Microcontroller based applied digital control (D. Ibrahim, 2006)

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Figure 2.7 Example mechanical system

Example 2.4

Figure 2.7 shows a mechanical system with three masses, two springs and a dashpot. A force is applied to mass m3 and a displacement is applied to spring k1. Drive an expression for the mathematical model of the system.

we can write (2.23)–(2.25) as

2.1.2 Rotational Mechanical Systems

The basic building blocks of rotational mechanical systems are the moment of inertia, torsion spring (or rotational spring) and rotary damper (Figure 2.8). The input to a rotational mechanical system may be the torque, T , and the output the rotational displacement, or angle, θ .

Figure 2.8 Rotational mechanical system components

A rotational spring is similar to a translational spring, but here the spring is twisted. The relationship between the applied torque, T , and the angle θ rotated by the spring is given by

where θ is known as the rotational stiffness constant. In our modelling we are assuming that the mass of the spring is negligible and the spring is linear.

The energy stored in a torsional spring when twisted by an angle θ is given by

A rotary damper element creates damping as it rotates. For example, when a disk rotates in a fluid we get a rotary damping effect. The relationship between the applied torque, T , and the angular velocity of the rotary damper is given by

In our modelling the mass of the rotary damper will be neglected, or will be assumed to be negligible. A rotary damper does not store energy.

Moment of inertia refers to a rotating body with a mass. When a torque is applied to a body with a moment of inertia we get an angular acceleration, and this acceleration rotates the body. The relationship between the applied torque, T , angular acceleration, a, and the moment of inertia, I , I is given by

Example 2.5

A disk of moment of inertia I is rotated (see Figure 2.9) with an applied torque of T . The disk is fixed at one end through an elastic shaft. Assuming that the shaft can be modelled with a

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Figure 2.9 Rotational mechanical system

rotational dashpot and a rotational spring, derive an equation for the mathematical model of this system.

Solution

The damper torque and spring torque oppose the applied torque. If θ is the angular displacement from the equilibrium, we can write

Example 2.6

Figure 2.10 shows a rotational mechanical system with two moments of inertia and a torque applied to each one. Derive a mathematical model for the system.

Solution

For the system shown in Figure 2.10 we can write the following equations: for disk 1,

Figure 2.10 Rotational mechanical system

2.1.2.1 Rotational Mechanical Systems with Gear-Train

Gear-train systems are very important in many mechanical engineering systems. Figure 2.11 shows a simple gear-train, consisting of two gears, each connected to two masses with moments of inertia I1 and I2. Suppose that gear 1 has n1 teeth and radius r1, and that gear 2 has n2 teeth and radius r2. In this analysis we assume that the gears have no backlash, they are rigid bodies, and the moment of inertia of the gears is assumed to be negligible.

The rotational displacement of the two gears depends on their radii and is given by the relationship

Figure 2.11 A two gear-train system

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and

It is clear from (2.52) that the moment of inertia of the load, I2, is reflected to the other side of the gear-train as n2 I2.

An example of a system coupled with a gear-train is given below.

Example 2.7

Figure 2.12 shows a rotational mechanical system coupled with a gear-train. Derive an expression for the model of the system.

Solution

Assuming that a torque T is applied to the system, we can write

Figure 2.12 Mechanical system with gear-train

2.2 ELECTRICAL SYSTEMS

The basic building blocks of electrical systems are the resistor, inductor and capacitor (Figure 2.13). The input to an electrical system may be the voltage, V , and current, i .

The relationship between the voltage across a resistor and the current through it is given by

where R is the resistance.

For an inductor, the potential difference across the inductor depends on the rate of change

Figure 2.13 Electrical system components

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The potential difference across a capacitor depends on the charge the plates hold, and is given by

The relationship between the current through the capacitor and the voltage across it is given by

The energy stored in a capacitor depends on the capacitance and the voltage across the capacitor and is given by

Electrical circuits are modelled using Kirchhoff’s laws. There are two laws: Kirchhoff’s current law and Kirchhoff’s voltage law. To apply these laws effectively, a sign convention should be employed.

Kirchhoff’s current law The sum of the currents at a node in a circuit is zero, i.e. the total current flowing into any junction in a circuit is equal to the total current leaving the junction.

Figure 2.14 shows the sign convention that can be employed when using Kirchhoff’s current law. We can write

i1 + i2 + i3 = 0

for the circuit in Figure 2.14(a),

−(i1 + i2 + i3) = 0

for the circuit in Figure 2.14(b) and

i1 + i2 − i3 = 0

for the circuit in Figure 2.14(c).

Kirchhoff’s voltage law The sum of voltages around any loop in a circuit is zero, i.e. in a circuit containing a source of electromotive force (e.m.f.), the algebraic sum of the potential drops across each circuit element is equal to the algebraic sum of the applied e.m.f.s.

Figure 2.14 Applying Kirchhoff’s current law

R

vR

vL L

vC

C

Figure 2.15 Applying Kirchhoff’s voltage law

It is important to observe the sign convention when applying Kirchhoff’s voltage law. An example circuit is given in Figure 2.15. For this circuit we can write.

vR + vL + vC = 0.

Some examples of the modelling of electrical circuits are given below.

Example 2.8

Figure 2.16 shows a simple electrical circuit consisting of a resistor, an inductor and a capacitor. A voltage Va is applied to the circuit. Derive an expression for the mathematical model for this system.

Solution

Applying Kirchhoff’s voltage law, we can write

For the capacitor we can write

i = C dvC . dt

R

C

Figure 2.16 Simple electrical circuit

Figure 2.17 Electrical circuit for the Example 2.9

Example 2.9

Figure 2.17 shows an electrical circuit consisting of a capacitor, an inductor and a resistor. The inductor and the capacitor are connected in parallel. A voltage Va is applied to the circuit. Derive a mathematical model for the system.

Solution

Applying Kirchhoff’s current law, we can write

Now, the potential difference across the inductor and also across the capacitor is vC . Similarly, the potential difference across the resistor is Va − vC . Thus,

Figure 2.18 Circuit for Example 2.18

Example 2.10

Figure 2.18 shows an electrical circuit consisting of two inductors, two resistors and a capacitor. A voltage Va is applied to the circuit. Derive an expression for the mathematical model for the circuit.

Solution

The circuit consists of two nodes and two loops. We can apply Kirchhoff’s current law to the nodes. For node 1,