- Gear Design -
Geometry, Power Ratings, Dynamic Loads, Fatigue
This section contains information on several important topics with respect to the properties of gears and the loads imposed on them in a geared transmission system. It will be worthwhile to take the time to read through these four pages if you are truly interested in knowing how a geared system functions.
CONTENTS
Introduction to Gears
This section explains the meaning of a few basic gear terms, including pitch diameter, diametral pitch, pressure angle, line of contact, involute profile, tangential force and separation force.
The purpose of gears is to transmit uniform rotary motion from one shaft to another, and often, to turn the driven shaft at a different speed than the driving shaft.
Imagine two circular wheels of different diameters, in contact with each other at their outside diameters, with one wheel turning at a constant speed, and driving the other wheel by friction, with no slippage between the mating surfaces. The motion imparted to the driven wheel would be uniform rotary motion.
Two gears in mesh behave like the two circular wheels described above. The pitch diameter of each gear is the outside diameter of its effective circular wheel. The two pitch circles touch at a point on the straight line between the two gear centers (the point of tangency).
The pitch circles of the two gears pictured below are marked PD1 and PD2 respectively, and touch each other at the point of tangency. The (vertical) centerline connects the center point of each gear. The point of tangency lies at the intersection of the centerline and the two pitch circles. The horizontal line marked TAN is perpendicular to the centerline and passes through the point of tangency.
Two gear teeth in mesh with each other are essentially two cams which have profiles developed to implement the required uniform rotary motion, while being able to transmit much greater forces than would be possible by friction contact between two wheels.
The profile which accomplishes that function is a mathematical function known as an involute. The nature of the involute profile is such that at any point from the beginning of mesh to the end of mesh, the contact point of the two tooth faces lies along a straight line (the line of contact, marked LOC in the picture).
The gear's pressure angle (PA) is the angle between the TAN line and the line of contact (LOC). Meshing gears must have the same pressure angle.
The the force which the tooth on the driving gear applies to its mating tooth on the driven gear is applied along the line of contact, and is known as the normal (perpendicular to the surface) force. Look at the picture closely and you can see two pairs of teeth in contact, and the points of contact are both on the LOC.
Because the involute profile maintains the contact point along the LOC, the effective contact point between the two gears remains at the point of pitch-circle tangency. That causes the motion imparted to the driven gear to be uniform angular motion.
The normal force (the force along the LOC) which the driving gear applies to the driven gear at the point of contact is generally handled as two perpendicular components: (a) the tangential force, and (b) the separation force.
The tangential force is applied by the driving gear along the line marked TAN. The value of the tangential force is the input torque divided by the pitch radius (half the pitch diameter).
The separation force is applied along the centerline between the centers of the two gears, and is trying to drive the two gears apart from each other.
The relationship between tangential and separation forces is determined by the pressure angle (PA) of the gears. From simple trigonometry and the picture above, it is clear that the normal force is the tangential force divided by the cosine of the pressure angle, and the separation force is the tangential force times the tangent of the pressure angle.
The term diametral pitch (DP) describes gear tooth size, and is defined as "the number of teeth per inch of pitch diameter".
As an example, a 6-DP gear with 21 teeth has a pitch diameter of 3.5 inches (21 ÷ 6 = 3.5). Now picture that 21-tooth gear meshed with a 6-DP, 47 tooth gear. The 21-tooth gear has a mean torque of 625 lb.-ft. applied to it. The force tangential to the pitch circle is 4286 pounds:
Tangential Force = (625 lb-ft x 12 inches per foot) ÷ (3.5 inches ÷ 2)
Tangential Force = 7500 ÷ 1.75 = 4286 lbs.
The separation force is 4286 x TAN(20) = 4286 x .364 = 1560 lbs.
The normal force is 4286 ÷ COS(20) = 4286 ÷ 0.94 = 4560 lbs.
The normal force must be counteracted by the bearings which support the gear shafts.
These loads are sometimes referred to as "static" gear loads, because they can be generated by statically loading the input shaft at the specified input torque, without any motion being imparted to the gears.
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PSRU Power Ratings
PSRU sellers often advertise the horsepower that their unit is claimed to handle. That’s of interest for cooling calculations, but essentially meaningless in terms of the loads applied to the internal components. Why? Because, as you know, TORQUE is the measure of the actual FORCE a rotating engine applies to its load. POWER is simply a mathematical expression of WORK-PER-UNIT-TIME (see Torque and Horsepower).
The TORQUE applied to the first driving gear in a gearbox is the value which determines the force applied to the faces of all other gears in a geartrain, as explained above in Introduction to Gears.
Consider this example. Suppose a certain engine produces a maximum POWER of 460 HP at 4800 RPM, and the engine's torque peak is 625 lb-ft. at 3800 RPM. Also suppose that engine is equipped with a PSRU which was designed for the peak engine power (460 HP at 4800 RPM). Since the engine torque at maximum power is 503 lb-ft, that PSRU would only be able to handle 75% of the engine's maximum torque (625 lb-ft) which occurs fairly close to the cruise operating RPM.
From the applied loads, the designer calculates static gear-tooth stresses: (a) the bending (tensile and compressive) stresses at the root of the tooth, and (b) the contact stress at the line across the tooth face where the teeth contact each other.
Because no piece of machinery is exactly perfect, provisions must be made for the fact that, due to tiny manufacturing inaccuracies, the contact force between two gear teeth will not be evenly applied across the entire face width of the tooth. In fact, this problem, known as edge-loading, can destroy a pair of otherwise-sufficient gears in a very short time. Therefore, special provisions must be made in the design of the gears to prevent edge-loading.
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Dynamic Gear Tooth Loads
An aspect of gear loading which is commonly ignored, but critically important, is the phenomenon known as dynamic load. This load occurs when a pair of teeth is just coming into mesh. Assume for the moment that we have a pair of gears which have been manufactured so accurately that the location of every tooth is absolutely perfect and the profile of each tooth is absolutely perfect. With such perfection, one might think that each successive tooth pair would pick up the load perfectly smoothly as the teeth enter the mesh. Not so, however.
Remember that metal objects have elasticity, or springiness. Visualize the tooth-mesh when a single pair of teeth is carrying all the load. If the load is significant, those teeth have deflected from their unloaded ("perfect") positions. That deflection allows the rest of the driving gear to move slightly ahead of its theoretical undeflected position and the rest of the driven gear to lag slightly behind its theoretical undeflected position. Those deflections move all the unloaded teeth on both gears slightly out of their correct positions with respect to the tooth-pair carrying the load.
Because of that deflection, when the next tooth-pair comes into mesh, they touch each other sooner than they would if there were no deflection, thus they pick up a disproportionate amount of the load very quickly. That sudden load application produces a hammer-like force (impact) which can cause the teeth to bounce apart (just as a hammer will rebound from striking a hard piece of steel) then re-contact later in the mesh, causing another impact. The forces seen by the teeth during these impacts can be significantly greater than the load applied by the transmitted torque.
Now we add another dash of reality. It is not possible to manufacture perfect gears. The AGMA Quality Number is a measure of the accuracy of the tooth locations and the tooth profiles. A gear with a lower Quality Number has greater errors in tooth location and tooth profile. As the errors in tooth location and profile increase, the magnitude of the dynamic loads increase as well.
The determination of the dynamic loads is a complex methodology. It takes into account factors which include gear tooth manufacturing tolerances, tooth stiffness, torsional rates of all loaded shafts and gears, and Jm values of the engine, propeller, shafts, and gears. That methodology, which includes the calculations for tooth bending stresses, contact stresses, stress concentrations as a function of tooth root radius, and depth of maximum shear, is implemented in a computer program which EPI developed for analyzing PSRU gear systems.
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Fatigue Loads
Having determined the stress levels and the properties of the material selected for the gears, the designer can estimate the fatigue life in the load model. The term estimate is used because the mathematics of fatigue are largely statistical and involve the use of safety factors derived from successful field experience (see Fatigue). In addition, case-hardened materials commonly used in highly-loaded gears (carburized 8620 and 9310) do not exhibit infinite life at bending stress levels suitable for 10 million load cycles (the point at which many designers expect infinite life).
Another critical fatigue issue is the use of an idler gear between the driving and driven gears. This arrangement is often used to make the output shaft rotate in the same direction as the input. If the same gear in the idler mechanism contacts both driving and driven gears, its teeth are subjected to the most severe kind of fatigue loading: fully-reversing loads. The idler teeth are loaded first in one direction by the driving gear, then in the opposite direction (with the same force) by the driven gear. If the dynamic loading characteristics of each mesh are the same (unlikely) then the loads will be equal and opposite. If the dynamic loading characteristics differ, then the loads will differ, sometimes significantly. Nevertheless, those loading characteristics must be determined and the design of the idler gear must implement a suitably lower endurance load limit.
Bottom line: the designer must determine the allowable working stress levels after taking into account a large number of loading, life, and metallurgy factors. Then the gears must be designed so that the working stress levels at the intended loadings are within those limits. (Note: at 4600 input RPM, it takes only 36 hours for the driving gear to reach 10 million fatigue cycles.)
