1、01FTM2The Ultimate Motion Graph for“Noiseless” Gearsby: H.J. Stadtfeld and U. Gaiser, The Gleason WorksTECHNICAL PAPERAmerican Gear ManufacturersAssociationThe Ultimate Motion Graph for “Noiseless” GearsHermann J. Stadtfeld and Uwe Gaiser, The Gleason WorksThestatementsandopinionscontainedhereinaret
2、hoseoftheauthorandshouldnotbeconstruedasanofficialactionoropinion of the American Gear Manufacturers Association.AbstractTheinnovationwastodevelopageargeometrythatreducesoreliminatesgearnoiseandincreasesthestrengthofgears.Gearnoiseisacommonprobleminallbevelandhypoidgeardrives. Avarietyofexpensivegea
3、rgeometryoptimizationsare applied daily in all hypoid gear manufacturing plants, to reduce gear noise. In many cases those efforts have littlesuccess. Additionalexpensive finishing operations (lapping after thegrinding) areapplied to achieve thegoal of quietandstronggearsets. Theultimatemotiongraphi
4、saconceptformodulatingthetoothsurfacesthatusesaphysicaleffecttocancel out the dynamic disturbances that are naturally generated by all up-to-date known kind of gears. The ultimatemotion graph also eliminates the sensitivity of gears against deflection under load or displacements because ofmanufactur
5、ing tolerances. Lower dynamic disturbances will also increase the dynamic strength.CopyrightGe32001American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 2001ISBN: 1-55589-781-91 The Ultimate Motion Graph for “Noiseless” Gears Dr. Hermann J. Stadtfeld V
6、ice President, Research and Development Uwe Gaiser, B. Sc. Manager, Application Engineering The Gleason Works Design Practice Today According to classic design, gearsets have three principal types of tooth form modifications. The type of tooth modification depends to the highest degree on the possib
7、le flank form modifications versus the conjugate flank form. The intent of flank modifications is to provide a limited contact area under no load or light load which provides insensitivity to gear housing tolerances, inaccuracies in the gear members and assembly, as well as deflections. The correcti
8、ons keep the contact pattern inside the boundaries of the teeth and, therefore, prevent edge contact. For cutting bevel gears there are three mechanisms to create modifications that have the intent to locate the bearing contact. Those modifications are called “crowning“. The first element is lengthw
9、ise crowning which is a circular modification along the face width. If a ring gear is cut to the conjugate theory and a circular modification is applied to the pinion the result of the interaction between pinion and gear after cutting is shown in Figure 1. Length crowning can be generated by modifyi
10、ng the cutter radius vs. the theoretical conjugate radius. For example, by reducing the length radius on a convex flank, clearance will be provided towards the heel or toe end of the tooth. Similarly, length crowning can be generated by tilting the face cutter head around the flank line tangent and
11、by a corresponding change of the blade angles. The so-called Ease-Off represents the interaction between pinion and gear roll position by roll position, across the whole flank surface. Pinion and gear are rotated from contact line to contact line about discrete angles with respect to their ratio. If
12、 we visualize this process, it becomes clear that a contact between the two mating flanks exists only in theoretical conjugate points. If no crowning were applied to the case shown in Figure 1, the Ease-Off topography would be a flat surface on top of the presentation plane with zero deviation in th
13、e ordinate direction 1. The Ease-Off presentation plane is an axial projection of the gear tooth like the outline of the gear tooth in a cross-sectional, two-dimensional blue print. In case of crowning, generally only one point or one line will remain, where the Ease-Off topography is zero. If it is
14、 one point, then this location is called the “mean point”. The mean point is the only location inside the entire flank area where the ratio is 2 Figure 1: Ease-Off with lengthwise crowning accurate and no acceleration or deceleration occurs; it is a conjugate point. In case of a contact in line betw
15、een Ease-Off and presentation plane, this line is called “mean line“. The mean line can cross many contact lines as shown in Figure 1, or it might be identical to one contact line. It can (but does not have to) be identical to the path of contact (see also 2). The second element to generate crowning
16、 is a profile modification on the tool. A concave curvature on the cutting edge of a blade (versus a straight edge) will take stock off on top and flank and cause a circular profile crowning. Figure 2 presents a profile crowning that results in a high bias contact if it is the only or the dominating
17、 correction. The third element of flank crowning is a flank twist from toe to heel. Figure 3 shows the classic flank twist. The most commonly used methods to obtain an Ease-Off as shown in Figure 3 is a cutter tilt around the root angle axis and a corresponding change of the machine root angle. Othe
18、r methods are the use of modified roll Figure 2: Ease-Off with profile crowning and helical motion 1. All real bevel and hypoid gear applications used in power transmissions (and many cylindrical gears) use a combination of all three types of crowning. Figure 3: Flank twist with bias out contact In
19、case of face milling, there is no basic machine set-up that provides conjugate flank geometry. Cutting in the root line plane and rolling in a plane that matches the pitch line can only be approximated for gears with a tapered tooth depth. The elements of flank form correction are used in the gear d
20、esign of tapered depth tooth systems to eliminate the systematic deviations from conjugate to some extent and leave a desired crowning characteristic. Principally, this 3 leads to similar flank corrections that were once applied to face hobbed gearsets with a uniform tooth depth design. Research Pro
21、ject to Develop Tooth Contact and Single Flank Characteristics for Quiet Rolling Gearsets The subject of a research project conducted at The Gleason Works was the development of a ground rear axle drive that has a noise characteristic similar to a very high quality lapped gearset. The selected vehic
22、le is one of the most sensitive passenger cars available. It has a rigid beam-style axle mounted basically without any isolation directly to the chassis. The one-piece propeller shaft transmits any kind of stimulation towards the front part of the cabin. Grinding developments done in the past for th
23、is vehicle never succeeded. In order to analyze the noise characteristic of the vehicle, airborne noise measurement instruments, speed pick-up as well as pressure gages to determine the exact load situation, were installed in the vehicle. The principle of a laboratory and vehicle measurement loop wa
24、s successfully applied in the past for grinding developments used in independent and isolated rear axles of expensive luxury vehicles. This closed loop system is shown in Figure 4. The same technique was applied during the actual research project to the most challenging conditions of the mentioned m
25、id-size sedan. Figure 4: Closed loop between tester and vehicle measurements The design for the gears was done by using the Universal Motion Concept (UMC) on the pinion. Higher order kinematic freedoms up to the 4th order were applied to achieve the best possible result in noise, sensitivity and adj
26、ustability. The grinding of the actual parts took place on a UMC capable free form grinding machine. During the development phase, highly accurate single flank and structure borne noise test equipment was used to pre-qualify the gearsets and to establish a correlation between vehicle and test machin
27、e. There are several reasons to prefer an advanced grinding process compared to the conventional lapping process. The heat-treat deflections, which are very difficult to control, basically do not have any influence on the final ground flank form. The parts do not need to be stored and built in pairs
28、 anymore, which reduces logistical effort and cost. The ground flank form is very easy to 4 control and also repeatable. When the process is developed once, the duplication of the job is almost only a matter of recalling the stored machine settings on the CNC-Controller. The development steps from l
29、ot to lot can be either eliminated completely or kept to a minimum. Surface studies on lapped gearsets showed that the lapping grain is attached to the actual flank, which means a continuous “light lapping“ takes place at all times when the gearset is in operation. Furthermore, the lapping grain get
30、s from the surface into the oil and amplifies the negative effect even more. Strength studies have proven a higher allowable bending stress of ground compared to lapped gearsets. The grinding of the edge radii in the root reduces the stress concentration tremendously. The lifeline of the ground gear
31、sets regarding bending strength can be increased by a factor of 2 or even more. The environmental aspect declares the grinding process as the much cleaner process. The application of modern filter techniques keeps the waste product at a very low level. All the above-described benefits combined with
32、good rolling and adjustability characteristics could make the grinding a very competitive process with increasing potential for the near future. The following chapters describe the optimization of crowning and motion transmission on the example of the selected vehicle and ground hypoid gearsets. The
33、 Physics of 1stHarmonic Noise The conflict associated with crowning is the motion error caused by non-conjugate members. If gearbox tolerances and shaft as well as bearing deflections were zero very little crowning would be needed to account for the tooth deflections under load. The small value of m
34、otion error would be appreciated for smooth and quiet roll behavior. With realistic part accuracy and deflections, a crowning of 0.1mm or more may be required. The high motion error will prevent a smooth rolling but also protect from damages by edge contact. Practically all real motion errors have a
35、 parabolic shape caused by the parabolic crowning element. Figure 5 discloses angles, velocities and accelerations during a tooth mesh of rolling gear members. The motion graph on top of Figure 5 is a second order function (see formula plotted over a time axis). 5 Figure 5: Relative motion, velocity
36、 and acceleration graphs The angular velocity change consequently is the first derivative of the motion graph function which result in dropping first order function (Figure 5). The RPM of the driven member is continuously dropping, although the pinion RPM is constant! This seems to be impossible but
37、 is realistic since the following tooth pair will start again at the high level of (left side of graph) and again drop continuously. The second derivative of the motion graph delivers the angular acceleration shown in the lower diagram of Figure 5. The acceleration is a constant with a negative valu
38、e. The step in the velocity function is physically defined as an infinitely high acceleration during an infinitely short amount of time. The result is the peak in the acceleration graph on the changeover point between teeth. A short infinite high acceleration 6 is physically an impulse and it reflec
39、ts the impact each pair of teeth causes at the moment of first contact. Reduction of Tooth Mesh Impact The result of the gear transmission is an audible noise of the tooth mesh frequency. After the physical discussion in the last paragraph, it seems unavoidable to allow the constant deceleration if
40、a second order crowning was applied to the gearset. The question is what kind of possibilities exist to reduce or eliminate the peak at the beginning of a new tooth mesh? One answer to this question is the flank-modification due to lapping. The highest removal of material takes place at the instant
41、of impact because of the peak-torque between the two mating members. In a physically natural way, the material that leads to disturbances will be removed during lapping. Figure 6 shows a parabolic shaped motion graph and its modification by lapping. The motion graph is lapped flat at the top and ind
42、icates a sinusoidal or third order modulation at the entrance area. Another possibility of reducing the peak in the acceleration was proposed in the Handbook of Bevel and Hypoid Gears 1 and deals with a fourth order crowning along the Path of Contact. In Figure 7 it is assumed that the motion graph
43、of a tooth pair under load is a horizontal straight line. The motion graph of an undeflected, not loaded pair of teeth needs to pick up the first contact with a horizontal tangent. The disadvantage of this method is the fact that there were no cutting or grinding methods in the past to apply such a
44、correction in a completing environment. Another problem presents the assumption of a horizontally straight motion graph under load. The assumption can be correct under high load but does not apply to the mostly noise critical low load condition. During 80% of the duty cycle, the exit area of the loa
45、d carrying tooth pair will cause a parabola shaped motion characteristic before the changeover occurs to the entrance area of the following pair of teeth. Figure 6: Motion graph modification due to lapping 7 Figure 7: Kinematic optimization using a fourth order function Theoretical Studies and Pract
46、ical Measurements of Higher Order Flank Modifications The law of physics and mathematics teach that the intersection between a steady monotonic dropping function with a steady monotonic rising function will not have any overlap but just the concrete defined crossing point. Applied to the motion grap
47、h this means that left of the intersection point the preceding pair of teeth transmitted torque and speed and to the right of the crossing point the actual teeth are the carrier of motion and load. In other words, a classic gearset with second order crowning elements will always have an effective co
48、ntact ratio of one (or slightly below) under light load conditions. This conclusion is completely independent from the calculated transverse and face contact ratio. Also, the study in Figure 7 of a kinematically optimized motion graph has no potential for more than one pair of teeth in contact at an
49、y time during a light load mesh. The motion graph of the lapped gearset (Figure 6) is not steadily monotonic at the entrance area. Here is a potential for overlap of two following pair of teeth even under zero load. The idea of multiple teeth mesh is not new, but the only solution so far is conjugate tooth contact. Since the conjugate tooth contact will not function under realistic conditions, multiple mesh under no load remained an academic desire and seemed impossible since it appeared to be in conflict with the applicable laws. The closest approximation to an overlapping c
