3.2 The Newtonian Clockwork Universe
61
3.2 The Newtonian Clockwork Universe
3.2.1 The mechanical clock
Feedback control in ancient water clocks
Measuring the passage of time was first related to the apparent motion of
celestial bodies. A calendar is a system that gives names to hierarchical
units of time. While years, months and days have natural (physical, biological)
basis, the week cycle seems to have a cultural origin. Clocks typically measure
time intervals within a day. Sundials were calibrated based on the direction
of shadows, gradually melting wax also was able to measure the continuously
passing time. Water clocks showed the time by measuring the regular outflow
of water from a container to some scaled vessel. These outflow clocks were
called clepsydra (“water-thief”). Since the velocity of the outflow of the water
depends on the level of the water, some regulatory mechanism was necessary
to keep the level at constant value. Ktesibios revolutionized the measurement
of time when he invented a new water clock, where the flow of water was
held steady by a feedback-controlled water supply valve. The appearance of
feedback control in the ancient technologies was propagated by Otto Mayr
[335], and a mathematical analysis of the Ktesibios’ water clock (and also for
some mechanical clocks discussed later) was given in [311]. A reconstruction
of Ktesibios’ clock is shown in Fig. 3.3, while Fig. 3.4 explains the operation
of the feedback loops. The goal was to ensure that the flow of water into a
measuring vessel should happen with constant velocity, independently of the
volume of the water in the “upper” vessel.
The control system ensures that the (small) deviation form the steady flow
of the water (which actually implies the increase of the hm at constant speed)
decays exponentially to the steady state.
Mechanical clock and feedback control: the “verge-and-foliot”
escapement
In the second half of the 13th century a new technology appeared in England, France and Italy: the mechanical clock. It showed time independently of
the season, the lengths of days etc. However, initially mechanical clocks had
disadvantages as well, since they were heavy and weight-driven. Also, they
were large, expensive, and in the beginning not very accurate. The real inno-
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3 FROM THE CLOCKWORK WORLD VIEW to IRREVERSIBILITY
H
qi
G
F
hf
E
A
D
qo
B
C
hm
Fig. 3.3. Reconstruction of Ktesibios’ clock: Adapted from [311].
−
hr
+
α
qi
+
R
m
√
qo
R
hm
−
Fig. 3.4. Block diagram from Ktesibios’ clock. hr is the reference level for the
canonical float in the feed vessel; hf is the actual level in the feed vessel; the α block
represents the relation between qi , the flow from the reservoir to the feed vessel and
the difference hr − hf . hm is the level of the float connected with the indicating
element; qo is the flow from the feed vessel to the measuring vessel, depending on
R
hf . The circles denote operators, + sums its two inputs, − negates its argument,
√
integrates, m is a monotonous function of the input. Finally,
represents Torrip
celli’s law: qo = kβ hf . Modified from Figure 2 of [311], using a somewhat different
convention than in control engineering.
3.2 The Newtonian Clockwork Universe
63
vative element in all mechanical clocks is the appearance of a new regulator,
a complicated mechanism called the escapement.4
The first escapement was the verge and foliot mechanism. The foliot is a
horizontal bar with weights on either end. It sits on a vertical rod, called a
verge. The verge has pallets to engage and release the main gear. The Fig. 3.5
shows the principle of the early clock escapement.
THREAD
ADJUSTABLE
WEIGHTS
FOLIOT
CROWN WHEEL
VERGE
PALLETS
Fig. 3.5. Verge-and-foliot. Adapted from www.elytradesign.com/ari/html/mechanicalclock.htm.
Conceptually, the verge-and-foliot escapement consists of two rotating
rigid bodies, which interact with collisions, and causes the rotatory movement of the foliot to one direction, or to the opposite one, depending whether
the upper or lower palletes were hit. (The name “foliot” may be associated
4
Joseph Needham (1900–1995) a legendary British biochemist, deeply interested in
Chinese scientific history, suggested [371, 372] that the escapement was invented
in China, already in the eighth century, and could be considered as the “missing
link” in the technological evolution of clocks between waterclocks and the European mechanical clock. However, the principle behind the European escapement
based on centrifugal force of a periodically moving object, seems to be unrelated
the Chinese implementation. David Landes, in his wonderful book about the revolutionary role of clocks in making the modern world [304] labels the Chinese
contribution as a “magnificent dead end”.
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3 FROM THE CLOCKWORK WORLD VIEW to IRREVERSIBILITY
to its “foolish” motion). Though the name of the inventor is unknown, there
is a consensus that the construction was ingenious. While it was intuitively
plausible to use some smooth, continuous movement for measuring the time,
the designer of the verge-and-foliot escapement adopted an entirely different
approach. He did not try to ensure a constant speed of the falling weight.
Alternatively, the speed was increased or decreased due to the collision of the
pallet and gear tooth.
Using a somewhat different terminology, the continuous time was decomposed into discrete time intervals defined by time periods between individual
collisions between a pallet and a tooth of the crown gear. Escapement makes
the gears move forward in small discrete jumps. The measurement of time
is reduced to define a unit, and a counter to count the impacts (“ticks and
tocks”). From the point of view of dynamics, the speed of the clock depends
on the interaction of its components, and it could be considered as a velocity
feedback control system [311]. While mechanics is generally considered as a
discipline dealing with smooth, continuous motions, the clock itself is a mixture of continuous and discrete models.5 In any case, feedback control proved
to be an invisible thread in the history of technology [57].
Huygens’s pendulum clock, feedback control, other escapements
The dynamics of the verge-and-foliot was determined by the interaction of
its components. Based on Galileo’s discovery Christian Huygens (1629–1695)
realized a mechanism, where its period does not depend on interaction, but is
given (almost, but not entirely) by a pendulum. Actually it turned out that
Galileo’s discovery is valid only for small displacements of the pendulum, in
this case the approximation leads to harmonic oscillation (see Section 3.2.3),
while the basic interaction between pallets and gear teeth has been conserved.
Since the pendulum is damped, its energy loss should be compensated by this
interaction. Feedback control was necessary to establish the appropriate phase
of the energy transfer (the phase is important, just like when a child is pushed
on a swing.)
A general equation of motion for a pendulum is
I θ̈ = −M g/l sin θ − C θ̇ + T (θ).
(3.1)
Here θ is the location of the pendulum (measured as the angular displacement), I is the moment of inertia, l is the given length of the pendulum, g is
5
A more appropriate mathematical framework of modeling the motion of the verge
and foliot escapement mechanism is the so-called impulsive differential equations.
The continuous differential equation describes the motion between impulses, an
impulse equation describes the jump to the impulse [448].
3.2 The Newtonian Clockwork Universe
65
the acceleration due to gravity, C is the coefficient of friction. The three terms
of the rhs of the equation describe the restoring torque due to the gravity, the
loss due to friction, and the escapement torques, respectively [276]. To a fix
friction loss C and T (θ) drive function the θ(t), the pendulum exhibits an
oscillatory motion with constant amplitude and frequency.6
The verge and foliot escapement has been substituted by “anchor escapement” by William Clement and Robert Hooke (1635–1703), and later an improved version (deadbeat escapement) by George Graham (1673–1751). A
pendulum-driven escape engaged and released gear teeth in the same plane
resulting in a reduction in the amplitude of the oscillation, and a strongly
improved accuracy (∼ 10sec per day). Anchor escapement established, what
is called recoil. While the escape wheel turned mostly in one direction, after
impact with the lever the escape wheel pushed it backwards (recoil). Graham
eliminated this recoil by his deadbeat escapement. Modern control technology
uses the deadbeat control, which means that the system is stabilized without
overshoot. Mark Headrick 7 labels John Harrison (1693–1776), as the “most
brilliant horologist of all time”. He adopted grasshopper escapement, (the
term was given after the discovery, and characterizes the jump of the lever).
The interaction between the wheel and the lever has a minimal friction only.
Harrison, an autodidact, spent decades to build several clocks in a competition to determine longitude at sea, which was a serious problem of marine
navigation.8
Since the local times change in East-West direction, the knowledge of local
times in two different points could be used to calculate the longitude distance
between them. Sailors wanted to calculate it to navigate more precisely. The
Royal Observatory in Greenwich was built in 1675 as reference point. Harrison
spent his whole life to build a series of portable clocks with highly precise
regulators. Some of his inventions (such as bimetallic strip to compensate the
effects of temperature changes, and the caged-roller bearing to reduce friction
are still used today). By these and other inventions he proved that longitude
could be measured from a watch.
6
7
8
For a mathematical analysis of the Huygens’s clock see [55].
There is an excellent website (http://www.abbeyclock.com/escapement.html,
June 11th, 2006), Mark Headrick’s Horology Page, what I read to learn the motion
of different escapements by looking their computer simulations.
Historical writings also might have self-similar structure. Chapter nine of David
Landes’s “Revolution in time” – Clocks and the Making of the Modern World was
of my main source about Harrison, his clocks, and his fight to get the Longitude
Prize. He writes [304], pp. 150: “. . . I shall not go into the detail of this mechanism,
which interested readers can learn about consulting Gould’s classic history. . . ”
(Gould restored Harrison’s timekeepers.)