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ME 375 System Modeling and Analysis

Section 1 – Introduction

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Spring 2009School of Mechanical Engineering

Douglas E. AdamsAssociate Professor

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What We Will DoExample: vehicle speed control

θθ

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MODEL: MODEL: Solid mechanics, dynamics, circuits, heat transfer, Solid mechanics, dynamics, circuits, heat transfer, fluid mechanics and measurements with a clear fluid mechanics and measurements with a clear understanding of the understanding of the inputs, outputs and assumptionsinputs, outputs and assumptions

Disturbance(Must model this)

Excitation / input(Must model this)

θθ

MgtfBvvMMgBvtfvM

e

e

−≅+−−=

)(sin)(

&

&

(Must model this)(Must model this)

gfe )(

© 2009 D. E. AdamsME 375 – Introduction

Assumptions

Response / output Who gives us this parameter?We must estimate it.

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What We Will DoExample: vehicle speed control

θθ

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ANALYZE: ANALYZE: Differential equations, Differential equations, Laplace, Laplace, time series, time series, frequency frequency spectra, spectra, MatlabMatlab and common senseand common sense

Speed___ Desired speed___ Actual speed

Steady stateerrorApply disturbance

Steady state response

pEncounter steep grade

Transient response

Time© 2009 D. E. AdamsME 375 – Introduction

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What We Will DoExample: vehicle speed control

θθ

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CONTROL: CONTROL: Feedback, Feedback, feedforwardfeedforward, , P, PD, PI, PID, etcP, PD, PI, PID, etc..control with clear understanding of whatcontrol with clear understanding of whatneeds needs to to be changed be changed and and performance performance metricsmetrics

Gravitational force

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due to inclineActualSpeedSpeed

ModelControl

Law- Ensure stability- Reduce response time- Reduce error

© 2009 D. E. AdamsME 375 – Introduction

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Course OutlineIntroduction

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Components/elements

Connections/interconnects

Mechanical

Thermal

Electrical

Electro- ____Hydraulic

Input/output Vs.state-variable

models

Time & frequencydomain tools of

systems analysis

Feedback andsystem design

© 2009 D. E. AdamsME 375 – Introduction

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Modeling ErrorsWhat can happen when we choose the wrong model?

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© 2009 D. E. AdamsME 375 – Introduction

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Modeling ErrorsExample of failure due to un-modeled aeroelastic oscillations

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Errors in model can be of two types:yp

First type (qualitative errors): Should be nonlinear when it is linear, should be time-varying when it is time invariant.Second type (quantitative errors): Model is correct but parameters (numerical, units, etc.) have random or bias errors.

© 2009 D. E. AdamsME 375 – Introduction

Wings should not fall off of aircraft Temperature variations are critical

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ModelingExample: stop-and-go traffic jam dynamics

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Consider the following generic modeling example with g g g pstop-and-go traffic jam dynamics:

k=1 k=2 k=3 1st order

Element law:

Connection law:

( )k k k k kM v B v f t+ =&

[ ]1 1 1( ) ( ) ( )k k k kf t G d t d t+ + += ⋅ −

© 2009 D. E. AdamsME 375 – Introduction

[ ]1 1

1

( ) ( )

( )k k kG v t v t dt

f t A+ += −

=∫What effect does the

neighboring lane of traffichave on these vehicles?

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Interdisciplinary nature of ME 375Analogous systems

Models are the same regardless of the physical domain of interest

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= = =

CurrentCharge

==

yy& velocityTransl

displ. Transl.==

yy& velocityAng

displ. Ang.==

yy& rateFlow

Volume ==

yy&

We only have to understand one mathematical model, but fourphysical systems!

VoltageCurrent

==

uy

ForcevelocityTransl.

==

uy

TorquevelocityAng.

==

uy

PressurerateFlow

==

uy

Input, u Output, y)(tuyyy =++ &&&

© 2009 D. E. AdamsME 375 – Introduction

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Gas pulsations reduce efficiency, reduce valve life and introduce operating noise. There is a trade-off between performance and

Influence of Gas Pulsations in CompressorMulti-cylinder compressor noise, efficiency and reliability

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p g pnoise – faster valves provide greater capacity but more noise.

Current approaches for designing multi-cylinder compressors are empirical involving many iterations (prototypes) prior to manufacture.

Suction Discharge

Suction valve

manifoldg

manifold Discharge valve

Cylinder wall

Piston

© 2009 D. E. AdamsME 375 – Introduction

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The analytical model consists of several parts- First, the fluid mechanics of the suction manifold are modeled.- Second the thermodynamics of the cylinders are modeled

Flow Chart for Analytical Modeling ApproachExample: Air conditioning compressor

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Second, the thermodynamics of the cylinders are modeled.- Third, the valve dynamics are modeled to estimate the mass flow rates.- Fourth, the first three models are combined.- Fifth, the temperature and muffler along the suction pipe are considered.

OperatingparametersValve Dynamics m&

( )tp ,θ

muffler

evapora

Manifold Thermodynamicstor

© 2009 D. E. AdamsME 375 – Introduction

Similar towater bottleresonance

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Analytical ModelsExample: Air conditioning compressor

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11 22 ∂∂ mpp &&

Constant cross section

)(11 *22

022 θθδ

θ−−=

∂∂

−∂∂

rAm

tp

cp

r

)(1 *22

θθδ∂∂∂∂ mphphph &&

Variable cross section

)(22222 θθδθθθ

−=∂

+∂∂

−∂

−rbt

pc

prr

p

512 Hz 503 Hz 512 Hz

© 2009 D. E. AdamsME 375 – Introduction

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Wake Turbulence on AA587Courtesy: Dr. Nick Lieven, Department of Aerospace Engineering, University of Bristol, UK

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Wind

JAL – 747

9/11Ground Zero

AA – A300

Eddy vortices

JFK Runway

© 2009 D. E. AdamsME 375 – Introduction

Courtesy: Dr. Nick Lieven, Department of Aerospace Engineering, University of Bristol, UK

Runway

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Wake Turbulence on AA587Courtesy: Dr. Nick Lieven, Department of Aerospace Engineering, University of Bristol, UK

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Wind

JAL – 747

JFK Runway

AA – A300

Eddy vortices

Runway

© 2009 D. E. AdamsME 375 – Introduction

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Wake Turbulence on AA587Courtesy: Dr. Nick Lieven, Department of Aerospace Engineering, University of Bristol, UK

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Wind

JAL – 747Eddy vortices

JFK RunwayAA – A300

Eddy vortices

© 2009 D. E. AdamsME 375 – Introduction

Aircraft damageto tail

Air Traffic Control

Prevailing Winds

PilotReaction

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Time and Frequency AnalysisRelationship between time and frequency response characteristics

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We utilize time and frequency domain analysis to estimate the q y yfree/forced and transient/steady state linear responses.

Time: Initial portion is transient, final portion is steady state

Frequency: Initial portion is attributed to steady state, final portion is attributed to transient

spon

se

pons

e

portion is attributed to transient

© 2009 D. E. AdamsME 375 – Introduction

Time

Res

Frequency

Res

p

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Feedback Control The concept of feedback to enhance system performance

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We utilize feedback control approaches to achieve better ppperformance than the system can provide on its own:

- Make certain the response is stable- Increase the speed of response (bandwidth)- Reduce the error in the response of the system- Reduce the error due to external disturbances- Reduce the error due to external disturbances

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ActualDesired

ErrorDeal with

error System

Disturbances

© 2009 D. E. AdamsME 375 – Introduction

Desired error y