In this lecture we continue with the modeling of systems discussion. We show by more examples how mathematical models for simple engineering systems can be developed.

Example Fluid System: Water Level Control

The system is a storage tank of cross-sectional area A whose liquid level or height is h. The liquid enters the tank from the top and leaves the tank at the bottom through the valve, whose fluid resistance is R. The volume flow rate in and the volume flow rate out are qi and qo, respectively. The fluid density p is constant. Please refer to Figure 1 for a schematic diagram of the system. In such a system it is desired to regulate the water level in the tank. Assume that the variable the we can change to control the water level is qi.

Figure 1: Diagram of the fluid system components and signals.

We first identify the input and output of the system
Input: volume flow rate in, qi, output: water level, h.

In order to obtain the differential equation of the system we use the conservation of mass principle which states that

The time rate of change of fluid mass inside the tank = the mass flow rate in - mass flow rate out

where, A is the cross sectional area of the tank, g is the acceleration due gravity and R is the fluid resistance through the valve (see Table 2.2, page 35 in the text book). Equation (1) is the differential equation that describes the dynamics of our thermal system. Note that the input and output appear in this equation. If we know the input, qi then we can solve equation (1) for the output, h. A block diagram representation of the fluid system is shown in Figure 2.

Figure 2: Block diagram representation of the fluid system.

Example Mechanical System: (2 mass)-spring-damper system

This system contains two masses, a spring, and a damper (refer to Figure 3). An external force is applied to the first mass and we would like to control the position of the second mass. The external force, f indirectly affects the motion of the second mass.

Figure 3: (a) Diagram of the mechanical system components. (b) Free body diagram of the system.
Click here to see an animation for the analysis.

We first identify the input and output of the system
input: extermal force, f, output: displacement of the second mass, z.
Next we find a set of differential equations describing this system. Newton's second law is applied to each mass to obtain these differential equations.

Notice how the system dynamics relate the input and output. Remember that f is the input and z is the output. A block diagram representation of the mechanical system is shown in Figure 4.

Figure 4: Block diagram representation of the mechanical system.

Example Electrical System: RLC Circuit (Parallel Connection)

The system below shows an electrical circuit with a current source i, resistor R, inductor L, and capacitor C. All of these parts are connected in parallel. It is required to regulate the capacitor voltage V.

Figure 4: RLC circuit (parallel connection).

In the next lecture we show how a state space representation of this mechanical system can be obtained.

We first identify the input and output of the system
input: current, i, output: voltage, V.

Next we find a set of differential equation that describes this system. Kirchoff's current law is applied. The sum of the three currents (R, L, and C) is equal to the overall source current i.

In the next lecture we show how a state space representation of this mechanical system can be obtained.