In determining the characteristics of coiled wire springs, if all the component forces, including those .of torsion, transverse shear, tension, and compression, are considered, the calculation may be complicated and involved, but for practical purposes of design all can be ignored except torsion. The calculation then becomes simple. The underlying principles of the formulas that express the torsional characteristics of an ordinary helical spring are the same as those that govern torsion in a straight shaft; and the fact that the result would be the same if the shaft or wire were twisted in the opposite direction makes it clear why a coiled spring has the same stiffness in either compression or extension so long as all the coils remain open.
In Begtrup's formula, as given in the handbooks, the only unknown factor is the modulus G, which is variously stated to be from 10,000,000 to 14,000,000 lb. An average of hundreds of tests, however, has shown that, for drawn wire in all grades of carbon steel, whether tempered or merely hard-drawn, for alloy-steel and for music wire, the true modulus is 11,500,000 lb. and will not vary from this value more than plus or minus 3 per cent. This value is entirely independent of the hardness or the heat-treatment.
In a spring with squared and ground ends, properly closed, the number of active coils is the total number of coils minus two. The accepted formula for deflection in springs of square material, as set forth by Begtrup, is said to be incorrect, due to the assumption that a square section retains its shape when stressed, which is not true. The constant K, which varies with the shape, is unity for a round section but has been shown by Saint-Venant to be 0.843 for a square section.
The main points to be considered in designing extension springs are that stress calculations for a given deflection are misleading unless the stress stored up as initial tension is included, and that deflection does not start from zero load, as in a compression spring, unless the coils are not under tension when coiled.
The spring of smallest diameter that can be produced commercially is that in which the mandrel size and the wire size are equal.
Special shapes of springs are infinite in number. Conical or barrel-shaped springs tend to decrease breakage. In torsion springs, the action within the wire is that of bending; for these, the derivation of the formula is partly experimental but is based on the fundamental formulas for beams. It is based on the assumption that the bar is straight when free; consequently, the load computed by it is greater than the actual load that a spring will develop. Expressing the deflection in terms of revolutions of the shaft on which the spring is operated and the load in pound-inches, and correcting the constant, enable a simple formula to be developed for the stresses in torsion springs, a subject not heretofore covered completely.
The designing of a spring involves a combination of a definite load with a definite deflection under that load; consequently, there is no such thing as a factor of safety in a spring, in the sense of that in a beam, but the use of proper judgment is urged.
Experiments show that the limiting factor in springs is the range of stress through which operation takes place from the initial to the maximum load, rather than the maximum stress, as has been supposed. If the test is conducted so that the spring is not allowed to relax to no load, the results will not be the same; it was found that a spring operated from the proverbial maximum of 80,000 lb., as an initial load, to about 86,000, as a final load, will run indefinitely. This range is not constant but varies with the analysis of material and with the temperature.
A uniformly sorbitic condition is the proper state for the best resistance to fatigue. A manganese content higher than that in the usual steels is best. Fatigue resistance depends on the condition of the surface. Cost must be considered and a compromise must be effected. Tolerances are a question of cost. What is a reasonable tolerance on a spring depends upon its proportions.
The conclusions reached are that the design of springs involves a knowledge of what the spring is required to do, the application of a few simple mathematical laws to determine its dimensions, a consideration of how accurately it must be held and the tempering of all these with ordinary common sense in keeping the stresses within reason.


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