This edition first published 2012

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Library of Congress Cataloging-in-Publication Data

Jelaska, Damir.

Gears and gear drives / Damir Jelaska.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-119-94130-9 (cloth)

1. Gearing. I. Title.

TJ184.J415 2012

621.8′33–dc23

2012014164

A catalogue record for this book is available from the British Library.

Print ISBN: 9781119941309

In memory of my mother

Preface

Since gear drives operate with a power efficiency significantly higher than any other mechanical drive, or any electrical, hydraulical or pneumatical power transmission, they have the widest use in transforming rotary motion from the prime mover to the actuator, and their importance is growing day by day. Although efficiency is not the only criterion for choosing the type of transmission, the gear drive, due to its robustness and operational reliability, presents an inevitable component of most mechanical engineering systems. Gear drives are known to be highly demanding in design, manufacture, control and maintenance.

The entire field is well provided with standards, books and journal and conference papers. Thus, why a necessity for this book? There are three main reasons:

This book is written with the presumption that the reader has a basic knowledge of mechanics and general mechanical engineering. It is primarily addressed to graduate and undergraduate students of mechanical engineering and to professionals dealing with the manufacturing of gears and gear drives. For all of these, it is supposed to be a primary text. Groups with an occasional need for this material are students of industrial engineering, technology, automotive engineering, students of marine engineering, aviation engineering and space engineering and professionals in control and maintenance. The objective of this book is to provide all of these with everything they need regarding the subject matter in a single book: (i) a background for dealing with gears and gear trains (classification, power, torque, transmission ratio distribution), (ii) a complete geometry and kinematics for almost any type of gears and gear drives, (iii) assessments of load capacities in accordance with recent standards, including the calculation of micro-pitting load capacity, (iv) directions and suggestions for the practical design of gears and gear drives, (v) detailed instructions and formulae for determining the tolerances and procedures for measuring and controlling the accuracy of drives and their members in accordance with the latest standards. The reading matter is accompanied with a large number of figures and every important formula is derived and discussed.

This book consists of seven chapters. The first chapter introduces the reader to the fundamental parameters of mechanical drives – transmission ratio, power, efficiency, torque and rotational speed – and explains the way for determining them. The classification of mechanical drives and gear drives is also included. The second chapter explains in depth the geometry of cylindrical gear toothing as the basis of the entire field of gear drives, beginning with the idea of rolling, through the manufacturing of gears, the mesh and interference of teeth, tooth modifications, to the gear tolerances. The third chapter deals with the integrity of cylindrical gears, presenting the ways of calculating the load capacities for pitting, tooth root strength, scuffing and micro-pitting. In the fourth chapter the cylindrical gear drive design process is suggested and the selection of gear materials and their heat treatment are explained in depth, as well as gear drive lubrication and the efficiency and temperature of the lubricant. The fifth chapter deals with bevel gear drives: geometry, manufacturing, control, tolerances and load capacity checks. Crossed gear drives are also explained. In the sixth chapter simple planetary gear trains are first presented: transmission ratio, torques, efficiency of power and branching. Special trains, like harmonic and composed trains and also coupled, closed and reduced coupled trains are explained, as well as planetary reducers. The seventh chapter deals with worm gear drives: their geometry, manufacture, deviation control and load capacity assessments for the wear, pitting, heating, wormwheel tooth root and worm shaft deflection.

The book assumes that the reader is familiar with the metric (SI) system of units. However, some remarks are given herein: since standard modules are given in millimetres, all gear dimensions should be expressed in millimetres as well. Hence, in all equations where only length units appear, all physical quantities are to be substituted in millimetres. The exceptions are the allowance equations, where gear dimensions are to be substituted in millimetres to obtain the allowance in microns. In other equations, where the dimensions of physical quantities are not only their lengths, the SI scale of units should be applied and gear dimensions should be substituted in metres, regardless of being marked in millimetres in the list of symbols at the end of each chapter. Relationship equations make an exception where the units of both sides of the equation are not the same. In each such equation the units of physical quantities (those which are to be substituted) are specified, as well as the unit of physical value obtained on the left side of the equation.

Acknowledgments

Boris Obsieger, Technical Faculty, University of Rijeka, Croatia

Irma R. Sharma, Hindustan Motors, Noida, India

Josip Obsieger, Tehnical Faculty, University of Rijeka, Croatia

Jože Flašker, Faculty of Mechanical Engineering, University of Maribor, Slovenia

Sran Podrug, Faculty of Mechanical Engineering, Electrical Engineering and Naval Architecture, University of Split, Croatia

Sreko Glodež, Faculty of Physical Sciences, University of Maribor, Slovenia

Stanislav Pehan, Faculty of Mechanical Engineering, University of Maribor, Slovenia

Zoran Ren, Faculty of Mechanical Engineering, University of Maribor, Slovenia

Chapter 1

Introduction

1.1 Power Transmissions and Mechanical Drives

Mechanical power transmissions¹ consist of units which, in distinction from electrical, pneumatic and hydraulic ones, transfer power from the prime mover to the actuator (operational machine or operational member) with the assistance of rotary motion. These units are called mechanical drives and are situated between the prime mover and the actuator (Figure 1.1). The drive is connected with both the prime mover and the actuator by couplings or clutches forming an entirety whose function is defined by the purpose of the actuator.

Figure 1.1 Schematic account of a mechanical drive application

The embedding of a power transmission to link the prime mover and the machine operating member can be due to a number of reasons:

• The required speed of the machine operating member very often differs from the speeds of the standard prime movers.
• One prime mover has to drive several actuators.
• The driven side speed has to be frequently changed (regulated), whereas the prime mover cannot be used to full advantage for this purpose.
• Certain periods of the driven side operation may require torques far from those obtained on the motor shaft.
• As a rule, standard motors are designed for uniform rotary motion, while operating members have sometimes to move with varying speed or periodic halts.
• If a resonant vibration of some member in the chain of power transmission cannot be solved in any other way, the frequency of rotary motion can be changed by building-in a drive.
• Sometimes considerations of safety, convenience of maintenance or the dimensions of the machine, especially if the prime mover and operational machine shaft axes are not coaxial, do not allow the direct coupling of the prime mover shaft with operating member.

The capital task of the designer is to select such an assembly ‘prime mover – transmission (drive)’ which should optimally meet the needs of the operational machine or member. This act of choosing is a complex task, whose solution depends on: (i) accessibility of the energy source and its price, (ii) efficiency of the entirety of prime mover – transmission – operational machine, (iii) investment costs, (iv) operational machine features, primarily the (v) variability of its speed of rotation, (vi) service conditions, (vii) drive maintainability and so on. Within the framework of this task, a particularly complex problem is defining the transmission: mechanical or some other? This question is beyond the scope of this book, but generally it may be affirmed that the basic advantage of mechanical drives in relation to all the others is their very high efficiency, which is becoming more and more important day by day.

The comparative advantages offered by possible transmissions and drives are outlined in Table 1.1 which gives only a general illustration. Recently, a prominent feature in power transfer has been the extensive employment of electric, hydraulic and pneumatic transmissions. Frequently, such transmissions together with mechanical drives are simultaneously used to actuate various mechanisms. The proper choice of a drive for each specific case can be made only by comparing the technical and economical features of several designs.

Table 1.1 Advantages of transmissions and drives.

The mechanical drive driving shaft receives power P₁ at speed of rotation n₁ from the prime mover driven shaft, and the mechanical drive driven shaft supplies power P₂ < P₁ at speed of rotation n₂ to the operational machine driving shaft. The difference P₁ − P₂ = P_L is called power loss and the ratio:

is called efficiency; it takes a special place amongst power transmission characteristics because it shows unproductive power expenditure and so indirectly characterizes the wear of the drive and its warming up – the capital problems in power transmissions. Warming up causes strength and lifetime decrease of drive parts. Their corrosion resistance and the functional ability of lubricant are also imperilled. The importance of efficiency is raised to a power by the global lack of increasingly expensive energy and its value also decisively affects the price of the drive.

The power loss consists of constant losses which on the whole do not depend on load, and variable losses which on the whole are proportional to the load. The value of constant losses approximates the power of idle run, that is, the power needed to rotate the drive at P = 0 on the driven shaft. It depends on the weight of the drive parts, the speed of rotation and the friction in the bearings and on other surfaces of contact.

The second fundamental parameter of a mechanical drive is the transmission ratio i defined as the ratio of its driving n₁ and driven n₂ shaft speeds of rotation or angular speeds:

(1.1)

If i > 1 (n₁ > n₂) the mechanical drive is called an underdrive and its member is called a reducer. It reduces the speed of rotation and the transmission ratio is also called a speed reducing ratio. If i < 1 the mechanical drive is called an overdrive, its member is called a multiplicator and the transmission ratio is also called a speed increasing ratio. It multiplies the speed of rotation. An overdrive usually works less efficiently than an underdrive. This is especially true for a toothed wheel gearing.

1.2 Classification of Mechanical Drives

The basic division of mechanical drives falls into:

• Drives with a constant transmission ratio.
• Drives with a variable transmission ratio.

In constant transmission ratio drives, the constant speed of driving shaft rotation results in a constant speed of driven shaft rotation, n₂ = n₁/i. Their design should, as a rule, include at least the following data: (i) transmitted power of the driving (P₁) or driven (P₂) shaft or related torques, (ii) speed of rotation (rpm) of the driving (n₁) and driven (n₂) shaft, mutual location of the shafts and distance between them, (iii) overall dimensions and drive operating conditions, especially the dependence of driven shaft rpm or torque on time. In general, this design has several solutions, that is, given conditions can be used to develop drives of various types. All possible designs should be compared according to their efficiency, weight, size, original and operational costs in order to select the most advantageous one. Some general considerations, mainly the available experience of design, manufacture and operation of various drives enables us to outline generally the limits of priority application of these drives.

Therefore, the designer must take into account the limit values of main parameters which the mechanical drive can reach. These parameters are the maximum values of transmission ratio, efficiency, power and speed of rotation. For underdrives, they are presented in Table 1.2.

Table 1.2 Limit parameters of mechanical drives.

However, these limits are of a temporary nature: as new materials are produced and manufacturing methods are improved, our knowledge of processes taking place in transmission becomes deeper and designs are perfected to suit broader fields of application.

For overdrives, these values are considerably smaller; such drives have a poor performance. It primary refers to efficiency and transmission ratio, but vibrations and noise levels are also much greater, especially in gear drives. It is due to the reasonable and experimentally approved fact that, with a similar error in the manufacture of the two meshing wheels, the driving wheel of greater diameter (in overdrive) causes larger angular accelerations (causing dynamic impacts) on the smaller driven wheel, while the reverse is applied to an underdrive.

Mechanical drives with a constant transmission ratio are divided into: (i) drives with immovable axes – classical mechanical drives – and (ii) drives with movable axes – planetary mechanical trains.

According to the mode in which they transmit motion from the driving wheel to the driven one, mechanical drives with immovable axes fall into the following types:

• Transmissions using friction: with direct contact of wheels (friction drives) or with a flexible connection (belt drives).
• Transmissions using mesh: with direct contact (toothed and worm gears) or with flexible connection (chain and gear-belts drives).

Table 1.3 Main classification of classical mechanical drives with constant transmission ratio and immovable axes.

This classification is clearly presented in Table 1.3.

Any individual drive is basically composed of a pair of wheels: gear wheels, rollers, belt pulleys and sprockets. It can operate on its own or can be built into mechanical trains of various machines and instruments and made in the form of an individual drive enclosed in a special housing. It can be also joined with other individual drives, making a multi-stage mechanical drive (see Section 1.4).

It is appropriate to mention that i = const classical friction drives are no longer in use. They are included in the classification because they can be still found in machinery plants.

Figure 1.2 presents simple examples of the three main types of mechanical drives with immovable axes: gear drive with a constant transmission ratio (Figure 1.2a), belt drive with a stepless change of transmission ratio by the help of an axially movable belt (Figure 1.2b), gear drive with a step by step change of transmission ratio by the help of an axially movable set of gears 1 and 3 (Figure 1.2c).

Figure 1.2 Sample types of mechanical drives with immovable axes: (a) constant transmission ratio, (b) stepless change of transmission ratio, (c) step by step change of transmission ratio

Planetary mechanical trains are mechanical drives where at least one wheel (planet) has movable axes and, when meshed with one or more (central) wheels with immovable (common) axes, it turns around them. The planets are joined together into the planet carrier which has its own shaft. By the mode in which they transmit the motion, that is, by the type of wheel, planetary trains are divided into planetary gear trains (see Chapter 6) and planetary friction trains. Both of them have the same kinematics.

Except for their high price and the noise they produce, gear drives have an advantage over all other drives in all features, especially in operational safety and endurance, efficiency and smaller dimensions. This is why gear drives, although very exacting in design and technology of production, make up approximately 80% of all mechanical drives (see diagram in Figure 1.3).

Figure 1.3 Global spread of mechanical drives towards the end of 2005

In variable transmission ratio drives, the constant speed n₁ of driving shaft rotation results in a variable speed n₂ of driven shaft rotation. If n₂ changes are stepless, such drives are called continuous variable speed drives or variators. The ratio drive is changed by a change in the mutual position of the wheels (in friction variators) or by a change in the belt position (in belt variators; see Figure 1.2b). Modern gear planetary variators are able to achieve a practically infinite range of regulation besides reverse.

Mechanical drives with a cyclically variable transmission ratio are realized mostly by means of gear pairs with gears of an elliptic or hypocycloid form (see Figure 1.6). A cyclic rotary motion can also be achieved by a segment gear drive (see Figure 1.7a); and an alternate rectilinear motion can be obtained by engaging two toothed racks with driving segment gear (see Figure 1.7b).

There are also gear drives with cyclically variable centre distance which, beside a variable transmission ratio, enable a change in the direction of rotation of the driven shaft (reverse) (see Figure 1.10).

Mechanical drives with a step by step ratio change are mostly appropriate for the gearboxes of automotive vehicles and machine tools. They function by the principle of changeable or axially movable gears which produce a transmission ratio depending on which of them is in mesh, that is, depending on which of them transfers the motion (see Figure 1.2c).

There are also planetary gear variators which, coupled with friction variators, can reach a practically limitless regulation range and change in the direction of revolution (reverse). Planetary gear drives with a step by step change of transmission ratio are known as planetary gear-boxes.

The application of mechanical drives with a changeable transmission ratio is always adequate when the prime mover is not able to change the speed of rotation in the manner required by the operational machine. An accustomed application is found in constant power drives at a larger range of speed of rotation and always when it is necessary to obtain the designed high precision transmission ratios. It is important to mention that regulation and control of speed at constant torque is more efficient by electric motor regulation than by mechanical drive.

1.3 Choosing a Mechanical Drive

A basic presumption in choosing a mechanical drive is the designer's good knowledge of all types of drives: their features and applications. Only in such a case he can correctly predict what type of drive should be the most appropriate one when the prime mover and operational machine are already determined.

The first step in the process of mechanical drive design is to select as much data as possible in order to outline the drive and to dimension its parts correctly. This information refers to nine basic fields: basic function data of the drive, prime mover data, actuator data, drive manufacture data, load data, user requirements, lubrication, environment and assembly and maintenance.

Basic function data of a drive:

• Driving and driven axes rotational speeds and nature of transmission ratio: constant or not; stepped or not?
• Transmission ratio variation.
• Reverse: yes or no?
• Types of prime mover and operational machine and their power and torque characteristics.
• Ordering party requests with regard to type of drive, built in position, rotational speed limits, prime mover, establishments and so on.
• Operational machine position in regard to prime mover: is it changeable; what are the limits?

Prime mover data:

• Speed of rotation.
• Consequences of machine damage or breakdown.
• Is there a need to take into account the machine overthrowing?
• What are the sizes of intensity and frequency and duration of service impact?
• What maximum torque can the prime mover reach; and could it cause a catastrophe? Is there an overload safeguard?

Operational machine (member) data:

• Speed of rotation and its change,
• Intensity, frequency and duration of service impact; load spectrum,
• Intermittency, that is, operational hours per day,
• Consequences of machine damage or breakdown.

Data needed for drive manufacture:

• Material selection limits,
• Tools and machines needed for drive manufacture,
• Limits in usage of machine tools, heat treatment furnaces and the like,
• Are there the technologies predicted that we do not master?

Load data:

• Is there any axial force on the driving or driven shaft?
• How is the drive fixed and what are the values of the forces arising from that?
• Other forces transferring to environment.

Ordering party requests, conditions of taking over:

• Driving and driven axes clutches,
• Design standards,
• Noise level, minimum efficiency, testing.
• Drive design requests: forged or welded gear body, cast or welded housing, joint between gear body and rim,
• Safety requests,
• Special directions in drive design,
• Conditions of drive transfer,
• Reserve price,
• Terms of delivery.

Lubrication:

• Is the designer limited in lubricant selection?
• Mineral or synthetic lubricant?
• Independent or central lubrication?
• Cooling requests: fresh- or sea-water, air or water emitted into the environment? What is the temperature of operation? Central cooling equipment or an independent unit?
• Drive operation control.

Environment, drive placing:

• Placing of the drive: open or enclosed space?
• Installation limits.
• Is there enough space around the drive?
• Substructures: are they taken together for the prime mover and actuator?
• Can there be difficulties in transferring the drive at the installation place?
• Room temperature.

Maintenance:

• Predicted type of maintenance.
• Lifetime of the drive and its components.
• Which diagnostic method is predicted?
• Is there a request for special access to certain components of the drive?
• Is there a request for a list of critical parts?
• Is it required to deliver the most necessary spare parts?
• How the drive should be coloured?

When there is agreement between the designer and the ordering party on the listed data and dilemmas, or at least on most of them, it is customary to write it in the form of a document. It is often only a morally required document but, by arrangement, it can also become a materially required document – a constituent part of the contract for the manufacture and delivery of the drive.

1.4 Multi-Step Drives

Simple mechanical drives with a single pair of wheels, like those depicted in Table 1.1, are called one-stage drives. Their transmission ratio is limited and a larger one is often required. The reduction is then obtained in a few stages in the way that the driven shaft of each particular stage is at the same time the driving shaft of the next stage. Such a drive is then called a multi-step drive.

It is understandable that each shaft can, in principle, drive one operational machine. In Figure 1.4 a fictitious, general multi-step mechanical drive is presented, which consists of prime mover PM, n wheels with corresponding shafts, that is, with n/2 stages and k operational machines. The first stage consists of the pair of spur gears 1, 2 and operational machine OM₁ connected to gear 2 shaft; the second stage is a pair of spur helical gears 3, 4 and operational machine OM₂ connected to gear 4 shaft and so on; the last stage is the V-belt drive with two V-belt pulleys, each with three V-belts transferring the power to the OM_k operational machine. The OM_k−1 operational machine is driven by the (n − 1)^th wheel.

Figure 1.4 Schematic account of a multi-step drive

When designing multi-step drives it is necessary to determine the required power of the prime mover for the given power of the operational machine, the power losses (i.e. efficiency) and a transmission ratio. Also, the torques of any shaft must be determined in order to dimension all the related components. For such a purpose, it is necessary to identify the relation between the total transmission ratio and the partial ones within each stage, the relation between the total efficiency and the partial ones within each stage and the power balance, that is, the relation between the powers of the prime mover and the operational machine.

Total transmission ratio i_tot can be written as:

(1.3)

where the angular speeds of particular wheels are signed with ω. It follows that:

(1.4)

where the partial transmission ratios are signed with i_j. It is clear now that the total transmission ratio equals the product of the particular stages' partial transmission ratios.

In the same way the total efficiency is obtained:

(1.5)

that is, the total efficiency equals the product of the particular stages' partial efficiencies.

From the mechanical drive energy balance, which declares that the energy produced in the prime mover equals the sum of energies spent by all operational machines plus their power losses, it follows that the power balance of a drive is the power of the prime mover P_PS which equals the sum of the operational machines powers enlarged for the power losses from driving to the single driven operational machine:

(1.6)

where P_OM1, P_OM2, ..., P_OMk are the powers of the operational machines, η_1,k is the efficiency at power transmission from the prime mover to the k^th (last) operational machine, η_1,k = η_u and k is the number of operational machines.

The power P_m at an arbitrary m^th wheel equals the power of the prime mover decreased for the powers spent by that prime mover for driving the operational machines and for the power losses from the prime mover to the m^th wheel:

(1.7)

where P_OM,i is the power of an i^th (arbitrary) operational machine located between the prime mover and the m^th wheel, η_1,i is the drive efficiency from the prime mover to the i^th operational machine located before the m^th wheel, N is the number of operational machines before the m^th wheel and η_1,m is the drive efficiency from the prime mover to the m^th wheel.

The torque T_m at an arbitrary m^th wheel is equal to the ratio of its power and angular speed:

(1.8)

(1.9)

As an example, the gear 4 torque will be determined. Its power is:

(1.10)

while the angular speed is:

(1.11)

The torque is now:

(1.12)

Just like the power, the torque is divided in two parts: left and right from gear 4. Left from gear 4 the power equals the operational machine OM₂ power:

(1.13)

the torque is T_4L = P_OM2/ω_4, while right from gear 4 the power P_4D equals

(1.14)

1.5 Features and Classification of Gear Drives

1.5.1 Features of Gear Drives

The gear is defined as a toothed member designed to transmit motion to or receive motion from another toothed member, by means of successively engaged teeth. The two gears are rotatable around axes whose relative positions are fixed, and they form a gear pair. The torque from the driving shaft to the driven one in a gear drive is transmitted due to the pressure of the teeth of the pinion (the gear in a pair which has the smaller number of teeth) on those of the wheel (the gear in a pair which has the greater number of teeth). To preserve a constant transmission ratio, the teeth of both pinion and wheel should have conjugate profiles. This condition is observed if the teeth of the mating gears are correctly meshed with the standard basic rack teeth which are used as a basis for defining the tooth dimensions.

Gear drives blaze the trail to other types of mechanical drives and by the frequency of building in they keep mastery. These are the reasons:

• High reliability and durability of its components,
• Less dimensions compared to transmitted loading,
• High efficiency: 98–99% for regular (one step) drive and even up to 99.6% for planetary drives,
• Power transfer is slideless and overloading is possible,
• Simple maintenance.

An essential advantage of (involute) gear drives is that the gears can be corrected to improve the characteristics of the drive with a minimum weight. The essence of correction lies in the fact that different portions of the involute of the given base circle are used to describe the active profile of the teeth if certain characteristics of engagement are to be changed.

The imperfections of gear drives are the noise and possible vibrations they produce and high costs of production.

Gears have to satisfy high requirements on the power they transmit, on the speed of rotation, on the precision of their manufacture and on the accuracy of their operation. Therefore, it is not easy to produce quality gear drives; for that, it is necessary to have a great knowledge and great experience. The unwritten rule is that the quality of the gear drives produced is implicit in the level of industrial development of the country or region.

1.5.2 Classification of Gear Drives

The basic classification of gear drives is: (i) those having immovable axes and (ii) those having rotationally movable axes – planetary gear drives.

Gear drives with immovable axes – classical gear drives, are classified into:

• Gear drives for parallel shafts,
• Gear drives for inclined shafts,
• Gear drives for shafts having skewed axes,
• Gear drives with step by step change of transmission ratio by help of axially movable set of gears (see sample drive in Figure 1.2c).

Gear drives for parallel shafts (Figure 1.5) are realized with:

• Pairs of cylindrical gears with external toothing: with straight teeth (spur gears; Figure 1.5a), with helical teeth (Figure 1.5b), with double helical (or sometimes herringbone) teeth (Figure 1.5c),
• Pairs of internal toothing gears which consist of a smaller gear (pinion) with an external and greater gear (wheel) with internal toothing (Figure 1.5d),
• Rack drive – external toothing (pinion) gear mated with toothed rack (infinity diameter gear) in order to transform rotary motion into translatory motion, or vice versa (Figure 1.5e).

Special, rarely applied types of gear drives with parallel axes are gear pairs with a cyclically variable transmission ratio (Figure 1.6). The gears have an elliptic form (Figure 1.6a, b), a coupling of an elliptic and eccentrically situated cylindrical gear (Figure 1.6c) or a hypocycloidal form (Figure 1.6d). Such drives are complex to manufacture and consequently their application is limited.

Figure 1.5 Gear drives for parallel shafts: (a) spur gear drive, (b) helical gear drive, (c) double helical gear drive, (d) internal gear drive, (e) rack drive

Figure 1.6 Types of drives with non-circular gears: (a) and (b) elliptic forms, (c) coupling of elliptic and eccentrically situated cylindrical gear, (d) hypocycloidal form

Segment gear drives can also be involved in this group of gear drives (Figure 1.7). Such drives enable a cyclic rotary motion (Figure 1.7a) or alternately a rectilinear motion of driven member 2 (Figure 1.7b) with continued rotary motion of driving member 1. In both cases, the driving gear is partially toothed (segments gear), and in the drive pursuant to Figure 1.7b the driven member 2 is composed of two toothed racks which are alternately mated with driving segment gear 1 which has continuous rotation. The result is rectilinear motion first on the left, then on the right side.

Figure 1.7 Schemes of gear drives with non-circular gears: (a) for cyclic rotary motion, (b) for alternate rectilinear motion

Gear drives for inclined shafts are realized with pairs of bevel gears having:

• Straight toothing (Figure 1.8a),
• Tangent toothing (Figure 1.8b),
• Curved toothing (to left or right; Figure 1.8c) in the form of an arc, Archimedes spiral, involute, epicycloid or sinusoid.

Figure 1.8 Gear drives for inclined shafts: (a) with straight toothing, (b) with tangent toothing, (c) with curved toothing

Gear drives for shafts having skewed axes are realized with:

• Pairs of helical gears with shafts skewed at a certain (mostly right) angle – so-called crossed gears drive (Figure 1.9a),
• A worm drive consisting of the worm screw, mostly called a worm, and a toothed wheel, mostly called a worm wheel (or worm gear; Figure 1.9b),
• Pairs of hypoid gears – hypoid drive. These are the gears whose axodes (rolling bodies) have a hyperboloid form. If the teeth of these hyperboloids are cut with the same normal pitch, a hyperboloid drive is obtained. In practice, only two portions of these hyperboloids are used: the central ones resulting in a crossed gear drive and the edge ones resulting in a hypoid drive (Figure 1.9c). This drive is mostly realized with specially designed bevel gears.

Figure 1.9 Gear drives for shafts having skewed axes: (a) crossed gear drive, (b) worm drive, (c) hypoid drive

Gear drives with movable axis – planetary gear drives, defined as such gear mechanisms having at least one movable axis rotating around some other (basic) axis. They are classified:

Gear drives with cyclically variable centre distance can also be involved in a group of drives with movable axes. Beside the small change in transmission ratio, they enable reverse, that is, alternate rectilinear or rotary motion of the actuator (see Figure 1.10). Here, driving gear 1 has three degrees of rotational freedom: (i) rotation around its own axis, (ii) rotation around actuator 2 with internal toothing and (iii) translation radially towards or opposite, from the axis of toothed actuator 2. Rotation around member 2 is achieved by guiding the roller at the end of gear 1 shaft along channel 3 whose form follows the member 2 crescent form. Radial translation is achieved by means of slider 4. When the driving gear reaches the end position of the ‘crescent’ (inflection point of the path), the actuator changes the direction of rotation.

Figure 1.10 Scheme of a gear mechanism for transforming uniform rotary motion into alternate rotary motion (oscillation)

This general course of ‘Gears and gear drives’ examines mechanical gear drives designed mostly for uniform rotary motion, while the ‘Planetary gear drives’ chapter describes both uniform and partially non-uniform rotary motion. Other types of mechanical drives, as well as electric, hydraulic and pneumatic transmissions are the subject of other courses.

1.6 List of Symbols

1.6.1 Subscripts to Symbols

Note

1. The power is a feature of some machine or device and cannot be transmitted. Actually, only the energy is transmitted, but it is globally common to say that the power is transmitted.