This document provides an overview of modeling mechanical and electrical systems using control system modeling techniques. It discusses modeling basic mechanical elements like mass, springs, and dashpots as well as their electrical analogues using circuits with resistors, inductors, and capacitors. The document also describes modeling multi-node mechanical systems and their corresponding electrical circuits. It introduces block diagram representation of systems and algebra for simplifying block diagrams, including rules for series, parallel, feedback and signal flow connections.

1. UNIT I
Control System Modeling

2. Mechanical Translational Systems

3.  Mass, Spring & Dash-Pot – Basic elements
 Mass- Weight of the mechanical system &
assumed to be concentrated at center of body
 Spring – Elastic deformation of the body
 Dash-Pot – Friction existing in mechanical
system. [Piston moving inside a cylinder filled
with viscous fluid]

5. Ideal mass element
The mass will offer an opposing force which is
proportional to acceleration of the body

6. The dash-pot will offering an opposing force which
is proportion to the velocity of the body

7. The spring will offering an opposing force
which is proportional to the displacement of
the body

8. Guidelines to determine the transfer function of
mechanical translational system

10. Solution:

14. Solution:

18.  Basic elements Moment of Inertia of Mass[J],
Torsional Spring with stiffness [K] & Dash-Pot
with rotational friction coefficient [B]
 Moment of Inertia of Mass[J] Weight of the
rotational mechanical system & assumed to be
concentrated at center of body
 Torsional Spring  Elastic deformation of the
body
 Dash-Pot  Friction existing in rotational
mechanical system. [Piston moving inside a
cylinder filled with viscous fluid]

29.  CURRENT VOLTAGE RELATION OF R,L & C:

48.  Systems remain ANALOGOUS – Differential
equations governing the systems or TFs are
in identical form.
 Electric Analogue - greater importance -
Easier to construct electrical models &
analyze them

49.  Input Force in mechanical system = voltage
or current source in electrical system
 Output velocity in mechanical system = Either
current or voltage in an element in electrical
system
 Electrical system – two types of inputs
[voltage or current source] – two types of
analogies i) Force-Voltage Analogy;
ii) Force-Current Analogy

57. PROBLEM

58. SOLUTION

60.  Mechanical system has 2 nodes – 2 meshes in
electrical circuit
 Force applied to M1 Voltage source in mesh-1;
 Elements M1,B1,K1 & B12 connected to node-1
have to be represented by analogous elements in
mesh-1 forming closed path.
 Elements K1,B12,M2,K2 and B2 connected to node-
2  have to be represented by analogous
elements in mesh-2 forming a closed path.
 Elements K1 & B12 are common b/w node1&2, so
they are represented by analogous element as
common elements b/w two meshes.

63.  Mechanical system has 2 nodes – 2 nodes in electrical
circuit
 Force applied to M1 current source connected to
node-1;
 Elements M1,B1,K1 & B12 connected to node-1 have
to be represented by analogous elements connected
to node-1.
 Elements K1,B12,M2,K2 and B2 connected to node-2
have to be represented by analogous elements
connected to node-2.
 Elements K1 & B12 are common b/w node1&2, so they
are represented by analogous element as common
elements b/w two nodes in analogous circuit.

74. PROBLEM

75.  Given mechanical rotational system has 3
nodes – Differential equations governing the
system are given by torque balance
equations.
 Let the angular displacements of J1,J2&J3 be
θ1, θ2 & θ3 respectively, corresponding
angular velocities be ω1, ω2 & ω3.

80.  Given mechanical system has 3 nodes – Torque Voltage analogous
circuit have 3 meshes
 Elements J1, K1 & B1 are connected to node-1  represented by
analogous element in mesh-1 forming a closed path
 Elements J2, B2, B1 & K1 are connected to node-2 represented by
analogous element in mesh-2 forming a closed path
 Elements J3, B2 & K3 are connected to node-3  represented by
analogous element in mesh-3 forming a closed path
 Elements K1 & B1 are common b/w nodes-1&2  represented by
analogous element common b/w mesh-1&2
 Element B2 is common b/w nodes-2&3  represented by analogous
element common b/w mesh-2&3

84.  Given mechanical system has 3 nodes – Torque Current
analogous circuit have 3 Nodes
 Elements J1, K1 & B1 are connected to node-1  represented
by analogous element in Node-1 forming a closed path
 Elements J2, B2, B1 & K1 are connected to node-2
represented by analogous element in Node-2 forming a closed
path
 Elements J3, B2 & K3 are connected to node-3  represented
by analogous element in Node-3 forming a closed path
 Elements K1 & B1 are common b/w nodes-1&2  represented
by analogous element common b/w Node-1&2
 Element B2 is common b/w nodes-2&3  represented by
analogous element common b/w Node-2&3

89.  A block diagram is a representation of a system
using blocks.
 For representing any system using block diagram,
it is necessary to find the transfer function of the
system which is the ratio of Laplace of output to
Laplace of input.
R(s) = Input
C(s) = output
G(s) = transfer functi
on
C(s) = R(s).G
(s)

90.  The basic elements of a block diagram are a
block, the summing point and the take-off point.
• Two blocks having transfer functions G(s) and H(s).
• One summing point and one take-off point.
• Arrows indicate the direction of the flow of signals.

91.  The transfer function of a component is
represented by a block.
 Block has single input and single output.
 Output of the block is obtained by multiplying
transfer function of the block with input.

92.  The summing point is represented with a circle
having cross (X) inside it.
 It has two or more inputs and single output.
 It produces the algebraic sum of the inputs.
 It also performs the summation or subtraction or
combination of summation and subtraction of the
inputs based on the polarity of the inputs.

93.  The take-off point is a point from which the same
input signal can be passed through more than one
branch.
 That means with the help of take-off point, we can
apply the same input to one or more blocks,
summing points.

94.  Consider a series of RLC circuit as shown in the
following figure, where, Vi(t) and Vo(t) are the
input and output voltages.
 Let i(t) be the current passing through the circuit.
This circuit is in time domain.

95.  By applying the Laplace transform to this circuit,
will get the circuit in s-domain.

97.  Block diagram algebra is nothing but the algebra
involved with the basic elements of the block
diagram.
 This algebra deals with the pictorial representation
of algebraic equations.
 Basic Connections for Blocks - three basic types of
connections between two blocks.

98. Compare this equation with the standard
form of the output equation,
• Can represent the series
connection of two blocks with a
single block.
• The transfer function of this single
block is the product of the transfer
functions of those two blocks.

99.  The blocks which are connected in parallel will
have the same input.

100.  can represent the parallel connection of
two blocks with a single block.
 The transfer function of this single block is
the sum of the transfer functions of
those two blocks.

101.  Two types of feedback — positive feedback and
negative feedback.
 figure shows negative feedback control system.

102.  Therefore, the negative
feedback closed loop
transfer function is

103. Similarly, the closed loop transfer function of the
positive feedback,

104.  There are two possibilities of shifting summing
points with respect to blocks −
• Shifting summing point after the block
• Shifting summing point before the block

105.  Consider the block diagram shown in the following
figure. Here, the summing point is present before the
block.

106.  Now, shift the summing point after the block. This
block diagram is shown in the following figure.

108.  Consider the block diagram shown in the following
figure. Here, the summing point is present after the
block.

109.  Now, shift the summing point before the block. This block
diagram is shown in the following figure.

111. Block Diagram Algebra for Take-off Points
 There are two possibilities of shifting the take-off
points with respect to blocks −
• Shifting take-off point after the block
• Shifting take-off point before the block

112.  Consider the block diagram shown in the following
figure. In this case, the take-off point is present before
the block.

114. Shifting Take-off Point Before the Block
 Consider the block diagram shown in the following figure. Here, the
take-off point is present after the block.

116.  Rules for simplifying (reducing) the block diagram, which is having
many blocks, summing points and take-off points
• Rule 1 − Check for the blocks connected in series and simplify.
• Rule 2 − Check for the blocks connected in parallel and simplify.
• Rule 3 − Check for the blocks connected in feedback loop and
simplify.
• Rule 4 − If there is difficulty with take-off point while simplifying,
shift it towards right.
• Rule 5 − If there is difficulty with summing point while
simplifying, shift it towards left.
• Rule 6 − Repeat the above steps till you get the simplified form,
i.e., single block.
Note − The transfer function present in this single block
is the transfer function of the overall block diagram.

117. Example
Consider the block diagram shown in the following figure.
Let us simplify (reduce) this block diagram using the block
diagram reduction rules.

123. Therefore, the transfer function of the system is

124.  Note − Follow these steps in order to calculate the
transfer function of the block diagram having
multiple inputs.
• Step 1 − Find the transfer function of block diagram
by considering one input at a time and
make the remaining inputs as zero.
• Step 2 − Repeat step 1 for remaining inputs.
• Step 3 − Get the overall transfer function by adding
all those transfer functions.
The block diagram reduction process takes more time for
complicated systems. Because, we have to draw the (partially
simplified) block diagram after each step. So, to overcome this
drawback, use signal flow graphs (representation).