Editing Gridiron pendulum (section)

===Compensation===
A gridiron pendulum is symmetrical, with two identical linkages of suspension rods, one on each side, suspending the bob from the pivot.   Within each suspension chain, the total change in length of the pendulum <math>L</math> is equal to the sum of the changes of the rods that make it up.  It is designed so with an increase in temperature the high expansion rods on each side push the pendulum bob up, in the opposite direction to the low expansion rods which push it down, so the net change in length is the difference between these changes  
:<math>\Delta L =  \sum \Delta L_\text{low} - \sum \Delta L_\text{high}</math>
From (1) the change in length <math>\Delta L</math> of a gridiron pendulum with a temperature change <math>\Delta\theta</math> is
:<math>\Delta L =  \sum\alpha_\text{low}L_\text{low}\Delta\theta - \sum\alpha_\text{high}L_\text{high}\Delta\theta</math>
:<math>\Delta L =  (\alpha_\text{low}\sum L_\text{low} - \alpha_\text{high}\sum L_\text{high})\Delta\theta</math>
where <math>\sum L_\text{low}</math> is the sum of the lengths of all the low expansion (steel) rods and <math>\sum L_\text{high}</math> is the sum of the lengths of the high expansion rods in the suspension chain from the bob to the pivot.  The condition for zero length change with temperature is
:<math>\alpha_\text{low}\sum L_\text{low} - \alpha_\text{high}\sum L_\text{high} = 0</math>
{{Equation box 1 |indent = |cellpadding = 0 |border = 2 |border colour = black |background colour = transparent
|equation = &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math>{\alpha_\text{high} \over \alpha_\text{low}} = {\sum L_\text{low} \over \sum L_\text{high}}</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(3)&nbsp;&nbsp;&nbsp;&nbsp;
}}
In other words, the ratio of the total rod lengths must be equal to the inverse ratio of the thermal expansion coefficients of the two metals<ref name="Britannica" /><ref name="Kater"/>{{rp|p.261}}<ref name="Glasgow1">"''The total lengths should be inversely proportional to the coefficients of expansion for the metals used''" Glasgow, David (1885) [https://archive.org/details/watchclockmaking00glas/page/288/mode/2up   ''Watch and Clock Making''], Cassell and Co., London, p.289</ref><br/>
In order to calculate the length of the individual rods, this equation is solved along with equation (2) giving the total length of pendulum needed for the correct period <math>T</math>
:<math>L = \sum L_\text{low} - \sum L_\text{high} = g\big({T \over 2\pi}\big)^2</math>
Most of the precision pendulum clocks with gridirons used a '[[seconds pendulum]]', in which the period was two seconds.  The length of the seconds pendulum was <math>L =\,</math>{{convert|0.9936|meter|inches|abbr=off}}.  

In an ordinary uncompensated pendulum, which has most of its mass in the bob, the [[center of oscillation]] is near the center of the bob, so it was usually accurate enough to make the length from the pivot to the center of the bob <math>L =</math> 0.9936 m and then correct the clock's period with the adjustment nut.  But in a gridiron pendulum, the gridiron constitutes a significant part of the mass of the pendulum.  This changes the [[moment of inertia]] so the center of oscillation is somewhat higher, above the bob in the gridiron. Therefore the total length <math>L</math> of the pendulum must be somewhat longer to give the correct period. This factor is hard to calculate accurately.  Another minor factor is that if the pendulum bob is supported at bottom by a nut on the pendulum rod, as is typical, the rise in center of gravity due to thermal expansion of the bob has to be taken into account.    Clockmakers of the 19th century usually used recommended lengths for gridiron rods that had been found by master clockmakers by trial and error.<ref name="Beckett" />{{rp|p.52}}<ref name="Glasgow"/>{{rp|p.289}}