Editing Gridiron pendulum (section)

==Mathematical analysis==
===Temperature error===
All substances expand with an increase in temperature <math>\theta</math>, so uncompensated pendulum rods get longer with a temperature increase, causing the clock to slow down, and get shorter with a temperature decrease, causing the clock to speed up.  The amount depends on the [[coefficient of thermal expansion|linear coefficient of thermal expansion]] (CTE) <math>\alpha</math> of the material they are composed of. CTE is usually given in parts per million (ppm) per degree Celsius. If a rod has a length <math>L</math> at some standard temperature <math>\theta_\text{0}</math>, the length of the rod as a function of temperature is 
:<math>L(\theta) = L + \alpha L(\theta - \theta_\text{0}) = L[1 + \alpha(\theta - \theta_\text{0})]</math>
If  <math>\Delta L = L(\theta) - L</math> and <math>\Delta\theta = \theta - \theta_\text{0}</math>, the expansion or contraction of a rod of length <math>L</math> with a coefficient of expansion <math>\alpha</math> caused by a temperature change <math>\Delta\theta</math> is<ref name="Baker2005">{{cite book
  | last1  = Baker
  | first1 = Gregory L.
  | last2  = Blackburn
  | first2 = James A. 
  | title  = The Pendulum: A Case Study in Physics
  | publisher = Oxford University Press
  | date = 2005
  | location =
  | pages = 
  | language = 
  | url = https://books.google.com/books?id=t4ISDAAAQBAJ&pg=PA250
  | archive-url= 
  | archive-date=
  | doi = 
  | id = 
  | isbn = 9780198567547
  | mr = 
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  | jfm =}}</ref>{{rp|p.250,eq.10.19}}
:<math>\Delta L = \alpha L \Delta\theta</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)

The [[Frequency|period of oscillation]] <math>T</math> of the pendulum (the time interval for a right swing and a left swing) is<ref name="Baker2005" />{{rp|p.239,eq.10.2}}
:<math>T = 2\pi\sqrt{L \over g}</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)
A change in length <math>\Delta L</math> due to a temperature change <math>\Delta\theta</math> will cause a change in the period <math>\Delta T</math>.  Since the expansion coefficient is so small, the length changes due to temperature are very small, parts per million, so <math>\Delta T << T</math> and the change in period can be approximated to first order as a linear function<ref name="Baker2005" />{{rp|p.250}}
:<math>\Delta T = {dT \over dL}\Delta L</math>
:<math>\qquad =  {d \over dL}\Big( 2\pi\sqrt{L \over g}\Big)\Delta L = \pi{\Delta L \over \sqrt{gL}}</math> 
Substituting equation (1), the change in the pendulum's period caused by a change in temperature <math>\Delta\theta</math> is
:<math>\qquad = \pi{\alpha L \Delta\theta \over \sqrt{gL}} = \alpha\pi\sqrt{L \over g}\Delta\theta</math>
:<math>\Delta T = {\alpha T\Delta\theta \over 2}</math>
{{Equation box 1 |indent = |cellpadding = 0 |border = 2 |border colour = black |background colour = transparent
|equation = <math>\quad{\Delta T \over T} = {1 \over 2}\alpha\Delta\theta\quad</math>
}}
So the fractional change in an uncompensated pendulum's period is equal to one-half the coefficient of expansion times the change in temperature.

Steel has a CTE of 11.5 parts per million per °C so a pendulum with a steel rod will have a thermal error rate of 
5.7 parts per million or 0.5 seconds per day per degree Celsius (0.9 seconds per day per degree Fahrenheit).  Before 1900 most buildings were unheated, so clocks in temperate climates like Europe and North America would experience a summer/winter temperature variation of around {{convert|25|F-change|C-change|order=flip}} resulting in an error rate of 6.8 seconds per day.<ref name="Kater">{{cite book
  | last1  = Kater
  | first1 = Henry
  | last2  = Lardner
  | first2 = Dionysus
  | title = A Treatise on Mechanics
  | publisher = Carey and Lea
  | date = 1831
  | location = Philadelphia
  | pages = 
  | url = https://archive.org/details/atreatiseonmech00lardgoog/page/n276/mode/2up
  | archive-url= 
  | archive-date=
  | doi = 
  | id = 
  | isbn = 
  | mr = 
  | zbl = 
  | jfm =}}</ref>{{rp|p.259}}  

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>
{{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}}

===Five rod gridiron===
In the 5 rod gridiron, there is one high expansion rod on each side, of length <math>L_\text{2}</math>, flanked by two low expansion rods with lengths <math>L_\text{1}</math> and <math>L_\text{3}</math>, one from the pivot to support the bottom of <math>L_\text{2}</math>, the other goes from the top of <math>L_\text{2}</math> down to support the bob.<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} \over L_\text{2}}</math>
Since to fit in the frame the high expansion rod must be equal to or shorter than each of the low expansion rods  <math>L_\text{1} \ge L_\text{2}</math> and <math>L_\text{3} \ge L_\text{2}</math> the geometrical condition for construction of the gridiron is
:<math>L_\text{1} + L_\text{3} \ge 2L_\text{2}</math>
Therefore the 5 rod gridiron can only be made with metals whose expansion coefficients have a ratio greater than or equal to two<ref name="Matthys" /><ref name="Baker2005" />{{rp|p.251}}
:<math>{\alpha_\text{high} \over \alpha_\text{low}} = {L_\text{1} + L_\text{3} \over L_\text{2}} \ge 2</math>
[[Zinc]] has a CTE of <math>\alpha</math> = 26.2 ppm per °C, versus the steel value of 11.5, so the ratio of <math>\alpha_\text{high}/\alpha_\text{low}</math> = 2.28.  Thus the zinc/steel combination can be used in 5 rod pendulums.<br/>  
The compensation condition for a zinc/steel gridiron is
:<math>{L_\text{1} + L_\text{3} \over L_\text{2}} = 2.28</math>

===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>

===Definition of variables===
{| class="wikitable" 
|+ 
!Symbol ||Unit ||Definition
|-
| <math>\alpha</math>  || degree Celsius{{sup|−1}}  || Coefficient of thermal expansion of the pendulum rod 
|-
| <math>\alpha_\text{high}</math>  || degree Celsius{{sup|−1}}  || Coefficient of thermal expansion of the high expansion (brass or zinc) rods
|-
| <math>\alpha_\text{low}</math>  || degree Celsius{{sup|−1}}  || Coefficient of thermal expansion of the low expansion (steel) rods
|-
| <math>\theta</math>  || degree Celsius || Ambient temperature 
|-
| <math>\pi</math>  || none || Mathematical constant (3.14159...)
|-
| <math>g</math>  || meter×second{{sup|−2}}   || Acceleration of gravity
|-
| <math>L</math>  || meter || Length of pendulum rod from the pivot to center of gravity of the bob 
|-
| <math>\sum L_\text{high}</math>  || meter || Sum of the lengths of the high expansion gridiron rods 
|-
| <math>\sum L_\text{low}</math>  || meter || Sum of the lengths of the low expansion gridiron rods 
|-
| <math>L_\text{n}</math>  || meter || Length of the n-th gridiron rod 
|-
| <math>T</math>  || second  || Period of the pendulum (time for a complete cycle of two swings) 
|-
|}