Mathematical Modelling
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Basic Types of Mechanical Systems
• Translational
– Linear Motion
• Rotational
– Rotational Motion
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Basic Elements of Translational Mechanical Systems
Translational Spring
i)
Translational Mass
ii)
Translational Damper
iii)
Translational Spring
• A translational spring is a mechanical element that
can be deformed by an external force such that the
deformation is directly proportional to the force
applied to it.
Translational Spring
i)
Circuit Symbols
Translational Spring
Translational Spring
• If F is the applied force
x1
x2
• Then x 1 is the deformation if x 2 0 F
• Or ( x1 x 2 ) is the deformation. F
• The equation of motion is given as
F k ( x1 x 2 )
• Where k is stiffness of spring expressed in N/m
Translational Spring
• Given two springs with spring constant k1 and k2, obtain
the equivalent spring constant keq for the two springs
connected in:
(1) Series (2)Parallel
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Translational Spring
• The two springs have same displacement therefore:
k1 x k 2 x F
(1) Series
( k1 k 2 ) x F
k eq x F
k eq k 1 k 2
• If n springs are connected in Series then:
k eq k 1 k 2 k n 7
Translational Spring
• The forces on two springs are same, F, however
displacements are different therefore:
(2) Parallel
k 1 x1 k 2 x 2 F
F F
x1 x2
k1 k2
• Since the total displacement is x x1 x2 , and we have F k eq x
F F F
x x1 x 2
k eq k1 k 2
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Translational Spring
F F F
k eq k1 k 2
• Then we can obtain
1 k1 k 2
k eq
1 1 k1 k 2
k1 k 2
• If n springs are connected in series then:
k1 k 2 k n
k eq
k1 k 2 k n
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Translational Mass
• Translational Mass is an inertia Translational Mass
element. ii)
• A mechanical system without
mass does not exist.
• If a force F is applied to a mass x (t )
and it is displaced to x meters
then the relation b/w force and F (t )
M
displacements is given by
Newton’s law.
F M x
Translational Damper
• When the viscosity or drag is not
negligible in a system, we often
model them with the damping
force.
• All the materials exhibit the Translational Damper
iii)
property of damping to some
extent.
• If damping in the system is not
enough then extra elements (e.g.
Dashpot) are added to increase
damping.
Common Uses of Dashpots
Door Stoppers
Vehicle Suspension
Bridge Suspension
Flyover Suspension
Translational Damper
B B
F B x F B ( x1 x2 )
• Where B is damping coefficient (N/ms-1).
Translational Damper
• Translational Dampers in series and parallel.
B1
B1 B2
B2
B1 B2
Beq B1 B2 Beq
B1 B2
Example
• Consider the following system (friction is negligible)
k
x
F
M
• Free Body Diagram
fk
M fM
F
• Where f k and f M are force applied by the spring and
inertial force respectively.
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Example
fk
M fM
F
F fk fM
• Then the differential equation of the system is:
F M x kx
• Taking the Laplace Transform of both sides and ignoring
initial conditions we get
2
F ( s ) Ms X ( s ) kX ( s )
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Example
2
F ( s ) Ms X ( s ) kX ( s )
• The transfer function of the system is
X (s) 1
F (s) Ms 2 k
• if
M 1000 kg
k 2000 Nm 1
X (s) 0 . 001
2
F (s) s 2
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Example
X (s) 0 . 001
2
F (s) s 2
• The pole-zero map of the system is
Pole-Zero Map
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Imaginary Axis
-10
-20
-30
-40
-1 -0.5 0 0.5 1
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Real Axis
Example
• Consider the following system
k
x
F
M
B
• Free Body Diagram
fk fB
M fM
F
F fk fM fB
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Example
Differential equation of the system is:
F M x C x kx
Taking the Laplace Transform of both sides and ignoring
Initial conditions we get
2
F ( s ) Ms X ( s ) CsX ( s ) kX ( s )
X (s) 1
F (s) Ms 2 Cs k
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Example
X (s) 1
F (s) Ms 2 Cs k
• if 2
Pole-Zero Map
1.5
M 1000 kg 1
1
k 2000 Nm 0.5
Imaginary Axis
1 0
C 1000 N / ms
-0.5
-1
-1.5
X (s) 0 . 001
2 -2
F (s) s s 1000
-1 -0.5 0
Real Axis
0.5 1
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Grounded chair representation
• General procedure for constructing grounded chair representation:
• 1). Draw coordinates ,such that the coordinate at which the force acts is at the top
and ground is at the bottom.
• 2). Insert each element in its correct orientation with respect to these coordinates.
• A). First isolate mass to extreme left with respect to the coordinate with whom it is
associated.(mass is to be represented in series and not in parallel between
coordinates because in determining the inertia forces , the acceleration of mass is
always taken with respect to ground.)
• B). Then consider elements between that coordinate and ground if any , position
them accordingly.
• C). Then consider elements between coordinates and position them in series
irrespective of their arrangement given in actual physical arrangement.
• D). Continue with next coordinates with the same methodology.
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Example
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Transfer function (f---x)
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• Construct grounded chair representation of the following
system and represent its mathematical modelling
k2
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TRANSFER FUNCTION
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TRANSFER FUNCTION
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Example
• Construct grounded chair representation of the following system and represent its
mathematical modelling
Example
x1 x2
k B3 B4
M1 M2
f (t )
B1 B2
• Construct grounded chair representation of the following system and represent its
mathematical modelling
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Example
x2 x3
x1
k1 B3 B4
u (t ) B1 M1 k2 M2 k3
B2 B5
• Construct grounded chair representation of the following system and represent its
mathematical modelling
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GROUNDED CHAIR REPRESENTATION
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MATHEMATICAL MODELLING
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• TRANSFER FUNCTION
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Basic Elements of Rotational Mechanical Systems
J J
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F-V &F-I ANALOG
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Two systems are said to be analogous to each other if
the following two conditions are satisfied:
•The two systems are physically different
•Differential equation modelling of these two systems
are same
Electrical systems and mechanical systems are two
physically different systems. There are two types of
electrical analogies of translational mechanical
systems. Those are force voltage analogy and force
current analogy.
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FORCE-VOLTAGE ANALOG
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FORCE-VOLTAGE ANALOG
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FORCE- CURRENT ANALOG
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FORCE- CURRENT ANALOG
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EXAMPLE:FORCE-CURRENT ANALOGY
REPRESENT GROUNDED CHAIR AND OBTAIN DIFFERENTIAL EQUATIONS OF MECHANICAL AND ELECTRICAL
SYSTEM AND DRAW ELECTRICAL ANALOGUS CIRCUIT BASED ON FORCE CURRENT ANALOGY
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GROUNDED CHAIR
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DIFFERENTIAL EQUTIONS FOR
MECHANICAL SYSTEM
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DIFFERENTIAL EQUTIONS FOR
ELECTRICAL SYSTEM
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F—I ANALOGUS CIRCUIT
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