MI*Net  
A washing machine balance problemEmail Ltd, Washing Products Division Email is using mathematical models of a spinning washing machine bowl to reduce the time it takes them to design a new machine. The market for washing machines is very competitive, which means that Email must continuously improve their products. One area of improvement readily noticed by consumers is vibration during the spin cycle. 

A ring around the rim of a washing machine bowl is used to control vibration during the spin cycle. The ring is partially filled with a liquid that moves to balance to rotating bowl. What is the ideal shape for the ring? How much fluid should it contain? 
An uneven load of wet clothes in a washing machine can lead to dangerous vibrations. Many machines use a ring around the top of the bowl to control the vibrations. The ring is partially filled with a saline solution; as the bowl spins the liquid moves to balance the bowl. Each time Email designs a new machine it must design a new balance ring. A well designed ring gives a relatively smooth motion of the machine over a wide range of unbalanced loads. But if the ring is shaped wrongly or is too full or empty it will not work properly. The present ring design process is based on experience, intuition and experimentation—which is time consuming and labour intensive. Email asked the MathematicsinIndustry Study Group 2000 workshop to help them find a better way of designing balance rings. They were after a model where measurable inputs such as spin speed, bowl diameter and depth of the bowl could translate into outputs such as the amount of solution required in the ring and the ideal ring shape. The team began by measuring the scale of the problem. Email provided a special machine with a transparent side and lid, and members of the team used it to establish an experimental program. They were able to document typical vibration behaviour, as well as measuring the impact of specific changes. One surprising result was that the same starting conditions could produced different patterns of vibration. Unbalanced loads lead to vibration because they move the centre of gravity of the bowl away from the centre of symmetry of the whole machine. Also, loads spread over different levels of the bowl tend to tilt the axis of rotation. Using a twodimensional model, the group was able to accurately predict the amplitude of the vibrations for various loads at various speeds, and then calculate the size of the ring and the mass of water needed to reduce the vibration to a specified maximum. The answer is simple—the ring should be half full. A more complex threedimensional model linked the angular momentum of rotation, the eccentricity of rotation (or deviation from a circular path), and the angle of tilt of the bowl from vertical. A system of 16 differential equations was used to describe the motion of the bowl. These equations fell into two categories:
Using this model, the group was able to show that while one balance ring around the top of the bowl could control vibration due to uneven distribution of the washing around the axis of the bowl, a second ring around the bottom of the bowl could reduce vibration due to uneven vertical distribution of the washing. Over the week, what was a complex problem, became a whole lot more complex for me, but is now described in a much better way. 
MI*Net consultant:  Bill Whiten University of Queensland 