Isochronism, The Balance Wheel, and the Balance Spring
I have been diving into the design aspects of the balance wheel and other elements of a watch movement from a very fundamental perspective. The first couple of blogs have set the groundwork for representing spinning objects in a mathematical model. I like to think of this as our watch avatar in space, and the mathematics allows us to find the most accurate re[presentation before we test it in real life.
For accurate timekeeping, the most crucial aspect of the watch is the balance wheel. The oscillation of the balance wheel, back and forth, creates the beats that set the rhythm for the whole watch. That is why we are starting here. It is also the most challenging part to model mathematically, or I think it is.
How To Describe The Balance Wheel For The Engineer
In order to model the balance wheel accurately, it is vital to understand what function it performs and how it achieves it. We know that it provides a regular beat from which the time can be calculated. For an engineer, though, this is not sufficient, and we need to understand what physical criteria the movement relies on to deliver accurate timekeeping.
A quick jump back in history, and let’s have a look at a grandfather clock. A couple of centuries ago, clocks relied on a pendulum that swung back and forth to measure the passage of time. The timing of these pendulums was governed by the action of gravity. It slows as it moves up the arc and then accelerates from the top of the arc to the neutral position.
What Needs To Be Balanced?
The balance wheel works differently; rather than rely on gravity, the balance wheel is designed to operate independently of gravity. The balance wheel works in conjunction with the balance spring, but what exactly is in balance?
All springs follow Hooke’s law. In the case of the balance spring, the force the spring exerts is proportional to the angular displacement from its neutral spring’s neutral position. The balance wheel and the balance spring set up opposing forces that ensure the balance wheel oscillates. As the balance wheel accelerates and imparts an angular displacement on the balance spring that will retard the acceleration of the balance wheel. Also, once the force of the spring stops the movement of the balance wheel, the balance spring will start to accelerate the balance wheel back to its neutral position.
Balance = Isochronism
So what balances? The balance wheel and the balance spring are tuned so that, regardless of the angular displacement of the balance wheel, the period, or time taken for an oscillation, is constant. In watchmaking, this is called “isochronism.” Stated more formally, the period of oscillation is independent of the oscillation amplitude.
Isochronism is fundamental in a watch for two key reasons. First, the mainspring cannot provide a constant driving force over its entire working range. Certainly, the modern mainsprings provide very constant force, but not perfectly stable. Second, there will always be variations due to friction in the drive train. This phenomenon is because as the watch ages and the lubricating oils age, and thus the movement’s friction changes. That is why it is crucial to have the balance wheel and balance spring operating in its isochronous range.
How Is Isochronous Oscillation Achieved?
So what we do know about the balance wheel and the balance spring that we can use to define the motion of our balance wheel avatar.
- The movement of the balance wheel must be isochronous. So regardless of the amplitude of the movement of the balance wheel, the period must be constant.
- We know that regardless of what is happening in the balance wheel, the force applied by the balance spring on the balance wheel is proportional to the angular displacement.
- Finally, if we ignore friction, the energy in the system is constant. An alternative way to state this is that the energy supplied through the pallet fork to the balance wheel compensates for friction losses. Therefore, at total amplitude displacement, the spring’s potential energy is equal to the kinetic energy of the balance wheel when the spring is in its neutral position.
These three specific aspects of the balance wheel and balance spring system will allow us to create a mathematical model. There is still the challenge of making the spring and wheel work in an isochronous manner, which is not a trivial problem.
How To Make It Isochronous
The problem was first solved through empirical investigations. It was not until 1861 that M. Phillips published a theoretical treatment of designing a spring for a wheel so that it would be isochronous. He proved that a spring with a center of gravity coincident with the wheel’s axis would be isochronous.
For our purposes, we will assume that our spring is isochronous, so the center of gravity of both the balance wheel and the balance spring are coincident. Explaining how the spring can have its center of gravity coincident with the center of gravity of the axis of the balance wheel would take an entire blog, or perhaps two. So I will set that aside for a while and look at the practical ramifications.
Practical Application Of The Balance Spring
The most common method of obtaining isochronism in a balance spring is by placing part of the outermost coil of the mainspring in a different plane to the rest of the mainspring. This style is called the Breguet overcoil. There are two common styles of overcoil, the gradual overcoil, and the Z-Bend. As the name suggests, the gradual overcoil gradually bends the spring so that over a 180-degree arc of the spring, the out end will gradually rise to a higher plane.
The Z-Bend is more dramatic. Two 45 degree bends are introduced into the balance spring over a distance of about three turns. The Z-Bend is only done for aesthetic reasons and is far more challenging to execute accurately and achieve isochronism, so it is less popular.
The Modern Implementation
The development has moved forward for many modern watches, driven mainly by economics. The “dogleg” method is a simpler but slightly less accurate implementation of the overcoil and thus a bit cheaper. Instead of moving the end of the spring vertically up or down to a different plane, in the dogleg method, the end is bent out further away but in the same plane as the balance spring. This keeps the end of the spring out of the way of the rest of the spring.
These are the key elements of the balance wheel and the balance spring. It provides us with sufficient understanding to describe our balance wheel/balance-spring system as a mathematical model or avatar. But please do not think this is the end of the challenge to create an accurate timekeeper. There are more layers to unpack when considering the balance spring and the balance wheel. What should the balance wheel be made of? What should the balance spring be made of? How does each element react with an increase or decrease in temperature?
The answer to these questions is essential for the movement’s accuracy but is for another time.