Editing Pendulum (section)

== Accuracy of pendulums as timekeepers ==
The timekeeping elements in all clocks, which include pendulums, [[balance wheel]]s, the [[crystal oscillator|quartz crystals]] used in [[quartz watch]]es, and even the vibrating atoms in [[atomic clock]]s, are in physics called [[harmonic oscillator]]s. The reason harmonic oscillators are used in clocks is that they vibrate or oscillate at a specific [[resonant frequency]] or period and resist oscillating at other rates. However, the resonant frequency is not infinitely 'sharp'.  Around the resonant frequency there is a narrow natural band of [[Frequency|frequencies]] (or periods), called the [[resonance width]] or [[Bandwidth (signal processing)|bandwidth]], where the harmonic oscillator will oscillate.<ref>{{cite web
  | title = Resonance Width
  | website = Glossary
  | publisher = Time and Frequency Division, US National Institute of Standards and Technology
  | year = 2009
  | url = http://tf.nist.gov/general/enc-re.htm
  | access-date = 2009-02-21
  | archive-url = https://web.archive.org/web/20090130083728/http://tf.nist.gov/general/enc-re.htm
  | archive-date = 2009-01-30
  | url-status = dead
  }}</ref><ref name="Jespersen">{{cite book
  | last = Jespersen
  | first = James
  |author2=Fitz-Randolph, Jane |author3=Robb, John
  | title = From Sundials to Atomic Clocks: Understanding Time and Frequency
  | publisher = Courier Dover
  | year = 1999
  | location = New York
  | pages = 41–50
  | url = https://books.google.com/books?id=Z7chuo4ebUAC&q=clock+resonance+pendulum&pg=PA42
  | isbn =978-0-486-40913-9 }} p.39</ref>  In a clock, the actual frequency of the pendulum may vary randomly within this resonance width in response to disturbances, but at frequencies outside this band, the clock will not function at all. The resonance width is determined by the [[damping]], the [[friction|frictional]] energy loss per swing of the pendulum.

=== ''Q'' factor ===
[[File:Shortt Synchronome free pendulum clock.jpg|thumb|250px|A [[Shortt-synchronome clock|Shortt-Synchronome free pendulum clock]], the most accurate pendulum clock ever made, at the [[NIST]] museum, [[Gaithersburg, MD]], USA. It kept time with two synchronized pendulums. The master pendulum in the vacuum tank ''(left)'' swung free of virtually any disturbance, and controlled the slave pendulum in the clock case ''(right)'' which performed the impulsing and timekeeping tasks. Its accuracy was about a second per year.]]

The measure of a harmonic oscillator's resistance to disturbances to its oscillation period is a dimensionless parameter called the [[Q factor|''Q'' factor]] equal to the resonant frequency divided by the [[resonance width]].<ref name="Jespersen" /><ref>{{cite book
  | last = Matthys
  | first = Robert J.
  | title = Accurate Pendulum Clocks
  | publisher = Oxford Univ. Press
  | year = 2004
  | location = UK
  | pages = 27–36
  | url = https://books.google.com/books?id=Lx0v2dhnZo8C&pg=PA27
  | isbn = 978-0-19-852971-2}} has an excellent comprehensive discussion of the controversy over the applicability of ''Q'' to the accuracy of pendulums.</ref>  The higher the ''Q'', the smaller the resonance width, and the more constant the frequency or period of the oscillator for a given disturbance.<ref>{{cite web
  | title = Quality Factor, Q
  | website = Glossary
  | publisher = Time and Frequency Division, US National Institute of Standards and Technology
  | year = 2009
  | url = http://tf.nist.gov/general/enc-q.htm
  | access-date = 2009-02-21
  | archive-url = https://web.archive.org/web/20080504160852/http://tf.nist.gov/general/enc-q.htm
  | archive-date = 2008-05-04
  | url-status = dead
  }}</ref>  The reciprocal of the Q is roughly proportional to the limiting accuracy achievable by a harmonic oscillator as a time standard.<ref>[https://books.google.com/books?id=Lx0v2dhnZo8C&pg=PA32 Matthys, 2004, p.32, fig. 7.2 and text]</ref>

The ''Q'' is related to how long it takes for the oscillations of an oscillator to die out. The [[Q factor|''Q'']] of a pendulum can be measured by counting the number of oscillations it takes for the amplitude of the pendulum's swing to decay to 1/''e'' = 36.8%  of its initial swing, and multiplying by ''π''.

In a clock, the pendulum must receive pushes from the clock's [[movement (clockwork)|movement]] to keep it swinging, to replace the energy the pendulum loses to friction. These pushes, applied by a mechanism called the [[escapement]], are the main source of disturbance to the pendulum's motion. The ''Q'' is equal to 2''π'' times the energy stored in the pendulum, divided by the energy lost to friction during each oscillation period, which is the same as the energy added by the escapement each period. It can be seen that the smaller the fraction of the pendulum's energy that is lost to friction, the less energy needs to be added, the less the disturbance from the escapement, the more 'independent' the pendulum is of the clock's mechanism, and the more constant its period is. The ''Q'' of a pendulum is given by:
<math display="block">Q = \frac{M \omega}{\Gamma} </math>
where ''M'' is the mass of the bob,  {{math|1=''ω'' = 2''π''/''T''}} is the pendulum's radian frequency of oscillation, and Γ is the frictional [[Damping ratio|damping force]] on the pendulum per unit velocity.

''ω'' is fixed by the pendulum's period, and ''M'' is limited by the load capacity and rigidity of the suspension. So the ''Q'' of clock pendulums is increased by minimizing frictional losses (Γ). Precision pendulums are suspended on low friction pivots consisting of triangular shaped 'knife' edges resting on agate plates. Around 99% of the energy loss in a freeswinging pendulum is due to air friction, so mounting a pendulum in a vacuum tank can increase the ''Q'', and thus the accuracy, by a factor of 100.<ref>[https://books.google.com/books?id=Lx0v2dhnZo8C&pg=PA81 Matthys, 2004, p.81]</ref>

The ''Q'' of pendulums ranges from several thousand in an ordinary clock to several hundred thousand for precision regulator pendulums swinging in vacuum.<ref name="Orologeria">{{cite web
  | title = Q, Quality Factor
  | website = Watch and clock magazine
  | publisher = Orologeria Lamberlin website
  | url = http://www.orologeria.com/english/magazine/magazine4.htm
  | access-date = 2009-02-21}}</ref>  A quality home pendulum clock might have a ''Q'' of 10,000 and an accuracy of 10 seconds per month. The most accurate commercially produced pendulum clock was the [[Shortt-synchronome clock|Shortt-Synchronome free pendulum clock]], invented in 1921.<ref name="Marrison" /><ref name="Jones2000" /><ref>Milham 1945, p.615</ref><ref>{{cite web
  | title = The Reifler and Shortt clocks
  | publisher = JagAir Institute of Time and Technology
  | url = http://www.clockvault.com/heritage/index.htm
  | access-date = 2009-12-29}}</ref><ref name="Betts">{{cite web
  | last = Betts
  | first = Jonathan
  | title = Expert's Statement, Case 6 (2008-09) William Hamilton Shortt regulator
  | website = Export licensing hearing, Reviewing Committee on the Export of Works of Art and Objects of Cultural Interest
  | publisher = UK Museums, Libraries, and Archives Council
  | date = May 22, 2008
  | url = http://www.mla.gov.uk/what/cultural/export/reviewing_cttee/~/media/Files/word/2009/RCEWA/Cases%202008-09/Case%206%202008-09%20Regulator/internet%20experts%20statement%20shortt.ashx
  | format = DOC
  | access-date = 2009-12-29
  | url-status = dead
  | archive-url = https://web.archive.org/web/20091025180404/http://www.mla.gov.uk/what/cultural/export/reviewing_cttee/~/media/Files/word/2009/RCEWA/Cases%202008-09/Case%206%202008-09%20Regulator/internet%20experts%20statement%20shortt.ashx
  | archive-date = October 25, 2009
  }}</ref>  Its [[Invar]] master pendulum swinging in a vacuum tank had a ''Q'' of 110,000<ref name="Orologeria" /> and an error rate of around a second per year.<ref name="Jones2000" />

Their Q of 10<sup>3</sup>–10<sup>5</sup>  is one reason why pendulums are more accurate timekeepers than the [[balance wheel]]s in watches, with ''Q'' around 100–300, but less accurate than the [[Crystal oscillator|quartz crystals]] in [[quartz clock]]s, with ''Q'' of 10<sup>5</sup>–10<sup>6</sup>.<ref name="Marrison" /><ref name="Orologeria" />

=== Escapement ===
Pendulums (unlike, for example, quartz crystals) have a low enough ''Q'' that the disturbance caused by the impulses to keep them moving is generally the limiting factor on their timekeeping accuracy. Therefore, the design of the [[escapement]], the mechanism that provides these impulses, has a large effect on the accuracy of a clock pendulum. If the impulses given to the pendulum by the escapement each swing could be exactly identical, the response of the pendulum would be identical, and its period would be constant. However, this is not achievable; unavoidable random fluctuations in the force due to friction of the clock's pallets, lubrication variations, and changes in the torque provided by the clock's power source as it runs down, mean that the force of the impulse applied by the escapement varies.

If these variations in the escapement's force cause changes in the pendulum's width of swing (amplitude), this will cause corresponding slight changes in the period, since (as discussed at top) a pendulum with a finite swing is not quite isochronous. Therefore, the goal of traditional escapement design is to apply the force with the proper profile, and at the correct point in the pendulum's cycle, so force variations have no effect on the pendulum's amplitude. This is called an ''isochronous escapement''.

=== The Airy condition ===
Clockmakers had known for centuries that the disturbing effect of the escapement's drive force on the period of a pendulum is smallest if given as a short impulse as the pendulum passes through its bottom [[Equilibrium point|equilibrium position]].<ref name="Marrison" />  If the impulse occurs before the pendulum reaches bottom, during the downward swing, it will have the effect of shortening the pendulum's natural period, so an increase in drive force will decrease the period.  If the impulse occurs after the pendulum reaches bottom, during the upswing, it will lengthen the period, so an increase in drive force will increase the pendulum's period.  In 1826 British astronomer [[George Airy]] proved this; specifically, he proved that if a pendulum is driven by an impulse that is [[Symmetry (mathematics)|symmetrical]] about its bottom equilibrium position, the pendulum's period will be unaffected by changes in the drive force.<ref>{{cite journal
|last=Airy
|first=George Biddle
|date=November 26, 1826
|title=On the Disturbances of Pendulums and Balances and on the Theory of Escapements
|journal=Transactions of the Cambridge Philosophical Society
|page=105
|url=https://books.google.com/books?id=xQEBAAAAYAAJ&pg=PA105
|access-date=2008-04-25
|volume=3 (Part 1)}}</ref>  The most accurate escapements, such as the [[Deadbeat escapement|deadbeat]], approximately satisfy this condition.<ref>[https://books.google.com/books?id=OvQ3AAAAMAAJ&pg=PA75 Beckett 1874], p.75-79</ref>