dynamic modelling of mechanical system
TRANSCRIPT
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Dynamic Modelling of Mechanical Systems
Dr. Bishakh Bhattacharya
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Hints of the Last Assi nment
TheGoverningEOMmaybewrittenas:
)()(...
.
11211
..
11 tfxBxxKxM a
Now,you
may
consider
the
following
states
for
the
system:
222212122 xxxxx
.
1
1
x
x
X
.
2
2
x
x
2
Covertt etwosecon or erODEsinto our irstor erODEs an o taint estatespace
representation.
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
MechanicalSystems
Mechanicalsystemsaregenerallymodeledasalumped
,
couldbe
considered
to
be
asystem
consisting
of
an
array
of
rigidinertiaelementslinkedbyacombinationof massless
springanddashpotelements.
Theinertia
elements
re resent
the
kinetic
ener
stored
in
thesystem;springsthepotentialenergyanddashpotsthe
energythatgetsdissipatedfromamechanicalsysteminthe
.
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Fortranslatorymechanicalsystems,inertiaisrepresentedby
massm,whileforrotationalsystemsthisisrepresentedby
momentofinertiaJ.
Considerarotor
of
mass
m,
rotating
about
its
centroidal
axis.Themomentofinertiawillbedefinedas:
dmrJm
2
Whererdenotesthedistanceofanelementalmassdm
fromthecentroidalaxis.Forarotationaboutanaxiswhichis
atadistance
d
from
the
centroidal
axis,
following
parallel
axistheoremthemomentofinertiacouldbeexpressedas:
mnew
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Fortranslator
mechanical
s stems,
stiffness
is
re resented
by springelementk,whileforrotationalsystemsthisis
representedbytorsionalspringelementkt.Forexample:
diameter
ame er
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
A few more translational spring constants
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Torsional S rin Constants
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Dam in Element
There are two common damping elements used to model energy dissipation from a
mechanical system. These are Viscoelastic Damping and Friction Damping.
Viscous damping model is most common; here, the damping force is taken to be
proportional to the velocity across the damper, acting in the direction opposite to
.
Linear damping force is represented by a viscous dashpot, which shows a piston
moving relative to a cylinder containing a fluid. The ideal linear relationship
between the force and the relative velocit holds ood so lon as the relative
Velocity is low, ensuring a laminar fluid flow.
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Friction Dam in ElementAnother type of common damping force is the so-called dry fr ict ion force between
two sol id interfaces. This is known as Coulomb damping. In this model, the
magnitude of damping force is assumed to be a constant, which is independent of
.
force is opposite to that of the relat ive velocity. In a physical model, a Coulomb
damper is represented by the symbol shown below. The nature of change of the frict ion
force with respect to displacement of the system is shown next. The area under this
.
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Conce t of De rees of Freedom
An important element in describing the dynamics of a system consisting of
mult iple lumped parameters is the Degrees of Freedom (DOF) for the system. This
is defined as the number of kinematically independent variables required to
describe completely the motion of the system.
It may be noted that the number of degrees of freedom of a particle/lumped mass
gets reduced if it is subjected to constraints. For example, a particle in three
dimensional s ace ma have 3 DOF, hence two such articles ma have total
6DOF. However, if they are connected together by a rigid link, this wi ll come down
to 6-1=5 DOF. Thus, the actual number of DOF of a system equals to the difference
between the numbers of unconstrained DOF and the constraining condit ions.
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Exam les and Assi nments
Consider the first two cases: there are two links of identical lengths but subjected
to different boundary constraints. Find out the DOF in each case.
(A)
ow, cons er e o ow ng ass gnmen s an n ou e govern ng o emechanical systems.
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
(B)
(C)
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
S ecial References for this Lecture
,
Fundamentals of Mechanical Vibrations S Graham Kelly, McGraw-Hill
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