Compact Design for Non-Standard Spur Gears

Volume, Issue, 0 Compact Design for Non-Standard Spur Gears Tuan Nguyen, Graduate Student, Department of Mechanical Engineering, The University of Memphis, tnguyen@memphis.edu Hsiang H. Lin, Professor, Department of Mechanical Engineering, The University of Memphis, hlin@memphis.edu Abstract This paper presents procedures for compact design of non-standard spur gear sets with the objective of minimizing the gear size. The procedures are for long and short addendum system gear pairs that have an equal but opposite amount of hob cutter offset applied to the pinion and gear in order to maintain the standard center distance. Hob offset has shown to be an effective way to balance the dynamic tooth stress of the pinion and the gear to increase load capacity. The allowable tooth stress and dynamic response are incorporated in the process to obtain a feasible design region. Various dynamic rating factors were investigated and evaluated. The constraints of contact stress limits and involute interference combined with the tooth bending strength provide the main criteria for this investigation. A three-dimensional design space involving the gear size, diametral pitch, and operating speed was developed to illustrate the optimal design of the non-standard spur gear pairs. Operating gears over a range of speeds creates variations in the dynamic response and changes the required gear size in a trend that parallels the dynamic factor. The dynamic factors are strongly affected by the system natural frequencies. The peak values of the dynamic factor within the operating speed range significantly influence the optimal gear designs. The refined dynamic factor introduced in this study yields more compact designs than those produced by AGMA dynamic factors. Introduction Designing gear transmission systems involves selecting combinations of gears to produce a desired speed ratio. If the ratio is not unity, the gears will have different diameters. The tooth strength of the smaller gear (the pinion) is generally weaker than that of the larger one (the gear) if both are made of the same material. In some designs, pinions with very small tooth numbers must be used in order to satisfy the required speed ratio and fit in the available space. This can lead to undercut of the teeth in the pinion which further reduces strength. One solution to the pinion design problem is to specify nonstandard gears in which the tooth profile of the pinion is shifted outward slightly, thus increasing its strength, while the gear tooth profile may be decreased by an equal amount. These changes in tooth proportions may be accomplished without changing operating center distance and with standard cutting tools by withdrawing the cutting tool slightly as the pinion blank is cut and advancing the cutter the same distance into the gear blank. This practice is called the long and short addendum system. When gears are cut by hobs, this adjustment to the tooth proportions is called hob cutter offset. Several studies of non-standard gears have been performed to determine the proper hob cutter offset to produce gear pairs of different sizes and contact ratios with similar tooth strength. Walsh and Mabie [] investigated the method to adjust the hob offset values for pinion and gear so that the stress in the pinion teeth was approximately equal to that in the gear teeth. They developed hob offset charts for various velocity ratios and for several changes in center distance. Mabie, Walsh, and Bateman [] performed similar but more detailed work about tooth stress equalization and compared the results with those found using either the Lewis form factor Y or the AGMA formulation. Mabie, Rogers, and Reinholtz [3] developed a numerical procedure to determine the individual hob offsets for a pair of gears to obtain maximum ratio of recess to approach action, to balance tooth strength of the pinion and gear, to maintain the desired contact ratio, and to avoid undercutting. Liou [4] tried to balance dynamic tooth stresses on both pinion and gear by adjusting hob offset values when cutting non-standard gears of long and short addendum design. Designing compact (minimum size) gear sets provides benefits such as minimal weight, lower material cost,

smaller housings, and smaller inertial loads. Gear designs must satisfy constraints, such as bending strength limits, pitting resistance, and scoring. Many approaches for improved gear design have been proposed in previous literature of references [5] to [6]. Previous research presented different approaches for optimal gear design. Savage et al. [] considered involute interference, contact stresses, and bending fatigue. They concluded that the optimal design usually occurs at the intersection point of curves relating the tooth numbers and diametral pitch required to avoid pitting and scoring. Wang et al. [3] expanded the model to include the AGMA geometry factor and AGMA dynamic factor in the tooth strength formulas. Their analysis found that the theoretical optimal gear set occurred at the intersection of the bending stress and contact stress constraints at the initial point of contact. More recently, the optimal design of gear sets has been expanded to include a wider range of considerations. Andrews et al. [4] approached the optimal strength design for nonstandard gears by calculating the hob offsets to equalize the maximum bending stress and contact stress between the pinion and gear. Savage et al. [5] treated the entire transmission as a complete system. In addition to the gear mesh parameters, the selection of bearing and shaft proportions were included in the design configuration. The mathematical formulation and an algorithm are introduced by Wang et al. [6] to solve the multiobjective gear design problem, where feasible solutions can be found in a three-dimensional solution space. However, none of them was for compact design study. Most of the foregoing literature dealt primarily with static tooth strength of standard gear geometries. These studies use the Lewis formula assuming that the static load is applied at the tip of the tooth. Some considered stress concentration and the AGMA geometry and dynamic factors. However, the operating speed must be considered for dynamic effects. Rather than using the AGMA dynamic factor, which increases as a simple function of pitch line velocity, a gear dynamics code by Lin et al. [5, 6, and 7] was used here to calculate a dynamic load factor. The purpose of the present work is to develop a design procedure for a compact size of non-standard spur gear sets of the long and short addendum system incorporating dynamic considerations. Constraint criteria employed for this investigation include the involute interference limits combined with the tooth bending strength and contact stress limits. Model Formulation Spur Gears Cut by a Hob Cutter The following analysis is based on the study of Mabie et al. [3]. Figure shows a hob cutting a pinion where the solid line indicates a pinion with fewer than the minimum number of teeth required to prevent interference. Figure Spur gear tooth cut by a standard hob or a withdrawn hob. The addendum line of the hob falls above the interference point E of the pinion, so that the flanks of the pinion teeth are undercut. To avoid undercutting, the hob can be withdrawn a distance e, so that the

addendum line of the hob passes through the interference point E. This condition is shown as dotted in Figure and results in the hob cutting a pinion with a wider tooth. As the hob is withdrawn, the outside radius of the pinion must also be increased (by starting with a larger blank) to maintain the same clearance between the tip of the pinion tooth and the root of the hob tooth. To show the change in the pinion tooth more clearly, the withdrawn hob in Figure was moved to the right to keep the left side of the tooth profile the same in both cases. The width of the enlarged pinion tooth on its cutting pitch circle can be determined from the tooth space of the hob on its cutting pitch line. From Figure, this thickness can be expressed by the following equation: p t = etanφ + ( ) Figure A hob cutter with offset e for enlarged pinion tooth thickness Equation () can be used to calculate the tooth thickness on the cutting pitch circle of a gear generated by a hob offset an amount e; e will be negative if the hob is advanced into the gear blank. In Figure, the hob was withdrawn just enough so that the addendum line passed through the interference point of the pinion. The withdrawn amount can be increased or decreased as desired as long as the pinion tooth does not become undercut or pointed. The equation that describes the relationship between e and other tooth geometry is Therefore, e = AB +OA - OP k = P d + Rbcosφ - R p e= k P - R p(- cos φ ) ( 3 ) d e= P (k- N d ( ) sin φ ) ( 4 ) Two equations that were developed from involutometry find particular application in this study: 3

cos = R A φb cos φ A ( 5 ) RB t B=R B( t A R A +inv φ -inv φ ) ( 6 ) A B By means of these equations, the pressure angle and tooth thickness at any radius R B can be found if the pressure angle and tooth thickness are known at a reference radius R A. This reference radius is the cutting pitch radius, and the tooth thickness on this cutting pitch circle can be easily calculated for any cutter offset. The reference pressure angle is the pressure angle of the hob cutter. When two gears, gear and gear, which have been cut with a hob offset e and e, respectively, are meshed together, they operate on pitch circles of radii R and R and at pressure angle φ. The thickness of the teeth on the operating pitch circles can be expressed as t and t, which can be calculated from Equation (6). These dimensions are shown in Figure 3 together with the thickness of the teeth t and t on the cutting pitch circles of radii R and R. Figure 3 Non-standard gears cut by offset hob cutters and operated at new pressure angle φ. To determine the pressure angle φ at which these gears will operate, we found ω = ω N = N R R ( 7 ) and t t πr = N πr = N + ( 8 ) Substituting Eq. (6) into Eq. (8) and dividing by R, 4

[ t R + ( inv R φ - invφ )] [ t + ( inv φ - invφ ) = R R π N + ( 9 ) Rearranging this equation gives, t R R t π R + = + ( + )( invφ inv ) ( 0 ) R R N R φ By substituting Eq. (7) and R = N/P d into the above equation and multiplying by N /P d, π t + t = + N + N P d d By substituting Eq. () for t and t, P (inv φ -inv φ ) () p p π N + N e φ + + e tan φ + = + (invφ - inv φ ) Pd Pd tan ( ) Simplifying the equation above gives, π (e + e )+ p= + N + N tan φ P d Pd By substituting p = π /P d and solving for inv φ, or inv = inv + P d( e + e ) φ φ N + N tan φ (inv φ -inv φ ) ( 3 ) ( 4 ) ( N+ N )(invφ - inv φ ) e + e = Pd tan φ ( 5 ) Expressing the above equation in metric unit with m as module, e + e = m ( N + N )(inv φ -inv φ ) tan φ In this study, we limit our investigation to the case that the hob cutter is advanced into the gear blank the same amount that it is withdrawn from the pinion, therefore, e = -e and, from Eq. (5) or (6), φ = φ. Because there is no change in the pressure angle, R = R and R = R, and the gears operate at the standard center distance. If the offsets are unequal (e -e ), then the center distance must change. This problem is not considered in the present investigation. Objective Function The design objective of this study is to obtain the most compact gear set of the long and short addendum system satisfying design requirements that include loads and power level, gear ratio and material parameters. The gears designed must satisfy operational constraints such as avoiding interference, pitting, scoring distress and tooth breakage. The required gear center distance C is the chosen parameter to be optimized. ( 6 ) where C = R + R p p (7) 5

R p pitch radius of gear R p pitch radius of gear Design Parameters and Variables The following table lists the parameters and variables used in this study: Table Basic gear design parameters and variables Gear Parameters Design Variables Bending strength and contact strength Number of pinion teeth limits Diametral pitch Operating torque Operating speed Gear ratio Face width Pressure angle Hob cutter offset Design Constraints Involute Interference Involute interference is defined as a condition in which there is an obstruction on the tooth surface that prevents proper tooth contact (Townsend et al. [8]); or contact between portions of tooth profiles that are not conjugate (South et al. [9]). Interference occurs when the driven gear contacts a noninvolute portion (below the base circle) of the driving gear. Undercutting occurs during tooth generation if the cutting tool removes the interference portion of the gear being cut. An undercut tooth is weaker, less resistant to bending stress, and prone to premature tooth failure. DANST has a built-in routine to check for interference. Bending Stress Tooth bending failure at the root is a major concern in gear design. If the bending stress exceeds the fatigue strength, the gear tooth has a high probability of failure. The AGMA bending stress equation can be found in the paper by Savage et al. [3] and in other gear literature. In this study, a modified Heywood formula for tooth root stress was used. This formula correlates well with the experimental data and finite element analysis results (Cornell et al. [0]): σ j Wj cos β j hf 6l f. h tan β 07. 07 L j = [ + 06. ( ) ][ + ( νtan β j ) ] F R h h l h h f f f f f f (8) where σ j root bending stress at loading position j. W j transmitted load at loading position j. β j load angle, degree F face width of gear tooth, inch ν approximately /4, according to Heywood []. R f fillet radius, inch other nomenclature is defined in Figure 4 and References [0] and []. To avoid tooth failure, the bending stress should be limited to the allowable bending strength of the material as suggested by AGMA [], St K L σ j σall = (9) K K where σ all allowable bending stress S t AGMA bending strength K L life factor K T temperature factor T R 6

K R reliability factor Kv dynamic factor Figure 4 Tooth geometry for stress calculations using modified Heywood formula. Surface Stress The surface failure of gear teeth is an important concern in gear design. Surface failure modes include pitting, scoring, and wear. Pitting is a gear tooth failure in the form of tooth surface cavities as a result of repeated stress applications. Scoring is another surface failure that usually results from high loads or lubrication problems. It is defined as the rapid removal of metal from the tooth surface caused by metal contact due to high overload or lubricant failure. The surface is characterized by a ragged appearance with furrows in the direction of tooth sliding (Shigley et al. [3]). Wear is a fairly uniform removal of material from the tooth surface. The stresses on the surface of gear teeth are determined by formulas derived from the work of Hertz [8]. The Hertzian contact stress between meshing teeth can be expressed as σ Hj = + W j cos β j ρ ρ π F φ ν ν + cos E E (0) where σ Hj contact stress at loading position j W j transmitted load at loading position j. βj load angle, degree F face width of gear tooth, inch φ pressure angle, degree ρ, radius of curvature of gear, at the point of contact, inch ν, Poisson s ratio of gear, E, modulus of elasticity of gear,, psi The AGMA recommends that this contact stress should also be considered in a similar manner as the bending endurance limit (Shigley et al. []). The equation is 7

σ σ = S C C Hj c, all L H C () CT CR where σ c,all allowable contact stress S c AGMA surface fatigue strength C L life factor C H hardness-ratio factor C T temperature factor C R reliability factor According to Savage et al. [], Hertzian stress is a measure of the tendency of the tooth surface to develop pits and is evaluated at the lowest point of single tooth contact rather than at the less critical pitch point as recommended by AGMA. Gear tip scoring failure is highly temperature dependent (Shigley et al. [3]) and the temperature rise is a direct result of the Hertz contact stress and relative sliding speed at the gear tip. Therefore, the possibility of scoring failure can be determined by Equation (0) with the contact stress evaluated at the initial point of contact. A more rigorous method not used here is the PVT equation or the Blok scoring equation (South et al. [8]). Dynamic Load Effect One of the major goals of this work is to study the effect of dynamic load on optimal gear design. The dynamic load calculation is based on the NASA gear dynamics code DANST developed by the authors. DANST has been validated with experimental data for high-accuracy gears at NASA Lewis Research Center (Oswald et al. [4]). DANST considers the influence of gear mass, meshing stiffness, tooth profile modification, and system natural frequencies in its dynamic calculations. The dynamic tooth load depends on the value of relative dynamic position and backlash of meshing tooth pairs. After the gear dynamic load is found, the dynamic load factor can be determined by the ratio of the maximum gear dynamic load during mesh to the applied load. The applied load equals the torque divided by the base circle radius. This ratio indicates the relative instantaneous gear tooth load. Compact gears designed using the dynamic load calculated by DANST will be compared with gears designed using the AGMA suggested dynamic factor, which is a simple function of the pitch line velocity. Gear Design Application Design Algorithm An algorithm was developed to perform the analyses and find the optimum design of non-standard gears of the long and short addendum system (LASA). The computer algorithm of the dynamic optimal design can start with the pinion tooth number loop. Within each pinion tooth number loop, all of the basic parameters are needed in dynamic analysis, including the diametral pitch, addendum ratio, dedendum ratio, gear material properties, transmitted power (torque), face width, and hob offset value. The DANST code also calculates additional parameters that are needed in dynamic analysis, such as base circle radius, pitch circle radius, addendum circle radius, stiffness, and the system natural frequencies, etc. One of the system natural frequencies (the one that results in the highest dynamic stresses) will be used as pinion operating speed. For this study, the diametral pitch was varied from two to twenty. Static analysis was first performed to check for involute interference and to calculate the meshing stiffness variations and static transmission errors of the gear pair. If there was a possibility of interference, the number of pinion teeth was increased by one and the static process was repeated. The number of gear teeth was also increased according to the specified gear ratio. Results from the static analyses were incorporated in the equations of motion of the gear set to obtain the dynamic motions of the system. Instantaneous dynamic load at each contact point along the tooth profile was determined from these motions. The varying contact stresses and root bending stresses during the gear mesh were calculated from these dynamic loads. If all the calculated stresses were less than the design stress limits for a possible gear set, the data for this set were added to a candidate group. At each value of the diametral pitch, the most compact gear set in the candidate group would have the smallest center distance. These different candidate designs can be compared in a table or graph to show the optimum design from all the sets studied. The analyses above are for gears operating at a single speed. To examine the effect of varying speed, the analyses can be repeated at different speeds. The program will stop when all the stresses are below the 8

design constraints. Thus, a feasible optimal gear set operating at the system natural frequency can be obtained. As the diametral pitch varies, the optimal gear sets determined from each diametral pitch value are collected to form the optimal design space. This design space, when transferred into the center distance space, which is our merit function of optimal design, will indicate all the feasible optimal gear sets to be selected from with given operating parameters. Design Applications Table shows the basic gear parameters for the sample gear sets to be studied. They were first used in a gear design problem by Shigley and Mitchell [9], and later used by Carroll and Johnson [3] as an example for optimal design of compact gear sets. The sample gear set transmits 00 horsepower at an input speed of 0 rpm. The gear set has standard full depth teeth and a speed reduction ratio of 4. In this study, the face width of the gear is always chosen to be one-half the pinion pitch diameter. In other words, the length to diameter ratio (λ) is 0.5. Table Basic design parameters of sample gear sets Pressure Angle, φ (degree) 0 Gear Ratio, M g 4.0 Length to Diameter Ratio, λ 0.5 Transmitted Power (hp) 00.0 Applied Torque (lb-in) 567.64 Input Speed (rpm) 0.0 Modulus of Elasticity (Mpsi) 30 Poisson s Ratio, ν 0.3 Scoring and Pitting Stress Limits, S S and S P (Kpsi) 79.3 Bending Stress Limit, S b (Kpsi) 9.8 Hob Cutter Offset (in) 0.03, 0.05, 0.07, 0.09 Cutting gears with offset hobs is an effective way to balance the dynamic tooth strength of the pinion and gear. It reduces the dynamic stress in the pinion and increases stress in the gear to achieve balance. In general, increasing the offset improves the balance in dynamic tooth strength. However, the best hob offset varies with the transmission speed and is limited by the maximum allowable offset that renders the pinion tooth pointed. The hob offset amount is varied from 0.03 inch to 0.09 inch with an increment of 0.0 inch in this investigation. Table 3 displays Carroll s [3] optimal design results for the sample gears of standard system without hob cutter offset. The optimal design is indicated in bold type and by an arrow. In the table, P d is the diametral pitch, NT and NT represent the number of teeth of pinion and gear, respectively, CD is the center distance, FW is the face width, CR is the contact ratio, and S b, S S, and S P are the stress limit for bending, scoring, and pitting, respectively. Table 3 Carroll s optimization results of sample standard gear set [3] (Using Lewis tooth stress formula) Pd NT NT CD FW CR S b S s S p.00 9 76 3.750 4.750.68.87 7.497 5.396.5 0 80. 4.444.69 3.553 7.6 55.853.50 84.000 4.00.70 4.75 7.53 59.950 3.00 3 9 9.67 3.833.77 5.80 74.00 67.3 4.00 7 08 6.875 3.375.745 9.0 77.876 78.860 6.00 40 60 6.667 3.333.805.703 66.97 78.309 8.00 53 6.563 3.33.840 6.9 63.30 78.47 0.00 66 64 6.500 3.300.863 9.670 6.463 78.85.00 86 344 7.97 3.583.887 9.77 53.9 69.603 6.00 3 58 0.65 4.5.97 9.76 4.459 57.37 0.00 85 740 3.5 4.65.934 9.74 35.708 48.760 The theoretical optimum for this example occurs at the intersection of bending stress and contact stress constraint curves at the lowest point of single tooth contact. This creates a gear set that has NT = 64 and P d = 9.8 for a theoretical center distance of 6.333 in. The minimum practical center distance (6.50 in.) is 9

obtained when NT = 66 and P d = 0.0. For comparison with the above results, we used the same AGMA dynamic factor K v (Equation 6) but with the modified Heywood tooth bending stress formula (Eq. ) in the calculations. Table 4 lists the optimization results obtained. Table 4 Optimization results of sample standard gears (Using modified Heywood tooth stress formula) Pd NT NT CD FW CR S b S s S p.00 9 76 3.750 4.750.68.847 66.873 5.390.5 9 76. 4..68 3.935 78.539 6.589.50 0 80 0.000 4.000.69 4.78 76.864 65.8 3.00 88 8.333 3.667.709 6.384 76.054 73.34 4.00 8 7.500 3.500.75 8.466 66.656 76.6 6.00 4 64 7.083 3.47.808.5 58.956 77.066 8.00 54 6 6.875 3.375.84 5.785 56.336 77.689 0.00 67 68 6.750 3.350.865 9.369 55.05 78.37.00 87 348 8.5 3.65.888 9.637 47.95 69.87 6.00 34 536 0.938 4.88.98 9.638 38.076 57.045 0.00 88 75 3.500 4.700.935 9.78 3.006 48.66 As can be seen from the table, the minimum practical center distance (6.750 in.) is obtained when NT = 67 and P d = 0.0. This is very close to Carroll s design but his optimal gear set will exceed the design limit of 9.8 Kpsi (from Table ) for maximum bending stress on the pinion according to our calculations. The differences between Carroll s results and those reported here are likely due to the use of different formulas for bending stress calculations. Figure 5 shows graphically the design space for the results presented in Table 4, depicting the stress constraint curves of bending, scoring, and pitting. The region above each constraint curve indicates feasible design space for that particular constraint. In the figure, the theoretical optimum is located at the intersection point of the scoring stress and the bending stress constraint. Figure 5 Design space for determination of optimized results of sample standard gears using modified Heywood tooth stress formula. 0

Table 5 displays the data for the optimum compact design of the sample non-standard gears (long and short addendum system) with a hob offset of 0.03 inch. The practical minimum center distance is equal to 5.000 inches and occurs at two values of P d = 0.00 and.00, with NT = 60 and 7. The practical minimum center distance decreases as the diametral pitch P d increases from.00 to 0.00; it then increases as the diametral pitch P d increases from.00 to 0.00. Table 5 Optimization results for non-standard gears (LASA) with 0.03 inch hob offset Pd NT NT CD FW CR S b S s S p.00 0 80 5.000 5.000.69.684 7.66 3.86 3.00 5 00 0.833 4.67.73 5.35 74.7 7.375 4.00 30 0 8.750 3.750.76 8.534 79.7 5.057 6.00 39 56 6.50 3.50.80.56 78.846 60.440 8.00 50 00 5.65 3.5.833 6.074 76.40 63.863 0.00 60 40 5.000 3.000.854.053 78.69 67.86.00 7 88 5.000 3.000.87 4.637 76.0 68.45 6.00 07 48 6.79 3.344.904 5.3 6.583 57.990 0.00 5 604 8.875 3.775.94 5.647 5.8 49.69 In Table 6, when the hob offset is increased to 0.05 inch, the optimum compact design occurs at P d = 0.0 with a practical minimum center distance of 5.750 inch and NT = 63,. As can be seen from the table, the minimum center distance decreases as the diametral pitch increases from.00 to 0.00. When the diametral pitch is greater than 0, the minimum center distance increases with the diametral pitch value. Table 6 Optimization results for non-standard gears with 0.05 inch hob offset P d NT NT CD FW CR S b S s S p.00 9 76 3.750 4.750.68 3.005 78.36 6.984 3.00 5 00 0.833 4.67.73 5.96 73.095 9.97 4.00 30 0 8.750 3.750.76 8.6 77.748 30.469 6.00 4 68 7.500 3.500.8 4.076 78.573 30.307 8.00 5 08 6.50 3.50.838 7.460 75.663 6.0 0.00 63 5 5.750 3.50.859 3.580 78. 65.03.00 79 36 6.458 3.9.880 5.68 7.70 6.58 6.00 7 508 9.844 3.969.95 5.93 54.8 49.583 0.00 96 784 4.500 4.900.936 5.309 4.570 38.476 The optimization results for hob offsets of 0.07 and 0.09 inches are very similar. For both cases the practical minimum center distance is equal to 7.500 and occurs at P d = 6.00 and 8.00, with NT = 4 and 56, respectively. Minimum center distance decreases as the diametral pitch increases from.00 to 6.00; it then increases as diametral pitch increases from 8.00. At P d = 6.00 and 8.00, the minimum center distance for both cases is unchanged. Some higher diametral pitch values do not produce optimization results due to the pointed teeth. Comparisons between non-standard gear designs at different hob offset values are shown in Figures 6(a) and 6(b). The compact design improves significantly as P d value increases from.00 to 4.00 for nonstandard gears and from.00 to 6.00 for standard gears.

Pinion Tooth Number 0 90 70 50 30 0 9 7 5 3 Feasible Design.00 3.00 4.00 6.00 8.00 0.00.00 6.00 0.00 Diametral Pitch Hob Offset 0.03 in 0.05 in 0.07 in 0.09 in (a) Center Distance (in.) 9 7 5 3 9 7 5 3 9 7 5 Feasible Design.00 3.00 4.00 6.00 8.00 0.00.00 6.00 0.00 Diametral Hob Offset 0.03 in 0.05 in 0.07 in 0.09 in (b) Figure 6 Design space and optimal gear sets for standard and non-standard gears, Gear ratio M g = 4, pressure angle φ = 0 0.

At diametral pitch values less than or equal to 3.00, non-standard gears with hob offset values of 0.07 inches and 0.09 inches appear to be more compact than standard gears while non-standard gears with hob offset values of 0.03 inches and 0.05 inches have the same center distance as standard gears (except nonstandard gears pair at P d =.00 with hob offset 0.05 inch that has slightly smaller center distance than standard gears pair). At a diametral pitch value of 4.00, non-standard gears and standard gears have the same compact design. At diametral pitch that is greater than 6.00, standard gears appear more compact than non-standard gears. The difference is extremely large at high diametral pitch values and high hob offset values. Comparisons between non-standard gears indicate that the curves in this study are almost identical to the one in Study 8. Therefore, similar conclusions are derived in this study. From Figure 6(b), hob offset values of 0.07 inches and 0.09 inches seem to result in better compact design than hob offsets of 0.03 inches and 0.05 inches when diametral pitch is less than or equal to 4.00. With diametral pitches that are greater than 4.00, non-standard gears with hob offset values of 0.07 inches and 0.09 inches become less compact when compared to non-standard gears at smaller hob offset values. Generally, at a diametral pitch value of 4.00 or less, increasing the hob offset value will result in a more compact design. When the diametral pitch is greater than 4.00, increasing the hob offset value will result in a larger gears center distance or a less compact design. When the diametral pitch value is high (greater than or equal to 0.00), non-standard gears at higher hob offsets (0.07 inches and 0.09 inches) seem very sensitive to subject load which result in a much larger gear pair. Thus, the degree of sensitivity increases with hob offset value at high diametral pitch values. As can be seen in the plots, because of the sensitivity in geometry, we could not obtain results for non-standard gears with hob offset values of 0.07 inches at P d = 0.00 and 0.09 inches at P d = 6.00 and 0.00. Conclusions This paper presents a method for optimal design of standard spur gears for minimum dynamic response. A study was performed using a sample gear set from the gear literature. Optimal gear sets were compared for designs based on the AGMA dynamic factor and a refined dynamic factor calculated using the DANST gear dynamics code. The operating speed was varied over a broad range to evaluate its effect on the required gear size. A design space for designing optimal compact gear sets can be developed using the current approach. The required size of an optimal gear set is significantly influenced by the dynamic factor. The peak dynamic factor at system natural frequencies can dominate the design of optimal gear sets that operate over a wide range of speeds. In the current investigation, refined dynamic factors calculated by the dynamic gear code developed by the authors allow a more compact gear design than that obtained by using the AGMA dynamic factor values. Compact gears designed using the modified Heywood tooth stress formula are similar to those designed using the simpler Lewis formula for the example cases studied here. Design charts such as those shown here can be used for a single speed or over a range of speeds. For the sample gears in the study, a diametral pitch of 0.0 was found to provide a compact gear set over the speed range considered. 3

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