Simulation and Modelling - Old Questions
3. Explain the analogy between Mechanical system and electrical system using Dynamic
Physical Model. Explain Dynamic mathematical model and static mathematical model.
Analogy between Mechanical system and electrical system
Consider the two systems shown in following figures i.e. Figure 1 and Figure 2.
Fig1: Mechanical System
Fig2: Electrical system
The Figure 1. represents a mass that is subject to an applied force F(t) varying with time, a spring whose force is proportional to its extension or contraction, and a shock absorber that exerts a damping force proportional to the velocity of the mass.It can be shown that the motion of the system is described by the following differential equation.
Where,
x is the distance moved, M is the mass, K is the stiffness of the spring & D is the damping factor of the shock absorber.
Figure 2. represents an electrical circuit with an inductance L, a resistance R, and a capacitance C, connected in series with a voltage source that varies in time according to the function E(t). If q is the charge on the capacitance, it can be shown that the behavior of the circuit is governed by the following differential equation:
Inspection of these two equations shows that they have exactly the same form and that the following equivalences occur between the quantities in the two systems:
a) Displacement x = Charge q
b) Velocity x’ = Current I, q’
c) Force F = Voltage E
d) Mass M = Inductance L
e) Damping Factor D = Resistance R
f) Spring stiffness K = Inverse of Capacitance 1/C
g) Acceleration x’’ = Rate of change of current q’’
The mechanical system and the electrical system are analogs of each other, and the performance of either can be studied with the other.
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.
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.