By imposing certain restrictions on the shape of a conically wound spring, it is valid to assume that the extension under a load is due entirely to torsion. It is found that the extension is then given to an approximation by the expression: $Z=\frac{2Mg{R}^{2}L}{3\pi n{r_1}^4}$ $\mathrm{Mg}$ is the load, $R$ is the radius of the end coil, $L$ and $r_1$ are respectively the length and radius of the wire of which the spring is made, and $n$ is its rigidity modulus. For a truncated conically wound spring, this expression becomes: $Z=\frac{2MgN(R+r)(R^2+Rr+r^2)}{3n{r_1}^4}$ $N$ is the number of coils of the spring, and $r$ is the radius of the small end coil. By making certain assumptions the period is shown to be: $T=2\mathrm{}\pi {\left(\left(1+\frac{\mathrm{xm}}{M}\right)\frac{z}{g}\right)}^{\frac{1}{2}}$ $m$ is the mass of the spring, and $X$ is a function of the ratio of the radii of the two end coils.

Close

PrevNext

View figure in article