mathematical modeling 1

8/3/2019 Mathematical Modeling 1

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Prepared by Mrs. Asma Adeel 1

Dynamic System Modeling

What is virtual environment? Time and frequency domain.

What is mathematical modeling?

Applications of mathematical modeling

Electrical Circuits

Translational Mechanical Systems

Rotational Mechanical Systems

( with or without lossless gears )

Electromechanical Systems


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What is a mathematical model?

Amathematical model is an abstractmodelthat uses mathematical language todescribe the behavior of a system.

A mathematical model as 'arepresentation of the essential aspects ofan existing system (or a system to beconstructed) which presents knowledge of

that system in usable form'.
http://en.wikipedia.org/wiki/Model_%28abstract%29http://en.wikipedia.org/wiki/Model_%28abstract%29http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Systemhttp://en.wikipedia.org/wiki/Systemhttp://en.wikipedia.org/wiki/Systemhttp://en.wikipedia.org/wiki/Systemhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Model_%28abstract%29http://en.wikipedia.org/wiki/Model_%28abstract%29


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Background of Mathematical Modeling:

Used in need of controlling and optimizing a system.

Engineers build the descriptive model of the system as ahypothesis how a system would work or try to estimate that howan unforeseeable event could affect the system.

Then different softwares could be used to simulate a system.

A mathematical model usually describes a system by a set ofvariables and a set of equations that establish relationships betweenthe variables.

The values of the variables can be practically anything; real or integer

numbers, boolean values or strings, The variables represent some properties of the system, for example,measured system outputs often in the form of signals, timing data,counters, event occurrence.

The actual model is the set of functions that describe the relations

between the different variables.
http://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Boolean_algebrahttp://en.wikipedia.org/wiki/Signal_%28information_theory%29http://en.wikipedia.org/wiki/Signal_%28information_theory%29http://en.wikipedia.org/wiki/Boolean_algebrahttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Real_number


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Building Blocks of the Mathematical Model:

There are six basic groups of variables:

Independent

Variables andDecision

variables Exogenousvariables orparametersand constants

Input

variables

State

Variables

Random

Variables

OutputVariables

Input VariableTypes

The variables arenot independent ofeach other.

Both represent

objective andconstraints on thesystem.


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A review of Laplace Transform

Clarify the difference

between time domain andfrequency domain.


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Laplace

T

ransform

P

roperties


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Transfer Function:

write a general nth order, linear, time invariant differential

equation,and

nc(t) /dtn+ an-1 dn-1 c(t) / dtn-1+ .. + aoc(t) =

bmdmr(t) / dtm+ bm-1 d

m-1r(t) / dtm-1+ . + bor(t)

Taking Laplace of both the sides and put all the initial conditions tozero.

The required ration

C(s) / R(s) = G(s) = bm sm + bm-1 s

m-1+. b0

ansn + an-1 sn-1+ a0

The transfer function is represented as a block

R(s) C(s)bm s

m + bm-1 sm-1+. b0

ansn + an-1 s

n-1+ a0

C(s)= R(s) G(s)


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Electrical System Modeling


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Electrical Networks

Impedance admittance


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Example 2.6

Transfer function- single loop via differential

equation


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Example 2.7

Repeat previous example using mesh analysis

and node analysis and transform methodswithout writing differential equations.

Replace the circuitelements by thefrequency domaincircuit parameters


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Example 2.10

Transfer functions with the multiple loopelectrical circuits.


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A quick way to write network equations

Writing Equations Via Inspection

Sum of Impedances

around Mesh 1 I1 (s) -Sum of Impedances

common to twomeshes I

2 (s)

= Sum of applied voltages around Mesh 1

Sum of Impedancescommon to twomeshes

- I1 (s) + Sum of Impedancesaround mesh 2

I2 (s)

= Sum of applied voltages around Mesh 2


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Writing Nodal Equations by inspection

(Sum of admittances to node 1) V1(s)(Sum of admittances common to twonodes) V2(s) = Sum of applied currentsat node 1

-(Sum of admittances common to twonodes)V1(s)+ ( Sum of admittances at

node 2 ) V2(s) = Sum of applied currentsat node 2


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Example 2.11

Transfer Function with multiple nodes

Example 2.12

Transfer Function with multiple nodes Do it Yourself!!!!!!!


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Example 2.19

Write Down the mesh equations of the system


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Skill Assessment 2.6

Find the Transfer Function


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Example 2.12



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Translational MechanicalSystem Modeling


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Translational Mechanical Systems

Number of Equations of Motion= Degree of

Freedom


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Example 2.16


Free BodyDiagram

Transformed FreeBody Diagram

G(s) = X(s)/ F(s) = 1 / Ms2 + fvs + K


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Equations of Motion by Inspection

(Sum of impedances connected to motion atx1) X1(s) (Sum of impedances between x1and x2) X2(s) = (Sum of applied forces at x1)

- (Sum of impedances between x1 and x2)X1(s) + (Sum of impedances connected to themotion at x2) X2(s) = (Sum of applied forcesat x2)


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Example 2.17

Write the motion equations but do not solve


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Example 2.18



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Skill Assessment 2.8


X2(s) / F(s) = (3s+1) / s( s3+ 7s2+ 5s + 1)


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Example 2.18



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Rotational MechanicalSystem Modeling

Torque angular displacement and angular


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Torque, angular displacement and angularvelocity relationships for rotational Mechanical

Systems

Number of Equations of Motion= Degree of

Freedom


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(Sum of impedances connected to motionat 1) 1(s) (Sum of impedancesbetween 1 and 2) 2(s) = (Sum of

applied torques at

1)

- (Sum of impedances between 1 and 2)

1

(s) + (Sum of impedances connected tothe motion at 2) 2(s) = (Sum of appliedtorques at 2)


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Example No. 1


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Example No. 2


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Example No. 3

G(s) = 1 / 2s2 +s+1


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Example No. 4


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Example No. 5


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Gears

Give Mechanicaladvantage to rotationalsystems.

gears allow to match the

drive system to load forappropriate mechanicalpower coupling.

a trade between torque

and speed

example moving uphill

More torque ----- lesser

speed

A non linearity feature

with gears is Backlash

Here Lossless gearsare considered wheredamping and moment

of inertia is zero
http://en.wikipedia.org/wiki/Image:Gears_animation.gif


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Transfer Functions for Systemswith Gears

r1/r2 = N1/N2 = T1/T2 = 2/1

r1 1= r2 2

T1 1 = T2 2

Translational energy

of force timesdisplacement equals

the rotationalenergy times

angulardisplacement


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Rotational Mechanical SystemsDriven with gears

Reflection of impedances


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Rotational MechanicalImpedances can be reflected

through gear trains by multiplying

the impedances by the ratioNumber of teeth of gear on

destinationshaft

Number of teeth of gear on

Source Shaft


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Example No. 4

One degree of freedom


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Cascading of Gear trains

Equivalent Gear Ratios is the product

of the individual gear Ratios


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Electro - mechanical

system

xmFext IM

lga

ext

0jlg,a

je0jlg,a

ji

KCL (or KVL) Newtons law

Translation Rotation

Here voltages are denoted by e to avoid confusion with velocities

Electromechanical Systems


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Electromechanical Systems are Hybrid ofelectrical and mechanical systems.

examples are

antenna azimuth position control system

robot controls sun and star

trackers

computer disk

drive position

controls.


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Antenna Azimuth Position control System (1)


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Antenna Azimuth Position control System (2)

Load

Effect of Mechanical motion and


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Effect of Mechanical motion andMagnetic field interaction with the

magnetic fieldA current-carrying wire of

length lplaced at right angle in a

magnetic field B experiences a

magnetic force F, perpendicularto the wire.

Law of motors:

BLiF

Effect of Motion & Magnetic Field


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Effect of Motion & Magnetic FieldInteraction on Electrical Field

A current-carrying wireof length L moving in amagnetic field Bat avelocity generates avoltage eacross thewire.

Law of generators:

x

e = B l V


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Armature Controlled DC Servo Motor


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DC motor Driving a rotationalmechanical load


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Prepared by Mrs Asma Adeel 52

Torque-speed curves with anarmature voltage ea, as a

parameter

mathematical modeling 1

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