TECHNICAL UNIVERSITY OF LODZ DESIGN BASIC OF INDUSTRIAL GEAR BOXES Calculation and Design Case Example Andrzej Maciejczyk Zbigniew Zdziennicki 2011 Department of Vehicles and Fundaments of Machine Design Table of contents Page Chapter 1: 6 BASIC KNOWLEDGE 1.1 Introduction 6 1.2 Basic size and selection 7 1.3 Torque selection 8 1.4 Materials and heat treatment 9 1.5 The size of the unit 12 1.6 Example 14 Chapter 2: 16 GEAR MESH 2.1 Ratios 16 2.2 Tooth-Pitch combinations 16 2.3 Pitch and module 16 2.4 Example 18 2.5 Face – widths 19 2.6 Detail of gears 20 2 Chapter 3: 24 SHAFT LOAD CALCULATION 3.1 Design description 25 3.2 Given data 26 3.3 Transmission torque 26 3.4 V-Belt pulley loads 26 3.5 Spur pinion loads 27 3.6 Free body diagram of the high speed shaft 27 3.7 Calculations and diagrams of bending moment (high speed shaft) 28 3.8 Torsion diagram 31 3.9 Critical section of the high speed shaft 32 3.10 Bearing loads of the high speed shaft 32 3.11 Minimal shaft diameter (for high speed one) 32 3.12 Simple method of shaft (minimal) diameter calculation 33 3.13 Minimal diameters of high speed shaft ends 33 3.14 Free body diagram of the low speed shaft 34 3.15 Calculations and diagrams of bending moment (low speed shaft) 34 3.16 Torque acting on the low speed shaft 35 3.17 Torsion diagram for the low speed shaft 35 3.18 Minimal shaft diameter (for low speed one) 36 3.19 Evaluation of minimal diameter for the low speed shaft with empirical method 36 3.20 Minimal diameters of slow speed shaft ends 37 3.21 Resume 37 Chapter 4 38 DEEP GROOVE BALL BEARINGS (Basis description and fundamentals calculation) 4.1 View 38 4.2 Application 38 4.3 Ball bearings description 39 4.4 Kinds of constructions 39 3 4.5 Theoretical basis 39 4.6 Life’s calculation basis 40 4.7 Example # 1 42 4.8 Example # 2 44 Chapter 5 47 NUMERICAL EXAMPLE OF BALL BEARING SELECTION Chapter 6 53 RADIAL SHAFT SEALS 6.1 Seals design 54 6.2 Type and destinations of materials 54 6.3 Materials recommendation 55 6.4 Temperature limits according to material types 55 6.5 Design types of radial seals 56 6.6 Radial shaft seals diameters in accordance with ISO – 6194 57 6.7 Mounting of radial shaft seal in housing 58 6.8 Mounting of radial shaft seal on shaft 59 6.9 Radial shaft seals under the pressure 59 6.10 Frictional loss 59 Chapter 7 60 V-BELT DRIVES (Basis data and calculation in accordance with PN-M-85203: 1967) 7.1 V – belt power capacity 61 7.2 Small pulley equivalent diameter 61 7.3 Transmission ratio factor k 61 i 7.4 V – belt length 62 7.5 Axis distance (recommended) 62 7.6 V – belt dimensions (in accordance with PN-ISO 4184: 2000) 62 4 7.7 V-belt length 63 7.8 Pulley Groove dimensions 64 7.9 Pulley diameters d 65 p 7.10 Duty power per belt (power transmitted with one belt) P 66 0 7.11 Belt length factor K 67 L 7.12 Belt contact factor Kφ 67 7.13 Service factors K (time and work conditions factor) 68 T 7.14 Example of V-belt drive calculation 70 Chapter 8 74 KEY JOINT (Keyway and key dimensions) 8.1. Key load sketch 74 8.2. Tension distribution 74 8.3. Durability calculation 75 8.4. Specification for metric rectangular keys and keyways 76 Chapter 9 79 GEAR – CASE DESIGN List of references 84 5 Chapter 1 BASIC KNOWLEDGE 8.3. Introduction Gear reducers are used in all industries, they reduce speed and increase torque. You will find them between the prime mover (i.e.: electric motor, gas, diesel or steam engine, etc.) and the driven equipment: conveyors, mills, paper machines, elevators, screws, agitators, etc.). An industrial gearbox is defined as a machine for the majority of drives requiring a reliable life and factor of safety, and with the pitch line velocity of the gears limited to below 25 m/s, as opposed to mass produced gearboxes designed for a specific duty and stressed to the limit, or used for very high speeds etc., e.g. automobile, aerospace, marine gearboxes. To the competent engineer, the design of a gear unit, like any other machine, may seem a fairly easy task. However without experience in this field the designer cannot be expected to cover all aspects of gearbox design. The purpose of this booklet is to set out the basic design for an industrial gearbox. It should help students not familiar with gearboxes, lay out a reliable working design. And it is intended for the reader to use his own experience in selecting formulae, stress values etc., for gearbox components. To avoid the situation presented in the picture below, you should design gearing carefully and correctly. Damage of helical teeth 6 1.2 Basic size and selection The two types of tooth that can be used for both parallel and angled drives are straight or helical (spiral). Spur gears are easier to manufacture and inspect than helical gears, and they can be rectified more easily at the assembly stage if required. The main disadvantage of a spur gear compared with a helical, is in the tooth engagement process. The whole of the spur tooth enters engagement at the same time, and therefore any pitch (spacing) error will cause interference and noise. Spur gears are generally used for pitch line speeds below 10 m/s in drives that are not loading the teeth to their maximum allowable limits. They are also used where gears are required to slide axially in and out of mesh. Helical gears can be manufactured on most modern gear cutting machines. They will probably take longer to machine because of the relative wider face, and hence be more expensive than an equivalent size spur gear. However, this is offset by the fact that the helical gear may be capable of carrying up to fifty per cent more load. Conversely, for a given power, helical gears can be made more compact than a spur set. Helical gears are superior to spur gears in most applications, especially where noise must be kept to a minimum, or the pitch line speed is in excess of 10 m/s. These gears are also easier to design to fit given centre distances because there are more parameters that can be re-arranged. The main disadvantage of the helical gear is the axial thrust generated by the gears when working. Double helical gearing has the same characteristics as the single helical but with the elimination of end thrust, as the two helices producing the thrust are cut with opposite “hands”. This type of gearing is also useful where the pinions are of small diameter, as the equivalent face to diameter ratio is only half that of a similar net face single helical gear. Bevel gears are used for drives requiring the input shaft to be at an angle,usually 90° to the output shaft. They can be cut with either straight teeth, where the same comments as for spur gears apply, or they can be cut spiral which correspond to the helical type of parallel gearing. Gearboxes can be designed using the same type of gearing throughout, or a combination depending on powers, speeds and application. TABLE 1.1. SUMMARY OF GEARING – COMMERCIAL GRADE GEARING Parallel Axis Angled Gears Finish cut ground Finish cut lapped Finish cut ground Finish cut lapped Spur Helical Spur Helical Straight Spiral Straight Spiral Gears Gears Gears Gears Bevel Bevel Bevel Bevel Max pitch 25 line veloc. 7 10 15 25 5 10 10 [m/s] Efficiency per mesh 97% 98-99% 97% 98-99% 97% 98% 97% 98% Power to weight Medium Medium Medium High Medium Medium Medium High ratio to high to high to high to high 7 1.3 Torque selection Before starting the preliminary design, the following factors must be known. • The type, powers and speeds of the prime mover. • The overall ratio of the gearbox. • The types of unit required – parallel or angled drive. • The application. • Any abnormal operating conditions. • The disposition of the input to output shaft. • The direction of rotation of the shafts. • Any outside loads that could influence the unit, e.g. overhung loads, brakes, outboard bearing etc. • The type of couplings to be fitted. • Any space restriction. To obtain the basic size of gearbox, the nominal torque at the output shaft is calculated, using the absorbed torque at the driven machine, or the prime mover torque multiplied by the gearbox ratio, if the absorbed torque is unknown. It may be possible to obtain a torque – time diagram of the drive, which will give a comprehensive result of the complete duty cycle. There are three important points to remember when calculating the nominal torque: 1. That if a brake is positioned anywhere before the gearbox output shaft, the unit should be sized on the brake torque, (assuming this torque is greater than the motor torque). This is because any external loads back driving the gearbox will be sustained by the unit until the brake slips. The above is also true of any form of back stopping (anti-reversing) device. A check should also be made on the kinetic energy that would have to be sustained by the unit if the brake is to be applied in an emergency. 2. That some prime movers, namely electric motors, can develop 2 or more times full load torque (FLT) on start up. If stop/start is a frequent occurrence then the gearbox must be sized accordingly. 3. Those rigid type couplings can transmit shock more easily to the gearbox than can flexible or gear type couplings, and the application factor selected accordingly. To select the basic size, the nominal torque must be multiplied by a service factor (see Table 2). These are based on field experience and take into account the working conditions for that particular application. It should also be noted that some motors can run at varying powers and speeds. The maximum torque is used for rating the gears for powerbased on an equivalent life to suit the duty cycles, while the maximum speed is used to ascertain the pitch line velocities. Most manufacturers of gearboxes produce excellent free catalogues from which can be gleaned a lot of useful information, including approximate size of units for a given power, thermal ratings, shaft sizes, calculations etc. 8 TABLE 1.2. APPLICATION FACTORS Example of Prime Driven Machine Load Classification Mover Uniform Moderate Shock Heavy Shock Uniform Electric Motor 1 1.25 1.75 Hydraulic Motor Turbine Moderate Shock Multi-cylinder Petrol 1.5 1.75 2.25 Engine Heavy Shock Single-cylinder Petrol 1.75 2 2.5 engine The above figures are based on 10 hrs/day duty. For 3 hrs/day duty multiply above by 0.85. For 24 hrs/day duty, multiply above by 1.25. NOTE – It is usual to equate a running time of 10 hrs/day to a total life of 22,000 hrs, and 24 hrs/day to 50,000 hrs. Examples of driven machine classifications Uniform: Generators, Constant Density Mixer. Moderate Shock: Bucket Elevators, Concrete Mixers. Heavy Shock: Stone Crusher, Sugar Mill, Steel Mill Draw Bench. 1.4 Materials and heat treatment The steel selected for gears must be strong to prevent tooth breakages. It must be hard to resist the contact stresses, and it must be ductile enough to resist shock loads imposed on the gears, due to any outside influence or dynamics built up in the system. The material selected for gears, solid with shaft, must also be capable of resisting any stresses imposed along the shaft. Through hardened pinions should be made approximately 40 BHN harder than their mating wheel to even out the life of the two parts with respect to fatigue and wear. Bar stock may be used for most industrial applications up to 300 mm dia., above this size forgings are usually used. In cases of high stresses it is advisable to purchase forgings as the structure is far superior to rolled bar. Stepped forgings can also be obtained and may offer a more economic alternative. Cast steel is often used for gear wheels but care must be taken to select a high quality material, devoid of blow holes etc. Steel for gears is usually treated in one of the following ways: Through hardened (including annealed or normalised) The material is heat treated before any machining is carried out. This avoids any heat treatment distortion, but because it has to be machined, there is a limit to the hardness, and therefore strength, to which it is possible to go. Most gear manufacturers dislike machining steel over 350 BHN, as not only does it reduce tool life, it must also have an effect on machine life as well. The most common steels (to PN-EN 10083-1+A1:1999) in this group is being C40, C45, C50, C55 and C60. 9 The final selection based on the allowable stress levels and the limiting ruling section involved. Flame or induction hardened The gear teeth are first cut into a gear blank, and then surface hardened. This retains the strong ductile core, while giving the tooth flanks a very hard wearing surface. On small teeth, of 4 module and under, the depth of hardening from both sides may converge in the middle and therefore make the whole tooth brittle (see Fig. 1). This is quite acceptable providing a slightly lower allowable bending stress is used for calculating the strength of the tooth, usually 80% of the allowable stress value of steel with hardness equal to that, of the root when in the unhardened condition. Spin hardening, where the component is spun inside an induction coil, has the same effects as above. See Fig. 1C. Fig.1A. Full contour hardened Fig.1B. Flank hardened Fig.1C. Spin hardened Because there is a certain amount of distortion due to the heat treatment, it is usual to leave a grinding allowance on the tooth flanks for grinding after hardening. Hardened gears can be left unground, but because of distortion, a certain amount of hand dressing of the teeth may be required to obtain an acceptable bedding mark when meshed with its mate. As hand dressing is a skilled, laborious job, it is best avoided if at all possible. Full contour hardening (Fig.1A) hardens the flank and the root of the tooth, and this avoids the abrupt finish of residual stresses in the critical area as in the case of flank hardened teeth (Fig.1B). For flank hardened teeth, use only 70% of the allowable bending stress of steel with the same root hardness in the unhardened condition. Flame or inductioned, hardened tooth flanks can, depending on the type of steel used, be expected to reach a hardness of 50-55 HRC at the surface and attain case 10