Module1-translational, mechl, FI, FV.pdf

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1. Basic Elements of Electrical Systems
•The time domain expression relating voltage and current for the resistor is given by Ohm’s lawRtitv
RR)()(=
•The Laplace transform of the above equation isRsIsV
RR )()(=
1.Resistor
1

2. 2.Capacitor
•The time domain expression relating voltage and current for the Capacitor is given as:dtti
C
tv
cc = )()(
1
•The Laplace transform of the above equation (assuming there is no charge stored in the capacitor) is)()(sI
Cs
sV
cc
1
=
2

3. 3.Inductor
•The time domain expression relating voltage and current for the inductor is given as:dt
tdi
Ltv
L
L
)(
)(=
•The Laplace transform of the above equation (assuming there is no energy stored in
inductor) is)()(sLsIsV
LL=
3

4. 4

5. ControlSystems
➢Fortheanalysisanddesignofcontrolsystems,weneedtoformulateamathematicaldescriptionofthesystem.
➢Theprocessofobtainingthedesiredmathematicaldescriptionofthesystemisknownas“modeling”.Thebasic
modelsofdynamicphysicalsystemsaredifferentialequationsobtainedbyapplicationoftheappropriatelawsofnature.
➢Theseequationsmaybelinearornonlineardependingonthephenomenabeingmodeled.
➢Thecontrolsystemscanberepresentedwithasetofmathematicalequationsknownasmathematicalmodel.
➢Thesemodelsareusefulforanalysisanddesignofcontrolsystems.Analysisofcontrolsystemmeansfindingtheoutputwhen
weknowtheinputandmathematicalmodel.
➢Designofcontrolsystemmeansfindingthemathematicalmodelwhenweknowtheinputandtheoutput.
Thefollowingmathematicalmodelsaremostlyused.
•Differential equationmodel
•Transfer functionmodel(valid only for linear time-invariant systems)
•State spacemodel
➢Althoughtheanalysisanddesignoflinearcontrolsystemshavebeenwelldeveloped,theircounterpartsfornonlinearsystemsare
usuallyquitecomplex.
➢Therefore,thecontrolsystemsengineeroftenhasthetaskofdeterminingnotonlyhowtoaccuratelydescribeasystem
mathematically,butalso,moreimportant,howtomakeproperassumptionsandapproximations,whenever
necessary,sothatthesystemmaybeadequatelycharacterizedbyalinearmathematicalmodel.
5

6. Example
Considerthefollowingelectricalsystemasshowninthefollowingfigure.Thiscircuitconsistsofresistor,inductorand
capacitor.Alltheseelectricalelementsareconnectedinseries.Theinputvoltageappliedtothiscircuitis????????????andthe
voltageacrossthecapacitoristheoutputvoltage????????????.
6
1. Differential EquationModel
Differentialequationmodelisatimedomainmathematicalmodelofcontrolsystems.
Followthesestepsfordifferentialequationmodel.
▪Applybasiclawstothegivencontrolsystem.
▪Getthedifferentialequationintermsofinputandoutputbyeliminatingthe
intermediatevariable(s).

7. 2.TransferFunctionModel
Transferfunctionmodelisans-domainmathematicalmodelofcontrolsystems.TheTransferfunctionofaLinearTime
Invariant(LTI)systemisdefinedastheratioofLaplacetransformofoutputandLaplacetransformofinputbyassumingallthe
initialconditionsarezero.
If�(�)and�(�)aretheinputandoutputofanLTIsystem,thenthecorrespondingLaplacetransformsare�(�)and�(�).
Therefore,thetransferfunctionofLTIsystemisequaltotheratioof�(�)and�(�).
i.e., Transfer function =
Y(s)
The transfer function model of an LTI system is shown in the followingfigure.
7
X(S)

8. 8

9. 9

10. V
i(S)
V
o(S)
Here, we show a first order electrical system with a block having the transfer
function inside it. And this block has an input ????????????
(�) & an output ????????????
(�). 10

11. ControlSystems
Mesh equation for this circuitis
13
Example
Considerthefollowingelectricalsystemasshowninthefollowingfigure.Thiscircuitconsistsofresistor,inductorandcapacitor.Allthese
electricalelementsareconnectedinseries.Theinputvoltageappliedtothiscircuitis????????????andthevoltageacrossthecapacitoristheoutputvoltage
????????????.
i(t)
1
2

12. The above equation is a second order differentialequation.
12

13. Where,
•????????????(�) is the Laplace transform of the input voltage????????????
•????????????(�) is the Laplace transform of the output voltage????????????
Theaboveequationisatransferfunctionofthesecondorderelectricalsystem.Thetransfer function model of this system is shownbelow.
Here, we show a second order electrical system with a block having the transfer function inside it. And this block has an input
????????????(�) & an output ????????????(�).
14

14. Analogous Systems:
14
➢Mostfeedbackcontrolsystemscontainmechanicalaswellaselectrical
components.
➢Fromamathematicalviewpoint,thedescriptionsofelectricalandmechanical
elementsareanalogous.Infact,wecanshowthatgivenanelectricaldevice,thereis
usuallyananalogousmechanicalcounterpart,andviceversa.
➢Theanalogy,ofcourse,isamathematicalone;thatis,twosystemsareanalogousto
eachotheriftheyaredescribedmathematicallybysimilarequations.
➢Themotionofmechanicalelementscanbedescribedinvariousdimensionsas
translational,rotational,oracombinationofboth.
➢Theequationsgoverningthemotionsofmechanicalsystemsareoftendirectlyor
indirectlyformulatedfromNewton’slawofmotion.

15. Themotionofmechanicalelementscanbedescribedinvariousdimensionsastranslational,
rotational,oracombinationofboth.Theequationsgoverningthemotionsofmechanicalsystemsare
oftendirectlyorindirectlyformulatedfromNewton’slawofmotion.
Letusdiscussthedifferentialequationmodelingofmechanicalsystems.
Therearetwotypesofmechanicalsystemsbasedonthetypeofmotion.
I.Translationalmechanicalsystems
II.Rotationalmechanicalsystems
15
MECHANICAL SYSTEMS

16. 16
I. ModelingofTranslational MechanicalSystems
➢Translationalmotiontakesplacealongastraightlineandthevariablesinvolvedindescribingastraight-line
motionaredisplacement,velocityandacceleration.
➢Newton’slawofmotiongovernsthelinearmotion.Accordingtothislaw,theproductofmassandacceleration
isequaltothealgebraicsumofforcesactingonit.
➢Newton’slawofmotionstatesthatthealgebraicsumofforcesactingonarigidbodyinagivendirectionisequal
totheproductofthemassofthebodyanditsaccelerationinthesamedirection.
➢ThelawcanbeexpressedasΣforces=Ma
whereMdenotesthemassandaistheaccelerationinthedirectionconsidered.

17. Translationalmechanicalsystemsmovealongastraightline.Thesesystemsmainlyconsistofthreebasic
elements:
Thoseare(1)mass(M),
(2)spring(K)and
(3)dashpotordamper(B).
➢Ifaforceisappliedtoatranslationalmechanicalsystem,thenitisopposedbyopposingforcesduetomass,
elasticityandfrictionofthesystem.
➢Sincetheappliedforceandtheopposingforcesareinoppositedirections,thealgebraicsumoftheforces
actingonthesystemiszero.
15
TranslationalmechanicalsystemsCont….

18. Translational Mechanical systems Cont….
18
Let us now see the force opposed by these three elements individually.
1.Mass:
➢The function of mass in linear motion is to store kinetic energy. Mass cannot store potential energy.
➢SupposeaforceisappliedtomassMasshowninFigure,themassstartsmovinginxdirectionasshown.
➢Forthetimebeing,wewillassumeotherforcessuchasfriction,etc.tobezero.Hence,accordingto
Newton’slaw,
•??????ismass
•�isacceleration
•??????isdisplacement
Note: Mass shouldn't be between two forces

19. 2.Spring:
➢Springisanelement,whichstorespotentialenergy.
➢IfaforceisappliedonspringK,thenitisopposedbyanopposingforceduetoelasticityofspring.
Thisopposingforceisproportionaltothedisplacementofthespring.
➢Assumemassandfrictionarenegligible.
F
k∝�
=> F
k= ??????�
??????= F
k= ??????�
Where,
•??????is the appliedforce
•????????????is the opposing force due to elasticity ofspring
•??????is springconstant
•??????isdisplacement
Two-mass–spring system:
16
Translational Mechanical systems Cont….
F
F
k= ??????(x
1–x
2)

20. 20
3.Friction :
➢Frictionis the force resisting the relative motion of solid surfaces, fluid layers, and material elements
sliding against each other.
➢Adashpotis a mechanical device, a damper which resists motion via viscous friction. The resulting force
is proportional to the velocity, but acts in the opposite direction, slowing the motion and absorbing energy.
➢It is commonly used in conjunction with a spring (which acts to resist displacement).
https://youtu.be/H877C_5BMkI
Translational Mechanical systems Cont….

21. 17
➢If a force is applied on dashpot B, then it is opposed by an opposing force due to friction
of the dashpot.
➢This opposing force is proportional to the velocity of the body. Assume mass and
elasticity are negligible.
????????????∝??????
Translational Mechanical systems Cont….

22. 22
Twoblocksm1andm2areconnectedbyadashpotwithdampingcoefficientB
F
The reaction force due to dashpot
between two masses, on either side of
dashpot is the displacements are
different i.e, x1 and x2
F
F
Translational Mechanical systems Cont….

23. 23
ForceF)-velocity(v), force(F)-displacement(x), and impedance
translationalrelationshipsforsprings(K),viscousdampers(B),and
mass(M):

24. 24
II.ModelingofRotationalMechanicalSystems
➢Rotationalmechanicalsystemsmoveaboutafixedaxis.
➢Thesesystemsmainlyconsistofthreebasicelements.
(1)momentofinertia(J)
(2)torsionalspring(K)
(3)dashpot(B)
➢Ifatorque(T)isappliedtoarotationalmechanicalsystem,thenitisopposedbyopposingtorquesdueto
momentofinertia,elasticityandfrictionofthesystem.
➢Sincetheappliedtorqueandtheopposingtorquesareinoppositedirections,thealgebraicsumoftorques
actingonthesystemiszero.
➢Letusnowseethetorqueopposedbythesethreeelementsindividually.

25. 25
1.MomentofInertia(J):
➢Intranslationalmechanicalsystem,massstoreskineticenergy.Similarly,in
rotationalmechanicalsystem,momentofinertiastoreskineticenergy.
➢IfatorqueisappliedonabodyhavingmomentofinertiaJ,thenitisopposedbyan
opposingtorqueduetothemomentofinertia.
➢Thisopposingtorqueisproportionaltoangularaccelerationofthebody.Assume
elasticityandfrictionarenegligible.

26. 26
2. Torsional Spring (K):
➢Intranslationalmechanicalsystem,springstorespotentialenergy.Similarly,inrotational
mechanicalsystem,torsionalspringstorespotentialenergy.
➢IfatorqueisappliedontorsionalspringK,thenitisopposedbyanopposingtorquedueto
theelasticityoftorsionalspring.
➢Thisopposingtorqueisproportionaltotheangulardisplacementofthetorsionalspring.
Assumethatthemomentofinertiaandfrictionarenegligible.

27. 27
J1
J2
Θ
1
Θ
2
Two inertia system with a torsional spring
)))

28. 28
3.Dashpot(B):
➢IfatorqueisappliedondashpotB,thenitisopposedbyanopposingtorqueduetothe
rotationalfrictionofthedashpot.
➢Thisopposingtorqueisproportionaltotheangularvelocityofthebody.Assumethe
momentofinertiaandelasticityarenegligible.
Where,
➢??????�is the opposing torque due to the rotational friction of the
dashpot
➢??????is the rotational friction coefficient
➢??????is the angular velocity
➢??????is the angular displacement.

29. 29
T
Two inertia system with a friction

30. 30
B
B
B
Torque(T)—angularvelocity(dθ/dt),
Torque(T)—angulardisplacement(θ),andimpedancerotationalrelationshipsfor
spring(K),viscousdampers(B),andInertia(J):

31. 31

32. 32
Procedure for writing differential equation for a complete mechanical system:
▪Identifynumberofdisplacements.
▪Markindependentnodesforeachdisplacement.
▪displacement=node
▪Markalltheelementsinparallelundertherespectivenodewhichareundertheinfluenceof
respectivedisplacement.
▪Now,drawthemechanicalnetworkandwriteForce/Torqueequationsateachnodebyequating
sumofforces/Torquesateachnodetozero.
Note:
➢NumberofMasses=Numberofdisplacements=NumberofNodes
➢(ApplicableifandonlyifwhenForce/Torqueisdirectlyappliedtomassorinertia
respectively)
➢Displacementoneithersideofdamper(dashpot)orspringmustbedifferentiftheyarebetween
movingbodies.
➢Mass/Inertiashouldnotbeconnectedbetweennodes.

33. 33
When f(t) is applied to mass M, it vibrates.
The displacement is x(t).
When f(t) is applied to mass M,itvibrates, due to
this there is a existence of friction between a body
with mass M and ground and frictional coefficient
is indicated by B.
The spring also experiences a displacement x(t)
and spring constant is K.
B
Note:Numberofdisplacementequaltonumber
nodesbecauseforceisappliedtoabodywithmassM

34. 34
For this problem, since force is directly
applied to a body with mass M,
Number of Masses = Number of
displacement
=Number of node(mechanical
network)= 1
i.e., x(t) and all three elements M,B,K
have same displacement x(t).
Therefore we have to connect all the
elements under node x(t) in parallel.
B

35. 35

36. ▪When Torque T(t) is applied to mass Inertia J,it rotates with
angular displacement is θ(t).
▪When T(t) is applied to J , it rotates, due to this there is a existence
of friction between a body with Inertia J and ground and frictional
coefficient is indicated by B.
▪The spring also experiences a angular displacement θ(t) and spring
constant is K.
For this problem, since force is directly applied to a body with Torque T,
▪Number of Inertia = Number of angular displacement
=Number of node(mechanical network)= 1
i.e., θ(t) and all three elements J,B,K have same displacement θ(t) .
Therefore we have to connect all the elements under node θ(t) in
parallel. 36
B

37. 37

38. 38
Number of displacement equal to number nodes
because force is applied to a body with mass M
When f(t) is applied to mass M1,it vibrates
and the displacement is x1(t), due to this there
is a existence of friction between a body with
mass M1 and ground and represented by
frictional coefficient is B1 and B1 is between
node x1 and reference.
K1 is between node x1 and reference.
Indirect force transferred to M2 through B3
and B3 connected between node x1 and x2.
M2,B2 and K2 have displacement of x2.
f(t),M1,B1 and K1 are under node x1.
M2 ,B2 and K2 are under node x2.
B3 is between x1 and x2.
B3

39. 39

40. 40

41. 41

42. 42

43. 43
Solution:
➢Number of displacement equal to number nodes because force is
applied to a body with mass M.
➢Therefore number of displacements= number of nodes= 2
i.e.,x1 (t) and x2 (t)
➢When f(t) is applied to mass M,it vibrates and the
displacement is xt), due to this there is a existence of friction
between a body with mass Mand ground and represented by
frictional coefficient is B and B is between node xt) and
reference.
➢K1 is between node xt)) and reference.
➢Indirect force transferred to M2through K2and M2
has displacement of x2 (t).
▪f(t),M1,B and K1 are under node x1.
▪M2 is under node x2.
▪K2 is between x1and x2.

44. 44

45. 45

46. 46

47. 47
➢NumberofdisplacementequaltonumbernodesbecausetorqueisappliedtoabodywithinertiaJ.
➢Therefore,
numberofangulardisplacements=numberofnodes=2
i.e.,ϴ1(t)andϴ2(t)
➢WhentorqueT(t)isappliedtoinertiaJ,itrotatesandtheangulardisplacementisϴ(t),duetothisthereisaexistenceoffriction
betweenabodywithinertiaJandgroundandrepresentedbyfrictionalcoefficientisB1andB1isbetweennodeϴ1(t)andreference.
➢IndirectforcetransferredtoJ2throughK1.
➢J2hasdisplacementofϴ2(t).
▪T(t),J,B1areundernodeϴ1(t)
▪J,B2,K2areundernodeϴ2(t).
▪K1isbetweennodesϴ1(t)andϴ2(t).
Mechanical
Network

48. 48
Thedifferentialequationsfornodesϴ1(t)andϴ2(t)arewrittenasfollows

49. 49

50. 50
➢Itisalwaysadvantageoustoobtainelectricalanalogousofthegivenmechanicalsystemas
wearewellfamiliarwiththemethodsofanalyzingelectricalnetworkthanmechanical
systems.
➢Therearetwomethodsofobtainingelectricalanalogousnetworksofmechanicalsystems,
namely,
1.Force-voltageAnalogy,i.e.DirectAnalogy.
2.Force-currentAnalogy,i.e.InverseAnalogy
➢Twosystemsaresaidtobeanalogoustoeachotherifthefollowingtwo
conditionsaresatisfied.
•The two systems are physically different.
•Differential equation modeling of these two systems are same.

51. 51
ConsidersimplemechanicalsystemasshownintheFig.
Duetotheappliedforce,massMwilldisplacebyanamountx(t)in
thedirectionoftheforcef(t)asshownintheFig.
AccordingtoNewton’slawofmotion,appliedforcewillcause
displacementx(t)inspring,accelerationtomassMagainst
frictionalforcehavingconstantB.
TheforceFandTorqueTbalancedequationsare
I
B
II

52. 52
Inthismethod,totheforcein
mechanicalsystem,voltageisassumed
tobeanalogousone.
Accordinglywewilltrytoderiveother
analogousterms.
Considerelectricnetworkasshownin
theFig.
TheequationaccordingtoKirchhoff’slawcanbe
writtenas,
We know that ʃ idt = q and
where q is charge.
Substituting the above values in V equation
III

53. 53
➢BycomparingEquationIandIIwithEquationIII,wewillgettheanalogousquantitiesof
thetranslational(Rotational)mechanicalsystemandelectricalsystembasedon
Force/Torque-Voltageanalogyasindicatedinthefollowingtable.
III
I
II

54. 54
In force current analogy, the mathematical equations of the translational and rotational mechanical system
are compared with the nodal equations of the electrical system.
Consider the following electrical system as shown in the following figure.
This circuit consists of current source i(t) , resistor( R), inductor(L) ,capacitor(C) and node voltage v(t). All
these electrical elements are connected in parallel.
The equation according to Kirchhoff’s current law for above system
is,
We know that and ψ(t) = ʃ v(t) dt where ψis flux.
Substituting v(t) and ʃ v(t) values in above equation ,we get
IV

55. 55
IV