DonCorson
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WatchTech 101.1 -- Motive Power
WatchTech 101.1 - Motive Power
A
mechanical wristwatch has many different functional groups that are
employed to fulfill different needs such as setting the time and keeping
the movement clean and dry. The first major need is one that is often
completely forgotten; it is the supply of motive power. You wind your
watch once a day, let's say, but it runs continuously. The mission of
the main spring is to store the power to run your watch.
In a mechanical watch the power is
stored in a steel spring. Springs work through the capability of many
materials, particularly metals, to reversibly change their form when a
force is applied. Reversibly means that when the force is removed the
spring will go back to its initial shape. Each material has its own
particular characteristics concerning how easy it is to deform, known as
the elastic module (E), and how much stress it will take before it will
no longer return to the original shape, known as the elastic limit (?max). These two characteristics are very important in determining the optimum size and shape for a main spring.
Spring
calculation is resumed as finding the best size and form for a barrel
spring to be able to store the energy needed (let's say, the maximum
energy possible in the space available for a long autonomy of the watch)
while taking into account such other things as the number of turns of
the barrel between fully wound and fully unwound and the maximum and
minimum force produced.
The force of a simple straight spring is
given by Hookes Law, named after the 17th century English physicist who
first formulated it. Simply stated Hookes law says that the force of a
spring is proportional to the deflection multiplied by a constant factor
depending on the material characteristics and the physical form of the
spring.
F=-k·d
F=Force
d=displacement
k=the spring constant
The
more you bend a spring the more it pushes back. This law is a
reasonable approximation for many materials as long as the elastic limit
is not exceeded. The spring constant depends on the material and the
physical shape of the spring.
Let's try and get an idea of what
is happening inside a spring as it is bent. In the diagram below we
can see a portion of a main spring in its relaxed state and when it is
bent to a smaller diameter. The length of the middle fiber of the
spring is the same (?0·r0=?1·r1).
When bent the length at the inner and outer surfaces changes. The
middle fiber length remains the same, but as we move more and more to
the surface the fibers get stretched (outside) or compressed (inside)
more and more with the maximum change in dimension at the surface of the
spring. This change in length of a fiber is called the strain of the
fiber. The spring can be bent up to the elastic limit of the material.
The elastic limit is the strain after which the fiber will not return
to its original form. While we will bend the spring up to the elastic
limit in order to store that maximum amount of energy, we will try to
remain below the elastic limit of the spring as otherwise the spring
will remain bent and not store the energy we need.
The moment of a barrel spring is given by the following formula:
using the characteristics of the spring:
E=elastic module
h=height
e=thickness
?=angle of winding
L=length.
As you can see the element with the greatest influence is the thickness of the spring as it goes into the calculation cubed.
The energy stored in the barrel spring is given by the following formula:
?max=elastic limit of the material
Here
we see that the energy stored is proportional to e·h·L which is nothing
other than the volume of the spring. There is no way around it; to
store more energy the volume of the spring will have to be greater, but
we also see here the possibilities that new materials can offer. If we
find a material with a higher elastic limit this will greatly increase
the stored energy as that goes into the calculation squared. Thus the
high interest in such materials as glassy metals at the moment. These
are metals that are solidified under special conditions to have an
amorphous structure which gives them very special properties. The first
use of glassy metals in watchmaking (that I know of), however, is in
the applied numbers and indexes on some dials because of their wonderful
properties for polishing, not because they make good springs.
But back to springs; how does the returning force of the spring turn the barrel anyway?
Let's
look at the spring in a barrel when it is fully wound. As we can see
in the picture below the spring is tightly wound around the axle in the
middle of the barrel. We can count that the spring is wound around the
axle 22 times.
Partially wound
When the spring is completely run
down it fills the space at the outer rim of the barrel. As the outer
rim has a greater diameter the spring needs less turns for its entire
length, in the picture below we can count 15 turns. This difference
22-15=7 is the number of turns that the barrel makes when unwinding.
As we have decided to not exceed the
elastic limit of the spring material the force increases linearly as
the barrel is wound until it reaches its maximum when the spring is
tightly wound.
Below is a graph of the torque curve of a barrel
spring. This is the torque curve of a 2824-2 barrel that I measured
when starting out my tourbillon project. At 7.2 turns of the barrel we
see the maximum torque. As this is a spring for an automatic watch it
has a bridle that slips when the spring is fully wound and one continues
winding. Main springs are usually identified by their force when
fully wound. The mainspring for an ETA 2824-2 is, for example, a 1300
g·mm (gram · millimeter) spring.
One needs a little more energy to wind a
spring up than it can deliver when unwinding. This is because of the
barrel losses. The loss of the barrel is mostly through friction; the
friction of the coils rubbing against each other. To reduce this modern
springs are lightly coated with teflon or similar materials. In
earlier times springs were oiled, but this has the disadvantage that
with the aging of the oils the coils can stick together, effectively
reducing the useful length of the spring. If the coils of the spring
are not perfectly flat they can also rub on the top and bottom of the
barrel.
Graphically the energy stored in the barrel is the area
below the curve between the wound and unwound state. For a barrel like
the 2824-2 that makes about 0.3 Joule stored in the volume of the barrel
or 1530 joules/liter.
Let's compare that to the energy stored in
gasoline. Gasoline contains about 9'700 watt hours per liter which
makes 35Mega joules/liter. We would only need about 116’666’666 watch
springs to store the same energy as one liter of gasoline.
On the
other hand the watch will run for about 40 hours on those 0.3 J of
energy. Let's compare that with a car, for example. Assuming the
average consumption of my Elise which is 6.3 liters/100km, the energy of
the main spring is enough to run the car for 0.14 mm. Say that
slowly, fourteen hundredths of a millimeter. A mechanical watch is
really quite efficient, isn’t it?
Next installment; why do mainsprings have such a strange form when they are taken out of the barrel?
Don
References:
For a more in depth treatment of these topics I am aware of two sources:
Traité de construction horlogère
Michel Vermot, Philippe Bovay, Damien Prongué, Sébastien Dordor, Vincent Beux
Presses polytechniques et universitaires romandes, Lausanne
2011
ISBN: 978-2-88074-883-8
1120 pages
Théorie general de l’horlogerie
Léopold Defossez
La Chambre suisse de la horlogerie, La Chaux-de-Fonds
1950
These references are both in French. If you know of others and in other languages, please let me know.
This message has been edited by DonCorson on 2012-09-04 01:34:49
WatchTech 101.1 -- Motive Power
Yes ! ...
I wish I'd known these springs can take off the tip of a finger
Spectacular! Thank you.
Thank you, Don.
According to Omega
