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ME 375 Handouts
1
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
! !
" #! ! ! ! !$ %& '
! ( ! ( !
! !!
!
School of Mechanical EngineeringPurdue University
ME375 Translation - 2
( 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)
School of Mechanical EngineeringPurdue University
ME375 Translation - 3
•• 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
School of Mechanical EngineeringPurdue University
ME375 Translation - 4
•• 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
School of Mechanical EngineeringPurdue University
ME375 Translation - 5
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
School of Mechanical EngineeringPurdue University
ME375 Translation - 6
ME 375 Handouts
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•• SpringsSprings in Parallelin Parallel
Parallel ConnectionParallel Connection
x xx1 x2
fS
KEQ
x1 x2
fS1fSfSK1
K2
School of Mechanical EngineeringPurdue University
ME375 Translation - 7
•• 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
School of Mechanical EngineeringPurdue University
ME375 Translation - 8
–– 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
School of Mechanical EngineeringPurdue University
ME375 Translation - 9
•• DampersDampers in Parallelin Parallel
Parallel ConnectionParallel Connection
x2x1
x1 x2
1
x2
fD
x1
fD
BEQ
fDfD B1
B2
School of Mechanical EngineeringPurdue University
ME375 Translation - 10
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
School of Mechanical EngineeringPurdue University
ME375 Translation - 11
•• 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
School of Mechanical EngineeringPurdue University
ME375 Translation - 12
–– 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
School of Mechanical EngineeringPurdue University
ME375 Translation - 13
+ ,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
School of Mechanical EngineeringPurdue University
ME375 Translation - 14
ME 375 Handouts
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Examples (Continued)Examples (Continued)
School of Mechanical EngineeringPurdue University
ME375 Translation - 15
Examples (Continued)Examples (Continued)
School of Mechanical EngineeringPurdue University
ME375 Translation - 16
ME 375 Handouts
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Examples (Continued)Examples (Continued)
School of Mechanical EngineeringPurdue University
ME375 Translation - 17
•• 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)
School of Mechanical EngineeringPurdue University
ME375 Translation - 18
ME 375 Handouts
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SDOF SuspensionSDOF Suspension–– Draw FBDDraw FBD –– Apply Interconnection LawsApply Interconnection Laws
School of Mechanical EngineeringPurdue University
ME375 Translation - 19
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.
School of Mechanical EngineeringPurdue University
ME375 Translation - 20
ME 375 Handouts
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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?
School of Mechanical EngineeringPurdue University
ME375 Translation - 21
•• Suspension SystemSuspension System
MDOF SuspensionMDOF Suspension–– Simplified Schematic (with tire model)Simplified Schematic (with tire model)
School of Mechanical EngineeringPurdue University
ME375 Translation - 22
ME 375 Handouts
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MDOF SuspensionMDOF Suspension–– Draw FBDDraw FBD –– Apply Interconnection LawsApply Interconnection Laws
School of Mechanical EngineeringPurdue University
ME375 Translation - 23
MDOF SuspensionMDOF Suspension–– Matrix FormMatrix Form
School of Mechanical EngineeringPurdue University
ME375 Translation - 24