Continuous System Simulation

Chapter 04(Continuous System simulation)

4-1:: Continuous System Models::

q  Question: How are continuous system simulated? Why the representation of time is not mandatory in simulating continuous systems? (’98)

Answer:

Answer: (1^st part):In a continuous system, the predominate activities of the system cause smooth changes in the attributes of the system entities. When such a system is modeled mathematically, the variables of the model representing the attributes are controlled by continuous functions. The model can be described through linear algebraic equations to describe how the attributes of the system were related to each other. More generally, we can represent continuous systems through differential equations, where the relationships describe the rates at which attributes change.

The simple differential equation models have one or more linear differential equations with constant coefficients. When it needs much labor to solve the model analytically, then simulation is preferable.

However, when non-linearities are introduced into the model, it frequently becomes impossible to, at least, very difficult to solve the models. The methods of applying simulation to continuous models can therefore be developed by showing their application to models where the differential equations are linear and have constant coefficients, and then generalizing to more complex equations.

(2^nd  part): The representation of time is not mandatory in simulating continuous systems, because continuous system usually used interval-oriented/fixed-increment time advance method to update simulation clock. In this method every time clock is updated by a fixed predefined time units.

Question: How can we model the continuous systems mathematically? Why, it is necessary to solve the model through simulation rather mathematically (or analytically)?

(OR)

Question: How can we model the continuous systems? When simulation is necessary to solve this continuous model?

(OR)

Question: How can we model the continuous system mathematically? (’01)

Answer:In a continuous system, the predominate activities of the system cause smooth changes in the attributes of the system entities. When such a system is modeled mathematically, the variables of the model representing the attributes are controlled by continuous functions. The model can be described through linear algebraic equations to describe how the attributes of the system were related to each other. More generally, we can represent continuous systems through differential equations, where the relationships describe the rates at which attributes change.

The simple differential equation models have one or more linear differential equations with constant coefficients. When it needs much labor to solve the model analytically, then simulation is preferable.

However, when non-linearities are introduced into the model, it frequently becomes impossible to, at least, very difficult to solve the models. The methods of applying simulation to continuous models can therefore be developed by showing their application to models where the differential equations are linear and have constant coefficients, and then generalizing to more complex equations.

4-2:: Differential Equations::

Question: Describe differential equations and its varieties to represent continuous system.

Answer:We can consider the following example:

A automobile wheel suspension system is a continuous system. It can represent by the following equations:

Here dependent variable x appears together with its first and second derivatives and , and that the terms involving these quantities are multiplied by constant coefficients and added. The quantity F(t) is an input to the system, depending upon the independent variable t. A linear differential equation with constant coefficients is always of this form, although derivatives of any order may enter the equation. If the dependent variable or any of its derivatives appear in any other form, such as being raised to a power, or are combined by multiplied together, the differential equation is said to be nonlinear.

When more than one independent variable occurs in a differential equation, the equation is said to be a partial differential equation. It can involve the derivatives of the same dependent variable with respect to each of the independent variables.

An example is an equation describing the flow of heat in a three dimensional body. There are four independent variables, representing the three dimensions and time, and one dependent variable, representing temperature. The general method of solving such equations numerically is to use finite differences to convert the equations into a set of ordinary (that is, non-partial) differential equations, which can be solved by the numerical methods.

Differential equations, both linear and nonlinear, occur repeatedly in scientific and engineering studies. Most physical and chemical processes involve rates of change, which require differential equations for their mathematical description. Since a differential coefficient can also represent a growth rate, continuous models can also be applied to problems of a social or economic nature where there is a need to understand the general effects of growth trends.

Question: Illustrate how differential equations can represent engineering problems with example.

Answer:We will show how differential equations describe an automobile wheel suspension system. The equation is derived from mechanical principles.

 

(Figure:1)

If we pick a point of the wheel as a reference point from which to measure the vertical displacement of the wheel, the variable x can represent the displacement of the point, taking x to be positive for an upward movement (figure: 1).

The velocity of the wheel, in the vertical direction, is the rate of change of position, which is the first differential, . The acceleration of the wheel, in the vertical direction, is the rate of change of the velocity, which is the second differential.

According to the mechanical law, acceleration of the body is proportional to the force. The coefficient of proportionality between the force and acceleration is the mass of the body, so in the case of the automobile wheel, where the mass is M and the applied fore is KF(t), the equation of motion in the absence of other forces would be as follows:

                                   

The shock absorber exerts a resisting force that depends on the velocity of the wheel: the force is zero when the wheel is at rest, and it increases as the velocity rises. If we assume the force is directly proportional to the velocity, it can be represented by

, where D is a measure of the dumping factor of the shock absorber. Similarly, the spring exerts a resisting force which depends on the extent to which it has been compressed. (Assume that x is defined so that it is zero when the spring is uncompressed.) Again, if the force is directly proportional to the compression, it can be represented by Kx, where K is a constant defining the stiffness of the spring. Since both these forces oppose the motion, they subtract from the applied force to give the following equation of motion:

                       

                                   

This is the mathematical model of automobile wheel suspension system.

Question: How are continuous system simulated? Why the representation of time is not mandatory in simulating continuous systems? (’98)

Answer: