机械设计的基本原理9.23
Kinematic Couplings: Three-Tooth
运动耦合：三齿耦合
To prevent thermal expansion, often it is desirable to make precision instrument components from Super Invar, which is an iron alloy with 36% nickel. However, Invar cannot be hardened, and hence it cannot support significant point-contact loads. Three-tooth couplings were invented by Dr. Layton Hale of the Lawrence Livermore National Laboratory to overcome this limitation1. The three-tooth coupling forms three theoretical lines of contact between the cylindrical teeth on one component and flat teeth on the other component. Practically, both members can be made identical with cylindrical teeth, which yields the best repeatability. Good performance for low cost can also be obtained if the members are made with flat teeth, which yields the greatest load capacity, least cost, and good repeatability. Each line of contact across two teeth represents a two-degree-of-freedom constraint, thereby giving a total of six constraints. Ideally manufactured with three identical cuts directly into each member, the teeth must be straight along the lines of contact but other tolerances may be relatively loose.
为了防止热膨胀，通常采用超因瓦合金来制作精密仪器，这种铁合金含36%的镍。但是，这种合金无法硬化，所以其无法承载太大的点载荷。劳伦斯国家实验室的Lyton博世发明了三齿耦合的设计来克服该合金的弱点。三个齿形耦合在圆柱形齿和另一个平面齿之间形成了理论上的线接触。实际上，两个零件都可以用圆柱形齿制造，从而达到最好的精度。如果都采用平面齿形的话，可以获得更低的成本和不错的性能，从而使得负载能力最大，成本最低，精度尚可。两个齿形之间的每对线接触都代表了两个自由度的约束，所以可以总共约束六个自由度。理论上，三个齿的形状一摸一样的话，可以直接相互啮合，线接触的地方齿必须是直的，但是其它部分尺寸的公差要求可以更低。
A three-tooth coupling can also be easily molded into plastic to enable very repeatable coupling and orientation of cylindrical members, such as required for lens assemblies. With added snap-fits, a very precise assembly can be obtained.
可以采用塑料模具来制造三齿耦合零件，从而获得非常高精度的耦合。诸如镜片装配应用中，通常需要定位圆柱形零件，就可以采用这种塑料三齿。通过额外增加的压入配合，可以获得非常好的装配准确度。
Compare the load capacity of a single Super Invar tooth-to-tooth line contact verses the load capacity of a hardened 20 mm diameter ball on a hardened flat steel contact. Using the spreadsheet Hertz_contact_line.xls a very high and unrealistic load capacity is predicted. This can be seen because the deflection is large compared to the critical dimensions. This indicates that the contact zone has also probably saturated and in effect, the teeth have flattened and face-to-face contact has occurred.
我们对比下单齿超级因瓦合金的线接触和20mm硬化小球与硬化平面接触的负载。通过计算表格Hertz_contact_line.xls，我们获得了非常不可思议的高负载预测。可以发现，变形的大小比关键尺寸大的多。这就意味着接触区域可能已经饱和了，事实上，齿面已经变平，变成了面对面的接触。
Contrast the output shown in this spreadsheet to a 20 mm diameter hard steel ball on a flat hard steel contact surface which can support three orders of magnitude less force:
对比如图20mm硬质钢球接触钢平面的情况，你可以看到其负载能力比三齿配合要少三个数量级：
Use the spreadsheets to load both the elements with the same load, and then compare the stresses. Does the three-tooth coupling’s apparent ease of machinability make it a more reasonable choice for coupling tasks that your machine requires? How will repeated cycles affect its repeatability? Can it be made from plastic or aluminum instead of steel?
用计算表格计算同一载荷情况下两种元件的应力。是否三齿耦合的易加工特性使得其更适合你的机器设计需要求呢？那么重复的循环载荷对三齿耦合的精度有何影响？可以用塑料或者铝合金来代替钢材么？
1. See http://www.llnl.gov/tid/lof/documents/pdf/235415.pdf for a full copy of Dr. Hale’s Ph.D. thesis.
