for all gears cut with circular tooth length curvatures according to Face Mill cutting method


Face Milling

Face Milling


Generating Plane Gear Plane Gear for pinion Plane Gear for gear

[T1e_v10_WIR]

Face Hobbing Plane Gear for gear + pinion

• Generating Plane Gears are different for pinion and gear

• Plane Gears are theoretically identical for pinion and gear

• axes of Gen. Plane Gear do not coincide  tooth depth tapered

• axes of Gen. Plane Gear do coincide  tooth depth parallel

• conjugate gear flanks not applicable

• allows to cut conjugate gear flanks



[T1e_v10_WIR]

63

Face Milling

Generated Cutting Lines

Face Hobbing

computed determination of generated cutting lines  both cases represent similar gear dimensions

heel

toe

• with circular tooth length form of single indexing F.M. cutting methods, the direction of generated cutting lines proceeds relatively steeply inclined along the height of tooth profiles • therefore a larger generating interval is required compared to Face Hobbing results in longer cutting time Klingelnberg AG · Training Center

toe

• caused by the simultaneous rotation of cutter and work with continuous indexing F.H. cutting methods, the direction of generated cutting lines proceeds relatively gently inclined along the height of tooth profiles

[T1e_v10_WIR]


Face Milling contact line

heel

Contact Lines cutting direction

cutting direction

Face Hobbing contact line cutting line

cutting line

cont. Index rotation generation roll

generation roll

• generated cutting lines are parallel with contact lines (they do coincide in direction) • cutting lines are in general shorter than with Face Hobbing • with large feed rates, generated cutting lines could appear as flats Klingelnberg AG · Training Center

• generated cutting lines are crossing the contact lines under an oblique angle • with large feed rates, generated cuttings lines could appear as flats on flanks of gener. gears /pinions, however, not as critical as with FM


[T1e_v10_WIR]

64

Face Milling

Face Hobbing

Contact Lines

contact line

contact line

generated cutting lines

generated cutting lines

• feed marks (generated cutting lines) are parallel to contact lines

• feed marks (generated cutting lines) are NOT parallel to contact lines; they are crossing each other

• in order to obtain good surface finish and good running behavior, blades must set and trued within very close tolerances (<1 m)

• larger tolerances can be allowed to still obtain good running behavior and low gear noise


Face Milling

[T1e_v10_WIR]


Blade System

• traditionally using standard pre-profiled blades  with ARCON-, RSR® or PENTAC-FM®

useful blade-width grinding stock

systems, however, bar blade tooling is available and widely established too)

grinding stock

useful blade-length

• grinding rake-face only • relatively large amount of stock required to remove wear-marks

Face Hobbing • using mainly bar blades • grinding rake- & clearance faces  small grinding stock • large useful length of bar allows to regrind blades up to 120 x • can be purchased from different suppliers of the user’s choice (... in case of rectangular cross section of sticks) Klingelnberg AG · Training Center


[T1e_v10_WIR]

65

Face Milling

Face Hobbing

Machine Capacity

RW

RW

radial distance

radial distance

RM

• radial distance is smaller for same dimension and gear parameters • requires less m/c capacity or allows larger max. dimensions with given m/c capacity Klingelnberg AG · Training Center

RM

• radial distance is larger for same dimensions and gear parameters • requires more m/c capacity or allows smaller max. dimensions with any given m/c capacity [T1e_v10_WIR]


Face Milling

Direction of Displacement

Face Hobbing RW

RW

RBM

+V

-V

V-

V+ +H

+H

-H +H

+H

-H -H

-H

V+

• for H+ of pinion, TCP moves to: - tip and heel for DRIVE - tip and toe for COAST Klingelnberg AG · Training Center

+V

V-

-V

• for H+ of pinion, TCP moves to: - tip and toe for DRIVE - tip and heel for COAST [Basics of Bevel Gears] · Slide 132

[T1e_v10_WIR]

66

6.

Basics of the Continuous Indexing System 6.1 The Epicycloide as Tooth Length Curvature 6.2 Position of Cutter Center 6.3 Cutters for the Continuous Indexing System 6.3.1

2-Blade/Group Cutter with z0 = 1

6.3.2

2-Blade/Group Cutters with higher z0

6.3.3

3-Blade/Group Cutters with higher z0

6.4 Conclusion


[T1e_v10_WIR]


Eb

Tooth Length Curvature: rW

of the Face Hobbing System is based on the development of the Elongated Epicycloide The Epycycloide develops as the path of the elongated cutter radius ( r = rW ) of a ROLLING CIRCLE ( r = Eb ) which rolls under the condition of no slip on a BASE CIRCLE ( r = Ey)

Ey

This condition is described with the equation:

Eb z 0  E y zp

Note: A tooth length curvature representing an Epicycloide is achieved by the simultaneous rotation of cutter and work. Klingelnberg AG · Training Center

Zp Z0 Eb

No. of teeth of Generating Plane Gear No. of Blade Groups on Cutter is proportional to Normal Modul and z0


[T1e_v10_WIR]

67

Tooth Length Curvature:

The appearance and the development of an Epicycloide can be nicely demonstrated in function of varying: - (a) base circle radius - (b) rolling circle radius (link to URL) - (c) elongated radius of rolling circle (= cutter radius) Klingelnberg AG · Training Center

[T1e_v10_WIR]


The Position of Cutter Center: the position of the cutter center relative to the center of the Generating Plane Gear is an important parameter of gear geometry. It is the major limiting factor of the required machine capacity. This distance is called 

Eb

Ey

S

(see p. 53)

Radial Distance

S

Eb

or is also referred to as:



Eccentric Distance

S

Ey

S = Eb + Ey Eb Ey

Radius of Rolling Circle Radius of Base Circle of Epicycloide



[T1e_v10_WIR]

68

Cutters for Face Hobbing:

roughing blade

the Face Hobbing cutting system is performed by rotating cutters, equipped with blade groups consisting of either 2 or 3 different blades:  RB roughing blade  OB outer finishing blade  IB inner finishing blade

1 blade group

outside finisher

• in order to maintain the previous condition, cutter and work rotate simultaneously and in timed relation • all blades of the same blade group are passing the same tooth gap • the next following blade group is passing the successive tooth gap Klingelnberg AG · Training Center

inside finisher

[T1e_v10_WIR]


The Continuous Indexing: RWOB

theoretical cutter z0 = 1 to explain the continuous indexing we consider the most simple cutter there is: 1 OUTSIDE finishing blade

+

=

RWIB

IB

OB

OB

( in opposite position )

1 INSIDE finishing blade

IB

180°

• both blades are arranged on the identical radius RW. • while rotating the cutter for 180° the former position of OB is taken by IB, meanwhile the gear has rotated by ½ of the Pitch = 180°/z Klingelnberg AG · Training Center

½  mn   180° / Z


[T1e_v10_WIR]

69

The Continuous Indexing: theoretical cutter z0 = 1 blade sequence angle:  OB  IB = 360°/ 2 / z0  IB  OB = 360°/ 2 / z0 = const.

Note: - radius of rolling circle is very small for z0 = 1 - rpm of work is relatively low compared to rpm of cutter - cutting direction from toe to heel


[T1e_v10_WIR]


The Continuous Indexing: 2-blade cutters with higher z0 • to provide high productivity with Face Hobbing cutters, the total number of blades; i.e. the number blade groups (z0) shall be high. • z0 can be increased as much as the cross section of blade shafts allows to arrange blades on the nominal cutter diameter  = 2  rW

EM

1 blade group RW

• for the number of blade groups z0 it is generally preferred to apply prime numbers in order to avoid the possibility of creating common factors with numbers of teeth to be cut Klingelnberg AG · Training Center

RW

z0 = 5


EM

z0 = 11

[T1e_v10_WIR]

70

The Continuous Indexing: 2-blade cutters with higher Z0 blade sequence angle:  OB  IB = 360°/ 2 / z0  IB  OB = 360°/ 2 / z0 = const.

Note: - radius of rolling circle is larger for higher number of blade groups; radius of rolling circle is proportional to number of blade groups - rpm of work is higher for larger number of blade groups



[T1e_v10_WIR]

The Continuous Indexing: 3-blade cutters with higher Z0 • with equidistant blade positions the sequence angle between finishing blades is wi = ½ w = 360°/2/ z0 • the displacement of a blade pitch point takes place along an Involute based at the rolling circle of the Epicycloid • with non-equidistant blade sequence angles, cutter blade radii need to be corrected in order to maintain the nominal pitch and tooth thickness  Note: multitudes of involutes of the same base circle feature constant distance along their entire length of curvature Klingelnberg AG · Training Center


[T1e_v10_WIR]

71

The Continuous Indexing: 360 z0

3-blade cutters with higher Z0

w 

• the displacement of blade cutter radii Rb can be derived according to the simple equations aside

 w 360    wi   w 2 2  z0 RB t   w w 2  t RB  w  2 w

• the Crowing Factor F = w x zo specifies the angular difference between the equidistant and the actual angular distance between the finishing blades of a blade group of a given cutter • the angular distance between the finishing blades can therefore be determined in function of the blade sequence angle and the number of starts: Blade Sequence angle wi = (180° - F ) / zo Klingelnberg AG · Training Center

  w mn    z 0  2 360     w  z0    mn 720  RB  F   mn 720 

[T1e_v10_WIR]


The Continuous Indexing: 3-blade cutters with higher Z0 • in order max. blade numbers with 3 blades / group can be achieved, the equidistant position of blades OB  IB  OB  ..... needs to be changed:  finishing blades OB - IB are to be positioned at closer proximity:

 wi < ½   w

(  wi < 180° / z0 )

rotation of cutter

RW

RB

EM

(L1/1) RB

OB

wi

(L1/2)

1/2

w

RB

• this allows to position RB blades • in order to maintain correct pitch of cut tooth slots, blade radii are corrected: RWOB = RW + RB RW IB = RW – RB Klingelnberg AG · Training Center


IB (L1/3)

[T1e_v10_WIR]

72

The Continuous Indexing: 3-blade cutters with higher Z0 blade sequence angle is not const.  OB  IB   IB  OB

Note: - radius of rolling circle is larger for higher number of blade groups; radius of rolling circle is proportional to number of blade groups - rpm of work is higher for larger number of blade groups


[T1e_v10_WIR]


3-blade cutters with higher Z0 • cutting pinion and gear, the combination of a L.H. + R.H. cutter designed to the previous principle results in a certain amount of length crowing • the so called Crowning Factor of cutters is defined as: F = wi  z0 = (½ w  wi)  z0 • concave flank pinion

B   R ex RW r conv gea B ve  R ca  con RWnion pi

RW + RB

is meshing with

convex flank gear

RW – RB

• convex flank pinion

RW – RB

is meshing with

concave flank gear Klingelnberg AG · Training Center

RB v e   nca o RW ar c ge

B x  R ve  con RWnion pi

The Continuous Indexing:

gear flanks

pinion flanks

RW + RB [Basics of Bevel Gears] · Slide 146

[T1e_v10_WIR]

73

The Continuous Indexing: 3-blade cutters SPIRAPID

IB (R2/6)

• the SPIROFLEX / SPIRAC cutting system requires a pair of cutters resulting in ZERO (0) – crowning in combination of the L.H.+ R.H. cutters • for this reason the normal blade sequence as applied in pinion cuter RB  OB  IB is changed to: OB  RB  IB

RB

rotation of cutter

RB (R2/5)

wi

w

(R2/4)

EM RB

RW

• this special arrangement of blades, applied on SPIRAPID cutters results in:  high blade density with 3 bl./gr.  zero–crowning for cutter tilt = 0°


1/2

OB


[T1e_v10_WIR]

The Continuous Indexing Method: CONCLUSION: • as the result of identical tooth length curvatures ( zero - crowning ) we would obtain with Face Hobbing: full-length contact pattern  conjugate teeth • for practical application conjugate gears are not useful ! • in order to allow tolerances in gear manufacturing & assembly and to allow reasonable deflections under loaded conditions, the following intentional corrections are introduced: modification

achieved with (in general):

 length crowning  profile crowning  BIAS (tooth twist)

cutter tilt curvature of blade profile m/c settings and/or modified motions



[T1e_v10_WIR]

74

Features of Continuous Indexing ( Face Hobbing ) :  short setup and change-over times, pinions and ring gears cut economically in one completing process  universal application of one (1) machine for both pinions + ring gears (compared to 5-cut method)  simplicity of gear design and development; parallel tooth depth  F.H. not appropriate as roughing method prior to grinding for mass production  manufacture of conjugate teeth is theoretically possible  high performance of gear-set as the result of small cutter design: – high contact ratio – pocketing effect – low sensitivity of contact to deflections under load – hence, long bearing patterns can be applied  contact lines & feed marks crossing each other under different directions; – larger tolerances for tool setting and surface roughness than F.M. Klingelnberg AG · Training Center

7.


[T1e_v10_WIR]

Particularities of Epicycloide as Length Curvature 7.1 Point of Involute (special case: N-type gear) 7.2 Point of Involute (general case) 7.3 Location of Point of Involute 7.4 Epicycloide Length Curvature: Basic Relations 7.5 Length Curvature for large cutter diameter 7.6 Length Curvature for small cutter diameter 7.7 Variations of Normal Module & Spiral Angle for different Cutter Diameter 7.8 High Flexibility of Face Hobbing



[T1e_v10_WIR]

75

Point of Involute (N-point)

z0

No. of blade groups

zp

No. of teeth of Gener. Plane Gear

Eb

radius of rolling circle = direction of orientation of blades

Eb =

½ x m p x z0

rB =

RP x sin p

condition:

Eb / Ey = z0 / zp


rotation of cutter RW

EB

rotation of work

RB RP RA

A_ BET

RI

P

EY

radius of curvature at calculation point

B

rb

N-Point (point of involute)

x 15

cutter radius

0.4

rw

special case : N-type gear

EX

RM B

[T1e_v10_WIR]



special case : N-type gear

Definition:

Epicycloide the N-point represents that position of the tooth length curvature, where the instantaneous radii of the Epicycloide and of the Involute of the same rolling circle are identical.

Involute

rolling circle inst. radius of curvature

base circle of epicycloide



[T1e_v10_WIR]

76

Point of Involute (N-point) cutter radius

rbm

radius of curvature at mean point

rotation of cutter

N-Point (point of involute)

(instantaneous radius of length curvature: graphical method according to Euler - Savary)

z0

No. of blade groups

zp

No. of teeth of Gener. Plane gear

Eb

radius of rolling circle

Em =

½  mnm  z0

Em

orientation of blades

RW

EBM RBM

_2

EX

rotation of work

Note:

EB / EY = z0 / zp

in general case the orientation EM  radius of rolling circle EB [T1e_v10_WIR]


Point of Involute (N-point) Definition:

EY

2 RM B TM

condition:


EB

rW

general case

general case

Epicycloide Involute

the N-point represents that position of the tooth length curvature, where the instantaneous radii of the epicycloide and of the involute of the same rolling circle are identical.

rolling circle base circle of epicycloide

inst. radius of curvature



[T1e_v10_WIR]

77


specification of location §

The position of Point of Involute is practically specified either as: Difference to the Outer Cone Dist. : Rinv - Re or as: Position Factor N-Point : Ni

(previously used notion by OERLIKON) (newly introduced notion in KIMoS)

or as:

Ratio of “Involute-to-Outer Cone” : Rinv / Re

(notion used by GLEASON)

Re

Re

Re

Rinv

Rinv

Rinv - Re < Rinv / Re < Ni =

0 1 0


Rinv

Rinv - Re = 0 Rinv / Re = 1 Ni = 1

Rinv - Re > 0 Rinv / Re > 1 Ni  2 [T1e_v10_WIR]


Epicycloide Length Curvature:

Basic Relations

from the rolling condition of the Epicycloide the radial displacement EM can easily be derived: RWM

RW

z0 m  Y  z0  n zp 2



RM

mn  tan  m 2 m Y  Rm  cos  m  zp  n 2 EM EB z0   Y EY zp

m

EY

RBM

X  Rm  sin  m  zp 

EM 

EM EBM EB

2  Rm  cos m zp

rot o f ation wo rk

mn 

rot of ation cu tte r

EB z0  EY zp

EX

Y

X

[T1e_v10_WIR]

78

Length Curvature large cutter Ø :

variation along face length of: Curvature Radius / Spiral Angle / Normal Module

• Radius of Length Curvature: at toe : RBi < RBm at heel : RBa > RBm

rotation of cutter

difference : RBa – RBi (in % of RW) is smaller than with small cutters rotation of work

• Spiral Angle: at toe : i < m at heel : a > m

a

difference : a – i is smaller than with small cutters

m i

• Normal Module: at toe : mn_i < mn_m at heel : m n_a > mn_m

difference : mn_a – mn_i is larger than with small cutters; max. normal module is off heel Klingelnberg AG · Training Center

[T1e_v10_WIR]


Length Curvature small cutter Ø:

variation along face length of: Curvature Radius / Spiral Angle / Normal Module

• Radius of Length Curvature: at toe : RBi < RBm at heel : RBa > RBm rotation of cutter

difference : RBa – RBi (in % of RW) is bigger than with large cutters • Spiral Angle: at toe : i < m at heel : a > m

rotation of work

a m

difference : a – i is bigger than with large cutters

i

• Normal Module: at toe : mn_i < mn_m at heel : mn_a > mn_m

difference : mn_a – mn_i is smaller than with large cutters; max. normal module near center of tooth width Klingelnberg AG · Training Center


[T1e_v10_WIR]

79

Variations of Normal Module and Spiral Angle for different Cutter Radii

Spiral Angle [°] 50 40 30 20

• note different development of toe-to-heel values of  spiral angle  normal module with reference to cutter radius

Cone Dist. [mm]

Normal Module [mm] 5.0 4.0 3.0

Cone Dist. [mm]

• Normal Module varies with a maximum at some point • the N–Point (point of Involute) represents the point of maximum Normal Module

Position of N-Point for cutter radius SMALL MEDIUM LARGE Klingelnberg AG · Training Center

[T1e_v10_WIR]


Developing of Normal Module and Width of Tooth Gap for different Tool Diameter (for continuous indexing methods) Design with relatively LARGE tool diameter

efa

• bending of length curvature is relatively small • the width of tooth gap from toe to heel is apparently increasing • sufficient overlap of blade point width might possibly not be provided at the heel

efi < efa efa

Design with relatively SMALL tool diameter • bending of length curvature is relatively large • the difference in tooth gap width from toe to heel is apparently small • sufficient overlap of blade point width can easily be provided from toe to heel Klingelnberg AG · Training Center


efi  efa [T1e_v10_WIR]

80

Epicycloide Length Curvature:

mnm 

dm  cos  m 2  Rm  cos  m  Z2 ZP

mnx 

dx  cos  x 2  Rx  cos  x  Z2 ZP

Spiral Angle at various Positions max. Normal Module rotation of cutter

RW

rotation of work

Rx  cos  x  Ey  sin  x

EBM

i

a

m a

m i

2  Ey mnx   sin  x zp

Ey

Ex

Ri Rm

mnmax. 

2  Ey  1.0 zp

condition at N - point : Klingelnberg AG · Training Center

Rx

Ra

x = 90° 

sin x = 1.0  point of max. normal module [T1e_v10_WIR]


High Flexibility of Face Hobbing

rw (x)

• different size of cutter can be applied for same size of gears with same or similar size of spiral angle

m

Rm

or: • the position of N-Point is determined by the cutter diameter

Note: the parameters R m , m and mnm are identical for the upper and the lower gear design Klingelnberg AG · Training Center


m

Rm [T1e_v10_WIR]

81

High Flexibility of Face Hobbing rw • same size of cutter can be applied for different size of gears with same or different size of spiral angle • Position of N-Point depends on – size of gears – size of spiral angle – size of rolling circle (i.e. No. of blade groups of cutter)

)

 m (x Rm (x)

rw

 Note: the parameters R m and m are different for the upper and the lower gear design Klingelnberg AG · Training Center


m

>

m

(x)

Rm > R m (x) [T1e_v10_WIR]

Different Length Curvature for same Cutter Diameter • application of cutters of identical Cutter Radius with different Number of Blades Group to the same size of gear result in differences for: – – – –

rolling circle of the Epicycloid orientation (EM) of the blades radius of curvature of the Epicycloid position of the Point of Involute

• EM(2) > EM(1)  RBm(1) > RBm(2)

NOTE: The parameters RW, Rm , m and mnm in upper and lower examples are identical Klingelnberg AG · Training Center


[T1e_v10_WIR]

82

8.

Face Hobbing: Generating - and FORM - cut Method 8.1 General Description 8.2 Length Crowning applied with Cutter Tilt 8.3 The Generating Cutting Method Configuration of Cutter in relation to Work 8.4 The FORM- or Plunge Cutting Method Configuration of Cutter in relation to Work


GENERATING cutting method for F.H.:


[T1e_v10_WIR]

FORM - or PLUNGE cutting method for F.H.:

General Note: • with Oerlikon continuous indexing system using FS - type cutters, above cutting methods have been referred to as  SPIROFLEX staying for GENERATING cutting method  SPIRAC staying for FORM - or PLUNGE cutting method • Notes and descriptions hereafter, however, are of some common information also for more recent cutting methods such as SPIRON (Klingelnberg-Oerlikon) TRIAC® or PENTAC® FH (Gleason) • axes of machines in simultaneous motion (as mentioned hereafter) for conventional m/c’s for 6-axes NC m/c’s GENERATING cut ring gear (no tilt) 3 4 GENERATING cut pinion 3 6 FORM cut ring gear 3 3 FORM cut pinion (generated) 3 6 Klingelnberg AG · Training Center


[T1e_v10_WIR]

83

Method of Length Crowning: General: applies for both systems • application of a variable degree of cutter tilt to achieve infinitely variable amount of length crowning • cutter tilt can be applied - on either gear or pinion - or on both members in general: cutter tilt is applied on pinion only in order to keep ring gear to theoretical dimensions • variation of cutter tilt without modification of blades can be used to control contact position Klingelnberg AG · Training Center

[T1e_v10_WIR]


Method of Length Crowning: General: applies for both systems

• increased tooth depth is providing a wider tooth gap towards both ends of tooth which results in mismatch i.e. length crowning

PHI_V

PH I _X

A

A H

H (tooth depth)

• the angle of cutter tilt needs to be compensated on blade flank angles in order to maintain nominal pressure angles of component to be cut

AL F_ N

V

F_ AL

NR

B (face width)

• tooth depth is cut slightly deeper towards both ends of face width due to cutter tilt

A-A Klingelnberg AG · Training Center


[T1e_v10_WIR]

84

Method of Length Crowning: General: applies for both systems • cutter tilt results in slightly increased tooth depth towards toe and heel of pinion • increased tooth depth is providing a wider tooth gap towards both ends of tooth which results in mismatch i.e. length crowning

 

• tilt angle of cutter requires to be compensated on flank angles of blades; as the tilt angle varies by variable amount, blades need to be ground to variable flank angles.





 necessity of individual stick blades Klingelnberg AG · Training Center

[T1e_v10_WIR]


Method of Length Crowning: General: applies for both systems • as the result of increased cutter tilt, the length of contact is reduced • the axes of the Generating Plane Gears for pinion and gear do not coincide any more

 no cutter tilt

 

 as a result of this - without any countermeasures - a certain amount of BIAS–IN would be introduced to the tooth flanks BIAS  BIAS 


 cutter tilt  no countermeasures

“tooth twist” “diagonal contact”


[T1e_v10_WIR]

85

Different Type/Reasons for Cutter Tilt cutter tilt in a horizontal plane:

cutter tilt in a horizontal plane:

cutter tilt in a vertical plane:

• to generate conjugate pinion flanks to an existing FORM cut ring gear

• to change BIAS conditions by means of “hollow cone” “

• to create length crowning • corrections of TCP in profile direction

• corrections of TCP in length direction


[T1e_v10_WIR]


GENERATING cutting method:

• generated Ring Gear with generated Pinion;

Ring Gear

• crowning applied with cutter tilt (or by other means) • Completing process; i.e. part is finished in one (1) cutting process • cutting time is approximately 25% longer compared to FORM cutting method • this cutting method to apply for low gear ratios i = z2/z1 < 2.5 ... 2.2


Pinion


[T1e_v10_WIR]

86

GENERATING cutting method:

• continuous indexing cutting method using bar blade cutters • cutter blades of 1 blade group represent 1 tooth of the generating pane gear • no tilt required in horizontal plane as cutter represents Generating Plane Gear • cutter tilt is applied in vertical plane to achieve length crowning


[T1e_v10_WIR]


GENERATING cutting method for Ring Gear: • 3 axes in simultaneous motion: • rotation of cutter (A-axis) and work (B-axis) in timed relation + either plunge cut (X-axis)

A-axis

B-axis

F,K

or

generation roll (W-axis) W-axis

• generation roll starts after X-axis has advanced to full depth of tooth • cycle is finished after end of roll



(generation roll)

X-axis

[T1e_v10_WIR]

87

GENERATING cutting method for Pinion:

machine root angle B-axis

• 3 axes in simultaneous motion: • rotation of cutter (A-axis) and work (B-axis) in timed relation + either plunge cut (X-axis)

A-axis t ro e n h c a m is x -a B

le g n a

i x -a W n e (g

l)s n tio ra -x A

or

generation roll (W-axis) W-axis (generation roll)

• generating the pinion with the theoretical plane gear


X-axis

[T1e_v10_WIR]


GENERATING cutting method for Pinion:

F,K

machine root angle

p

B-axis


-a tB ro e in h c a m

is x le g F,Kp

x -a W n e (g

l)s n tio ra e -x A

A-axis

or


• generating the pinion with a modified plane gear to control BIAS condition (tooth twist)  referred to as “Hollow Cone”



X-axis

[T1e_v10_WIR]

88

FORM - or PLUNGE cutting method: • plunge - cut ring-gear with generated Pinion;

Ring Gear

• length crowning applied with tilt (or by other means) • Completing process; i.e. part is finished in one (1) cutting process • more economical cutting method due to shorter cutting times on ring-gears • this cutting method to apply for medium to high gear ratios i = z2/z1 > 2.2 ... 2.5


Pinion


[T1e_v10_WIR]

FORM - or PLUNGE cutting method:

• continuous indexing cutting method using bar blade cutters • cutter blades of 1 blade group represent 1 tooth of generating gear (= ring gear) • cutter tilt required in horizontal plane as rotating blades of cutter represent teeth of Generating Gear • additional tilt of cutter required in vertical plane in order to achieve length crowning



[T1e_v10_WIR]

89

FORM- or PLUNGE cutting method for Ring Gear: • 3 axes in simultaneous motion: • rotation of cutter (A-axis) and work (B-axis) in timed relation + plunge cut (X-axis)

A-axis

F,K B-axis

• cutting cycle is finished after X-axis has advanced to full depth of tooth

W-axis (no generation roll)

X-axis

• no generation-roll is applied • cutting time is approximately 25% shorter compared to the GENERATED cutting method Klingelnberg AG · Training Center

[T1e_v10_WIR]


FORM- or PLUNGE cutting method for Pinion:

machine root angle = 0°

B-axis


A-axis

or


• generating the pinion with the theoretical (actual mating) gear



X-axis

[T1e_v10_WIR]

90

FORM- or PLUNGE cutting method for Pinion:

machine root angle

B-axis


A-axis

or


• generating the pinion with gear of modified number of teeth to control BIAS condition (tooth twist)


9.

X-axis


[T1e_v10_WIR]

Ease Off 9.1 Crowning of Tooth Flanks 9.2 Definition 9.3 Ease Off Analysis / - Synthesis 9.4 Description of Flank Deviations 9.5 Major Parameters of Influence to the Ease Off 9.6 Ease Off Parameters and Side Effects 9.7 Effect of Profile Crowning and Twist to Path of Contact



[T1e_v10_WIR]

91

Ease Off

Crowning of Tooth Flanks:

Spur - & Helical Gears

Spiral Bevel Gears

• Involute profile in direction of tooth height; involutes are self-equidistant

• Octoide profile in direction of tooth height

• displacements occur in distance between axes

• displacements occur in 3D direction of offset and mounting distance D/P & R/G

• displacements have no influence to contact position

• displacements have influence to contact position >> crowning is required


Ease Off

[T1e_v10_WIR]


Definition:

• ... is the minimum distance of a pair of flanks for an ideal meshing operation • ... is the graphical representation of all kind of mismatch and flank modifications applied to gear and pinion

Tooth Contact Analysis ( TCA ) computation of all 4 flanks point by point (15  15 or 25  25 lattice points) • • • •

simulation of meshing operation Ease Off Contact Pattern and Path of Contact Transmission Error



pinion flank

“gear” flank [T1e_v10_WIR]

92

Ease - Off

Ease Off Synthesis :

• describes the correlation between Ease Off and the parameters of machining operation

EaseOff of conjugate gears

• enables user driven modifications of individual parameters of machining operation Ease Off is specified with 5 characteristics • profile crowning

EaseOff with profile modifications

• length crowning • pressure angle deviation • spiral angle deviation • tooth twist Klingelnberg AG · Training Center

Ease – Off

[T1e_v10_WIR]


Ease Off (graphical) Description of Flank Deviations

Flank Form or Profile deviations in Ease Off are specified with a parameter and may either occur as ...

HB Profile Crowning

LB Length Crowning

EaseOff

d

Pressure Angle Deviation

d

Spiral Angle Deviation

dv

Flank Twist

Tooth Contact Path-of-Contact



[T1e_v10_WIR]

93

Ease – Off

Ease Off (graphical) Tooth Contact: Ease Off Synthesis large (+)

The sign and the amount of the EaseOff parameters specify the position and shape of the tooth contact

HB

small ()

large (+) small ()

LB

EaseOff plus

(+)

d

minus ()

+/   toe/heel d

Tooth Contact Path-of-Contact dv


Ease - Off


+/ direction of P.o.C. [T1e_v10_WIR]

Major Parameters of Influence to Ease Off

Profile Crowning: Shape of the tool Work offset Lengthwise crowning: Diameter of the tool Tilt and flank angle of tool Pressure angle difference: Flank angle of tool Mounting distance Tilt Ratio of roll Helical Motion 1.Order Spiral angle difference: Radial distance Machine root angle Tilt Twist: Modified Roll 2. , 4. & 6. Order Helical Motion 2. ,4. & 6. Order Klingelnberg AG · Training Center


[T1e_v10_WIR]

94

Ease - Off

Ease Off Parameters + side Effects (individual flanks)

Profile Crowning: Shape of the tool Work offset Lengthwise crowning:

--Twist , Spiral Angle, Pressure Angle

Diameter of the tool Tilt and flank angle of tool Pressure angle difference:

-----

Flank angle of tool Mounting distance Tilt Ratio of roll Helical Motion 1.Order Spiral angle difference:

--Twist , Spiral Angle Twist, Length Crowning Twist Twist

Radial distance Machine root angle Tilt & Swivel Twist:

--Twist, Pressure Angle Twist, Length Crowning


Ease - Off

Length Crowning Length Crowning [Basics of Bevel Gears] · Slide 189

Ease Off Parameters + side Effects (Completing)

Profile Crowning: Shape of the tool Work offset Lengthwise crowning:

independently adjustable for cx + cv flank different reaction for cx + cv flank

Diameter of the tool Tilt and flank angle of tool Pressure angle difference:

different reaction for cx + cv flank similar reaction for cx + cv flank

Flank angle of tool Mounting distance Tilt Ratio of roll Helical Motion 1.Order Spiral angle difference:

independently adjustable for cx + cv flank similar reaction for cx + cv flank different reaction for cx + cv flank different reaction for cx + cv flank similar reaction for cx + cv flank

Radial distance Machine root angle Tilt & Swivel Twist:

different reaction for cx + cv flank similar reaction for cx + cv flank similar reaction for cx + cv flank

Modified Roll 2.Order Helical Motion 2.Order Klingelnberg AG · Training Center

[T1e_v10_WIR]

see influence of individually corrected machine settinngs to the position of contact and to flank deviations  additional document

Modified Roll 2.Order Helical Motion 2.Order

different reaction for cx + cv flank similar reaction for cx + cv flank [Basics of Bevel Gears] · Slide 190

[T1e_v10_WIR]

95

Ease - Off no Bias

Effect of Profile Crowning and Tooth Twist to the Path-of-Contact

straight profile

curved profile

• direction and shape of Path of Contact depend strongly on the amount of profile crowning

straight profile Bias In

• this fact makes it quite difficult to judge the nature of any Bias Condition

curved profile

• therefore the Ease Off analysis value for tooth twist (dv) allows easier judgment and control of Bias Condition

straight profile Bias Out curved profile



[T1e_v10_WIR]

10. BIAS Condition - Tooth Twist 10.1

Definition of Vertical- and Horizontal Displacement

10.2

Reaction of TCP for Displacement in Vertical Direction

10.3

Reaction of TCP for Displacement in Horizontal Direction

10.4

Checking Bias Conditions with V / H - check

10.5

Definition of BIAS

10.6

Conditions of Bias due to Distortion

10.7

Intentional Determination of Bias Conditions of Ease Off

10.8

Methods to Control Bias



[T1e_v10_WIR]

96

Definition:

Vertical and Horizontal Displacement

H+ increase Mounting Distance Pinion

H+ increase Mounting Distance Pinion

V+ Ring Gear up or Pinion down

V+ Ring Gear up or Pinion down

 Offset (av) is increased L.H. pinion R.H. r.gear

 Offset (av) is decreased

H+

MD

V+

R.H. pinion L.H. r.gear

MD

H+

V+ av

av

V+ H+ Klingelnberg AG · Training Center

Definition:

H+

V+

[T1e_v10_WIR]


Vertical- and Horizontal Displacements

Denomination of axes V – H – J

Denomination of axes E – P – G

(KLINGELNBERG – OERLIKON)

(GLEASON)

pinion LEFT gear RIGHT

V+

Hypoid Offset is increasing

E+


E+


pinion RIGHT gear LEFT

V+

Hypoid Offset is decreasing



[T1e_v10_WIR]

97

BIAS:

Contact Pattern Reaction because of V displacement

The inclination of the direction of displacement of the tooth contact does not depend on the relative size of tool diameter DRIVE (R.H. Gear) Heel 1

3

2

4 Toe

Root ring gear

V = E = change in offset:

Note:

1. E = -0.2mm

The Tooth Contact Position (TCP) moves mainly in direction of tooth length

2. E = -0.1mm 3. E = +0.1mm 4. E = +0.2mm


BIAS:

[T1e_v10_WIR]


Contact Pattern Reaction because of H displacement

The inclination of the direction of displacement of the tooth contact does very much depend on the relative size of tool diameter! DRIVE (R.H. Gear) 4 Heel

3 Toe

2 1 Root ring gear

H = P = change in pinion MD:

Note: The Tooth Contact Position (TCP) moves mainly in direction of tooth profile

1. P = -0.2mm

 displayed direction of H displacement applies for large cutter diameter F. Milling

3. P = +0.1mm



2. P = -0.1mm 4. P = +0.2mm [T1e_v10_WIR]

98

BIAS:

checking Bias Conditions with V/H - check

V/ H- check is used when developing gears for production and monitoring the quality of production gears (soft, hard or finished lapped or ground) With a V/ H-check, the amount and direction of axial displacements are controlled with a rolling test machine. Conditions / Limitations for V/H-check 1. TCP is to be displaced to Toe / Heel position on Drive and Coast flank 2. TCP’s at Toe/Heel positions must be positioned at central profile 3. TCP’s shall not exceed Toe / Heel and the original length of TCP must be maintained at Toe/Heel positions (  to avoid “boxing-up” of tooth contact) 4. calculated “center” shall be central-central Klingelnberg AG · Training Center

BIAS:

[T1e_v10_WIR]



V/ H-checks can be recorded in a matrix ...

Contact Pattern V H

... or can be plotted into a V/ H - chart

0 0

+V TOE Toe DRIVE

+ 10 - 10

- 35 + 32

45 43

- 12.5 1.05 +11.0

COAST V H

- 12 + 09

+ 38 - 35

50 44

+13.0 1.14 - 13.0

+H

Center Position DRIVE


Heel DRIVE

HEEL Total Ratio Center

DRIVE V H

units are typically either: 1/100 of mm or 1/1000 of inch Note: following is alternatively used for V & H E = offset (V) P = pinion axial displacement (H) G = gear axial displacement


[T1e_v10_WIR]

99

BIAS:


Note: - distance H indicates the amount of “lameness” of tooth contact at V=0 - the ratio of total V/H displacements indicates the amount of BIAS +V

Heel COAST

+V

Center Position COAST

Toe DRIVE

+H

+H

H Center Position DRIVE


BIAS:

Toe COAST

Heel DRIVE

[T1e_v10_WIR]


Gear Design / - Development based on V/H Displacements of the Carrier

Due to design specific features and stiffness each carrier or axle respectively deflects under load. The distortion of the carrier, the gear members, bearings and shafts are the reason for the displacement of tooth contact The displacements for different load conditions need either to be measured on an axle rig or to be calculated with FEM.

Relative Ritzelabdrängungen Zahnflankenmitte Zug 0.8000 0.6000

 hereby shall be ensured that the contacts under load spread evenly and optimally on the whole tooth flanks Klingelnberg AG · Training Center

0.4000 Abdrängung (mm)

Development of a Gear Set:  the finished lapped or ground gear set is supposed to feature the same or a similar BIAS condition as the carrier

0.2000 0.0000 0.00

0.50

1.00

1.50

2.00

2.50

-0.2000 -0.4000 -0.6000 -0.8000


Dreh mo me nt (1.000 Nm) E = -y

P= x

G=z

Da

Linear (E = -y)

Linear (P = x)

Linear (G = z)

Linear (Da)

[T1e_v10_WIR]

100

BIAS:

Definition Variant 1 DRIVE

Bias IN Heel

Bias OUT Root ring gear

Toe

Bias IN if the center of contact moves towards tip/heel when increasing the mounting distance and decreasing the offset in the same amount.

Bias Out if the center of contact moves towards root/heel when increasing the mounting distance and decreasing the offset in the same amount. Klingelnberg AG · Training Center

BIAS:

[T1e_v10_WIR]


Definition Variant 2

Heel

Root ring gear

Toe

the change in the offset E and the change in the Mounting Distance P are adjusted as such that the center of TCP moves parallel to the pitch line towards the heel.

Bias IN Bias OUT

 V  E  1 H P  V  E  1 H P


Note: Definitions 1+2 apply only for Hypoid pinions with spiral angle m1 in the order of 45° to 50°


[T1e_v10_WIR]

101

BIAS:

Definition Variant 3

m 1

assuming that the carrier shows about equal stiffness in horizontal and vertical direction, deflections under load take place in a direction perpendicular to a tangent to the pinion tooth at mean point.

m 1 V H

Bias conditions therefore depend on the spiral angle of the pinion

 V  E   c tan  m1 H P  V  E   c tan m1 Bias OUT H P

Bias IN


BIAS:

Heel Toe Root ring gear [T1e_v10_WIR]


General Definition

Bias IN light load contact

with reference to the ring gear the contact develops from - tip at heel to the root at toe on the convex (Drive) side - tip at toe to the root at heel on the concave (Coast) side

heavy load contact

Bias NEUTRAL

Bias OUT with reference to the ring gear the contact develops from - root at heel to the tip at toe on the convex (Drive) side - root at toe to the tip at heel on the convex (Coast) side Klingelnberg AG · Training Center


Bias IN

Bias OUT

[T1e_v10_WIR]

102

BIAS Conditions: Any changes of Bias Conditions can occur due to heat treatment distortions.

Bias 0 or Bias NEUTRAL

Unequal distribution of blank material in areas of tooth- and tooth root typically results in unequal stresses of the case hardened structure of gear materials.  this effect is sometimes referred to as “unwind of gear teeth”

Note: along the Path of Contact there appear - more lines of contact for Bias In - less lines of contact for Bias Out Klingelnberg AG · Training Center

simplified representation of Bias or Tooth Twist

Bias In

Bias Out [T1e_v10_WIR]


BIAS Conditions: Bias 0

without Profile Crowning

Bias conditions i.e. a certain amount of tooth twist can intentionally be considered in designing the Ease Off

dv = 0°

flank twist

dv = 0°

Characteristic of the EaseOff: • profile crowning HB = 0 … 5 m • flank twist dvdrive = 0° dvcoast = 0° • length crowning LB = 40 … 45 m Note: - practically no twist of the EaseOff - path of contact : > in profile direction vertical within contact - without profile crowning the contact can easily be recognized to be Bias Neutral - transmission error ca. 18 rad Klingelnberg AG · Training Center


path of contact

[T1e_v10_WIR]

103

BIAS Conditions: Bias IN

without Profile Crowning


dv > 0°

flank twist

dv < 0°

Characterisitc of the EaseOff: • profile crowning HB = 0 … 5 m • flank twist dvdrive = 0.7° dvcoast = 0.7° • length crowning LB = 45 … 50 m Note: -  twist of opposite flanks - path of contact : > in profile direction diagonal within contact - Bias IN tendency can easily be recognized - with same length crowning the contact is longer along the Path of Contact - transmission error smaller ca. 7.5 rad Klingelnberg AG · Training Center

path of contact

[T1e_v10_WIR]


BIAS Conditions: Bias OUT without Profile Crowning Bias conditions i.e. a certain amount of tooth twist can intentionally be considered in designing the Ease Off

dv < 0°

flank twist

dv > 0°

Characteristic of the EaseOff: • profile crowning HB = 0 … 5 m • flank twist dvdrive =  0.7° dvcoast = 0.7° • length crowing LB = 40 … 45 m Note: -  twist of opposite flanks - path of contact : > in profile direction inverse diagonal (Z-Form) - Bias OUT tendency can easily be recognized - with same length crowning the contact is shorter along the Path of Contact - transmission error larger ca. 40 rad Klingelnberg AG · Training Center


path of contact

[T1e_v10_WIR]

104

BIAS Conditions: Bias 0

with Profile Crowning


dv = 0°

flank twist

dv = 0°

Characteristic of the EaseOff: • profile crwning HB = ca. 15 m • flank twist dvdrive = 0° dvcoast = 0° • length crowning LB = ca. 40 m Note: - practically no twist of the EaseOff - path of contact : > in profile direction diagonally oblique (although no apparent flank twist !) - transmission error ca. 65 - 70 rad


path of contact

[T1e_v10_WIR]


BIAS Conditions: Bias IN

with Profile Crowning


dv > 0°

flank twist

dv < 0°

Characteristic of the EaseOff: • profile crowing HB = ca. 15 m • flank twist dvdrive = 0.7° dvcoast = 0.7° • length crowing LB = ca. 44 m

Note: -  twist of opposite flanks - path of contact : > in profile direction diagonally oblique (hardly distinguishable from Bias 0 ) - transmission error smaller ca. 40 rad Klingelnberg AG · Training Center


path of contact

[T1e_v10_WIR]

105

BIAS Conditions: Bias OUT with Profile Crowning Bias conditions i.e. a certain amount of tooth twist can intentionally be considered in designing the Ease Off

dv < 0°

flank twist

dv > 0°

Characteristic of the EaseOff: • profile crowning HB = ca. 15 m • flank twist dvdrive =  0.7° dvcoast = 0.7° • length crowning LB = 40 … 45 m

Note: -  twist of opposite flanks - path of contact : > in profile direction slightly oblique (hardly distinguishable from Bias 0 - transmission error larger ca. 96 rad Klingelnberg AG · Training Center

path of contact


Methods to control BIAS:

[T1e_v10_WIR]

General

• A number of machine setting modifications have direct influence or side effects to tooth twist (>> see sections 9.5 / 9.6) • The most effective parameters to control Bias or Tooth Twist, however, are Modified Roll and Helical Motion Note:

a modification of Modified Roll 1st order corresponds to a modification of the Decimal Ratio. (Modifications to the pressure angle are undesired side effects in changing the Bias with the Decimal Ratio)

• All modifications are effective mainly along the Path of Contact and therefore appear obliquely across the tooth flank • for Face Hobbing there are alternative methods to control Bias conditions such as: - modified Generating Gear (“Hollow Cone”) or - Hook Angle of Blades • these methods are particularly applied with conventional gear cutting machines; i.e. without the ability of applying Modified Motions Klingelnberg AG · Training Center


[T1e_v10_WIR]

106


Principle of Modified Roll

Modified Roll changes the constant Ratio of Roll into a to a polynomial function eccentric function (E) and approximated polynomial function (P)

modified ratio 1/u

Principle of a Modified Roll Ratio for a mechanically controlled m/c

generating cradle

E  ca. 0.35 (=  20°) generating interval practically applied to generate Spiral Bevel- or Hypoid pinions

worm drive generating angle

E

eccentric function (E)

approx. by a polynominal function of 4th order (P)


example of EaseOff of Mod.Roll 2nd order [T1e_v10_WIR]



Definition of Modified Roll

Modified Roll  With Modified Roll the relationship (ratio) of motions between the generating cradle to the work spindle is not constant  the generation roll motion is either accelerated or decelerated during the travel within the interval of the generating roll angle Generating Roll Angle:  =  start-of-roll … mean … end-of-roll

e.g. roll positions for this example … - Start of Roll at heel tip (convex flank)  = 65° - Start of Roll at heel root (concave flank)  = 69° (see position of green arrow Klingelnberg AG · Training Center

)

- anywhere during generating roll (see position of green arrow

- End of Roll at toe root (convex flank) - End of Roll at toe tip (concave flank)


 = 89°

)

 = 97°  = 100° [T1e_v10_WIR]

107


General Facts of Modified Roll

• in case of completing cut gears flanks, as well as in case of inconstant generating ratios (= modified roll) the action of generating both concave and convex flanks simultaneously results in different effects at opposite flank areas • i.e. depending on modified generating motions running either ahead or lacking behind the constant roll motion, certain amount of material is either “removed” (-) or “left over” (+) on the opposite flanks respectively

–

+

Note: •

any flank modifications of generating roll appear in a direction tangential to the flank

•

consequences to both flanks are therefore different

•

effects caused by different pressure angles of the tool convex/concave are therefore not as significant as they appear with modifications of Helical Motions


[T1e_v10_WIR]



General Facts for Modified Roll

 corrective effects with Completing systems for DRIVE / COAST flank occur : ... along the Path of Contact in OPPOSITE direction for Modified Roll  corrections for cv/cx flanks are slightly staggered due to the fact that the tool is cutting a tooth gap; i.e. generating opposite flanks in slightly different positions toe

heel

 = b ( – m) without correction

 = e ( – m)4 Modified Roll 4th order Klingelnberg AG · Training Center

toe

heel

 = c ( – m)2 Modified Roll 2nd order

 = f ( – m)5 Modified Roll 5th order [Basics of Bevel Gears] · Slide 216

toe

heel

 = d ( – m)3 Modified Roll 3rd order

 = g ( – m)6 Modified Roll 6th order [T1e_v10_WIR]

108


Modified Roll Example shown for Face Milling Completing

following sequence of images display a flank comparison of pinion flanks modified with different modifications of roll compared with a reference tooth  no corrections Modified Roll 1. order (b = +0.005)

Modified Roll 2. order (c = 0.025)

Modified Roll 3. order (d = 0.1)

note: modifications of convex / concave flanks appear in opposite direction

Modified Roll 4. order (e = +0.4)

Modified Roll 5. order (f = +1.5)


Modified Roll 6. order (g = +6.0) [T1e_v10_WIR]



Modified Roll


Face Hobbing

10:41-200/25 ARCON(II)14-6“

10:41-200/25 SPIRON(II)13-76 different directions of generating lines

e.g. Modified Roll 2nd order

flank modifications develop for both cutting methods: • along the Path of Contact • in direction of generating lines e.g. Modified Roll 4th order Klingelnberg AG · Training Center


[T1e_v10_WIR]

109


General Facts for Helical Motions

 corrective effects with Completing systems for DRIVE / COAST flank occur : ... along the Path of Contact in SAME direction for Helical Motion  modifications for Helical Motion to cv/cx flank depend on pressure angles: small modifications for small - / large modifications for larger pressure angles toe

heel

toe

heel

toe

heel

linear Helical Motion (H.M.)

linear H.M. and 2nd order

linear H.M. and 3rd order

linear H.M. and 4th order




[T1e_v10_WIR]



General Facts for Helical Motions

• during the generating roll, Helical Motion performs an additional feed (advance or withdrawal) in direction of tooth depth; this additional motion could be performed either linear or in form of a Polynomial function of higher order • Helical Motion in case of completing cut gears, results in similar effects in areas of opposite flanks concave/convex • i.e. depending on the direction of the additional Helical Motion, on both flanks certain amount of material is either “removed” (-) or “left over” (+)

–

Note: •

in case of Hypoid gears, pressure angles of concave/convex flanks are usually considerably different

•

Modifications caused by Helical Motion are of different amount on both flanks, however, they result in the same direction



–

[T1e_v10_WIR]

110


Helical Motion Example shown for Face Milling Completing

Helical Motion 1. order (b = +0.15)

Helical Motion 2. order (c = +0.7)

Helical Motion 3. order (d = +3)

note: only very small modifications of concave flank (  small pressure angle : Hypoid)

Helical Motion 4. order (e =+15)

Helical Motion 5. order (f =+75)


Helical Motion 6. order (g =+350) [T1e_v10_WIR]



Helical Motion


Face Hobbing

10:41-200/25 ARCON(II)14-6“

10:41-200/25 SPIRON(II)13-76 different directions of generating lines

e.g. Helical Motion 2nd order

flank modifications develop for both cutting methods: • along the Path of Contact • in direction of generating lines e.g. Helical Motion 4th order Klingelnberg AG · Training Center


[T1e_v10_WIR]

111


Corrective Effects (Summary)

Corrective Effects along Path of Contact for Modified Roll / Helical Motion below indications are valid for positive coefficients

Modified Roll

Hand of Spiral

LH Pinion

+ –

more material removal less material removal

Helical Motion

1./3./5. order

2./4./6. order

1./3./5. order

2./4./6. order

Δ Ease-off on Gear flank




DRIVE

DRIVE

–

+ –

+

+ –

+

+

– –

–

–

– –

+

–

RH Gear

RH Pinion

COAST

OPPOSITE Effects on

SIMILAR Effects on

COAST

DRIVE and COAST flanks!

DRIVE and COAST flanks!

+

+

– + LH Gear

–

+

–

–

–

– –

DRIVE


COAST

–

–

+ +

COAST

– DRIVE



Example: Modified Motions

Modified Roll 2. order POSITIVE

Helical Motion 2. order NEGATIVE

[T1e_v10_WIR]

+ with every Duplex- or Completing method, a wanted correction on the concave side will have some unwanted side effects on the convex side

in case only 1 flank shall be corrected, a second correction effect is to be superimposed to compensate unwanted corrections of convex side

= Klingelnberg AG · Training Center


[T1e_v10_WIR]

112


Hollow Cone

for Face Hob Generating Cutting System a condition of Bias OUT is introduced in generating the pinion using a Generating Plane Gear of reduced number of teeth. Hence, the Generating Plane Gear is not “plane” anymore; it is therefore referred to as “Hollow Cone” Note: • method is somehow limited for high number of blade groups and large Hypoid offset • any modification of “Hollow Cone” modifies Bias conditions on both flanks Drive / Coast similarly • profile crowning will be introduced as side effect of modification in Bias


[T1e_v10_WIR]



Hollow Cone

for Face Hob FORM Cutting System a condition of Bias OUT is introduced in generating the pinion using a Generating Gear of reduced number of teeth. Hence, the Generating Gear does not represent the actual mating gear anymore; it is also referred to as “Hollow Cone” Note: • method is somehow limited for high number of blade groups and large Hypoid offset • any modification of “Hollow Cone” modifies Bias conditions on both flanks Drive / Coast similarly • profile crowning will be introduced as side effect of modification in Bias



[T1e_v10_WIR]

113


Hook Angles of Blades

for Face Hobbing Cutting System a condition of Bias is introduced in case the cutting edge is running in a plane NOT perpendicular to the cutting direction. as there is some cutting action at the tips of blades, any amount of blade hook angle should be positive; this condition introduces Bias Out to the flanks of pinion and gear

u ()

Note: • with 2-face blades, a relatively small hook angle results from the blade slot tilt angle of cutter and from the actual blade pressure angle • with 3-face blades & large blade slot tilt angles, hook angles can be selected intentionally ( limited by grinding restrictions) • in case of Face Milling methods the amount of hook angle has got no influence at all to flank twist Klingelnberg AG · Training Center

[T1e_v10_WIR]



Lapping of Gears

4 2

3

1

control of lapping motions

resulting contact on test m/c

• increased ratio of values V / H  Bias-In ratio for lapping motions

• Bias-In tooth contact

• increased lapping of : toe root + heel tip of gear for COAST toe tip + heel root of gear for DRIVE Klingelnberg AG · Training Center

• more material is lapped off in H positions 1-, 2+, 3+, 4• smoother tooth meshing, smaller transmission error


[T1e_v10_WIR]

114


Lapping of Gears

4 2

3

1

control of lapping motions

resulting contact on Test m/c

• decreased ratio of values V / H  Bias-Out ratio for lapping motions

• Bias-Out tooth contact • more material is lapped off in H positions 1+, 2-, 3-, 4+

• increased lapping of : toe tip + heel root of gear for COAST toe root + heel tip of gear for DRIVE Klingelnberg AG · Training Center

• harsher tooth meshing, larger transmission error


[T1e_v10_WIR]

11. Calculation of Axial - and Radial Forces 11.1

General

11.2

Calculation of Components

11.3

Example:

11.4

Examples: Hypoid Gears

11.5

Main Direction of Deflections “Drive” Conditions

11.6

Main Direction of Deflections “Coast” Conditions

11.7

Determination of Gear Deflections under Load

Spiral Bevel Gears W R

W )R (x M T B



[T1e_v10_WIR]

115

Axial- & Radial Forces:

General

based on the pinion input torque Md1 the following forces result :

1

Note:

- positive values in direction of arrow - for equations see following page

Fu1 Fu2

Tangential Force Pinion Tangential Force R.Gear

Fa1

Axial Force Pinion for DRIVE and COAST

Fr1

Radial Force Pinion for DRIVE and COAST

Fa2

Axial Force R.Gear for DRIVE and COAST

Fr2

Radial Force R.Gear for DRIVE and COAST

Md

Fa2

Fr2

Fa1 Fu1 [T1e_v10_WIR]



Calculation

PINION (1) (cx) (cv)

RING GEAR (2)

n1(cx)  n  

n2 (cx)  n  

n1(cv)  n  

n2 (cv)  n  

2  1000  Md Fu1  dm1

Tangential Force

Fa1

Fr2

Fu2


Pressure Angle Pressure Angle

Fr1

2

Fu2  Fu1 

cos  m2 cos  m1

Axial Force (Drive) sin 1  tan m1  cos 1 ) cos  m1

Fa2 (Drv)  Fu2  (tan n2 (cx) 

sin 2  tan m 2  cos 2 ) cos  m2

sin 1  tan m1  cos 1 ) cos  m1

Fa2 (Cst)  Fu2  (tan n 2 (cv) 

sin  2  tan m 2  cos 2 ) cos  m 2

cos 1  tan m1  sin 1 ) cos  m1

Fr2 (Drv)  Fu2  (tan n 2 (cx) 

cos  2  tan m2  sin 2 ) cos  m2

cos 1  tan m1  sin 1 ) cos  m1

Fr2 (Cst)  Fu2  (tan n2 (cv) 

cos 2  tan m2  sin 2 ) cos  m2

Fa1(Drv)  Fu1  (tan n1(cv) 

Axial Force (Coast) Fa1(Cst)  Fu1  (tan n1(cx) 

Radial Force (Drive) Fr1(Drv)  Fu1  (tan n1(cv) 

Radial Force (Coast) Fr1(Cst)  Fu1  (tan n1(cx)  Klingelnberg AG · Training Center


[T1e_v10_WIR]

116


Example 1 for Spiral Bevel Gear

pinion

ring gear

Number of Teeth Offset / Pitch Diameter Normal Module Pitch Cone Angle Mean Diameter Spiral Angle Nominal Pressure Angle Pr. Angle Modification Pressure Angle concave fl. Pressure Angle convex fl.

z av / d mn  dm m n   n (cv)  n (cx)

Input Torque Circumferencial Force

Md Pu

650.00 32737.35

32737.35

[ Nm ] [N]

AXIAL force

DRIVE

Fa (Drv)

21526.15

9119.79

[N]

Fa1(Drv) = Fr2 (Drv)

AXIAL force

COAST

10

43 [ -- ] 200.00 [ mm ] [ mm ] 76.91 [°] 170.77 [ mm ] 30.00 [°] [°] [°] 20.00 [°] 20.00 [°]

0.00 3.44 13.09 39.71 30.00 20.00 0.00 20.00 20.00

Spiral Bevel with m=30°

= for info only = input

= result

Note: for Spiral Bevel gears there is:

Fa (Cst)

-15293.17

17682.45

[N]

Fa1(Cst) = Fr2 (Cst)

RADIAL force DRIVE

Fr (Drv)

9119.90

21526.19

[N]

Fr1(Drv) = Fa2 (Drv)

RADIAL force COAST

Fr (Cst)

17682.38

-15293.09

[N]

Fr1(Cst) = Fa2 (Cst)



Example 2 for Hypoid Gear

pinion

gear




Md Fu

650.00 27524.88

32540.41

[ Nm ] [N]

AXIAL force

DRIVE

Fa (Drv)

28175.46

5748.55

[N]

AXIAL force

COAST

Fa (Cst)

-19619.23

20192.07

[N]

10 20 3.46 18.21 47.23 42.90 20.00 1.72 18.28 21.72

43 [ -- ] 200 [ mm ] [ mm ] 71.34 [°] 171.57 [ mm ] 30.00 [°] [°] [°] 21.72 [°] 18.28 [°]

RADIAL force DRIVE

Fr (Drv)

3797.33

21770.88

[N]

RADIAL force COAST

Fr (Cst)

22211.29

-13010.65

[N]


[T1e_v10_WIR]



Hypoid with ... – small offset: 20mm – spiral angle gear: 30° = for info only = input

= result

Note: max. thrust = +30% of Spiral Bevel of same ratio and dimensions (m=30°)

[T1e_v10_WIR]

117



pinion

gear




Md Fu

650.00 23506.01

31264.99

[ Nm ] [N]

AXIAL force

DRIVE

Fa (Drv)

29706.01

3626.89

[N]

AXIAL force

COAST

10 30 3..4707 21.55 55.31 49.38 20.00 2.34 17.66 22.34

43 [ -- ] 200 [ mm ] [ mm ] 67.28 [°] 172.32 [ mm ] 30.00 [°] [°] [°] 22.34 [°] 17.66 [°]

Fa (Cst)

-20034.73

20659.62

[N]

RADIAL force DRIVE

Fr (Drv)

622.76

21088.23

[N]

RADIAL force COAST

Fr (Cst)

23866.50

-10918.51

[N]


Hypoid with ... – small offset: 30mm – spiral angle gear: 30° = for info only = input

= result

Note: max. thrust = +38% of Spiral Bevel of same ratio and dimensions (m=30°)

[T1e_v10_WIR]




pinion

ring gear




Md Fu

650.00 22727.27

32263.15

[ Nm ] [N]

AXIAL force

DRIVE

Fa (Drv)

28834.42

2447.79

[N]

AXIAL force

COAST

Fa (Cst)

-18056.96

20620.75

[N]

10 40.00 3.68 24.61 57.20 50.00 20.00 4.05 15.95 24.05

43 [ -- ] 200.00 [ mm ] [ mm ] 63.03 [°] 173.26 [ mm ] 24.15 [°] [°] [°] 24.05 [°] 15.95 [°]

Hypoid with relatively – large offset – large spiral angle on pinion = for info only = input

= result

RADIAL force DRIVE

Fr (Drv)

-2087.69

17475.96

[N]

Note: • max. thrust = +34% of Spiral Bevel based on same ratio & dimensions (Spiral angle gear is here smaller than the previous Sp. Bevel Gear

RADIAL force COAST

Fr (Cst)

25621.22

-5736.15

[N]

• radial thrust on pinion might get negative for certain conditions



[T1e_v10_WIR]

118


Effects of Spiral and Pressure Angles

Pinion concave (DRIVE)

Baseline (100%):

Pinion convex (COAST)

Hypoid gearset g.pitch dia. 200 mm offset 40 mm press. angle 22° gear sp.ang. 25.0° pinion sp.ang. 51.5° torque :

650 Nm

Axial Force

(>> alike to example 4)

Radial Force

Gear Spiral Angle beta_m2 [ ° ]

legend:




Effects of Spiral and Pressure Angles

Gear convex (DRIVE)

Baseline (100%):

[T1e_v10_WIR]

Gear concave (COAST)

Hypoid gearset g.pitch dia. 200 mm offset 40 mm press. angle 22° gear sp.ang. 25.0° pinion sp.ang. 51.5° torque :

650 Nm

Axial Force

(>> alike to example 4)

legend:


Radial Force

Gear Spiral Angle beta_m2 [ ° ]


[T1e_v10_WIR]

119


“Driving” ( L1/R2)

Case 1: L.H. pinion + R.H. ring gear meshing flanks : pinion CONCAVE with : r.gear CONVEX resulting effects:  pinion displacement in direction perpendicular to tangent of flanks  decreasing offset (V-)  increasing pinion mounting distance  axial thrust is pushing pinion in direction towards the main bearing  favorable condition! Klingelnberg AG · Training Center



V– H+ [T1e_v10_WIR]

“Coasting” ( L1/R2 )

Case 2: L.H. pinion + R.H. ring gear meshing flanks : pinion CONVEX with : r.gear CONCAVE resulting effects:  pinion displacement in direction perpendicular to tangent of flanks  increasing offset (V+)  decreasing pinion mounting distance  axial thrust is pulling pinion in direction off the main bearing  unfavorable condition! Klingelnberg AG · Training Center

V+ H– [Basics of Bevel Gears] · Slide 240

[T1e_v10_WIR]

120


“Driving” ( R1/L2 )

Case 3: R.H. pinion + L.H. ring gear meshing flanks : pinion CONCAVE with : r.gear CONVEX resulting effects:  pinion displacement in direction perpendicular to tangent of flanks  decreasing offset (V+) !  increasing pinion mounting distance  axial thrust is pushing pinion in direction towards the main bearing  favorable condition! Klingelnberg AG · Training Center



V+ H+ [T1e_v10_WIR]

“Coasting” ( R1/L2 )

Case 4: R.H. pinion + L.H. ring gear meshing flanks : pinion CONVEX with : r.gear CONCAVE resulting effects:  pinion displacement in direction perpendicular to tangent of flanks  increasing offset (V-)  decreasing pinion mounting distance  axial thrust is pulling pinion in direction off the main bearing  unfavorable condition! Klingelnberg AG · Training Center

V– H– [Basics of Bevel Gears] · Slide 242

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121

Determination of Deflections of the Gear Set under Load Deflections of gears, shafts, bearings and carrier itself due to forces appearing under load are generally not known. This information therefore needs either to be established with relatively elaborate methods of measurement (A) or to be calculated (B) A) Deflection Test The effective or relative displacements of pinion and gear can be measured on an axle test rig in loaded, quasi static conditions (i.e. at extremely low rpm) For this purpose the gear members as well as the carrier need to be modified in a way that the displacements and distortions can be measured at points significant for respective distortions in all 3 directions V, H and J


[T1e_v10_WIR]


Determination of Deflections of the Gear Set under Load With an experimentally performed tests the displacements of 2 passenger car gears of identical ratio and dimensions have been compared in 2 different carriers Deflection Test

carrier in GTS

carrier in ALU

Lines of deflection

read-out of measurement values for load conditions: Md = 0, 25, 50, 75 & 100% pinion input torque [Nm] Klingelnberg AG · Training Center


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122

Determination of Deflections of the Gear Set under Load Deflection Test

200 150

The collected and compiled results allow to conclude the effective amount of deflections and the required BIAS characteristic of the gear set respectively

100

SCHUB (GTS)

50

ZUG

(GTS)

SCHUB (ALU)

0

ZUG

-50

(ALU)

-100 -150 0

25

50

75

100

Drehmoment in %

relative displacements V (vertical) Ausführung GTS ZUG SCHUB

V -60 80

H 45 -30

J -50 490

V/H 1.33 2.67

ZUG SCHUB

-75 105

60 -40

-55 670

1.25 2.63

in m

Drehmoment am Ritzel 200

540 Nm

150

670 Nm

100

ZUG

50

Ausfühurung ALU ZUG SCHUB

V -100 130

H 85 -80

J -50 660

V/H 1.18 1.63

ZUG SCHUB

-125 150

140 -80

-55 855

0.89 1.88

in m

-50

540 Nm

-100


ZUG

0

Drehmoment am Ritzel

(ALU)

SCHUB (ALU)

0

670 Nm

(GTS)

SCHUB (GTS)

25

50

75

100

Drehmoment in %

relative displacements H (horizontal)


[T1e_v10_WIR]

Determination of Deflections of the Gear Set under Load B) Calculation of Displacements acc. to BECAL (KIMoS) The TCA (Tooth Contact Analysis) under load acc. to BECAL offers the opportunity to calculate the deflection under load in directions V, H and J in relation to the entered torque. In addition to the specification of the gear set and the respective EaseOff, for this purpose the so-called “environment” needs to be known and specified. The calculations are based on BEM or so-called “Boundary Elements Method.” The term “Environment” stands for the description or modeling of the following: - diameters and lengths of stepped segments of shafts - dimensions, specifications, arrangements and pre-load of roller type bearings Note:

the influence of the stiffness of gear carrier is NOT considered with this method !



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123

Determination of Deflections of the Gear Set under Load Calculation of Displacements acc. to BECAL (KIMoS)

there are calculated: • lines and total amount of deflections • flank pressures of the loaded tooth contact • root bending stresses



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Determination of Deflections of the Gear Set under Load C) Calculation of Displacements with ROMAX DESIGNER Software Starting with version v2.10 KIMoS offers the opportunity of an interface to export all relevant dimensions Spiral Bevel- or Hypoid gear design to ROMAX. ROMAX DESIGNER then is used to complete the modeling of the complete gear set including shafts, bearings, differential and the arrangement of the carrier. Based on established forces, displacements under load are calculated and re-imported back into KIMoS for the purpose of L – TCA



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124

12. Influence of Cutter Diameter 12.1

Application of Small Cutter Diameter

12.2

Application of Large Cutter Diameter

12.3

Displacement of TCP for Small Cutter Diameter

12.4

Displacement of TCP for Medium Cutter Diameter

12.5

Displacement of TCP for Large Cutter Diameter

12.6

Example: Displacement of TCP in comparison



[T1e_v10_WIR]

Application of SMALL Cutter Diameter: Face Milling • rectangular case: the Radial S and the tangent to tooth at mean point are parallel • this case is possible but rarely applied as it is difficult to manufacture; smallest recommended cutter radius is rc > 1.1  Rm  sin m view to Generating Gear (vertical tangent to tooth)

view to cutter (cutting Hypoid pinion L.H.) Note: cutter is located “behind” the pinion)

S m Rm rc = Ds/2 = Rm x sinm



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125

Application of SMALL Cutter Diameter: Face Milling • for horizontal H+ displacement TCP moves towards: tip and heel for Coast tip and toe for Drive • for vertical V+ displacement TCP moves towards: tip and heel for Coast tip and toe for Drive • the directions of vertical and horizontal displacement almost converge in the rectangular case


-V

-H

+H +V

+H -H -V +V


[T1e_v10_WIR]

Application of Small Cutter Diameter: Face Hobbing

• rectangular case can more easily be applied than with Face Milling methods, however, special location of contact position is required to allow TCP to utilize full length of flanks under load. view to Generating Gear (tangent to tooth vertical)

m

S

view to cutter (cutting Hypoid pinion L.H.) Note: cutter is “behind” pinion)

rbm = Rm x sin m

Rm



[T1e_v10_WIR]

126

Application of SMALL Cutter Diameter: Face Hobbing • very low sensitivity, stable contact even for large deflections of pinion & gear under heavy load • V- / H+ or V+ / H- directions compensate each other

R.H. gear / L.H. pinion

• strong tooth due small length curvature with max. normal module near to the calculation point • lapping abilities somewhat limited as lapping takes place in diagonal direction along the face width only

-V

-H

+V

4

V+

V-

2

+H -H +V


+H

V+ 3

-V

V-

1

[T1e_v10_WIR]


Application of SMALL Cutter Diameter : Face Hobbing

Comparison of directions of displacement: pinion: L.H. / gear: R.H.

-H

V-

+H

V+

pinion: R.H. / gear: L.H.

4

V+

V-

2

4

2

-V

-V

+V

+H

+V

+H

+H -H V+

-H

V-

V-

V+ 3

3

1

+V

-H

-V 1

+V

note: to perform a V/H – check is NOT possible! Klingelnberg AG · Training Center


[T1e_v10_WIR]

127

Face Milling

Displacement Directions small cutters

10:41-200/25 Arcon(II)12-5“ Rinv/Re= 0.931

Face Hobbing

10:41-200/25 Spiron(II)11-51

rw = 63.6 mm

Rinv/Re= 0.868 rbm = 47.2 mm

DUPLEX Completing

The tool radius for this example design is merely rw = 1.25  R m  sin m and hence afar of the 90° case.

The tool radius for this example design is rw = 1.1  Rm  sin m corresponds practically to 90° case!!

Though a problem is getting apparent with the tendency of large and small length crowing between tip and root

The EaseOff allows to design without problems & additional correction effects


note: smaller crowning LB & HB ! [T1e_v10_WIR]


Face Milling

Displacement Direction V small cutter

10:41-200/25 Arcon(II)12-5“ R inv/Re= 0.931

Face Hobbing

10:41-200/25 Spiron(II)11-51

R inv/Re= 0.868

V+0.30

V+0.30

V+ V-

V

V

V-

V-0.30



V+

V-

V+

V-0.30

[T1e_v10_WIR]

128

Face Milling

Displacement Directions H small cutter

10:41-200/25 Arcon(II)12-5“ R inv/Re= 0.931

Face Hobbing

10:41-200/25 Spiron(II)11-51

H+0.20

H+0.30

H+ H-

H-

H+

H+

H+ H-

H-

H-0.30


H-0.20

[T1e_v10_WIR]


Face Milling

Displacement Directions V+H small cutter

10:41-200/25 Arcon(II)12-5“ R inv/Re= 0.931

Face Hobbing

10:41-200/25 Spiron(II)11-51

H+

V+ V

H-

H-

V

H+

H- V-

V-0.30 H+0.3


R inv/Re= 0.868

V+0.30 H-0.3

V+0.30 H-0.3

V-

R inv/Re= 0.868

H+ V+

H-V-

H+ V+

V-0.30 H+0.3


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129

Application of LARGE Cutter Diameter: Face Milling view to Generating Gear (tangent to tooth vertical)



[T1e_v10_WIR]


Application of LARGE Cutter Diameter: Face Milling • for horizontal H+ displacement TCP moves towards: tip and toe for Coast tip and heel for Drive • for vertical V+ displacement TCP moves towards: tip and heel for Coast tip and toe for Drive

-V

• with large cutter diameter, the directions of vertical and horizontal displacement diverge

+H


-H

-H

+V Klingelnberg AG · Training Center

+V +H

-V [T1e_v10_WIR]

130

Application of LARGE Cutter Diameter : Face Milling

Comparison of directions of displacement: pinion: L.H. / gear: R.H.

-V

pinion: R.H. / gear: L.H. VH+

+V

V+ H-

4

+H

-V 2

+H 2

+H

+V

V+ H+

VH-

-H

4

-H +H

-H 3

+V

-V


V+ H-

VH+

-H

1

1

3

V+ H+

VH-

+V


-V

[T1e_v10_WIR]

Application of LARGE Cutter Diameter: Face Hobbing view to Generating Gear (tangent to tooth vertical)




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131

Application of LARGE Cutter Diameter: Face Hobbing • sensitivity of contact for V/H deflections increases with the cutter radius • Point of Involute (i.e. point of largest normal module) is located off the heel

R.H. gear / L.H. pinion

-V

VH+

+V 4

• V- / H+ or V+ / H- directions are adding to each other

2

+H

-H

+H

• good lapping abilities as lapping area can be moved to either tip-/rootposition at toe and heel

-H 3

V+ H-

+V -V


V+ H-

VH+

1

[T1e_v10_WIR]


Application of LARGE Cutter Diameter : Face Hobbing Comparison of directions of displacement: pinion: L.H. / gear: R.H.

VH+

+V

-V

pinion: R.H. / gear: L.H. V+ H-

4

2

+H

-H

4

V+ H+

VH-

-V

2

+V +H

+H

-H +H

-H

-H 3

+V -V


V+ H-

VH+

1 3


V+ H+

VH-

1

+V

-V

[T1e_v10_WIR]

132

Face Milling

Face Hobbing

direction of displacement

r c = Ds/2

rb m

+V

-V -V

+V +H

+H

-H

+H +H

-H

-H

-H +V

+V

-V

-V

Note: different direction of contact displacement for  H (horizontal) “actual” radius = Ds/2 Klingelnberg AG · Training Center

Face Milling

Note: “actual” radius rbm < Ds/2 (= instantaneous radius of length length curve at mean point)


Displacement Directions large cutters

10:41-200/25 Arcon(II)16-7.5“ Rinv/Re=1.24

[T1e_v10_WIR]

Face Hobbing

10:41-200/25 Spiron(II)17-88 Rinv/Re =1.08

rw = 95.25 mm

rbm = 66.4 mm

In order to allow a direct comparison of tooth contact displacements between the two cutting methods, EaseOff parameters for an example design are calculated identically Exemplary the tooth contact for both designs is placed in the center of tooth width and profile; the amount of length- & profile crowning as well as twist are assumed identical note:

this does not represent a generally or practically required position of a NOMINAL tooth contact for this application



[T1e_v10_WIR]

133

Face Milling

Displacement Directions V large cutters

10:41-200/25 Arcon(II)16-7.5“ Rinv/Re =1.24

Face Hobbing

10:41-200/25 Spiron(II)17-88 Rinv/Re=1.08

V +0.35

V +0.35

V+

V+ V-

V-

V+

V+ V-

V-

V -0.35


Face Milling

V -0.35

[T1e_v10_WIR]


Displacement Directions H large cutters

10:41-200/25 Arcon(II)16-7.5“ Rinv/Re =1.24

Face Hobbing

10:41-200/25 Spiron(II)17-88 Rinv/Re=1.08

H +0.35

H+

H +0.35

H+ H-

H+ H-

H-

H -0.33



H+ H-

H -0.35

[T1e_v10_WIR]

134

Face Milling

Displacement Directions V+H large cutters

10:41-200/25 Arcon(II)16-7.5“ Rinv/Re =1.24

Face Hobbing

10:41-200/25 Spiron(II)17-88 Rinv/Re=1.08

V+0.30 H-0.30

H+ V-

V+ H-

H+

V+0.30 H-0.30

V+

H+

V-

H-

V+

VH-

H-

V-0.30 H+0.30

V-0.30 H+0.30


H+

V+

V-


Direction of TCP Displacement: general

[T1e_v10_WIR]

Face Hobbing

The directions of tooth contact displacements shown in previous pages do not only depend on the relative tool diameter or the position of the Point of Involute respectively, TCP displacements depend also on the initial position of TCP. This characteristic can apparently be recognized for gears designed with medium size tool diameter like Rinv /Re = ca. 0.85 … 1.05 • The direction of displacement V (vertical) is generally independent from the position of N-point as well as from the initial TCP; the vertical directions stretch for all applications at almost identical inclinations slightly diagonal across the flanks • The inclination of the direction of displacement H (horizontal) becomes increasingly more leveled the closer the initial TCP is positioned towards the heel. With a TCP in close proximity to the Point of Involute, the directions of displacement resembles that of a gear designed with relatively small tool diameter • the described tendency for the direction of displacement H (horizontal) appears more distinctively on the Coast flank as it does on the Drive flank >> see page 270



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135

Direction of Displacement: V+H medium size cutters 10:41-200/25 Spiron(II)13-76 Rinv/Re= 0.99

Face Hobbing

Tooth Contact in MEAN Position

V+0.30

V+ V-

H+0.30

V+

H+

V-

H-

H+ H-

MEAN position

V0.30


H0.30

[T1e_v10_WIR]


Direction of Displacement: H medium size cutters 10:41-200/25 Spiron(II)13-76 Rinv/Re= 0.99

Face Hobbing

TCP in TOE or HEEL position

H+0.30

H+ H-

H+ H-

H+

H+

H-

TOE position

HHEEL position

H0.30



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136

Displacement under Load (schematically): no load condition:

small cutter 

deflection under load:  position of contact is stable


[T1e_v10_WIR]


Displacement of TCP under Load :

small cutter 

contact at no load condition:

deflection of contact under load:

DRIVE: center position COAST: center-to-heel position

DRIVE: contact spreads evenly COAST: contact spreads more to toe position of contact is stable



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137

Displacement under Load (schematically): medium cutter  no load condition:

deflection under load:  contact moves slightly to heel


Displacement under Load (schematically): no load condition:

[T1e_v10_WIR]


large cutter 

deflection under load:  pos. of contact moves to heel



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138

Displacement of TCP under Load:

large cutter 

contact at no load condition:

deflection of contact under load:

DRIVE: toe position COAST: toe position

DRIVE: contact spreads to heel COAST: contact spreads to heel position of contact moves to heel



[T1e_v10_WIR]

thank you for your attention !



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139

.... and finally let’s repeat the very, very basics of gearing !



[T1e_v10_WIR]

... an amusing introduction to the world of gearing expressions: M Z P.C.D. C.P. D.P. HK HF c

Gear Rhymes : Bramley-Moore Sanderson Brothers Pty.Ltd. Thomastown, Australia Illustrations Klingelnberg AG · Training Center

: : : : : : : :

Module Number of Teeth Pitch Circle Diameter Circular pitch Diametral Pitch Addendum Dedendum Clearance

published : Charles Cooper

Robert Wirthlin

GEARTECHNOLOGY Magazin / Sept.2000 Oerlikon Geartec


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140

Those who belong to the trade Engineering and wish for success, they m u s t understand gearing; Wherever you go where machinery’s fixed, you are bound to find gear wheels, all sizes, all mixed.


Diameters then shall be called letter D. It shortens the word, so I hope you agree. Big D is measured right over the teeth, Pitch D is measured a little beneath.


[T1e_v10_WIR]



D P.C.D.

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141

From one tooth to the next, if measured it be, along the Pitch Circle and not on Big D will give us the Circular Pitch of the gear, a word you will probably frequently hear. The number of teeth in a gear wheel, you see, depends on the Circular pitch and Pitch D.

C.P. 

P.C.D.    M  Z

Eva’s showing the Pitch of the gear! Klingelnberg AG · Training Center

[T1e_v10_WIR]


If two are but known, you can find out the third with the help of a rather peculiar word. PI it is called, a valuable key, three-point-one-four-one-and-six it must be!

M If you are given Circular Pitch an the Teeth, put these on top and put PI underneath. Work out this fraction and you will obtain the answer Pitch D. Now let me explain that if you require any other relation, it’s easily got from this simple equation.



P.C.D. 

P.C.D. C.P.  Z  C.P.  Z 

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142

If Pitch D and PI are both multiplied, to get Circular Pitch, by Teeth you divide. Reverse the last two, and the answer will be the Number of Teeth in the gear wheel, you see.

P.C.D.   Z P.C.D.   Z C.P.

C.P. 

The height from Pitch D to the top of the tooth is called the Addendum, it’s really the roof. To reckon Addendum you just specify the Circular Pitch and divide it by PI.

HK 


[T1e_v10_WIR]


Now the Circular Pitch should not be confused with more simple method more frequently used. Diametral is better than Circular Pitch, the figures are shorter, no chance of a hitch. Let us call it D.P., it saves waste of time, it’s not only correct but it is easier to rhyme. It gets over the use of those troublesome PI’s ; moreover its value once signifies the Number of Teeth for each inch of Pitch D. Large D.P. means size of teeth becomes wee.

C.P. M 

D.P. 

Z P.C.D.

D.P. D.P.



[T1e_v10_WIR]

143

The Number of Teeth over D.P. will at once give the answer Pitch D, unless you’re a dunce. The other way round, Teeth - over Pitch D, will obviously give you the answer D.P.

P.C.D. 

Z D.P.

Z  P .C.D.  D .P . 1 D.P . 0 .25 ... c D.P .

HK  For Number of Teeth, now kindly take heed, use Pitch D and D.P. it’s their product you need. For Addendum you take one, and divide by D.P. from this you can easily work out Big D. If it is the clearance you’re anxious to know, write point-two-five-and-something, with D.P. below.


To convert D.P. in circular measure it so easily done that it’s really a pleasure. Divide PI by D.P. , that is all you need do. The thing is so simple it hardly seems true!

If you want to convert these the other way round, the answer is quickly and easily found. Divide PI by the Circular Pitch and you then get the answer D.P. with the stroke of a pen.


[T1e_v10_WIR]



D.P. 

 C.P.

C.P. 

 D.P.

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144

APUNTES_GLEASSON_Basics of Spiral Bevel & Hypoid Gears

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