translational mechanical system

ME 375 Handouts

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Translational Mechanical SystemsTranslational Mechanical Systems

•• Basic (Idealized) Modeling ElementsBasic (Idealized) Modeling Elements•• Basic (Idealized) Modeling ElementsBasic (Idealized) Modeling Elements•• Interconnection Relationships Interconnection Relationships --Physical LawsPhysical Laws•• Derive Equation of Motion (EOM) Derive Equation of Motion (EOM) -- SDOFSDOF•• Energy TransferEnergy Transfer•• Series and Parallel ConnectionsSeries and Parallel Connections

School of Mechanical EngineeringPurdue University

ME375 Translation - 1

•• Derive Equation of Motion (EOM) Derive Equation of Motion (EOM) -- MDOFMDOF

VariablesVariables

•• xx :: displacementdisplacement [m][m] d x x v!•• xx :: displacement displacement [m][m]•• vv :: velocity velocity [m/sec][m/sec]•• aa :: acceleration acceleration [m/sec[m/sec22]]•• ff :: forceforce [N][N]•• pp :: power power [Nm/sec][Nm/sec]•• ww :: work ( energy ) work ( energy ) [Nm] [Nm]

2

2

x x vdtd d d dv v x x x adt dt dt dt

dp f v f x wdt

! !

" #! ! ! ! !$ %& '

! ( ! ( !

! !!

!



( gy )( gy ) [ ][ ]1 [Nm] = 1 [J] (Joule)1 [Nm] = 1 [J] (Joule)

1

0

1

0

1 0

0

( ) ( ) ( )

( ) ( )

t

t

t

t

w t w t p t dt

w t f x dt

! )

! ) (

*

* !

ME 375 Handouts

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Idealized Modeling ElementsIdealized Modeling Elements

•• Inertia (mass)Inertia (mass)•• Inertia (mass)Inertia (mass)•• Stiffness (spring)Stiffness (spring)•• Dissipation (damper)Dissipation (damper)



•• SpringSpring–– Stiffness ElementStiffness Element

Basic (Idealized) Modeling ElementsBasic (Idealized) Modeling Elements

–– RealityReality•• 1/3 of the spring mass may be1/3 of the spring mass may be

–– IdealizationIdealizationM lM l

•• 1/3 of the spring mass may be 1/3 of the spring mass may be considered into the lumped considered into the lumped model.model.

•• In large displacement operation In large displacement operation springs are springs are nonlinearnonlinear..

x2

K

x1

fS fS

+ ,2 1Sf K x x! -fS



•• MasslessMassless•• No DampingNo Damping•• LinearLinear

–– Stores Energy Stores Energy (x2 - x1)

ME 375 Handouts

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Practical Nonlinear SpringPractical Nonlinear Spring

Engine Mount:Engine Mount:#T062 VERTICALExperimental Anal tical

-4000

-2000

0

2000

4000

-20 -15 -10 -5 0 5

DISP (mm)

LOAD

(N)

Experimental Analytical

1

2



DISP (mm)

+ , xxK ..)! 2force Restoring /

K0isolationfor

motions Small12

loads staticfor motions Large

xK .)! /2

•• SpringsSprings in Seriesin Series

Series ConnectionSeries Connection

x1 x2 x1 x2

K1 K2

fSfSKEQ

fSfS1



ME 375 Handouts

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•• SpringsSprings in Parallelin Parallel

Parallel ConnectionParallel Connection

x xx1 x2

fS

KEQ

x1 x2

fS1fSfSK1

K2



•• DamperDamper–– Friction ElementFriction Element

Basic (Idealized) Modeling ElementsBasic (Idealized) Modeling Elements•• MassMass

–– Inertia ElementInertia Elementxx

–– Dissipate EnergyDissipate Energy

+ , + ,2 1 2 1Df B x x B v v! - ! -! !

x2x1

fDfD

fD

x

f1

f2

f3

1 2 3ii

M x f f f f! ! - -2!!

M



–– Stores Kinetic EnergyStores Kinetic Energy

+ ,2 1x x-! !

ME 375 Handouts

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•• DampersDampers in Seriesin Series

Series ConnectionSeries Connection

x1 x2 x2x1

fDfD fDfD1B1 B2 BEQ



•• DampersDampers in Parallelin Parallel

Parallel ConnectionParallel Connection

x2x1

x1 x2

1

x2

fD

x1

fD

BEQ

fDfD B1

B2



ME 375 Handouts

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•• Newton’s Second LawNewton’s Second Law

Interconnection LawsInterconnection Laws

+ ,d M v M x f! !2!!

•• Newton’s Third LawNewton’s Third Law–– Action & Reaction ForcesAction & Reaction Forces

+ ,"

LinearMomentum

EXTii

M v M x fdt

! !2

x

KM

M



•• Displacement LawDisplacement Law

MK

Modeling StepsModeling Steps

•• Understand System Function, Define Problem, and Understand System Function, Define Problem, and Identify Input/Output VariablesIdentify Input/Output Variablesy p py p p

•• Draw Simplified Schematics Using Basic ElementsDraw Simplified Schematics Using Basic Elements•• Develop Mathematical Model (Diff. Eq.)Develop Mathematical Model (Diff. Eq.)

–– Identify reference point and positive direction.Identify reference point and positive direction.–– Draw FreeDraw Free--BodyBody--Diagram (FBD) for each basic element.Diagram (FBD) for each basic element.

Write Elemental Equations as well as InterconnectingWrite Elemental Equations as well as Interconnecting



–– Write Elemental Equations as well as Interconnecting Write Elemental Equations as well as Interconnecting Equations by applying physical laws. (Equations by applying physical laws. (Check: # eq = # unk)Check: # eq = # unk)

–– Combine Equations by eliminating intermediate variables. Combine Equations by eliminating intermediate variables. •• Validate Model by Comparing Simulation Results Validate Model by Comparing Simulation Results

with Physical Measurements with Physical Measurements

ME 375 Handouts

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•• EOM of a simple MassEOM of a simple Mass--SpringSpring--Damper SystemDamper System

Energy DistributionEnergy Distribution

" " " "( )M x Bx K x f t) ) !!! !

xK

M f

We want to look at the energy distribution of the system. How should we start ?

•• Multiply the above equation by the velocity termMultiply the above equation by the velocity term vv : : 34What have we done ?

•• Integrate the second equation w.r.t. time: Integrate the second equation w.r.t. time: 34What are we doing now ?

TotalContribution Contribution ContributionApplied Forceof Inertia of the Damper of the Spring

+ ,1 1 1 1t t t tM x x dt Bx x dt K x x dt f t v dt) ) !* * * *!! ! ! ! !

B



+ ,0 0 0 0

0 1

212 20 Total work done by the

applied force ( ) from time to

1 102 2

tt

t t t t

f tt t

Bx dtKE M x PE K x E

M x x dt Bx x dt K x x dt f t v dt

5*. ! . ! .

( ) ( ) ( ! (

66 6

* * * *!!

#$%$& #$%$& #$%$& #$%$&

ExamplesExamples



ME 375 Handouts

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Examples (Continued)Examples (Continued)






ME 375 Handouts

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•• Suspension SystemSuspension SystemMinimize the effect of the surface Minimize the effect of the surface roughness of the road on the drivers’ roughness of the road on the drivers’ comfortcomfort

Example Example ---- SDOF SuspensionSDOF Suspension–– Simplified Schematic (neglecting tire model)Simplified Schematic (neglecting tire model)

Define the reference position for the displacement of the Define the reference position for the displacement of the car as the position when the spring does not have any car as the position when the spring does not have any comfort. comfort. p p g yp p g ydeflection (i.e., the neutral position)deflection (i.e., the neutral position)



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SDOF SuspensionSDOF Suspension–– Draw FBDDraw FBD –– Apply Interconnection LawsApply Interconnection Laws



Q:Q: Since gravity is always present, is there a Since gravity is always present, is there a way to represent the suspension system by way to represent the suspension system by subtracting the effect of gravity? subtracting the effect of gravity?

SDOF Suspension (II)SDOF Suspension (II)•• Relative Displacement ApproachRelative Displacement Approach

Define the reference position as the position of the Define the reference position as the position of the car when the system is at rest in the gravity field,car when the system is at rest in the gravity field,

–– FBDFBD

car when the system is at rest in the gravity field, car when the system is at rest in the gravity field, i.e., the spring force balances the car’s weight.i.e., the spring force balances the car’s weight.



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SDOF Suspension (II)SDOF Suspension (II)–– Interconnection Laws & Interconnection Laws &

SimplificationSimplificationQ:Q: What are the differences between the two What are the differences between the two

models?models?

Q:Q: Do the two models represent the same Do the two models represent the same physical system? If they do, why are they physical system? If they do, why are they different?different?



•• Suspension SystemSuspension System

MDOF SuspensionMDOF Suspension–– Simplified Schematic (with tire model)Simplified Schematic (with tire model)



ME 375 Handouts

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MDOF SuspensionMDOF Suspension–– Draw FBDDraw FBD –– Apply Interconnection LawsApply Interconnection Laws



MDOF SuspensionMDOF Suspension–– Matrix FormMatrix Form



translational mechanical system

Documents

sdof suspension

me375 translation

x2

reference

energy

system

displacement

eom