CHAPTER 4 



Stability 



4/1. The words l stability ', ' steady state ', and ' equilibrium ' 

 are used by a variety of authors with a variety of meanings, 

 though there is always the same underlying theme. As we shall 

 be much concerned with stability and its properties, an exact 

 definition must be provided. 



The subject may be opened by a presentation of the three 

 standard elementary examples. A cube resting with one face 

 on a horizontal surface typifies ' stable ' equilibrium ; a sphere 

 resting on a horizontal surface typifies 'neutral' equilibrium; 

 and a cone balanced on its point typifies ' unstable ' equilibrium. 

 With neutral and unstable equilibria we shall have little concern, 

 but the concept of ' stable equilibrium ' will be used repeatedly. 



These three dynamic systems are restricted in their behaviour 

 by the fact that each system contains a fixed quantity of energy, 

 so that any subsequent movement must conform to this invari- 

 ance. We, however, shall be considering systems which are 

 abundantly supplied with free energy so that no such limitation 

 is imposed. Here are two examples. 



The first is the Watt's governor. A steam-engine rotates a pair 

 of weights which, as they are rotated faster, separate more widely 

 by centrifugal action; their separation controls mechanically 

 the position of the throttle; and the position of the throttle 

 controls the flow of steam to the engine. The connexions are 

 arranged so that an increase in the speed of the engine causes a 

 decrease in the flow of steam. The result is that if any transient 

 disturbance slows or accelerates the engine, the governor brings 

 the speed back to the usual value. By this return the system 

 demonstrates its stability. 



The second example is the thermostat, of which many types 

 exist. All, however, work on the same principle: a chilling of 

 the main object causes a change which in its turn causes the 

 heating to become more intense or more effective; and vice versa. 



44 



