Balancing The Balance Wheel Equals Accuracy
It has been a winding road to get here. We have looked at the history of the balance wheel and the early days of the escapement and balance wheel. There has been plenty of mathematics too, looking at how we can create a model (or should I say avatar) for the mechanical movements. We looked at the energy in the spring and also the energy in the balance wheel as it spins back and forth. Before all that, we had to dive into how we can define the rotational movement of a body in the mathematical world. At the end of the day, everything in the back of your watch rotates, so this is very important to understand. All of this is to construct a model of the isochronous movement of the beating heart of every mechanical watch.
Today I am going to bring all this together so that we have a model we can use to design a balance wheel and balance spring system that will keep good time. I will use a few of the mathematical equations that we have created through the previous blogs and I hope this helps to illustrate how the balance spring and balance wheel work together. So let’s get started.
The Balance System
Up to this point, I have spoken of the balance spring and the balance wheel as separate items and analyzed them as individual units. The fact is that they work together, the balance wheel is no use without the balance spring and vice versa. This is a system at work and more specifically the balance wheel and balance spring together is described, in engineering terms, as an undamped mass-spring system. It is this system we are seeking to model today.
To start with the challenge is to link the motion of the balance spring to the motion of the balance wheel. If we go back to the earlier post The Energy In A Spring – A Watchmakers Guide we were able to calculate the potential energy in a coiled spring as:
Where A is the modulus of elasticity for the spring and Θ is the angle through which the spring is rotated.
Then in the blog post about calculating the energy in the balance wheel, I did just that, derive an equation to calculate the energy in a rotating disc. The kinetic energy in a rotating disc can be calculated by the equation
Where m is the mass of the rotating disc, R is the radius of the rotating disc and ω is the rotational velocity of the disc.
Where Is the Energy?
So how can these two equations be linked together and provide us with useful insight into the design of the balance wheel and balance spring system? The best place to start is to consider the movement of the balance spring and the balance wheel and consider what is going on at various points through the cycle of the balance wheel speeding up and slowing down.
When the balance wheel is stationary all the energy of the system (both the balance wheel and the balance spring) is held in the spring as potential energy. We know this is true as if the rotational velocity of the balance wheel is zero then from the equation above, the kinetic energy is zero.
The Energy Is Somewhere
Second, when the balance spring is at its neutral position all the energy of the system is in the balance wheel as kinetic energy from its spinning. This is evident from the equation above as for there to be potential energy to be stored there has to be a displacement of the balance spring.
If the complete system is considered then at any given time, the total energy of the system is the sum of the kinetic energy in the balance wheel plus the potential energy stored in the spring. We can write this as follows:
This equation is correct instantaneously as the balance wheel and the balance spring are interacting as a single system, with some assumptions. These assumptions are that there are no losses in the system due to air resistance as the balance wheel rotates, friction in the arbor of the balance wheel is negligible and hysteresis in the contraction and relaxation of the spring is also negligible. For our purposes, this is correct, but in reality, energy is being lost to all these inefficiencies. These losses are compensated for in the real world by energy from the mainspring being applied to the balance wheel through the impulse pin.
The Balance Equation
If we rearrange the equation a little bit we know that the potential energy in the spring when it is at maximum rotation is equal to the kinetic energy of the balance wheel when it is moving at maximum velocity, or:
This can be simplified so that all the constants are on the left-hand side of the equation and the variables (omega and theta) are on the other side of the equation. We can see the speed of the balance wheel and the maximum displacement of the balance spring are in ratio to the physical constants of the balance wheel and the spring.
This equation is particularly powerful because we now have a relationship between the physical properties of the spring and the balance wheel such as the weight, radius, and the spring constant and the motion of the balance wheel, and the displacement of the spring. I consider this the “Balance Equation” as it balances – although in the world of watches they may consider it differently!
Clocking The Oscillations
This is all very interesting but at the end of the day, we are trying to design the balance system for an isochronous beat. The isochronous frequency for a watch movement is, in engineering terms, referred to as the natural frequency of the system which in this case is the natural frequency of the balance system, a nondamped mass-spring system.
A Short Cut
At this point, I would love to take you through the derivation of how to calculate the natural frequency of a nondamped mass-spring system. The trouble is that this would require some serious mathematics so for the sanity of 99.9% of my readers I am going to just present it here.
Where the natural frequency is the angular frequency of the movement, A is the spring constant of the spring and I is the moment of inertia of the rotating mass.
So what should we do with these two equations?
If we start by considering a typical balance spring and balance wheel system, the balance wheel will rotate approximately 270 degrees for a single beat. This will allow the peak rotational speed to be calculated. Generally the rotational speed should be as high as possible so that the gyroscopic effects of the rotation will provide more balance to the rotation, it will also reduce the effect of gravity on the balance wheel which will all improve the accuracy.
But What About Keeping Time?
On a watch movement you typically see that the number of “beats per hour” is indicated. A common number is 28,800 beats per hour and this is 8 beats per second or 4 complete cycles which is equivalent to 4 Hertz. This number can vary, the higher the beats per second the higher the accuracy of the watch, and the smaller the time period that can be measured in general, but there are limits. When stopwatches were all mechanical increasing the beats per hour was important so that timing could be more accurate.
If we are designing for the balance system to have a natural frequency of 4Hz (28,800 beats per hour), we can represent this as an angular frequency 8π (4Hz x 2π). The inertia of the balance wheel can be calculated so that it is the correct size for a given spring or vice versa.
From these two equations, we can determine the physical properties of all the components to achieve the desired design, calculate the maximum amplitude and the maximum velocity of the balance wheel.
It is now over to the material engineer to work out how to make a spring of amazing precision and ensure that temperature variations do not affect the accuracy of the balance system.
