Wundervölker, Monstrosität und Hässlichkeit im Mittelalter (German Edition)

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Asymmetric spur gear Direct gear design High contact ratio Load sharing ratio Non-dimensional stress. Submitting the report failed. Please, try again. If the error persists, contact the administrator by writing to support infona. You can change the active elements on the page buttons and links by pressing a combination of keys:. I accept. Polski English Login or register account. Kapelevich, Asymmetric modified gear drives: reduction of noise, localization of contact, simulation of meshing and stress analysis, Computer Methods in Applied Mechanics and Engineering, ,.

Kapelevich and Roderick E. Kleiss This paper presents an alternative method of analysis and design of spur and helical involute gears. Formulas for gear calculation external gears Contents: Relationship between the involute elements Determination of base tooth thickness from a known thickness and vice-versa. Cylindrical spur gears with. A gear is a wheel with teeth that mesh together with other gears. Gears change the : speed torque rot. Page 1 1. Qi Fan and Dr.

Direct Gear Design for Optimal Gear Performance - PDF

Lowell Wilcox This article is printed with permission of the copyright holder. The tooth. Hanover Massachusetts Tel. Alexander Kapelevich It has been documented that epicyclic gear stages provide high load capacity and compactness to gear drives. This paper will. Abstract: Two methods for determining tip factor of external. Brown, S. Davidson, D. Hanes, D. Weires and A. Sachs, P. Neville W. Plastic Threads Technical University of Gabrovo Yordanka Atanasova Threads in plastic products can be produced in three ways: a by direct moulding with thread punch or die; b by placing a threaded metal.

Gear Trains Introduction: Sometimes, two or more gears are made to mesh with each other to transmit power from one shaft to another. Such a combination is called gear train or train of toothed wheels. Module 7 Screw threads and gear manufacturing methods Lesson 32 Manufacturing of Gears. Instructional objectives At the end of this lesson, the students will be able to i State the basic purposes of.

Berg manufactures several styles of gears. Each gear has and serves its own particular application. Listed below are brief descriptions and application notes for the variety of available styles. Chapter 2 Lead Screws 2. It has a history that stretches back to the ancient times. A very interesting. The input pinion. Drawing an Approximate Representation of an Involute Spur Gear Tooth Project Description Create a solid model and a working drawing of the 24 pitch gears specified below.

It is only necessary to create. Page 47 Heron of Alexandria The word teamster today means a truck driver, but originally it meant a wagon driver, someone who drives a team of horses, mules or oxen. Over years ago teamsters in the. Andrzej Kawalec e-mail: ak prz. Screw Thread Design Screw Thread Fundamentals A screw thread is defined as a ridge of uniform section in the form of a helix on either the external or internal surface of a cylinder.

Internal threads refer. Natti S. Rao Nick R. Stanar Spur Gear 0 0 Figure - shows the meshing of stanar spur gears. With the patented InvoMilling process it is. The following. Starting from the related acceleration. Company Introduction M-7 Technologies is a privately owned, ISO certified company offering engineered solutions for.

Introduction 1. Industry Automation Industry automation is the term that describes a vital development programme of a production community where the project engineers build up automated manufacturing. Training Objectives After watching the video and reviewing this printed material, the viewer will gain knowledge and understanding of the design and use of various thread types and how they are produced.

Woods and Timothy G. SLS is one of the most accurate 3D printing processes. The process has a layer thickness of 0. This is the thickness with which a new layer is added to each part. In any direction therefore the maximum. Back to Elements - Tetrahedra vs. This file aims to introducing basic concepts of gears and pulleys. Litvin and W. Tsung University of Illinois at Chicago Chicago,. Measurement of Residual Stress in Plastics An evaluation has been made of the effectiveness of the chemical probe and hole drilling techniques to measure the residual stresses present in thermoplastic.

Wood Dale, IL www. Common Mechanical Engineering Terms Ball and Detent n A simple mechanical arrangement used to hold a moving part in a temporarily fixed position relative to another part. The ball slides within a bored. Ales Ricar Autodesk, Inc. Find out. November Content Motivation, state.

The majority of engineers in both the fastener manufacturing. The purpose of this introduction to dynamic simulation project is to explorer the dynamic simulation environment of Autodesk Inventor Professional. This environment allows you to perform rigid body dynamic. High speed machining and conventional die and mould machining Reprint from HSM - High Speed Machining There are a lot of questions about HSM today and many different, more or less complicated, definitions.

You should judge. Tacitius, Annals. Bello-Gomez 1, M. Guerrero-Mata 1, L. Leduc Lezama 1, T. Berber- Solano 1, L. Nieves 2, F. Gonzalez 2, H. The software for all applications. Bud Graham, Contributing Writer Ok, were going to talk about curves. Not the one seen in a ball game but the. A powerful alternative for additional higher radial and axial loads. Low backlash High. Some of the most common power trains include: Flexible.

A spring. Hosch, P. Dorner Mfg. Vehicle Systems Modelling and Testing, Vol. Elements are used for. ME Ahmad A. Abbas Sr. Analysis Engineer Ahmad. Abbas AdvancedCAE. Therefore, a force has the attributes. Log in Registration. Search for. Size: px. Start display at page:. Cornelius Nicholson 2 years ago Views:. Similar documents. More information. Formulas for gear calculation external gears. Contents: Formulas for gear calculation external gears Contents: Relationship between the involute elements Determination of base tooth thickness from a known thickness and vice-versa.

It allows reducing the contact and bending stress, increas- ing load capacity and power transmission density.


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This indicates potential advantages of the asymmetric gears over the symmetric ones for gear drives that transmit load mostly in one direction. The pitch factor analysis is the additional Direct Gear Design analytical tool that can be used for comparison of different gear geometry solutions, helping the designer better understand available options and choose the optimal one. However, it is applicable to any other type of involute gears: helical, bevel, worm, face gears, etc. This normal plane tooth proile can be considered a tooth proile of the virtual spur gear.

Virtual numbers of teeth are usu- ally real numbers with the decimal parts. The tooth geometry of virtual spur gears is optimized by means of Direct Gear Design. Then the optimized tooth proiles are considered the normal section tooth proiles of the actual gears. If a block contour of some generating rack does not contain an optimal gear geometry for a partic- ular gear application, the generating rack parameters pressure angle, adden- dum, whole depth proportions, etc.

Then the altered rack block contour may include the desired optimal gear geometry. However, gear pair mesh geometry selection is also limited by an area of existence. This area deines parameter limits for a pair of spur gears. It is also used for any kind of involute gears, by their conversion to the virtual spur gears see Section 2. In this chapter areas of existence are deined and analyzed considering the gear tooth tip radii, mesh backlash, and all tolerances equal to zero.

Vulgakov introduced areas of existence of involute gears in his theory of generalized parameters [7]. Unlike traditional design block contours, where the tooth thicknesses at the tip diameters vary depending on the X-shifts, the area of existence is constructed for selected constant relative tooth tip thicknesses ma1,2. These isograms do not depend on the relative tooth tip thicknesses ma1,2. The practical range of the pressure angle varies depending on the type of involute gears and their application. The most common range is This allows realizing a signiicantly higher transverse pres- sure angle level.

For example, the self-locking helical gears Section 6. They are described by the system of Equations 3. Although the interference isograms present the borders of the area of existence, this does not mean that the gear meshes do not exist beyond these borders. However, those gear combinations have the tooth root undercut in at least one of the mating gears, and they are typically undesirable. Equations 2. Its isogram is described by the system of Equations 3.

Its condition and isogram are deined by Equations 2. For external gears this means that the operating pitch diameter is larger than the form diameter and smaller than the tooth tip diameter. For internal gears this means that the operating pitch diameter is smaller than the form diameter and larger than the tooth tip diameter.

Normally a gear mesh has the approach and recess actions, while the contact point is moving along the contact line. An approach action is when the contact point C lies between point A2 and pitch point P of the contact line, or the driving pinion dedendum is in contact with the driven gear addendum.

Accordingly, the driven gear or rack tooth has only the addendum without dedendum. In the recess action gearing the driving gear tooth has only the addendum without dedendum. Accordingly, the driven gear or rack tooth has only the dedendum without addendum. Considering Equations 3. For external gearing it is a solution of a system of Equations 3. Most gear applications use the conventional action gearing, because it provides better performance parameters, such as high mesh eficiency minimal tooth proile sliding , tooth surface durability, bending stress balance, etc.

However, the approach and recess action gearings also may have rational areas of applications. For example, the recess action gearing is used for the self-locking gears Section 6. This section presents a few of them, which deine gear pairs with certain constant performance characteristics. Gear transmission power density and its load capacity in many cases are deined by the tooth surface durability. Maximization of gear eficiency is critically important for many gear appli- cations. Hs and Ht are ratios of the sliding velocity to the rolling velocity.

Alternatively, from 3. Then from 3. This means that a maximum of the gear mesh eficiency Emax can be deined for the external and internal gearings from 3. Gear tooth geometry, including the tooth lanks and root illet, affect the maximum bending stress level. The bending stress balance equalizes the tooth root strength of mating gears. If gears are made of similar materials and have a relatively close number of load cycles, the maximum bending stresses of mating gears should be equalized. Then during inal gear design the mating gear face widths can be adjusted also considering a number of load cycles of each gear to achieve more accurate bending strength equalization.

In traditional gear design the tooth illet proile is typically a trajectory of the generating tooling gear rack.

However, the tooth illet proile optimization is a time-consuming process that is used for the inal stage of gear design. It is not practical for browsing the area of existence analyzing many sets of gear pairs. For preliminary construction of the interference-free tooth root illet proile that also provides relatively low bending stress concentration, the virtual ellipsis arc is built into the tooth tip that is tangent to the involute proiles at the tip of the tooth [39].

The ellipsis arc is chosen because it its both symmetric and asymmetric tooth proiles and results in lower bend- ing stress level. When the gear tooth with the root illet is deined, the bending stress is calculated by the inite element analysis FEA method.

Gear Terminology

This isogram is typical for the external gearing and the rack and pinion gearing. The internal gear tooth with the equal face width with its external mating gear usually has signiicantly lower bending stress, because its root tooth thickness is typically much greater. Every point of the area of exis- tence presents the gear pair with a certain set of parameters and gear tooth proiles. Some of those gear pair tooth proiles have a kind of exotic shape and pres- ent rather theoretical interest.

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However, even they may ind practical applica- tions for some unconventional gear drives. Pressure angle and area of existence coordinates at point A are deined by the combined solu- tion of Equations 2. This pressure angle and the tooth tip proile angles at point B for exter- nal gearing are deined from a combined solution of Equations 2. Then the gear mesh parameters at point A are deined by Equations 2. Its gear mesh parameters are deined from the com- bined solution of Equations 2.

It is much greater than any generating rack block contour.


  • Direct Gear Design for Spur and Helical Involute Gears;
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  • It actually includes any gear pairs, generated by all possible racks, and also the gear pairs, which require two different dedicated racks to generate the mating gears. This comparison demonstrates how many gear pair combinations available with Direct Gear Design are not even considered by the traditional gear design approach. It basically presents an overlay of two areas of existence: one for the drive lanks and another for the coast lanks of asymmetric gears. It is built with preselected values of the relative tooth tip thicknesses ma1,2 and the asymmetry factor K.

    Most of the isogram equations for asymmetric gears are the same as the equations for the symmetric gears. They deine constant parameter values or mesh conditions separately for the drive and coast gear lanks. Solid isograms are for the drive lanks, dashed isograms are for the coast lanks; a for external gears, b for internal gears, c for rack and pinion.

    For this reason, their area of existence contains only reversible gear combinations. Helical gears can have a contact ratio lower than 1. Solid line isograms are for the drive lanks, dashed line isograms are for the coast lanks. When the K factor increases, the pressure angle also increases but the area of existence decreases. Here points A and points B coincide. It demonstrates potentials of asymmetric gears in increasing the drive pressure angle. Rational selection of the asymmetry factor K are considered in Section 5. This type of area of existence of involute gears deines only the drive lank gear meshes.

    The gears with symmetric teeth are always reversible. For the drive lanks of the external gears the pressure angle isogram equa- tion is deined from Equation 2. The pressure angle equation at point A is deined as a solution of Equations 3. This maximum contact ratio is deined as a solution of Equations 3. Points of the area of existence with the constant drive lank pitch factor do not deine complete mating gear teeth, but just their drive lanks. This allows independent selection of the tooth tip thicknesses and the coast tooth lank parameters of asymmetric gears.

    The proile angles at the lowest points of contact near the root illet for the drive and coast tooth lanks are described by Equations 2. It also allows locating gear pairs with certain characteristics. Its practical purpose is to deine the gear pair parameters that satisfy speciic performance require- ments before detailed design and calculations. This involute gear research tool is incorporated into the preliminary design program with the inite ele- ment analysis FEA subroutine.

    Such software allows limits of parameter selection of involute gears to be deined quickly, feasible gear pairs to be located and animated, and their geometry and stress levels to be reviewed. This chapter is dedi- cated to deinition of the gear parameter limits for the spur and helical gears with symmetric and asymmetric teeth. Some of the tooth shapes and gear mesh combinations presented below look rather unusual and may not have rational practical applications.

    A goal of this chapter is to show boundaries of the involute gearing parameters. First, it provides the required gear ratio. Second, when the gear ratio and center distance are speciied, selected numbers of teeth deine the gear tooth size that is described by a module in the metric system of a diametral pitch in the English system.

    The gear tooth size is a main parameter in the deini- tion of the bending stress. Third, the number of gear teeth is a major factor in the deinition of gear mesh eficiency. Along with the gear mesh geom- etry parameters pressure angle, contact ratio, etc. The maximum number of gear teeth is limited by application practicality and manufacturing technology. Most mechanically controlled gear hobbing machines can produce gears with a number of teeth up to using one-start hobs. Usage of multistart hobs increases this limit accordingly. Some com- puter numerical control CNC gear hobbing machines can produce gears with a number of teeth up to using one-start hobs.

    Other gear fabrica- tion technologies like, for example, proile cutting or injection molding can provide gears with an even greater number of teeth. From the application point, the maximum number of gear teeth may also be limited by tolerance sensitivity and operating conditions. If the gear pitch diameter is constant, an increase in the number of teeth leads to their size reduction, to the point where the size of the tooth becomes comparable with tolerance values. The tooth root undercut occurs when the generating rack addendum trajectory line A-A is below the tangent point N, where the normal to the rack proile at the pitch point touches the base diameter db.

    The spur gears with a number of teeth as low as 6 are used in the external gear pumps. Directly designed spur symmetric gears are not constrained by limita- tions imposed by the generating rack and its X-shift. The pinion and gear tooth tip proile angles, and the pressure angle for these gears are from Equations 3. Work of this gear stage is also quite unusual. Normally in an epicyclic gear stage all idler gears transmitting motion from the sun gear to the planet gear are constantly and simultaneously engaged with both of them.

    The right idler gear does not transmit motion, but it is moved by the ring gear. Minimum numbers of teeth of such reversible spur asymmetric gears are the same as for the spur symmetric gears. Such gears may not have the coast lank involute proiles at all. The main parameters of these gears are shown in Table 4. One of the main tooth proportion parameters is a pressure angle. The term pressure angle, in this case, is actually related not to the gear mesh, but to the basic or generating rack that is used for design or as a cutter proile, accordingly. The gear except the gear rack involute proile angle varies from the form diameter to the tooth tip diameter.

    For external spur gears it can be found from Equation 3. The tooth tip proile angles in this case are from Equations 3. It depends on the number of teeth of mating gears z1,2 and also on the relative tooth tip thicknesses ma1,2. The general solution for the maximum pressure angle for both symmetric and asymmetric gears is presented in Equation 3. If the tooth asymmetry is deined by the factor K, the minimum and maximum drive pressure angles are deined in points B and A of the area of existence, accordingly.

    In this case the active involute proiles of the mating gear lanks are shrunk to the point. Helical gears with theoretically pointed contact have found practical application in noninvolute Wildhaber-Novikov gears [43, 44]. In traditional gear design the maximum transverse contact ratio is deined by the selected basic or generating rack and its X-shifts for a pinion and gear. Its maxi- mum value depends on the type of gearing external, internal, or rack and pinion , tooth proile symmetric or asymmetric , number of teeth, and rela- tive tooth tip thicknesses.

    The highest contact ratio for gear pairs with a particular number of teeth and the relative tooth tip thicknesses is achieved at point B of area of exis- tence at the intersection of the interference isograms. The lower the relative tooth tip thicknesses ma1,2, the higher the contact ratio at point B. For external symmetric gears it is deined from Equation 3. The highest contact ratio is achieved at point B of area of existence at the intersection of the drive tooth lank interference isograms.

    Below this point the coast lank interference occurs, resulting in involute proile undercut. Therefore application of asymmet- ric reversible gears for drive contact ratio maximization is not practical. Irreversible asymmetric gears present more theoretical rather than practi- cal interest, because beneits of their applications are not apparent.

    It indicates that irreversible asymmetric gears allow realization of a signiicantly higher contact ratio. However, in most cases the gear tooth and mesh geometry parameters do not reach their theoretical limits, because of, irst, speciic gear application per- formance requirements and, second, some material and technological con- straints. For example, application of gears with a very low number of teeth is limited by increased speciic sliding velocities, resulting in low mesh efi- ciency, higher gear mesh temperature, and tooth lank scufing probability.

    At the same time, this reduces tooth delection under the operating load and lank impact damping, resulting in higher noise and vibration. On the con- trary, gears with a given pitch diameter and very high number of teeth have a very small tooth size. A practical maximum pressure angle and transverse contact ratio are lim- ited by the minimum tooth tip thickness.

    For a case of hardened teeth, it should be suficient to avoid the hardening through the tooth tip. For gears out of soft metals and plastics it should be suficient to exclude tooth tip bend- ing. These conditions also identify the practical maximum pressure angle. Application of gears with asymmetric teeth allows increasing the drive lank pressure angle in comparison with gears with symmetric teeth by the coast lank pressure angle reduction.

    If the coast lanks are not normally used for load transmission and may just occasionally be engaged in contact as a result of tooth bouncing, inertial load during gear drive deceleration, etc. These slanted tooth tips can be produced by the special topping gear cutter hob or by the secondary after the tooth hobbing tooth tip milling operation. If the coast lanks are used in normal operating load transmission, as in, for example, idler or planet gears in epicycling drives , asymmetry factor K and practical range of pressure angles and contact ratios are deined based on speciic application requirements see Section 5.

    Its tooth geometry boundaries are consid- erably expanded in comparison to the traditional gear design method. This allows deinition of an optimal tooth shape for speciic custom gear application. Gear tooth geometry optimization is a part of gear drive optimization that also includes optimization of the gear arrangement of multistage gear drives, rational material and manufacturing technology selection, choice of lubrication system, etc.

    A starting point of gear tooth geometry optimiza- tion is to establish a set of priorities for speciic gear drive applications. As a part of the gear drive design it should be done in combination with other gear transmission component optimization. Both types of tooth surface defects depend on contact stress and tooth proile sliding velocity. This also makes gear teeth stubby, with the reduced whole depth and increased thickness at the root area reducing the bending stress.

    Drawbacks of a high-pressure angle gear application are higher separating load taken by bearings and higher stiffness of symmetric teeth tooth that reduces tooth engagement impact absorbing and leads to higher noise and vibration. In asymmetric teeth with high drive pressure angle the coast lank is designed independently, which allows reduction of the tooth stiffness, noise, and vibration.

    Another way for the tooth proile optimization to reduce gear pair size is application of gears with high contact ratio HCR. Conventional spur involute gears have a transverse contact ratio 1. As a result, they have increased tooth delection, providing a better load sharing between engaged tooth pairs and allowing reduction of contact and bending stresses, and also a noise and vibration level that makes them applicable for aerospace gear transmissions [5, 45]. However, the HCR gears must be accurate enough to have the base pitch variation lower than the tooth delection under operating load to pro- vide load sharing.

    These gears also have some drawbacks. Long tooth addendum and low operat- ing pressure angle result in higher sliding velocity that increases scufing probability and mesh power losses. This could be explained by high stiffness of the buttress teeth that have low drive pressure angle and high coast pressure angle. Application of this approach to the asymmetric HCR gears allows design of the coast lanks independently to reduce the gear tooth stiffness for better tooth load sharing. This allows reduction of drive lank contact stress and sliding velocity.

    As a result, the drive lanks of asymmetric gears have higher tooth surface endurance to pitting and scufing, providing maxi- mized transmission density. Selection of the asymmetry factor K depends on the gear pair operating cycle that is deined by RPM and transmitted load in the main and reversed directions, and life requirements [47].

    These data allow calculation of num- bers and magnitude of the tooth load cycles in each regime in both load transmission directions. If the gear tooth is equally loaded in both the main and reversed rotation directions, asymmetric tooth proiles should not be considered. This should provide better lank tooth surface pitting or scoring resistance.

    At the same time, the contact stress and slid- ing velocity of the coast lanks are close to these parameters of the baseline gears and should provide a tooth surface load capacity similar to that for the baseline gears. This type of gear may ind applications for drives with one main load transmission direction, but it should be capable to carry a lighter load for shorter periods of time in the opposite load transmission direction.

    These types of gears are only for unidirectional load transmission. They may ind applications for drives with only one load transmission direction that may occasionally have a very low load coast lank tooth contact, like in the case of a tooth bouncing in high-speed transmissions. As a result, the bearing load is signiicant. In this case, the asymmetry factor K for a gear pair is deined by equalizing potential accumulated tooth surface damage deined by operating contact stress and number of tooth lank load cycles.

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    In other words, the contact stress safety factor SH should be the same for the drive and coast tooth lanks. Then from 5. Example 1 The drive pinion torque T1d is two times greater than the coast pin- ion torque T1c. The drive tooth lank has load cycles, and the coast tooth lank has load cycles during the life of the gear drive.

    This coast tooth lank load can be signiicant and should be taken in consideration while deining the asymmetry factor K. If the gear drive is completely irreversible and the coast tooth lanks never transmit any load Case 4 , the asymmetry factor is deined only by the drive lank geometry. In this case, increase of the drive lank pressure could be lim- ited by a minimum selected contact ratio and a separating load applied to the bearings.

    Application of a very high drive lank pressure angle results in the reduced coast lank pressure angle and possibly its involute proile undercut near the tooth root. Another limitation of the asymmetry factor of the irre- versible gear drive is growing compressive bending stress at the coast lank root. Usually for conventional symmetric gears compressive bending stress does not present a problem, because its allowable limit is signiicantly higher than for the tensile bending stress. However, for asymmetric gears it may become an issue, especially for gears with thin rims.

    This arrangement seems unsuitable for asym- metric gear application. This allows equalizing contact stresses on opposite lanks of the asymmetric teeth to achieve maximum load capacity. Equation 5. Numbers of the idler gear tooth load cycles and permissible contact stresses in this case are equal in both meshes, and Equation 5. Then considering Equation 2.

    In this case, the asymmetry factor K is also deined by Equation 5. This allows simpliication of Equation 5. This gear ratio is always greater than 1. For spur gears, the percent of mesh losses is deined in Equation 3. Then Equation 3.

    How is it possible to operate electric drives without a gearbox?

    Here the triangle linear inite elements are used, although other kinds of inite ele- ments can be used for this purpose. An area at the illet proile, where high stress is expected, has a higher inite element node density and the elements are smaller than in the rest of the tooth proile to achieve a more accurate stress calculation result within a short period of time. For conventional gears with the transverse contact ratio 1. Typically the force application point is located between two inite element nodes. In this case the force is replaced with two force components applied to these two closest nodes.

    F - applied force; n - number of FEA nodes on the tooth proile. In fact, tooth bending strength is usually provided by material and heat treatment improvement rather than gear geometry enhancement. But the root illet proile is basically a by-product of the cutter edge motion. The illet proile and, as a result, bending stress level are dependent on the cutter radial clearance and tip radius.

    The standard radial clearance usually is 0. The standard cutter tooth radius for the coarse pitch gears is 0. For ine pitch gears the standard cutter tooth radius may not be speciied and could be equal to zero [38]. Unlike the contact Hertz stress, the bending stress typically does not deine the major gear pair dimensions such as pitch diameters or center distance.

    If the calculated bending stress is unacceptably high, the coarser diametral pitch larger module can be applied. The number of teeth, in this case, is proportionally reduced to keep original pitch diameters and center distance, and the same or close gear ratio. There are two general approaches for reducing bending stress for the given tooth size. The most common application of such an approach is the tooling rack with the full tip radius.

    Another approach is to alter the gear tooth illet pro- ile [53—56]. The most common application of such an approach is using the circular arc root illet proile. Further development of both these approaches is based on a mathematical function itting technique where the cutter tip radius or the gear tooth trochoid illet proile is replaced by a parabola, ellip- sis, chain curve, or other curve, reducing the bending stress.

    The resulting tooth illet proile must be checked for interference with the mating gear tooth tip. A goal is to achieve a minimum of stress concentration on the tooth illet proile. In other words, the maximum bend- ing stress should be evenly distributed along the large portion of the root illet proile. The initial illet proile is a trajectory of the mating gear tooth tip in the tight zero-backlash mesh. This allows interference with the mating gear tooth tip to be avoided. The illet optimization method that is used in Direct Gear Design was devel- oped by Dr.

    Shekhtman [57, 58]. Such a set may contain the trigonometric, polynomial, hyperbolic, exponential, and other functions and their combinations. Parameters of these functions are deined during the optimization process. This kind of inite element allows achievement of satisfactory opti- mization results within reasonable time.

    In [59] the boundary ele- ment method BEM is used for stress analysis of the optimized tooth root illet. It is deined as the center of the best-itted circular arc, and it is connected to the inite element nodes located on the initial illet proile [57, 58]. The irst and last inite element nodes of the initial illet proile located on the form diameter circle cannot be moved during the optimization process.

    The rest of the initial illet inite element nodes are moved along the straight lines beams that connect through the illet center. Variable parameters of mathematical functions that describe the illet proile for the next iteration are deined depending on stress calculation results of the previous iteration. If stress was increased, the nodes are moved in the opposite direction. After the speciied number of iterations the optimization process stops, resulting in the optimized illet proile.

    The more inite element nodes that are placed on the illet proile, the more accurate are the stress calculation results, but this requires more itera- tions and the illet proile optimization takes more time. During the optimi- zation process the illet nodes cannot be moved inside the initial illet proile because this may cause interference with the mating gear tooth tip.

    One of them is a minimum radial clear- ance. The optimized illet typically results in low radial clearance, much less than in conventional standard or custom gear drives. The concern here is that the radial clearance can be so small that the lubricant could be trapped in the illet space, resulting in additional power losses and gear eficiency reduc- tion. In this case, the root diameter providing the acceptable radial clear- ance should be established and the optimized illet proile must be tangent to this root diameter. There are three different rim design options that can be considered.

    This is typical for idler gears when the rim surface is used as the roll or ball bearing race, or for gears with the spokes connecting the rim with the hob. The second option of rim design is typical for gears that have the sliding it or the glue it on the rim surface with very small clearance. This restrains the rim radial delections, but does not create additional hoop stress.

    It imposes additional hoop stress depending on press it interference. If press it interference is signiicant or rim thickness is low, resulting in high hoop stress, the root illet optimization may not be possible. One more additional illet optimization constraint is related to manufactur- ability of the gears with the optimized illet. The illet optimization process tries to cre- ate minimum curvature maximum radius at the maximum stress illet area to minimize its concentration. For gears with a low number of teeth this may result in a small illet radius near the form diameter creating the undercut.

    Unlike the undercut that occurs in conventional gears, this one is made for a purpose, and it does not affect the active involute lank proile. However, to make such an optimized illet proile could be dificult or even impos- sible by some gear fabrication methods, such as proile cutting, hobbing, etc. These additional illet optimization constraints compromise root stress concentration reduction in comparison to the optimized illet constrained only by the initial illet pro- ile. The involute lanks, face widths, tooth load, and its application point are the same for all illet proile options.

    Parameters of other illet proile options are deined relative to the option 1 parameters. At the maximum tensile stress point the optimized illet has a signiicantly larger illet radius Rf, and a smaller distance H and root clearance C. It has the lowest maximum bending stress, which is evenly distributed along the large portion of the illet proile. Other illet proiles have signiicantly greater and sharply concentrated maximum stress.

    Analysis of the illet optimization results has indicated that the optimized illet proile practically does not depend on the force value and its application point on the involute lank, except in the case where the application point is located very close to the form diameter. In this case, compression under the applied force may affect the optimized illet proile. Such load application should not be considered for illet optimization, because it induces minimal tensile stress in the root illet in comparison to other load application points along the tooth lank.

    If the load capacity of gears with conventional trochoidal or circular root illet proiles is limited by maximum tooth bending stress, illet proile optimization increases gear load capacity pro- portionally to the bending stress reduction. However, quite often, gear load capacity, and consequently possible gear drive size and weight reduction, is limited by the tooth surface durability deined by pitting and scufing resistance, which greatly depends on the contact stress, proile sliding, and contact lash temperature. Then potential bending stress reduction pro- vided by the illet optimization can be used to improve other gear drive performance parameters.

    The number of teeth varies from 12 to 75, and module varies accordingly from 5 to 0. For example, the bend- ing stress level of MPa is considered acceptable. This level is achievable for the tooth gears with the standard illet or for the tooth gears with lower module with the optimized illet. However, the tooth gear pair has a higher contact ratio and, as a result, lower contact stress. Potential beneits of the bending stress concentration reduction by the tooth illet proile optimization can be extended.

    Direct Gear Design Drives Performance

    For example, this also allows use of gears with a greater number of teeth and lower module for noise and vibration reduction and allows increase of the elastohydrodynamic lubricant EHL ilm thickness also because of the reduced lash temperature and proile sliding. In order to provide equally strong teeth of a pinion and gear, their maximum bending stress should be balanced or safety factors should be equalized [57].

    Direct Gear Design uses the stress balance approach that utilizes a combination of an iteration method with the FEA stress calculation to satisfy the bending stress balance condition 5. This bending stress balance procedure should work in combination with the tooth proile and root illet optimization. In this case, the tooth load value is deined at different contact points of the involute tooth lank.

    Positions of these points are affected by the tooth bending delection and the contact delection of the involute lank. The FEA models of mating gears loaded by the driving torque at different angular positions allow load and stress distributions to be deined during the tooth pair engagement. Gear material properties like modulus of elastic- ity and Poisson ratio are used for delection calculations.

    The FEA proce- dure deines bending stress. The Hertz equation is used for contact stress deinition. This approach allows maximum contact and bending stresses to be deined. Conventional spur gears with the contact ratio 1. Rp1 and R a1 are radii at beginning and end of driving gear tooth engagement in contact with driven gear tooth. Size and weight reduction often also accompanies cost reduction. This chapter presents an approach that allows optimizing gearbox kinematic arrangement and gear tooth geometry to achieve high gear transmission den- sity.

    This approach uses dimensionless gearbox volume functions, which can be minimized by the gear drive internal gear ratio optimization [61]. The gear pair transmission density coeficient Ko statistically varies about 0. The volume utilization coeficients Kv1 and Kv2 depend on the gear body shape solid body or with central or lightening holes, rim, web, spokes, etc. Their values for driving pinions sun gears statistically vary in a range of 0.