Mechanical Response of Metal Solenoids Subjected to Electric Currents

Abstract

Solenoids, ubiquitous in electrical engineering applications, are devices formed from a coil of wire that use electric current to produce a magnetic field. In contrast to typical electrical engineering applications that pertain to their magnetic field, of interest here is their use as actuators by studying their mechanical deformation. An analytically tractable model of parallel, coaxial circular rings is used to find the solenoid’s axial deformation when subjected to a combined electrical (current) and mechanical (axial force) loading. Both finite and infinite solenoids are considered and their equilibrium configurations as well as their stability are investigated as functions of their geometry and applied current intensity.

Appendices

Appendix A: Naked and Tube-Encased Coil Stiffnesses

The stiffness of the naked solenoid depends on its geometry. For the helicoidal coil the stiffness $k_{s}$ is given by (3.2). For the parallel ring naked solenoid, the stiffness of the separating ligament is

$$ \mbox{parallel ring solenoid stiffness:} \quad k_{s} = {\frac{E_{s} \pi \rho ^{2}}{\ell}} \; . $$

(A.1)

Assuming a minimum encasing tube thickness of $h=2\rho$, we conclude from the definitions in (3.1) of the dimensionless tube stiffness parameter $S$ and the aspect ratio $R$ that

$$ \mbox{helicoidal solenoid:} \quad S=8 \pi R \left ({\frac{2a}{\rho}} \right )^{3} {\frac{G_{t}}{G_{s}}} \ ; \quad \mbox{parallel rings:} \quad S = {\frac{2a}{\rho}}(1+\nu _{s}) {\frac{G_{t}}{G_{s}}} \; . $$

(A.2)

For soft polymeric tubes and metallic solenoids (typically copper) the ratio is of the order $G_{t}/G_{s} \approx 10^{-3}$ while a typical value⁹ of the ratio $\rho /2a \approx 10^{-2}$, thus justifying the range of values for $S$ used in the calculations of Sect. 3.

Appendix B: Electric Current Limitations

Finally, an interesting question can be raised on the subject of any physically imposed limits to the magnitude of the applied currents. Two physical mechanisms are here at play: the dielectric resistance of the medium in which the solenoid is encased and the ohmic heating of the solenoid wire. We bypass the first issue under the hypothesis of good electrical isolating properties of the encasing tube and address here the limitations due to the second mechanism.

Assuming a complete conversion of ohmic losses to heat in a circular section of radius $\rho$ wire of conductivity $\gamma$, mass density $m$ and specific heat $c_{p}$ we have that the maximum current $I_{\text{max}}$ allowed by an increase of temperature rate $\dot{\theta}_{\text{max}}$

$$ I_{\text{max}} = \pi \rho ^{2} (\gamma m c_{p} \dot{\theta}_{ \text{max}})^{1/2} \; . $$

(B.1)

For a coil made of copper wire (in MKSA units: $m = 8.94 \times 10^{3},\; \gamma = 5.96 \times 10^{7},\; c_{p}= 0.385 \times 10^{3}$) of radius $\rho = 10^{-3}$ (the dimensions considered in Sect. 3.2) we have the following maximum current

$$ I_{\text{max}} \approx 45 (\dot{\theta}_{\text{max}})^{1/2}\; \text{A} \; . $$

(B.2)

Consequently assuming $\dot{\theta}_{\text{max}} = 1^{\circ}/$sec the maximum current can be $I_{\text{max}} \approx 45$ A while for $\dot{\theta}_{\text{max}} = (10^{-2})^{\circ}/$sec the corresponding $I_{\text{max}} \approx 4.5$ A, thus giving an estimate of how fast the three different solenoids of Sect. 3.2 are heating.