Bearings Questions and Answers Guide This guide is intended to help users with common issues related to Bearings. There are separate guides on the subject of Shafts and Gears. The sections covered are: 30.1 Bearing External Data The bearing external data covers the bore, outer diameter and width of the bearing. This is the basic bearing geometry and all other bearing details reference this in some way. Bore must be smaller than outer diameter, and all values must be greater than 0. Basic mechanical properties such as mass, material, running speeds and stated load capacity are also defined here. 30.2 Bearing Internal Data The bearing internal data covers the number and dimensions of the rolling elements. The pitch circle diameter is the reference point for the centre of each bearing. The element radius is limited by a combination of this diameter, the bore and OD and the number of elements. For taper roller or angular contact ball bearings, the contact angle is also specified, as is the row separation for double row bearings. For roller bearings, the rib height and width is specified. RomaxDesigner allows a single value of height, width and chamfer for all the ribs on a bearing. The rib clearance can be set independently for the inner and outer ribs. The Radial Internal Clearance is the distance the bearing rings can be displaced radially. With no clearance, the bearings are in tight contact, and may not run smoothly. With a high clearance, the bearings will be in a very loose contact, and may cause other running problems. The Element Offset from centre is the distance that the elements are offset from the centre of the bearing. This is common practice in taper roller bearings. The value defaults towards the right, so will be towards the right if the bearing is right-mounted and towards the left if it is left mounted. The Roller Taper Angle is the angle that the edge of a roller is from the axis of that roller. This is only found in taper roller bearings, not in other types of bearings. The full length of the roller is the input value for the total actual roller length. This is used to calculate the values for the crowning and other profile modifications. In the classical analysis of bearing force it is possible to define an axial position for the bearing support that is coincident with the resultant load and axis of rotation. This has the advantage of eliminating the bearing?s resisting moment from static equilibrium considerations. The idealised sketch below shows how the resultant bearing reaction at the outer race contact position can be resolved into vertical and axial components. Bearing reactions can then be determined directly from force and moment equilibrium for a statically determinate system using the beam model if the operating contact angle, and hence shaft support span, is known. However, all ball bearings have a contact angle that changes with axial, radial, centrifugal or pre-loads and thus the effective span is, in reality, unknown. For some instances, this change is small, although in other cases, the change can be substantial. Thus, classical analysis cannot be used for accurate bearing analysis, since additional information on displacement and tilt is required along with force equilibrium to obtain the resultant bearing reaction vector and the actual contact angle. Romax has adopted a ?general-case? strategy to avoid discontinuities in the software when dealing with bearings. The Romax model places the bearing node coincident with the mass centre of its element and includes its tilt as well as radial stiffness in the global system formulation. With the coupled system and bearing iterative solution scheme, force and moment equilibrium is established, which is compatible with the bearing?s displacement and misalignment and is applicable to all statically indeterminate systems. The detailed transmission path of the load from the raceway to the shaft and housing is not considered in RomaxDesigner (shaft and housing nodes are located at the sections cross-sectional centre) hence the reported local stresses within the bearing envelope should only be regarded as nominal values. As far as the software structure is concerned, this is a natural approach to take since the ?Object Oriented? approach that makes RomaxDesigner so easy to use relies on the fact that a bearing on a shaft can quickly and easily be replaced by another, regardless of whether it is a cylindrical roller, radial ball, taper roller pr angular contact bearing. 31.1. Validation in RomaxDesigner The parallel axis theorem can be used to demonstrate that the forces generated by RomaxDesigner are an accurate translation of the forces that would be calculated if the forces were resolved to a point where the bearing moment was equal to zero; the point P. A RomaxDesigner model has been created to accompany this document, it is titled ?Taper Bearing Validation Model?. Open this and run the static analysis for the default load case. Inspection of the shaft results show that there is a moment acting at each bearing equal to 24.71Nm. This is in addition to the vertical reaction force of 5000N and the axial load of 1517.7N: To check that the moment is correct we need to establish the axial position of P. From the internal details window of the bearings the PCD of the bearing is 39.538, and from the edit window the contact angle is 14.04. This gives a moment arm of 4.94mm and a resulting moment of 24.72Nm, which is within acceptable accuracy limits given the number of significant places available for the variables. 31.2. Conclusion It is common for engineers to ask why RomaxDesigner does not use the theoretical centre of effort in the analysis of taper roller bearings. Romax seems to work in a different way from most textbooks and bearing catalogues, which describe the ?classical? approach. It turns out that the classical approach can only be used for simple analysis. If the true operating condition of the bearing is to be considered, the contact angle of the bearing alters and the theoretical centre of effort moves. Hence, there is no benefit in using this fixed point as a reference. Romax uses the centre of mass of the elements as the reference point since this assists the object oriented structure of the software. It can be seen that, in simple cases, this approach gives answers that are identical to those obtained by the classical method. The advantage is that more accurate analysis and the analysis of statically indeterminate systems can be achieved. Bearing Stiffness refers to the force/deflection characteristic of the bearing. The relationship between bearing force and displacement is non-linear (i.e. the stiffness is not constant), and an iterative solution is required. The diagram below shows four different loading conditions on the same bearing. The number of elements loaded has an effect on the overall stiffness, as does the overall load. 32.1. Ball Bearings. For Ball Bearings the relationship between force and displacement is calculated assuming Hertzian Contact at two interfaces - between the outer race and the ball and between the inner race and the ball. This is performed for each ball, summing the forces from each ball to take into account the give the total force transmitted by the bearing. This relationship is calculated by an iteration of assuming an initial displacement, then calculating the forces at that displacement, linearising the force-displacement graph at that point and moving to a more accurate point on the graph. This repetition eventually results in a compatible set of forces and displacements for the entire model. When the solution is found, the force transmitted by the bearing is compatible with the displacement of the outer ring to the inner ring. Within the bearing the force created by this displacement is the sum of the forces on all the individual elements, taking into account the bearing internal geometry (roller geometry, raceway geometry, internal clearance etc.) The solution takes into account, not only the ?diagonal? terms of the stiffness matrix (the relationship between x-displacement and x-force), but the ?off-diagonal? terms (e.g. the relationship between the x-displacement and the y-force, the y-moment etc.). The diagram below shows the full bearing stiffness matrix. Note that the terms relating to Z-rotation are all zero, as the bearing is designed to allow free rotation along the z-axis. The result is that due to the inter-linking between the different directional stiffnesses, the direction of displacement can be different from the direction of force. The effect of this is demonstrated in the diagram below. 32.2. Roller Bearings. For Roller Bearings the calculation of moments is more complex. Each roller is split into strips and the Hertzian force/displacement relationship calculated for each strip individually in the same way as demonstrated above. 32.3. Advanced Bearing Analysis Calculation of Stiffness. The Advanced Bearing Analysis uses the value of Bearing Stiffness calculated by the RomaxDesigner Bearing Model in calculating the bearing misalignment. It was found that there was no benefit in using the Advanced Bearing Analysis to re-calculate the Bearing Stiffness, since the results obtained were so close to those calculated by the RomaxDesigner model. 33 How do ISO and Romax Bearing Life Calculations Compare The ISO Standard is a simplified model of a bearing, which does not calculate the effect of many of the real world influences on bearing life. Instead it uses adjustment factors, derived from empirical testing, to take account of these. The influences which are neglected by ISO are listed below: The bearing operates with no internal clearance There is no tilt misalignment between the inner and outer races The operating contact angle does not change with loading The inner race rotates whilst the outer is stationary Speeds are low enough to ignore inertial forcing on the elements Speeds are high enough to establish adequate lubrication Element and raceway profiles produce ideal stress distributions
The RomaxDesigner Adjusted Life calculates the effect of all but the lubrication conditions on the L10 life, and combines them into a Load Zone Factor. This will usually have the effect of decreasing the ideal life of the bearing, in some cases by a very substantial amount, and thus it provides a much more accurate design parameter. However, the calculation of the L10 life in the core of RomaxDesigner still uses the quoted dynamic capacity of the bearing provided by the manufacturer. For a greater level of precision it is possible to use the Advanced Bearing Analysis (ABA) to calculate the dynamic capacity of the bearing from the actual internal geometry of the bearing according to equations given in ISO 281. This means that the calculation is free from any "fudge-factors" that may have been applied to the Dynamic Capacity by the supplier, as it based on first principles. Additionally, the roller bearings are split up into a larger number of strips for the stress distribution analysis and the use of an Influence Coefficient Method means that the effect of edge stresses is accurately taken into account.
34.1. Limitations of the ISO Life Calculation ISO bearing rating approximates the loads on a bearing by use of X and Y factors to take account of axial and radial forces. RomaxDesigner uses a fundamentally more valid approach by examining the actual normal element load and the load distribution around the bearing. This produces a true value for the combined axial and radial loads. From this the equivalent radial load is calculated which is used directly in the life calculation to give the L10 life. 34.2. When can the Load Zone Factor be Above 1? In certain circumstances it is possible for the Load Zone Factor to be greater than unity. This can occur because of the assumptions of ISO. Because the ISO standard assumes zero internal clearance and no misalignment the absolute maximum amount of contact is one half of the outer race. If a small amount of axial load is applied than the contact area will extend beyond this and so the mean load on the elements will be reduced. This will produce a longer bearing life which is represented by a Load Zone Factor greater than unity. 34.3. What is the Effect of Stationary Inner/Outer Races on Bearing Life? The standard assumption within ISO is that the inner race is rotating relative to the direction of load whilst the outer race is stationary. This clearly has implications on the life of the race in the zone of maximum stress directly in line with the load. RomaxDesigner takes account of the relative rotations in the radial load calculations and alters the radial equivalent load accordingly. 34.4. Does Romax Account for Bearing Ring Stiffness? To the best of Romax?s knowledge there is no analysis that includes the effect of bearing ring deflection in bearing life calculation, and RomaxDesigner is no exception. The principle justification for this is that in the vast majority of cases the effect is negligible and so including it would add unnecessary complication to the calculations. Furthermore, it is necessary to assume infinite stiffness in order to calculate the contact deformation from geometrical considerations arising from relative displacements between the inner and outer ring. Other second order effects like ring ovality and tolerance changes due to interference fitting the inner race are not calculated either. However, it is possible to alter the mounted internal clearance to model the changes due to an interference fit. Gears Questions and Answers Guide This guide is intended to help users with common issues related to Gears. There are separate guides on the subject of Bearings and Shafts. The sections covered are: 40 How do you Calculate Forces for Loading, Concept and Detailed Gears There are three types of gear definition in RomaxDesigner. In order of increasing complexity, these are: 1. Loading Gear 2. Concept Gear Pair 3. Detailed Gear Pair
All three can be used to calculate shaft loads, but as the data associated with each definition varies the calculation is slightly different. This document outlines the assumptions made and how the gear forces are calculated. 40.1. Calculation of Gear Forces The gear forces for each gear pair are calculated from their input data. The complexity of the calculation increases as the amount of input data increases. 40.1.1. General Equations Tangential Force = Ft Axial Force = Fax Separating Force = Fs S = Helix Angle = Pressure Angle
40.1.2. Loading Gears Ft = Fax = Fs =
40.1.3. Concept and Detailed Gear Pairs Now there are changes in the equations. NNbecomes b?, the working normal pressure angle ? becomes ?, the helix angle on the pitch cylinder The calculation of the working pressure angle and the pitch cylinder helix angle can be found in most books on gearing. The standardisation of gear tooth forms is achieved by fixing the basic rack proportions. This has the advantage that the tooling used to produce the gear designs can also be standardised. The figure below shows a typical basic rack. All the dimensions are specified as a proportion of module. To find the actual dimensions of the rack or tool the numbers must be multiplied by the module. 41.1. Addendum Modification When addendum modification is applied to the generated gears this is effectively a shift of the cutter reference line. A positive addendum modification coefficient will shift the rack further from the gear centre and thus the further from the gear reference diameter, a negative addendum modification coefficient will shift the rack closer. The rack proportions still however remain unchanged although the generated gear addendum and dedendum will change. The table clearly shows that addendum modification has no effect on the dimensions of the Rack, but alters the dimensions of the gear, and that there is no need to alter the dimensions of the rack in order to provide addendum modification. Modifications through the tooth normal or transverse thickness will also have the same effect.
41.2. Alternative Way to Describe Basic Rack Proportions Many companies describe the rack proportions relative to the reference diameter of the gear to be generated, rather than an offset cutter reference line. This causes the rack proportions to be exactly the same as the addendum and dedendum of the generated gear in proportion to the module. The problem with this definition is that the same tool will have different proportions depending on the actual gear generated. Positive addendum modification coefficient defining rack reference line the same as reference diameter: 41.3. Converting Basic Rack Proportions To convert the rack proportions relative to the reference diameter to the Romax definition use the following formulae: hfp* Romax= hfp*reference +x hap* Romax= hap*reference - x Mesh misalignment is an important part of both the ISO 6336 and AGMA 2001 rating methods. This document explains how the defining factor, called FBetaX in ISO 6336, is calculated in RomaxDesigner. Of note, is that RomaxDesigner employs a more sophisticated method than these standards to obtain a more accurate solution. 42.1. Why Deviate From Standard Methods? In the ISO 6336 standard, the gear is modelled as a series of points across its face width. Further simplifying assumptions are made about the load distribution of the gear, the shaft bending and the shaft twist that allows the user to estimate the resulting mesh misalignment. The standard makes these simplifications because it assumes that the user is unable to calculate the actual shaft bending and bearing deflection. By modelling the complete system in RomaxDesigner however, this information is made available and the mesh misalignment can be calculated more precisely. There are still some lesser assumptions made by RomaxDesigner and if the user wishes to take additional factors into account please contact Romax Technology Limited. 42.2. The Calculation Method ISO calculates FBetaX as the largest gap between two points on the face width, once the two gears have been brought into contact (reference ISO6336-1:1996(E) Annex C). RomaxDesigner defines FbetaX in the same way: The procedure is explained below: 1. RomaxDesigner records the x- and y- displacements of both shafts at five equally spaced points across the face width of the gear due to the shaft deflections. 4. The direction of the line-of-action of the gear pair is calculated from the mesh angle of the gear pair, the transverse working pressure angle, the direction of rotation of the gears and which is the driver/driven gear. For example: Mesh angle = -90 degrees, pinion rotating anti-clockwise and driving, working transverse pressure angle = 26.32 degrees. Direction of line-of-action = -156.32 degrees. 5. The displacements of the five points on each gear are resolved into this line-of-action. For Example: Point 1 on pinion, displacement in line of action = (39.19*cos(-156.32))+(36.51*sin(-156.32)) = -51.35 um 6. The displacement of each point on the driven gear is subtracted from the displacement of the corresponding point on the driving gear. The displacements are then "normalised" by subtracting the lowest value for the five pairs of points. e.g. displacement of point 1 on wheel in line-of-action = 28.65 um. Thus the relative displacement = -73.96 um. The relative displacement at point 5 is the lowest for all five at -58.84 um. Thus this is the point which will contact first. When this happens, the gap at point 1 will be 15.11 um.
42.3. Sign Convention for FBetaX It can be seen in the above example that the final answer has come out as a negative value for FBetaX. Clearly it is possible for one end of the tooth or the other to be in contact, and a sign convention was necessary to distinguish between the two. The convention that was arbitrarily chosen is one where a positive mesh misalignment has the teeth separating as you move along the z-axis in the positive direction along the pinion of the gear set. This is calculated in the pinion"s co-ordinate system, not the shaft or world co-ordinate systems. Conversely a negative mesh misalignment has the teeth separating as you move along the pinion z-axis in the negative direction. The two are demonstrated in the diagrams below: The form diameter is the diameter of a circle at which the trochoid produced by the tooling intersects, or joins, the involute or specified profile. It is where the usable part of the flank begins. This diameter cannot be less than the base circle diameter. The form diameter can be calculated from the basic rack details and the addendum modification coefficient. In general a decrease in root diameter causes a decrease in the form diameter and a reduction in the tool tip radius causes a reduction in the form diameter. However since the form diameter cannot exist below the base circle diameter, any reduction in the root diameter or decrease in the tool tip radius further than that which would result in a form diameter equal to the base circle will lead to the gear being undercut. This then leads to the form diameter increasing rather than decreasing with a decrease in root diameter and a reduction in the tool tip radius. Since the form diameter is dependent upon the finishing process and tooling it may not be sufficient to calculate it using the basic rack definition. The value of the form diameter will be highly dependent upon the manufacturing tolerances machine settings and tooling. If its value is known the user can override the calculated for diameter and input the actual value if this is known through measurement or manufacturing simulation. The gear blank definition serves two purposes. 44.1. Gear Inertia For gears with a defined gear blank, the gear blank"s inertia is calculated, and this is used in all calculations that require it. 44.2. AGMA Rim Thickness Factor The thickness of the rim on the gear blank is used in the AGMA gear rating standard, as the Rim Thickness Factor. Where the thickness is not sufficient to provide full support for the tooth root, the location of bending fatigue failure may be through the gear rim, rather than at the root fillet. Published data suggest the use of a stress modifying factor, KB, in this case. The rim thickness factor, KB, adjusts the calculated bending stress number for thin rimmed gears. It is a function of the Backup Ratio, mB. where ht = gear tooth whole depth, in (mm) tR = gear thickness, in (mm) The effects of webs and stiffeners are ignored. The effect of tapered rims has not been investigated. When previous experience justifies, lower values of KB may be used. Ref. AGMA 2101-C95 AnnexB There is a large quantity of useful data produced by the RomaxDesigner gear modeller. This can be split into two groups, one concerning the gear geometry and the other dealing with the rating results. The interpretation of these two groups of results will be considered separately. There are two gear rating standards available in RomaxDesigner, ISO 6336 and AGMA 2001. The output from each will be considered and compared here. 45.1. Results from the ISO 6336 Standard The diagram below shows the ISO S-N (Stress-Number of Cycles) bending fatigue life curve for a material. Each material has two curves, one for bending and one for contact. All the fatigue properties of any material in RomaxDesigner are calculated from this curve. The equations for the ISO 6336 standard can be found in the Romax Knowledge Base help file.
45.1.1. Results for Bending Stresses The Bending Stress Number is used to define the gear material properties. It specifies one point on the S-N curve from which the standard determines the rest of the points on the S-N curve. In this instance the Bending Stress Number is defined at 1e7 cycles, although it may be different depending on the material being studied. It corresponds to a specific point on the S-N curve, where the slope of the curve changes. This can be seen in the graph, below. If the bending stress number is changed on a gear material, all the points on the S-N curve will be shifted up or down accordingly. The Permissible Bending Stress is the value of the actual stress that would lead to bending failure for the actual number of cycles. It is calculated from the Bending Stress Number by equation 3.04. from ISO 6336/3. It takes the Bending Stress Number (eFLim) and applies a number of factors that are related to the gear geometry, surface roughness, minimum safety factor, actual number of cycles etc. The Actual Bending Stress is the bending stress that occurs under the actual loading conditions. It is calculated from ISO 6336/3 equations 3.01 and 3.02. It includes a number of Bending Stress Factors (Y..) and also some General Influence Factors (K..). These factors are explained in the RomaxDesigner Knowledge Base online help file. The Nominal Bending Stress is the Bending Stress calculated from the applied load and the Bending Stress Factors (Y?), but not including the General Influence Factors (K?). Thus the Actual Bending Stress is the Nominal Bending Stress multiplied by the General Influence Factors. 45.1.2. Bending Damage Results The Permissible Bending Stress is the stress that would lead to failure under the given number of cycles. The Actual Bending Stress is the stress that occurs. The ratio of the Permissible to the Actual stress is the Safety Factor. For a Permissible stress which is greater than the Actual stress, the Safety Factor is greater than 1. The Actual Bending Stress occurs at the Number of Cycles specified by the loadcase. RomaxDesigner also calculates the number of cycles to failure under these conditions. The ratio of the two (expressed as a percentage) is the Duty Cycle Damage. If the number of cycles to failure is less than the Number of Cycles, the percentage damage is less than 100%. 45.1.3. Results related to the S-N curve The Safety Factor and Duty Cycle Damage can be related to the S-N graph as follows: Safety Factor = Permissible Bending Stress / Actual Bending Stress Duty Cycle Damage = 100 % x Actual Number of Cycles / Number of Cycles to Failure Both are ratios derived from the graph. One is found by drawing a horizontal line from the operating point to the S-N curve and reading off the intersection point, the other is found by drawing a vertical line. 45.1.4. Results for Contact The results for are derived in the same way, with the relevant formulae being as follows: Permissible Contact Stress - derived from the Contact Stress Number using ISO 6336/2 equation 3.04. Actual Contact Stress - derived from the loading conditions and the gear geometry etc. See ISO 6336/2 equation 3.01 and 3.02. Nominal Contact Stress - the relationship between Nominal and Actual Stress is the same for Contact as Bending (see above). The Nominal Contact Stress takes into account the Contact Stress Factors (Z..), but not the General Influence Factors (K..). 45.2. Verification of Results for ISO 6336 The expressions used to derive the results for the gear rating are all listed in the Romax Knowledge Base help file. These can be used to verify the results that are obtained. This can have a number of uses: To verify that the software is working correctly To help you develop a greater understanding of the standards. To allow you to compare and contrast the results obtained from different standards and from you own (in-house or bought-in) gear rating programmes. To help you with this process, all the factors used in the Gear Rating calculation are also included. As can be seen in the above explanation, there are three main groups of factors in the ISO Rating (K, Y and Z factors). These are all output in the Gear Design Window and can be seen by highlighting the Details view. 45.3. Results from AGMA 2001 The following is the output from the AGMA Gear Rating: The diagram below shows the AGMA S-N (Stress-Number of Cycles) contact fatigue life curve for a material. Each material has two curves, one for bending and one for contact. All the fatigue properties of any material in RomaxDesigner are calculated from this curve. The equations for the AGMA 2001 standard can be found in the Romax Knowledge Base help file. This standard approaches the problem in a slightly different way to the ISO standard. 45.3.1. Results for Contact As before, the Contact Stress Number is used to define the gear material property. In AGMA both the Contact and Bending Stress Numbers are defined at 1e7 cycles, and the standard assumes that this is the place where the slope of the S-N curve changes (Note that this is different from ISO). The Permissible (or Allowable) Contact Stress is calculated from the Contact Stress Number by equation 5.4 from AGMA 2001. It takes the Contact Stress Number (Sac) and applies a number of factors. The output is the value of the actual stress that would lead to contact failure for the actual number of cycles. The Actual Contact Stress is the contact stress that occurs under the actual loading conditions. It is calculated from AGMA equation 5.1. The Allowable Contact Power is the power that, if applied to the gear for the given number of cycles, would just result in failure. It is the Power that would result in the Actual Contact Stress being equal to the Permissible Contact Stress for the given number of cycles. It is calculated using equation 5.5 in AGMA 2001, which is derived from equations 5.1 and 5.4. The Projected Contact Life is the Number of Cycles that would result in the Actual Contact Stress being equal to the Permissible Contact Stress for the given Power. 45.3.2. Other Results Some industries use the Contact Load Factor as a quick Gear Rating method. It is calculated from the equation 5.6. It is simpler than the full rating method, is not as accurate, and is rarely used in the Automotive Industry. The Permissible Contact Stress is the stress that would lead to failure under the given number of cycles. The Actual Contact Stress is the stress that occurs. The ratio of the two is the Safety Factor. The Projected Contact Life is the number of cycles that would lead to failure for the given stress. The load case defines the actual number of cycles. The ratio of the two (expressed as a percentage) is the Duty Cycle Damage. 45.3.3. Results related to the S-N curve The two last results (Safety Factor and Duty Cycle Damage) can be related to the above graph as follows: Safety Factor = Permissible Contact Stress / Actual Contact Stress Duty Cycle Damage = 100 % x Actual Number of Cycles / Projected Contact Life Both are ratios derived from the graph. One is found by drawing a horizontal line from the operating point to the S-N curve and reading off the intersection point, the other is found by drawing a vertical line. 45.3.4. Results for Bending The results for bending are derived in the same way, with the relevant formulae and factors being defined and explained in Help file. 45.4. Verification of Results As with ISO, the expressions used to derive the AGMA results for the gear rating are all listed in the Romax Knowledge Base help file. These can be used to verify the results that are obtained. 46 What do the Options on the Gear Addendum Division Mean?
The following equations are used by the optimisation process to calculate the pinion addendum modification coefficient dependent on the option for addendum division selected. They are taken from the British Standard publication PD 6457. 46.1 Zero Pinion Addendum Modification When this option is selected, the software will set the Pinion modification coefficient to zero and adjust the Wheel Addendum Modification Coefficient so as to give zero backlash. 46.2 Pinion addendum modification for general applications This is for general reduction applications. The addendum modification makes slightly more than half the pinion tooth action take place during recess. 46.3 Approximately equal pinion and wheel bending strength This adjusts addendum modification coefficients so that the bending strengths of the pinion and wheel are approximately the same. The approximation comes because the strength of a gear is different depending on what Standard (with various factors and assumptions) is used. Thus it will be seen that the Bending Stress Results are NOT the same when this option is selected. 46.4 Pinion addendum modification for balance of slide roll ratios This changes the gears so that the action at the extremes of approach and recess have approximately equal slide/roll ratios. This would be done to avoid scuffing in high speed gears. where: 46.5 Pinion addendum modification for speed increasing gear pairs This changes the gears for speed increasing purposes. In this case the majority of action takes place in recess. This prevents the pinion tooth tip gouging the wheel?s flanks during approach. 46.6 Pinion addendum modification to avoid undercut This adjusts the pinion?s addendum modification coefficient so that it is not undercut, the wheel is then adjusted to give zero backlash. This equation is always used by the optimisation process to check the calculated values of addendum modification coefficient. Any value that is calculated which is less than that which will give undercut is replaced by the minimum value calculated by this equation. 46.7. What do the Options on the Gear Addendum Division Mean? The following equations are used by the optimisation process to calculate the pinion addendum modification coefficient dependent on the option for addendum division selected. They are taken from the British Standard publication PD 6457. The x1 multiplication factor is a special value that does not relate directly to the geometry of the gear pair. The x1 multiplication factor is used to allow the optimisation process to imitate the behaviour of the gear geometry designer dialog?s slider bar. The value calculated by a standard equation (e.g. general applications ) is slightly modified by the factor in the same way as moving the pinion addendum slider bar to the left or right does when the gear geometry designer is used. Using this factor, the optimisation process is able to vary the value of x1 in a range around the calculated value. For example, an x1 multiplication factor of 0.5 will lead to half the calculated value of x1 being used in the creation of a candidate geometry, and a value of 1.1 will lead to 1.1 times the calculated value of x1 being used in the creation of a candidate geometry. RomaxDesigner allows a user to define a Power Load in terms of Speed, Torque, Power, all three, or any combination of the three. When a Planetary Gear Set is defined, the software needs to check that the inputs (torque, speed and power at each of the three outputs) are compatible. The problem is complex as there are many possible combinations of inputs, and the software has to be able to cope with each. This document discusses when the analysis is capable of coming to a suitable solution, and when it identifies a set of inputs as being over-constrained or under-constrained. 48.1. Basic Equations of a Planetary Gear Set There are a number of basic equations that govern the behaviour of a Planetary Gear Set. These can be worked out from first principles or obtained from textbooks. They are combined in the RomaxDesigner powerflow analysis to derive the power conditions of the Planetary Gear Set. 48.1.1. Kinematic Constraint There is a relationship between the rotational speeds of the three components of a planetary gear set that must be observed. Thus two speeds need to be defined for the third to be calculable. 48.1.2. Torque Balance The torque balance at the planetary gear pin relates the annulus torque to the sun torque. A further torque balance allows the planet carrier torque to be calculated. Hence defining any one of the torques allows the other two to be calculated. 48.1.3. Power = Torque x Angular Velocity This fundamental relationship for rotating machinery can also be used to calculate the operation of a planetary gear set. 48.1.4. Power In = Power Out No machinery can violate this fundamental law and it can be used to calculate the operation of a planetary gear set. 48.2. Over and Under-Constrained Planetary Systems The above equations are combined to see if a system is over- or under-constrained. If not the power, speed and torque of each component in the gear set is calculated. The following table gives examples of each condition: Under-Constrained | Calculable | Over-Constrained | 1 Speed, 1 Torque | 2 Speeds, 1 Torque | 3 Speeds, 1 Torque | 2 Speeds, 0 Torque | 1 Speed and Torque, 1 Power | 2 Speeds, 2 Torques |
Under-constrained systems cannot be analysed. Over-constrained systems can be analysed if the constraints are compatible with each other. For example, if all three speeds are defined, it could be possible that they are compatible for the given geometry and a solution could be found. However, it is best to let RomaxDesigner find the third solution to avoid problems with rounding errors and numbers not quite matching up. | 
