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Automatic Control System
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Automatic Control System
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- 29 -CHAPTER – 3
Electrical Analogy of Physical System.
3.1. INTRODUCTION:
The concept of an analogous system is very useful in practice since one type of system may be
easier to handle experimentally than others. Systems remain analogous as their differential equations or
transfer functions are of identical form. The electric analog of any other kind of system is of great
importance because of our acquaintance with electrical engineering systems. Here our special emphasis
is on electric analog of mechanical systems.
Once the electrical analogous circuit is obtained, it is possible to visualize and even predict the
system behavior by inspection. The case of changing the values of electrical components is an
invaluable aid in the proper design of the actual system.
A property common to all basic laws of physical systems is that certain fundamental quantities
can be defined by numerical values. The physical laws define relationships between these fundamental
quantities and are usually represented by mathematical equations and functions. These mathematical
models are then combined to produce a composite mathematical model of the system which usually
takes the form of differential equations with time as independent variable.
Any physical system can be represented by any one of the following mathematical models
depending on their internal characteristics.
1. Partial differential equations involving one or more independent variables together with
partial derivatives of the dependent variables with respect to the independent variable.
2. Time varying differential equations are those in which one or more terms depend explicitly
on the in dependent variable, time t.
3.2. MECHANICAL SYSTEM ELEMENTS:
An electrical engineer familiar with the electrical system can easily analyses the system under
study and predict the behaviour of the system. Many circuit theorems, impedance concept can be
applicable and the electric analog system is easy to handle experimentally.
Hence when we deal with a system other than electrical, if the electric analog can be obtained,
then there is clear advantage. The motion of mechanical elements can be described in various
dimensions as translational, rotational or a combination of both. The equations governing the motion
of mechanical systems are often directly or indirectly formulated from Newton’s Law of motion.
3.2.1. Translation System:
The motion takes place along a straight line is known as translational motion. There are three
types of forces that resist motion .
1. Inertia Force : Consider a body of mass ‘M’ & acceleration ‘a’ then according to Newton’s 2nd
law of motion the inertia force will be equal to the product of mass ‘M’ & acceleration ‘a’. Shown in
figure 3.2.1(a).
FM (t ) Ma(t ) …………………………………………….……..… (3.1)
In terms of velocity the equation (3.2.1.1) becomes
dv(t )
FM (t ) M ………………………..………………………… (3.2)
dt
In terms of displacement the equation (3.1) can be expressed as
d 2 x(t )
FM (t ) M ………………………………………..…………… (3.3)
dt 2
2. Damping Force: For viscous friction we assume that the damping force is proportional to the
velocity.
, - 30 -
d
x(t ) ………………………………………………...… (3.4)
FD (t ) Bv(t ) B
dt
Where, B = Damping coefficient, unit of B = N/m/sec.
We can represent ‘B’ by a dashpot, consists of piston and cylinder.
3. Spring Force: A spring stores the potential energy. The restoring force of a spring is
proportional to the displacement.
FK (t ) x(t ) Kx(t )
FK (t ) K v(t )dt
t
= K [ vdt x(0)] ……………………………………………........ (3.5)
0
Where, K = spring constant or stiffness
Unit of K =N/m. The stiffness of a spring can be defined as restoring force per unit displacement.
x(t)
x(t) x(t)
FKt)
FD(t)
M F(t) K
Fig 3.2.1(a) Fig 3.2.1(b) Fig 3.2.1(c)
3.2.2. Rotational System:
The rotational motional of a body can be defined as the motion of a body about a fixed axis.
There are three types of torques resist the rotational motion.
1. Inertial Torque: Inertia (J) is the property of an element that stores the kinetic energy of
rotational motion. The inertia torque T1 is the product of moment of inertia J and angular
acceleration (t ) .
T1 (t ) J (t )
d d2
T1 J (t ) J 2 (t ) ……………………………………………………... (3.6)
dt dt
Where (t ) = angular velocity.
(t ) = angular displacement. Unit of torque = N-m.
2. Damping Torque: The damping Torque TD (t ) is the product of damping coefficient B and
angular velocity .
d
TD (t ) B (t ) B (t ) ……………………………………………….… (3.7)
dt
3. Spring Torque: Spring torque T (t ) is the product of torsional stiffness and angular
displacement.
T (t ) K (t ) ………………………………….………..…………..…… (3.8)
Unit of K = N-m/rad.
B T (t )
T1(t) K
(t) TD(t)
J
(t)
Fig.3.2.2.1(a) Fig.3.2.2.1(b) Fig.3.2.2.1(c)
, - 31 -
We get an analogous system. The analogous quantities are tabulated as follows:
Translation Rotational
Force, F Torque, T
Acceleration, a Angular acceleration,
Velocity, v Angular velocity,
Displacement, x Angular displacement,
Mass, M Moment of inertia, J
Damping coefficient, B Rotational damping coefficient, B
Stiffness(or Compliance), K Torsional stiffness(or Torsional Compliance), K
3.3. FREEBODY DIAGRAM:
The first step in obtaining mathematical model of a mechanical system is to draw a free body
diagram indicating the various forces acting on it. In free body diagram each mass is isolated from the
rest of the system and the forces acting on the mass that causes displacement, forces opposing the
displacement are represented pictorially.
Procedure to draw free body diagram:
1. Assume the direction of displacement of the mass as positive direction.
2. Find all the forces with direction acting on the mass. (For example, force applied in the
direction of displacement and opposing forces due to spring, dash-pot and mass are opposite to
the direction of displacement).
3. Using Newton’s laws of motion express all the forces interns of displacement or velocity of
the mass.
Once the free body diagram is obtained we can easily write the differential equation by equating the sum
of applied forces to te sum of opposing forces.
3.4. D’ALEMBERT’S PRINCIPLE:
This principle states that ” For any body, the algebraic sum of externally applied forces and the
forces resisting motion in any given direction is zero”.
D’Alembert principle is useful in writing the equation of motion of mechanical system.
Consider, a system shown in fig.3.4.1 consisting of a mass M, spring & dashpot.
First choose a reference direction. All the forces in the direction of reference direction
considered as positive & the forces opposite to the reference direction taken as negative.
External Force: F(t).
d2 d
Resisting Forces : Inertia Force, Fm (t ) M 2
x(t ) ; Damping Force, FD (t ) B x(t )
dt dt
Spring Force, FK (t ) Kx(t )
According to D’Alembert’s principle
d2 d
F (t ) Fm (t ) FD (t ) FK (t ) 0 ; F (t ) - M 2 x(t ) B x(t ) Kx(t ) = 0
dt dt
2
d d
Or, F (t ) M 2 x(t ) B x(t ) Kx(t ) . ………………………………………………....... (3.9)
dt dt
Fig. 3.4.1 Fig. 3.4.2
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