Solved: The figure shows a conical compression helical

Chapter 10, Problem 10-29

(choose chapter or problem)

The figure shows a conical compression helical coil spring where \(R_1\) and \(R_2\) are the initial and final coil radii, respectively, d is the diameter of the wire, and \(N_a\) is the total number of active coils. The wire cross section primarily transmits a torsional moment, which changes with the coil radius. Let the coil radius be given by

\(R=R_1+\frac{R_2-R_1}{2 \pi N_a} \theta\)

where \(\theta\) is in radians. Use Castigliano's method to estimate the spring rate as

\(k=\frac{d^4 G}{16 N_a\left(R_2+R_1\right)\left(R_2^2+R_1^2\right)}\)