Simulation and Modelling - Old Questions
1. Describe different types of mathematical simulation models. Develop a mathematical model (differential equation) for any dynamic system.
Two types of mathematical model:
1. Static Mathematical Model:
- Static mathematical model is the mathematical model that represents the logical view of the system in equilibrium state.
- Such models are time-invariant.
- It is generally represented by the basic algebraic equations.
- E.g. An equation relating the length and weight on each side of a playground variation, supply and demand relationship model of a market and so on.
2. Dynamic Mathematical Model:
- Dynamic mathematical model is the mathematical model that accounts for the time dependent changes in the logical state of the system.
- Such models are time-variant.
- It is generally represented by differential equations or difference equations.
- E.g. The equation of motion of planets around the sun in the solar system.
The derivation may be made with an analytical solution or with a numerical computation, depending upon the complexity of the model. The equation that was derived to describe the behavior of a car wheel is an example of a dynamic mathematical model; in this case, an equation that can be solved analytically.
It is customary to write the equation in the form
Where 2 ζ ω =D/M and ω 2=K/M
Expressed in this form, solutions can be given in terms of the variable wt. Figure below shows how x varies in response to a steady force applied at time t = 0 as would occur, for instance, if a load were suddenly placed on the automobile.
Solutions are shown for several values of ζ , and it can be seen that when ζ is less than 1, the motion is oscillatory.
Fig: Solutions of second order equations
The factor ζ is called the damping ratio and, when the motion is oscillatory, the frequency of oscillation is determined from the formula:
Where f is the number of cycles per second.