thumb|upright=0.6|Wall clock with a 5-rod gridiron pendulum
A gridiron pendulum was a temperature-compensated clock pendulum invented by British clockmaker John Harrison around 1726. It was used in precision clocks. In ordinary clock pendulums, the pendulum rod expands and contracts with changes in temperature. The period of the pendulum's swing depends on its length, so a pendulum clock's rate varied with changes in ambient temperature, causing inaccurate timekeeping. The gridiron pendulum consists of alternating parallel rods of two metals with different thermal expansion coefficients, such as steel and brass. The rods are connected by a frame in such a way that their different thermal expansions (or contractions) compensate for each other, so that the overall length of the pendulum, and thus its period, stays constant with temperature.
The gridiron pendulum was used during the Industrial Revolution period in pendulum clocks, particularly precision regulator clocks
:<math>\Delta L = \alpha L \Delta\theta</math> (1)
The period of oscillation <math>T</math> of the pendulum (the time interval for a right swing and a left swing) is
Although this seems like a small error, it should be kept in mind that in the 1700s and 1800s pendulum clocks were primary standards used for exacting tasks like keeping trains on schedule, and that outside big cities there were no time standards, so it was a difficult process to set a clock accurately to the correct time. A transit telescope instrument was required to observe the exact moment when the sun or a star passed overhead, then almanac tables were consulted to determine the time the clock should be set to. So clocks in rural areas typically had to run for long periods between being set. A 6.8 second per day temperature error accumulates a 21 minute error over 6 months.
Wood has a smaller CTE of 4.9 ppm per °C thus a pendulum with a wood rod will have a smaller thermal error of 0.21 sec per day per °C or 2.9 seconds per day for a 14°C seasonal change, so wood pendulum rods were often used in quality domestic clocks. The wood had to be varnished to protect it from the atmosphere as humidity could also cause changes in length.
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>
= {\sum L_\text{low} \over \sum L_\text{high</math> (3)
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<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>.
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.