Editing Gridiron pendulum (section)

===Nine rod gridiron===
[[Image:Gridiron pendulum on stand.png|thumb|upright=0.7|Demonstration 9 rod brass/steel gridiron on stand for use in education]] 
To allow the use of metals with a lower ratio of expansion coefficients, such as brass and steel, a greater proportion of the suspension length must be the high expansion metal, so a construction with more high expansion rods must be used.  In the 9 rod gridiron, there are two high expansion rods on each side, of length <math>L_\text{2}</math> and <math>L_\text{4}</math>, flanked by three low expansion rods with lengths <math>L_\text{1}</math>, <math>L_\text{3}</math> and <math>L_\text{5}</math>.<ref name="Matthys" /> So from equation (3) the condition for compensation is
:<math>{\alpha_\text{high} \over \alpha_\text{low}} = {L_\text{1} + L_\text{3} + L_\text{5} \over L_\text{2} + L_\text{4}}</math>
Since to fit in the frame each of the two high expansion rods must be as short as or shorter than each of the high expansion rods, the geometrical condition for construction is
:<math>L_\text{1} + L_\text{3} + L_\text{5}  \ge  {3 \over 2}(L_\text{2} + L_\text{2})</math>
Therefore the 9 rod gridiron can be made with metals with a ratio of thermal expansion coefficients exceeding 1.5.<ref name="Matthys" /><ref name="Baker2005" />{{rp|p.251}}
:<math>{\alpha_\text{high} \over \alpha_\text{low}} = {L_\text{1} + L_\text{3} + L_\text{5} \over L_\text{2} + L_\text{4}} \ge 1.5</math>
[[Brass]] has a CTE of around <math>\alpha</math> = 19.3 ppm per °C, giving a ratio of <math>\alpha_\text{high}/\alpha_\text{low}</math> = 1.68.  So while brass/steel cannot be used in 5 rod gridirons, it can be used in the 9 rod version.<ref name="Matthys" />
So the compensation condition for a brass/steel gridiron using brass with the above CTE is
:<math>{L_\text{1} + L_\text{3} + L_\text{5} \over L_\text{2} + L_\text{4}} = 1.68</math>