In the previous pages, we introduced the basics of gears, including 'Module', 'Pressure Angle', 'Number of Teeth' and 'Tooth Depth and Thickness'. In this section we introduce the basic parts of Spur Gears (Cylindrical gears) and dimensional calculations.
Spur gears with helicoid teeth are called Helical Gears.The majority of calculations for spur gears can be applied to helical gears too. This type of gear comes with two kinds of tooth profiles in accordance with the datum surface. (Figure 2.9)
Crack Gear Generator 3 14
Reference diameter (d) of the helical gear with transverse system can be calculated from Equation (2.8).Reference diameter (d) of the helical gear with normal system can be calculated from Equation (2.14).
The following is a calculation for the Reference Diameter of a helical gear with:Transverse module mt = 2, Number of teeth z = 30, Helix angle β = 15 (R)Reference Diameter d = zmt = 30 2 = 60The following is a calculation for the Reference Diameter of a helical gear with:Normal module mn = 2, Number of teeth z = 30, Helix angle β = 15 (R)Reference Diameter d = zmn / cos β = 30 2 / cos 15 = 62.117
When the gear surface is repeatedly subjected to load and the force near the contact point exceeds the material's fatigue limit, fine cracks occur and eventually develop into separation of small pieces, thereby creating pits (craters).
The initial cause comes from small convex portions of the gear surfaces contacting each other and the local load exceeding the fatigue limit. As gears are driven and surfaces become worn in, local convex portions disappear and the load is equalized and pitting stops.
Even after gear surfaces are worn in and load is equalized, with time more pitting starts to occur and pits get enlarged.(1) When an overload condition exists and the gear surface load exceeds the fatigue limit of the material.(2) While being driven, the load distribution could become uneven across the gear face due to various parts' deflection causing the fatigue limit to become exceeded.These are some of the possible reasons of progressive pitting.
This is the condition in which the lubricant coating breaks down due to overheating of local contact areas causing the deterioration of the gear surface from metal to metal contact. It is possible for this condition to progress from moderate to break down.
In the direction of gear sliding, groove like condition appears. This is part of abrasive wear and the following causes are possibilities.(1) Wear from a solid foreign object larger than the oil film thickness getting caught in the gear mesh.(2) Wear from a solid foreign object buried for some reason in the opposing gear tooth. (3) Wear from the hard convex portion of the opposing gear tooth digging into the meshing gear.
Wear that looks like an injury from abrasion or has the appearance of lapping. Below are some of the causes.(1) Possible wear occurring from solid foreign objects mixed in the lubricant (such as metal wear debris, burr, scale, sand, etc.).(2) Wear from the difference in hardness of two meshing gears in which the hard convex portion digs into the softer gear surface.
This refers to the symptom of relatively large metal chips falling off from the gear surface due to material fatigue below the surface from high load. The gear surface's concave part is large and the shape and the depth are irregular. Because the applied shear force exceeds the material's fatigue limit, fatigue cracks appear and grow leading to possible breakage of the tooth.
Wear from the gear surface being subjected to intense repeated metal to metal contact which occurs when the oil film is thin and the lubrication is insufficient relative to the load and surface roughness of the gear. This condition tends to occur when operating at very low speed and high load.
Breakage that comes from an unexpectedly heavy load for one or several action cycles (Normally, mistakes in design or manufacturing are not included). The fracture surface spreads fibrously from a starting point and indicates a sudden splitting. The cause is due to the load exceeding the tensile strength of the gear material. This may come from the prime mover, driven mechanism or breakage of bearings or other gears which could cause biting of teeth, sudden stop, or concentration of load due to irregular tooth contact.
This is the case in which the root portions of gear are subjected to a repeated load exceeding the material's fatigue limit. A fracture that starts in the corner of the gear root propagates until the tooth breaks. The fractured surface is relatively smooth and the starting point can often be recognized by the beach mark (shell pattern) around it.
This describes when a tooth separates from the body by shearing due to a one time extreme overload. The breakage is straight in the circumferential direction and appear flat as if machined. The nearby area shows plastic deformation. It happens when the applied force exceeds the shear strength of the material. It happens when a high stiffness and strength gear is meshed with a gear which has a relatively low modulus of elasticity and weak material.
The use of bulk metallic glasses (BMGs) as the flexspline in strain wave gears (SWGs), also known as harmonic drives, is presented. SWGs are unique, ultra-precision gearboxes that function through the elastic flexing of a thin-walled cup, called a flexspline. The current research demonstrates that BMGs can be cast at extremely low cost relative to machining and can be implemented into SWGs as an alternative to steel. This approach may significantly reduce the cost of SWGs, enabling lower-cost robotics. The attractive properties of BMGs, such as hardness, elastic limit and yield strength, may also be suitable for extreme environment applications in spacecraft.
The operation of a steel SWG is shown in Fig. 1. Although SWGs can be constructed using a variety of geometries, the three components of a standard cup-type SWG are shown disassembled in Fig. 1a, for a CSF-8 purchased from Harmonic Drive Systems, Inc., Tokyo, Japan. They are (1) a stiff outer spline, also called a circular spline, with internal gear teeth, (2) a thin-walled flexspline with external teeth numbering two less than the outer spline, and (3) an elliptical wave generator with steel ball bearings confined in an elliptical race by a steel band. When assembled, the wave generator forces the wall of the flexspline to expand and engage the teeth of the outer spline. The output torque is generally provided by the base of the flexspline while the outer spline stays fixed. The typical operation of a SWG is shown schematically in Fig. 1c. The wave generator forces the teeth on the flexspline to engage the outer spline and when the wave generator is rotated, the flexspline elastically deforms to maintain contact. After a 180 degree rotation, the flexpline has moved by one tooth relative to the outer spline. After a full rotation, the flexspline and the outer spline have been offset by two teeth. Unlike spur and planetary gears, the reduction ratio of a SWG is not a function of the size of the gears, but rather by the number of teeth. The reduction ratio, i, which is defined as the ratio of the input speed to the output speed is:
(a) A disassembled SWG showing the three components: an outer spline, a wave generator, and a flexspline. (b) An assembled CSF-8 flexspline from Harmonic Drive, LLC. (c) A schematic showing the operation of a SWG where each 180 revolution of the wave generator moves the flexspline by one tooth. (d) A schematic of a load torque versus number of cycles plot for a SWG showing the various failure mechanisms and how to design for them.
In addition to wire-EDM for the fabrication of the gear teeth, we also directly cast the gear teeth against a mold that had previously been made using wire-EDM, shown in Fig. 2e for Zr35Ti20Cu8.25Be26.75. Using both techniques, we were successful at fabricating BMG flexsplines with very similar dimensions to standard steel flexsplines (the measured dimensions are shown in Table 2). Figure 2f shows the top and bottom of the first BMG flexspline prototype compared directly to the steel version. After manufacturing, the flexsplines were integrated into a CSF-8 SWG using the standard steel outer spline and wave generator. Figure 2g shows the BMG flexspline from Fig. 2f installed in the SWG. A video showing the operation of this flexspline in the SWG is shown in the Supplementary Material. Figure 2(h,i) shows differential scanning calorimetry (DSC) traces and X-ray diffraction (XRD) traces for three BMG flexsplines after manufacturing. The alloy Zr35Ti30Cu8.25Be26.75 (GHDT), is known for its high toughness and large thermoplastic forming region (which can be seen in the DSC image as the distance between the arrows indicating the glass transition temperature and the crystallization temperature). The Ti-based BMG (Ti40Zr20Cu10Be30) and the Zr-based BMG (Zr44Ti11Cu10Ni10Be25, LM1b)21 have smaller thermoplastic forming regions but are both shown to be mostly amorphous. Figure 2i shows XRD traces from flexsplines made from all three BMGs, showing mostly amorphous microstructures. The LM1b flexsplines are shown later in the text and were commercially cast, showing only small evidence of partial crystallization.
In addition to the CSF-8 BMG flexsplines that were cast, shown in Fig. 4(a,b), several other alloys were also manufactured into these flexsplines to investigate the castability and properties of different BMG alloys. Figure 4i shows the four different BMGs that were commercially cast and their properties appear in Table 1. Two well-known non-Be alloys were cast into the flexspline, Vitreloy 106 (Zr57Nb5Cu15.4Ni12.6Al10) and Vitreloy 105 (Zr52.5Ti5Cu17.9Ni14.6Al10). Both alloys formed an amorphous part but their higher melting point and higher viscosity produced a lower quality casting. The BMG GHDT, which was used for the prototypes shown in Figs 2 and 3, was also cast into a flexspline to compare the commercial casting process with the prototype made at JPL using lab-grade material. Lastly, the alloy LM1b was used predominantly for testing and characterization of the casting process. Figure 4(j,k) shows a selection of optical micrographs from both machined steel and cast BMG flexsplines. A Keynance large depth-of-field microscope was used to perform both optical imaging and profilometry on the surface and the gear teeth. Figure 4j shows a comparison between the CSG-50 flexspline cast from BMG and machined from steel. The teeth have virtually the same profile but the surface of the BMG has a random texture compared to the steel, which exhibits horizontal machining lines. The tips of the teeth in the BMG alloy is rounder than the steel, due to the high viscosity of the BMG during casting. Figure 4k shows the smaller CSF-8 flexspline in steel, a BMG prototype that was created using wire EDM and two BMG cast parts. Due to the smaller feature size compared to the larger part, the replication of the gear teeth is not quite as good. The teeth are slightly rounder and the surface is rougher. The BMG sample that was cut using wire EDM shows a very rough surface on the gear teeth as compared to the cast samples, indicating that casting provides a better finish, and likely, improved wear performance. Measurements of cast BMG gears are shown in Table 3. 2ff7e9595c
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