dynamic modelling of mechanical system

8/12/2019 dynamic modelling of mechanical system

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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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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.

JointInitiativeofIITsandIIScFundedbyMHRD


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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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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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Fortranslator

mechanical

s stems,

stiffness

is

re resented

by springelementk,whileforrotationalsystemsthisis

representedbytorsionalspringelementkt.Forexample:

diameter

ame er



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A few more translational spring constants

7JointInitiativeofIITsandIIScFundedbyMHRD


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Torsional S rin Constants



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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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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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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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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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(B)

(C)



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S ecial References for this Lecture

,

Fundamentals of Mechanical Vibrations S Graham Kelly, McGraw-Hill

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dynamic modelling of mechanical system

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