A helical compression spring is wound using $2\mathrm{.}75-$mm$-$diameter High Density $($HD$)$ Carbon Steel A$\mathrm{.}227$ wire material. The material

is $($cold$)$ hard$-$drawn. The spring has an inner diameter of $32$ mm with plain ground ends, and $15$ total coils. The spring operates between a preload

and maximum operating load of $5$ N and $45$ N $.$ The length of the spring at the maximum operating load is $52\mathrm{.}5$ mm $.$ Assume a clash allowance of $15$

$\%,$ and linear operating range. Further, assumed material is peened for fatigue calculations $($with Zimmerli data and the modified Goodman criterion$).$

a$.$ Estimate total number of coils, solid length, force needed to bring spring to its solid length, pitch, and free length.

b$.$ Does the spring yield statically? Why, or why not? If it does yield statically, what processing operations or design parameters $($tensile

strength, which is a function of wire size, etc.$)$ would you modify to prevent static yielding?

c$.$ If the spring is operated between the preload and maximum operating range under fatigue conditions, obtain the factor of safety guarding

against fatigue failure. Use the modified Goodman criterion for fatigue failure.

d$.$ Using equations that will be presented in Lecture $3$ slides, obtain the spring rate. Now, using the deflections you have obtained from the

load analysis, obtain the spring rate. Is the difference between the spring rates from the two calculations acceptable $($i$.$e$.,$ within $0\mathrm{.}5\%)?$

e$.$ Is there a possibility that the spring might buckle in service? Assume both spring ends to be pivoted. We will look at critical length to prevent

buckling during Lecture $4,$ but you are welcome to consult Lecture $3$ notes for the equations involved.

f$.($ME $2020$ material$)$: If the spring were a solid column with a circular cross$-$section, material properties being the same as it were, use the

Euler buckling formula to find the critical load at which the spring will buckle: $P_{cr}=\frac{{}^{2}El}{L^2}$