Discontinuities

Discontinuities What is a discontinuity? A good example occurs in the modeling of a hydraulic jack. The movement of the piston is limited by the physical dimensions of the jack body. If the piston is moving within the jack body and reaches the limit of its travel, it must be brought to rest. Often devices are incorporated to ‘cushion’ the motion so that the piston is brought to rest over a finite distance and in a finite time. If no such device is incorporated, the piston will still be brought to rest over a finite distance and in a finite time due to deformation of the parts involved in the collision.

Modeling of this situation is possible and extremely useful for detailed design of hydraulic jacks. However, the transient behavior would be extremely rapid and complex. Use of a hydraulic jack model incorporating these features in general hydraulic simulation is unnecessary and would lead to unacceptable run times.

Hydraulic Jack For most general-purpose simulations of a hydraulic jack, it is better to assume that at the limits of its travel the jack comes instantaneously to rest. This is an example of a discontinuity. A discontinuity is characterized by a physical quantity jumping from one value to another instantaneously. In this example the physical quantity is the piston velocity.

Submodels incorporating discontinuities in physical quantities are quite rare. However, there are related phenomena which are more common. A simple example occurs in control system, which include saturation elements. There is an input and an output. The output is limited as follows:

A typical example is shown graphically in the figure.

Figure 1: Discontinuities

The graph of yagainst x is continuous. However, the derivative of y is discontinuous at two points. This function has a discontinuity in its first derivative. If the output from this saturation is passed to an integrator, the corresponding state variables will have a discontinuity in its second derivative. These are not artificially constructed examples but rather examples that do occur in modeling work.

Discontinuities, examples

Here are some other examples of discontinuities:

• any form of backlash,

• transition from linear to turbulent characteristics in a pipe or an orifice,

• frictional stick-slip,

• any form of dead-band,

• a valve such as a check valve that has characteristics when open which are completely different to when closed,

• any form of hysteresis,

• any sort of duty cycle involving steps or ramps,

• submodels which employ linear interpolation for empirical data.

At discontinuity points the eigenvalue spectrum and hence the overall characteristics of the equations may change. Thus equations which were non-stiff before the discontinuity might become very stiff after it.

The consequences of discontinuities in model equations on the integration methods employed will be discussed in a later section.