# On multi-metal thermostats

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## Summary

An elastic analysis of bi-metal thermostats has been made by Timoshenko. The effect of having more than two metals, on the performance characteristics of a thermostat of the same total thickness as the bi-metal one, has not received attention. The performance characteristics of a thermostat such as temperature of cut-off, the temperature of cutting-in and sensitivity depend upon the radius of curvature to which a composite strip made of different materials will bend on being uniformly heated. A general expression for radius of curvature to which a multi-metal strip will bend when heated uniformly is derived. Using this expression and expressions obtained by Timoshenko for temperatures of buckling of a metal strip contained between supports and heated uniformly, a complete analysis of a tri-metal thermostat of equal metal thickness is made. From the analysis of the latter it is concluded that a multi-metal thermostat made of materials having different coefficients of linear expansion but of nearly equal moduli of elasticity and equal thicknesses, has a greater sensitivity over a practical range of temperature difference than one made of two metals, subject to the following conditions: that it has the same total thickness as the multimetal one, that the thickness of each of the materials if equal and the difference in the coefficients of linear expansion of the metals is equal to the difference of the coefficients of linear expansion of the extreme metals of the multiple metal one. Consideration of intersurface stresses in the case of a tri-metal thermostat, indicate that they can be kept to a lower level than in the case with bi-metal strips.

## Keywords

Total Thickness Linear Expansion Metal Strip Uniform Heating Initial Deflection## Nomenclature

*h*total thickness of thermostat

*a*_{1},*a*_{2}...*a*_{n}thickness of each material

- α
_{1}, α_{2}, ... α_{n} coefficients of linear expansion of the materials (α

_{1}> α_{2}> > ... > α_{ n }).*E*_{1},*E*_{2}...*E*_{n}moduli of elasticity of the materials

*I*_{1},*I*_{2}...*I*_{n}second moments of area of the materials

*L*length between supports

- ρ
^{0} initial radius of curvature of thermostat

- ρ
radius of curvature of thermostat with the ends free or simply supported for the temperature rise of t

_{0}to*t*- δ
_{0} initial upward deflection corresponding to ρ

^{0}- δ
downward deflection corresponding to ρ

- δ
_{1} upward deflection at the cut-off temperature “

*t*” for the thermostat held between fixed supports*t*_{0}initial temperature

*t*temperature of cut-off

*t*_{1}temperature of cutting-in

*t*_{2}temperature rise for the curved strip with ends free to deflect downwards by an amount 2δ

_{0}*E**E*_{1}+*E*_{2}+ ... +*E*_{n}/*n*average modulus of elasticity*I*1 ×

*h*^{3}/ 12 second moment of area of thermostat about a horizontal line through the centroid*S**t*−*t*_{0}/*t*−*t*_{1}sensitivity- β
3δ

_{0}^{2}/*h*^{2}*P*_{E}π

^{2}*EI*/*L*^{2}*y**P/P*_{E}*x*δ

_{0}/δ_{1}*s*_{1}*a*_{1}*E*_{1}*s*_{2}*a*_{2}*E*_{2}...*s*_{n}*a*_{n}E_{n}

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## References

- 1).Timoshenko, S., J. Opt. Soc. A.,
**11**(1925) 233.ADSCrossRefGoogle Scholar - 2).Vasudevan, M. and W. Johnson, J. Roy Aeron. Soc.
**65**(1961) 507.Google Scholar - 3).Mahrenholtz, O. and Johnson, W., Inter. J. Mech. Sci. (to be published).Google Scholar