ANNA UNIVERSITY,REGULATION 2017, EEE, IC8451-CONTROL SYSTEM

1. IC 8451&CONTROL SYSTEMS
Department of Electrical and Electronics Engineering
Approved by AICTE | Affiliated to Anna University | Accredited by NAAC | Accredited NBA | Recognized by UGC under 2(f) and 12(B)
Chennai Main Road, Kumbakonam- 612 501.
ARASU ENGINEERING COLLEGE
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2. Prepared by
Mrs .K.Kalpana.,M.E.,(Ph.D).,
Department of Electrical and Electronics Engineering
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3.  TRANSFER FUNCTION
Transfer function of a system is defined as the ratio of
the Laplace Transform of the output variable to the Laplace
Transform of the input variable assuming all the initial
condition as zero
TRANSFER FUNCTION =
with zero initial conditions
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input
of
transform
Laplace
output
of
transform
Laplace

4.  Mechanical Translational systems
 The model of mechanical translational systems can be
obtained by using three basic elements mass, spring and
dashpot.
 When a force is applied to a translational mechanical system,
it is opposed by opposing forces due to mass, friction and
elasticity of the system.
 The force acting on a mechanical body is governed by
Newton's second law of motion.
 For translational systems, it states that the sum of forces acting
on a body is zero (or the sum of applied forces is equal to the
sum of opposing forces on a body)
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List of symbols used in mechanical translational system
x = Displacement, m
v = dx /dt = Velocity, m/s
a = d2x /dt2 = Acceleration, m/s2
f = Applied force, N
fm = Opposing force offered by mass of the body, N
fk = Opposing force offered by elasticity of the body(spring),N
fb = Opposing force offered by friction of the body(dash pot),N
M = Mass, kg
K = Stiffness of spring, N/m
B = Viscous friction co efficient, N-s/m

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Force balance equations of idealized elements:
Consider an ideal mass element shown in fig. which has negligible
friction and elasticity.
Let a force be applied on it. The mass will offer an opposing force which
is proportional to acceleration of a body.
Let f = applied force
fm =opposing force due to mass
Here fm α (d2x / dt2) or f m= M (d2x / dt2)
By Newton's second law, f = f m= M (d2x / dt2)

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Consider an ideal frictional element dash-pot shown in fig. which has
negligible mass and elasticity.
 Let a force be applied on it. The dashpot will be offer an opposing
force which is proportional to velocity of the body.

8.  When the dashpot has displacement at both the ends as
shown in fig, the opposing force is proportional to differential
velocity.
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9.  Consider an ideal elastic element spring shown in fig ,which
has negligible mass and friction.
 Let a force be applied on it. The spring will offer an opposing
force which is proportional to displacement of the body.
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10.  When the spring has displacement at both the ends as shown in
figure, the opposing force is proportional to differential
displacement.
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11.  Guidelines to determine the transfer function of mechanical
translational system
 In mechanical translational system the differential equations governing the
system are obtained by writing force balance equations at nodes in the
system.
 The nodes are meeting point of elements. Generally the nodes are mass
elements in the system. In some cases the nodes may be without mass
element.
 The linear displacement of the masses are assumed as x1,x2,x3,....etc. and
assign a displacement to each mass(node). The first derivative of the
displacement is velocity and the second derivative of the displacement is
acceleration.
 Draw the free body diagram of the system. The free body diagram is
obtained by drawing each mass separately and then marking all the forces
acting on that mass. Always the opposing forces acts in a direction opposite
to applied force. The mass has to move in the direction of the applied force.
Hence the displacement ,velocity and acceleration of the mass will be in the
direction of applied force.
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12.  For each free body diagram write one differential equation by
equating the sum of applied forces to the sum of opposing
forces.
 Take Laplace transform of differential equations to convert
them to algebraic equations. Then rearrange the s domain
equations to eliminate the unwanted variables and obtain the
ratio between output variable and input variable. This ratio is
the transfer function of the system.
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13.  EXAMPLE PROBLEM -1
Write the differential equations governing the mechanical
system shown in fig and determine the transfer function.
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The required transfer function is
The system has two nodes and they are mass M1 and M2
)
(
)
(
S
F
S
X

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Let the displacement of mass M1 is x1.The opposing forces acting on mass M1 are
marked as fm1,fb1,fb,fk1,fk
1

15.  The free body diagram of mass M2 is shown in fig. The opposing forces
acting on M2 are marked as fm2,fb2,fb and fk
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2

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Substituting x1(s) in equation 2

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EXAMPLE PROBLEM -2
Determine the transfer function
)
(
)
(
2
s
F
s
Y

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21.  HOME WORK PROBLEM
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