Control Systems

Introduction to Control Systems

Unit 1: Mathematical Models of Different Types of Systems

Content:

S.No. Concept Page No.1 Learning Objectives 2 Mathematical Models of Different Types of Systems3 Introduction4 Mechanical Elements5 Mechanical Systems6 Electrical Systems7 Analogous Between Electrical and Mechanical Systems8 Solved Examples9 Exercises

10 Review Questions 11 Match The Following12 True or False13 Drag and Drop14 Model Questions

Title: Learning ObjectivesContent Instruction to Graphics

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By the completion of this unit the learner will able to

Understand Mathematical Modeling

Know about different types of Mechanical Elements

Know about different types of Mechanical Systems

Translational Systems Rotational Systems

Understand Electrical Systems

Know Analogous Between Electrical and Mechanical Systems

Convert Mechanical to Electrical Systems

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Concept Mathematical Models of Different Types of SystemsTitle: Introduction

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Mathematical model of a system is nothing but the mathematical representation of the physical model through the use of appropriate physical laws.

Two classes of equations with broad application in the description of systems are differential equations and difference equations

A differential equation is any algebraic equality which involves either differentials or derivatives.

Differential equations are useful for relating rates of change of variables and other parameters.

A difference equation is an algebraic equality which involves more than one value of the dependent variable (s) corresponding to more than one value of at least one of the independent variable (s).

Difference equations are useful for relating the evolution of variables (or parameters) from one discrete instant of time (or other independent variable) to another.

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Concept Mathematical Models of Different Types of SystemsTitle: Mechanical Elements

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Mechanical Elements

Mechanical systems and devices can be modeled by means of three ideal translator and three ideal rotary elements.

Translational Elements:

Newton’s second law states that the sum of applied forces is equal to the sum of opposing forces on a body.

1) The mass element

The force (F) applied to the mass (M) is proportional to the rate of change of its velocity v(t).

By Newton’s second law,

2) The spring element

Spring with one end fixed

The force (F) applied to the spring is proportional to its displacement x(t).

By Newton’s second law,

Fig. 1

Fig. 2

Fig. 3

Spring with two ends open

By Newton’s second law,

where K is spring stiffness ( Newton / m )

3) The damping element

Damper with one end fixed

The force (F) applied to the spring is proportional to the rate of change of its displacement x(t).

By Newton’s second law,

Damper with two ends open

By Newton’s second law,

where f is viscous friction coefficient ( Newton per m/sec )

where x is in m, v is in m/sec, M is in kg

Insert figures 1 ,2,3, beside the statements “1. The mass element, 2……, 3……,“ resp

Insert figures 4 , 5, 6, beside the statements “1. The inertiaelement, 2……, 3……,“ resp

Fig. 4

Fig. 5

Fig. 6

Rotational Elements:

1) The inertia element

The torque (T) applied to the rotational mass is proportional to the rate of change of its angular velocity .

By Newton’s second law,

2) The torsional spring with one end fixed

The torque (T) applied to the torsional spring is proportional to its angular displacement .

By Newton’s second law,

The torsional spring with two end open

where K is spring stiffness ( Newton / m )

3) The damping element with one end fixed

The torque (T) applied to the torsional spring is proportional to its angular velocity .

By Newton’s second law,

The damping element with two ends open

By Newton’s second law,

where f is viscous friction coefficient ( Newton per m/sec )

Mass/Inertia and the two kinds of springs are the energy storage elements where in energy can be stored without loss and so these are called conservative elements.

Energy stored in these elements is expressed as:

Mass:

Inertia:

Spring (translatory):

Spring (torsional):

Damper is a dissipative element and power it consumes (lost in form of heat) is given as

(Translational)

(rotational)

Concept Mathematical Models of Different Types of SystemsTitle: Mechanical Systems

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Translational Systems

Consider the mechanical system shown in figure.

The way of analyzing such a system is to draw a free - body diagram as shown in below figure.

By applying Newton’s law of motion to the free - body diagram, the force equation can be written as

………. (1)

Rotational Systems

Consider now, the rotational mechanical system as shown in figure.

Let T be the applied torque which tends to rotate the disc. The free - body diagram is shown in below figure.

The torque equation obtained from the free - body diagram is

…………… (2)

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Concept Mathematical Models of Different Types of SystemsTitle: Electrical Systems

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The resistor, inductor and capacitor are the three basic elements of electrical circuits.

These circuits are analyzed by the application of Kirchhoff’s voltage and current laws.

Let us analyze the RLC series circuit shown in figure.

By applying KVL

we know, electric charge

…………… (3)

Now, consider the RLC parallel circuit shown in figure.

By applying KCL

We know,

Insert fig. 1 above the sentence “ By applying KVL”

Insert fig. 2 above the sentence “ By applying KCL”

Fig. 1

Fig. 2

…………………… (4)

Concept Mathematical Models of Different Types of SystemsTitle: Analogous Between Electrical and Mechanical Systems

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The force balance equations from the mechanical system are

……………… (1)

……………… (2)

The voltage, current equations from the series and parallel RLC circuits are

……………… (3)

……………… (4)

Force (Torque) – Voltage analogy

By comparing eqn. (1) and eqn. (2) with eqn. (3), it is seen that they are of identical form. Such systems whose differential equations are of identical form are called analogous systems.

The force F (torque T) and voltage ‘e’ are analogous variables here. This is called Force (Torque) - Voltage analogy.

A list of analogous variables in this analogy is given in below table.

Note: while drawing tables 1st take electrical systems.

Reminder:

In the table write viscous….f (or B)

Force (Torque) – Current analogy

Similarly, by comparing eqn. (1) and eqn. (2) with eqn. (4), it is seen that they are of identical form.

In this case force F (torque T) and current are analogous variables. This is called the Force (Torque) - Current analogy.

A list of analogous variables in this analogy is given in below table.

Steps to obtain Electrical analogous of Mechanical Systems

1. Identify all the displacements due to the applied force. The elements spring and friction between two moving surfaces cause change in displacement.

2. Draw the equivalent mechanical system based on node basis. The elements under same displacement will get connected in parallel under that node.

3. Each displacement is represented by separate node. Elements causing change in displacement is always between the two nodes.

4. Write the algebraic equations. At any node algebraic sum of all the forces acting is equal to zero.

5. In F - V analogy, use following replacements and rewrite equations,

6. Simulate the equations using loop method. Number of displacements equal to number of loop currents.

7. In F - I analogy, use following replacements and rewrite equations,

8. Simulate the equations using node basis. Number of displacements equal to the number of node voltages.

Note: The system will be exactly same as equivalent mechanical system obtained in step2 with appropriate replacements.

Example:

1) Draw the equivalent mechanical system of the given system. Hence write the set of equilibrium equations for it and obtain electrical analogous circuits using,

i) F - V Analogy and (ii) F - I Analogy.

Insert fig. 1 below Question

Insert fig. 2 above the sentence ” At node 1”

Fig. 1

Fig. 2

Solution:

There are two displacements in the given figure.

There is no element at node 1; hence the force is directly applied to spring K1. So it will store energy and cause to change the force applied to .

Hence displacement of is x2 and as B2 and K2 are connected to fixed

supports both are under only as shown in the equivalent system.

At node 1,

At node 2,

By applying Laplace transform to the above equations, we get

…………… (1)

………… (2)

i) F - V analogy:

From equations (1) and (2)

……………. (3) Insert fig. 3 at the end of “F-V analogy”

…………… (4)

By putting in the equations (3) & (4), we get

By applying inverse Laplace transform, we get

……………… (5)

……………… (6)

The electrical circuit representing above equations is given below.

ii) F –I analogy:

From equations (1) & (2)

……………… (7)

…………… (8)

Insert fig. 4 at the end of “F-I analogy”

Fig. 3

By putting in the equations (7) & (8), we get

By applying inverse Laplace transform, we get

…………… (9)

………… (10)

The electrical circuit representing above equations is given below.

Fig. 4

Concept Mathematical Models of Different Types of SystemsTitle: Solved ExamplesContent Instruction to

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1) Consider Rotational system shown in the following figure,where J = Moment of inertia of disk, B = Friction constant, K = Torsional spring constantand disk is subjected to the torque T (t) as shown.

Draw its analogous network based on (i) T - V analogy (ii) T - I analogy.

Solution:

As K and B are with respect to fixed support, all J, K and B are under only.

Hence equivalent system is,

By applying Laplace transform,

Insert fig. 1.1 below Q1

Insert fig. 1.2 below the sentence” Hence equivalent system is,”

Fig. 1.1

Fig. 1.2

Fig. 1.3

i) Torque - Voltage Analogy:

By substituting , we get

By applying inverse Laplace transform,

The electrical network representing above equation is given below.

ii) Torque - Current Analogy:

By replacing

By applying inverse Laplace transform,

Insert fig. 1.3 Above “Torque- Current Analogy:”

Fig. 1.4

The electrical network representing above equation is given below.

2) Draw the equivalent mechanical system of the given system. Hence write the set of equilibrium equations for it and obtain electrical analogous circuits using,

i) F - V Analogy and (ii) F - I Analogy.

Solution:

The displacement of M1 is and as B1 is between M1 and

fixed support hence it is also under the influence of .

B2 changes the displacement from to as it is between two moving points. M2 and K are under the displacement .At node1,

…………… (1)

At node2,

…………… (2)

i) F - V analogy:

From equations (1) & (2)

Insert fig. 1.4 above Q2.

Insert fig. 2.1 below Q2

Insert fig. 2.2 above the sentence “ At node 1”

Fig. 2.1

Fig. 2.2

By replacing in the above equations

By applying inverse Laplace transform, we get

The electrical network representing above equation is given below.

ii) F - I analogy:

From equations (1) & (2)

Insert fig. 2.3 at the end of F-V analogy

Fig. 2.3

By replacing in the above equations

By applying inverse Laplace transform, we get

The electrical network representing above equation is given below.

Insert fig. 2.4 at the end of F-I analogy

Fig. 2.4

Concept Mathematical Models of Different Types of SystemsTitle: ExercisesContent Instruction to

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1) Draw the equivalent mechanical system of the given system and obtain electrical analogous circuit using, Force - Current Analogy.

Answer:

The equivalent mechanical system is,

Electrical analogous circuit is,

2) Draw the equivalent mechanical system of the given system. And obtain electrical analogous circuits using,

i) F - V Analogy and (ii) F - I Analogy.

Answer:

The equivalent mechanical system is,

Electrical analogous circuit is using F - V analogy is,

Insert fig. 1.1 below Q1

Insert fig. 1.2 below the sentence “The equivalent mechanical system is,”

Insert fig. 1.3 below the sentence ” Electrical analogous circuit is,”

Insert fig. 2.1 below Q2

Insert fig. 2.2 below the sentence “The equivalent mechanical system is,”

Insert fig. 2.3 below the

Fig. 1.1

Fig. 1.2

Fig. 1.3

Fig. 2.1

Electrical analogous circuit is using F - I analogy is,

3) Draw the equivalent mechanical system of the given system. And obtain electrical analogous circuits using,

i) T - V Analogy and (ii) T - I Analogy.

Answer:

The equivalent mechanical system is,

Electrical analogous circuit is using T - I analogy is,

Electrical analogous circuit is using T - V analogy is,

sentence ” Electrical analogous circuit using F-V analogy is,”

Insert fig. 2.4 below the sentence ” Electrical analogous circuit using F-I analogy is,”

Insert fig. 3.1 below Q3

Insert fig. 3.2 below the sentence “The equivalent mechanical system is,”

Insert fig. 3.3 below the sentence ” Electrical analogous circuit using T-V analogy is,”

Insert fig. 3.4 below the sentence ”

Fig. 2.2

Fig. 2.3

Fig. 2.4

Fig. 3.1

Fig. 3.2

Fig. 3.3

Fig. 3.4

Electrical analogous circuit using T-I analogy is,”

Concept Review Questions Title: Review QuestionsContent Instruction to

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1. Which of the following forces resist motion?

a) Inertia force

b) Spring force

c) Damping force

d) All of the above

2. In linear systems damping force is considered to be proportional to

a) Displacement

b) Velocity

c) Acceleration

d) Any of the above

3. The restoring force of a spring is considered proportional to

a) Displacementb) Velocityc) Accelerationd) Coulomb friction

4. Under torque- voltage analogy, moment of inertia is analogous to

a) Voltageb) Resistancec) Inductanced) Charge

5. The mechanical system is shown in the given figure.

The system is described as

a)

b)

c)

d)

6. Which one of the following represents the linear mathematical model of the physical system shown in the figure?

a)

b)

c)

d)

7. Under torque- voltage analogy, angular velocity is analogous to

a) Charge b) Currentc) Resistanced) Reciprocal of capacitance

8. Under torque- voltage analogy, Charge is analogous to

Insert fig. q5 below q5

Insert fig. q6 below q6

a) Viscous friction co- efficientb) Coulomb’s friction co- efficient c) angular velocityd) angular displacement

9. Under torque- voltage analogy, viscous friction co- efficient is analogous to

a) resistanceb) reciprocal of resistancec) capacitanced) reciprocal of capacitance

10. Under torque- voltage analogy, torsional spring stiffness is analogous to

a) capacitance b) reciprocal of capacitancec) inductanced) reciprocal of inductance

Concept Mathematical Models of Different Types of SystemsTitle: Match The Following

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Match the following:

1) Match the correct list of analogous quantities in Force - Voltage Analogy given below

Mechanical Translational System Electrical Systems

1. Force F a) Inductance L

2. Spring stiffness K b) Resistance R

3. Displacement c) Voltage e

4. Viscous friction coefficient f d) Charge q

5. Mass e) 1/C

6. Velocity f) Current i

Concept True or False Title: True or FalseContent Instruction to

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1. Under force- current analogy, capacitance is analogous to displacement

False

2. Under force- voltage analogy, mass is analogous to inductance

True

3. Under force- voltage analogy, resistance is analogous to displacement

False

4. Under force- current analogy, magnetic flux linkage is analogous to displacement

True

5. Under force- voltage analogy, velocity is analogous to current

True

6. Under force- voltage analogy, velocity is analogous toreciprocal of inductance

True

Concept Drag and Drop Title: Drag and Drop Content Instruction

to aniamtorGraphics

For the mechanical translational system elements shown below find the equivalent analogous electrical elements for Force – Current analogy.

1. Force F a) Current i

2. Spring stiffness K b) 1/L

3. Displacement c) Magnetic flux linkage

4. Viscous friction coefficient f d) 1/R

5. Mass e) Capacitance C

6. Velocity f) Voltage e

Concept Mathematical Models of Different Types of Systems Title: Model Questions

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1) For the mechanical system shown below obtain the electrical analogous systems.

Solution:

There are two displacements are there in the given system. The mechanical equivalent circuit for the given system is shown below.

At node 1,

……………… (1)

At node 2,

……………… (2)

i) F - V analogy:

From equations (1) & (2)

Insert fig. 1.1 below Q1

Insert fig. 1.2 below 1st line of solution

Fig. 1.1

Fig. 1.2

By replacing in the above equations

By applying inverse Laplace transform, we get

The electrical network representing above equation is given below.

ii) F - I analogy:

From equations (1) & (2)

By replacing in the above equations

Insert fig. 1.3 at the end of F-V analogy

Fig. 1.3

By applying inverse Laplace transform, we get

The electrical network representing above equation is given below.

2) For the mechanical system shown below obtain the torque - voltage and torque current electrical analogous system.

Solution:

The mechanical equivalent to the given system is shown below.

At node1,

At node 2,

Insert fig. 1.4 at the end of F-I analogy

Insert fig. 2.1 below Q2

Insert fig. 2.2 below the 1st line of solution

Fig. 1.4

Fig. 2.1

Fig. 2.2

By applying Laplace transform,

……………. (1)

…………….. (2)

i) Torque - Voltage Analogy:

From equations (1) and (2)

By substituting , we get

By applying inverse Laplace transform,

The electrical network representing above equations is given below.

ii) Torque - Current Analogy:

Insert fig. 2.3 at the end of T-V analogy

Fig. 2.3

From equations (1) and (2)

By substituting , we get

By applying inverse Laplace transform,

The electrical network representing above equation is given below.

Insert fig. 2.4 at the end of T-I analogy

Fig. 2.4