ME375 Dynamic System Modeling and Control

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MESB 374 System Modeling and AnalysisTranslati
onal Mechanical System
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Translational Mechanical Systems

• Basic (Idealized) Modeling Elements
• Interconnection Relationships -Physical Laws
• Derive Equation of Motion (EOM) - SDOF
• Energy Transfer
• Series and Parallel Connections
• Derive Equation of Motion (EOM) - MDOF

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Key Concepts to Remember

• Three primary elements of interest
• Mass ( inertia ) M
• Stiffness ( spring ) K
• Dissipation ( damper ) B
• Usually we deal with equivalent M, K, B
• Distributed mass ? lumped mass
• Lumped parameters
• Mass maintains motion
• Stiffness restores motion
• Damping eliminates motion

(Kinetic Energy)
(Potential Energy)
(Eliminate Energy ? )
(Absorb Energy )
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Variables

• x displacement m
• v velocity m/sec
• a acceleration m/sec2
• f force N
• p power Nm/sec
• w work ( energy ) Nm
• 1 Nm 1 J (Joule)

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Basic (Idealized) Modeling Elements

• Spring
• Stiffness Element

• Reality
• 1/3 of the spring mass may be considered into the
  lumped model.
• In large displacement operation springs are
  nonlinear.

Linear spring ? nonlinear spring ? broken spring
!!

• Idealization
• Massless
• No Damping
• Linear

• Stores Energy

Hard Spring
Potential Energy
Soft Spring
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Basic (Idealized) Modeling Elements

• Damper
• Friction Element

• Mass
• Inertia Element

• Dissipate Energy

fD

• Stores Kinetic Energy

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Interconnection Laws

• Newtons Second Law

Lumped Model of a Flexible Beam

• Newtons Third Law
• Action Reaction Forces

Massless spring
E.O.M.
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Modeling Steps

• Understand System Function, Define Problem, and
  Identify Input/Output Variables
• Draw Simplified Schematics Using Basic Elements
• Develop Mathematical Model (Diff. Eq.)
• Identify reference point and positive direction.
• Draw Free-Body-Diagram (FBD) for each basic
  element.
• Write Elemental Equations as well as
  Interconnecting Equations by applying physical
  laws. (Check eq unk)
• Combine Equations by eliminating intermediate
  variables.
• Validate Model by Comparing Simulation Results
  with Physical Measurements

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Vertical Single Degree of Freedom (SDOF) System
g

• Define Problem

The motion of the object

• Input
• Output

• Develop Mathematical Model (Diff. Eq.)
• Identify reference point and positive direction.

• Draw Free-Body-Diagram (FBD)

• Write Elemental Equations

From the undeformed position
From the deformed (static equilibrium) position

• Validate Model by Comparing Simulation Results
  with Physical Measurement

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Energy Distribution

• EOM of a simple Mass-Spring-Damper System
• We want to look at the energy distribution of
  the system. How should we start ?
• Multiply the above equation by the velocity term
  v Ü What have we done ?
• Integrate the second equation w.r.t. time
  Ü What are we doing now ?

f
Change of kinetic energy
Change of potential energy
Energy dissipated by damper
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Example -- SDOF Suspension (Example)

• Simplified Schematic (neglecting tire model)

• Suspension System
• Minimize the effect of the surface roughness of
  the road on the drivers comfort.

From the absolute zero
From the path
From nominal position
g
M
x
K
B
x
x
p
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Series Connection

• Springs in Series

fS
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Series Connection

• Dampers in Series

Û
fD
fD
BEQ
B1
B2
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Parallel Connection

• Springs in Parallel

fS
fS
Û
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Parallel Connection

• Dampers in Parallel

Û
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Horizontal Two Degree of Freedom (TDOF) System

• DOF 2

• Absolute coordinates

• FBD

K
K

• Newtons law

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Horizontal Two Degree of Freedom (TDOF) System
Static coupling

• Absolute coordinates

• Relative coordinates

Dynamic coupling
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Two DOF System Matrix Form of EOM
K
K
Input vector

• Absolute coordinates

Output vector
Mass matrix
Damping matrix
SYMMETRIC

• Relative coordinates

Stiffness matrix
NON-SYMMETRIC
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MDOF Suspension

• Simplified Schematic (with tire model)

• Suspension System

TRY THIS
x
p
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MDOF Suspension

• Simplified Schematic (with tire model)

• Suspension System

Assume ref. is when springs are Deflected by
weights
Car body
Suspension
Wheel
Tire
Road
Reference
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Example -- MDOF Suspension

• Draw FBD

• Apply Interconnection Laws

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Example -- MDOF Suspension

• Matrix Form

Mass matrix
Damping matrix
Stiffness matrix
Input Vector