A return spring of infinite life (ie. 10⁷ cycles) is required for a cam follower which moves through 25 mm, 400 times a minute. The spring must fit over a Φ 15 mm shaft and inside a Φ 65 mm hole, and it must exert a force which varies between 300 and 600 N. Design a suitable spring with closed and ground ends, made from ASTM A229.

Although the material specified is not optimum for fatigue applications, the solution to the problem will be a useful benchmark for other solutions. A small safety factor of 1.1 will be selected for reasons given previously and presuming the consequences of failure are not severe.

Completing the characteristic, given   δ_hi - δ_lo = 25 mm,   F_lo = 300 N and   F_hi = 600 N, it follows that the spring stiffness must be   k = ( 600 -300)/25 = 12 N/mm, and further   δ_lo = 25 mm,   δ_hi = 50 mm.
Assuming a 12% clash allowance, then   δ_s - δ_hi ≥ 0.12 δ_hi, so take   δ_s = 56 mm and hence the solid load is   F_s = kδ_s = 672 N.
This completes the characteristic as sketched.

The mean and alternating load components are   F_m = 450 N and   F_a = 150N, so, from ( 5b) with   S_us/S_ut = 0.63 and   S_es/S_ut = 0.13 for A229 from Table 2, and taking forces in N :

( a)       F_e =   ( 2 x 450/0.63) C K_s + ( 2 x 150/0.13) C K_h
                    =   C ( 1430 K_s + 2310 K_h )       - a function of C only, from ( 1).

In order to obtain a ball-park estimate of the wire diameter necessary, assume   C = 7.5 which lies midway in the usual recommended range. From ( a) it follows that   F_e = 7.5 ( 1430 x 1.07 + 2310 x 1.19) = 32.1 kN, so that the tensile capacity of a wire suitable for this fatigue application is   F_ut = nF_e = 1.1 x 32.1 = 35.3kN.

From Table 3, a suitable wire diameter lies somewhere between 5 and 6.3 mm; so a table of candidate solutions is prepared for solutions based on wire diameters in this range from the R20 series of AS 2338. Succeeding rows of the table assess the practicability of each candidate wire diameter by addressing :

• Calculation of the maximum spring index via ( a) corresponding to the selected safety factor (1.1 in fatigue here), with subsequent checks for spring overall geometry - candidates A and E do not meet the present geometric constraints, so these candidates are dropped.
• Calculation of the corresponding number of turns necessary for the target spring stiffness (12 N/mm here).
• Verification that the candidate is close-coiled ( helix angle α ≤ 12^o ) - all remaining candidates conform here.
• Determination of whether the candidate is absolutely stable, conditionally stable or unstable. Candidate B is unstable here, so is passed over.
• Computation of the candidate's fundamental natural frequency which ideally should be at least 12 times the running frequency. In the present case, the ratio of natural / running frequencies for the two remaining candidates is over 14, so neither candidate should lead to resonance problems.
• Assessing whether the candidate is likely to yield if solidified during assembly. The shear stress of both remaining candidates when solid is much less than their shear yield, so yielding is unlikely - though the effect of tolerance on both wire and coil diameters would have to be taken into account in a real design.