ME 375 Handouts 1 Translational Mechanical Systems Translational Mechanical Systems • Basic (Idealized) Modeling Elements Basic (Idealized) Modeling Elements • Basic (Idealized) Modeling Elements Basic (Idealized) Modeling Elements • Interconnection Relationships Interconnection Relationships -Physical Laws Physical Laws • Derive Equation of Motion (EOM) Derive Equation of Motion (EOM) - SDOF SDOF • Energy Transfer Energy Transfer • Series and Parallel Connections Series and Parallel Connections School of Mechanical Engineering Purdue University ME375 Translation - 1 • Derive Equation of Motion (EOM) Derive Equation of Motion (EOM) - MDOF MDOF Variables Variables • x : displacement displacement [m] [m] d x x v ! • x : displacement displacement [m] [m] • v : velocity velocity [m/sec] [m/sec] • a : acceleration acceleration [m/sec [m/sec 2 ] • f : force force [N] [N] • p : power power [Nm/sec] [Nm/sec] • w : work ( energy ) work ( energy ) [Nm] [Nm] 2 2 x x v dt d d d d v v x x x a dt dt dt dt d p f v f x w dt ! ! " # ! ! ! ! ! $ % & ’ ! ( ! ( ! ! !! ! School of Mechanical Engineering Purdue University ME375 Translation - 2 1 [Nm] = 1 [J] (Joule) 1 [Nm] = 1 [J] (Joule) 1 0 1 0 1 0 0 () ( ) () ( ) ( ) t t t t wt wt p t dt wt f x dt ! ) ! ) ( * * !
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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]
•• 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
7
•• 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
8
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
9
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)
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?
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.
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?