TAN Activity2 Section5.1

8/18/2019 TAN Activity2 Section5.1

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Name: Ceferino Kevin A. Tan Course and Year: BSECE - 4EE 179.1 Section: 4! "a#orator$ Sc%edu&e: onda$ 1'-(

Activit$ ': at%ematica& ode&in) of *%$sica&S$stems

Abstract+#vious&$, t%e oeration of %$sica& s$stems often invo&ve a &ot of

non&inearit$ and unredicta#i&it$. e norma&&$ refer to t%ese uncertainties as

nonidea&ities. /nderstandin) t%e e0ects of t%ese nonidea&ities can contri#ute

to si)nicant imrovement in %$sica& s$stems in terms of accurac$ and

&on)-term re&ia#i&it$.

To create e0ective mac%ines or %$sica& s$stems, understandin) t%e %$sica&

nonidea&ities and %o2 2e can miti)ate t%ese 2i&& $ie&d muc% more accurate

and muc% more e3cient s$stems. e t%en create mat%ematica& mode&s of

t%ese nonidea&ities to tae account of t%eir e0ects in our %$sica& s$stems.

e norma&&$ tae account of t%e nonidea&ities of %$sica& s$stems into our

ca&cu&ations #$ t%e use of t%e ordinar$ di0erentia& e5uations, 2%ic% tae

mode&ed nonidea&ities, sa$, t%e corio&is e0ect-on a &on)-ran)e #u&&et, or t%e

srin) constant of a nonidea& srin).

6n t%is activit$, 2e 2i&& mae use of t%e caa#i&ities of Sci"a# to erform

mat%ematica& oerations invo&vin) ordinar$ di0erentia& e5uations, 2%ic% 2i&&

#e crucia& in #ui&din) e0ective contro& s$stems of %$sica& s$stems andor

mac%iner$.

1 Objectives

• To )ras t%e imortant ro&e of mat%ematica& mode&s of %$sica&

s$stems in t%e desi)n and ana&$sis of contro& s$stems.• To &earn %o2 Sci&a# %e&s in so&vin) suc% mode&s.

2 List of equipment/software

• *ersona& Comuter

• Sci&a#

3 Deliverables

• Sci&a# scrits and t%eir resu&ts for a&& t%e assi)nments and e8ercises

roer&$ discussed and e8&ained.• Ana&$tica& conc&usion for t%is &a# activit$.


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4 Modelin !"#sical $#stems

4%1 Mass&$prin $#stem Model

Consider t%e fo&&o2in) ass -Srin) s$stem s%o2n in i)ure 1 2%ere ' is t%e

srin) force, f v is t%e friction coe3cient, ()t* is t%e dis&acement and f)t* is

t%e a&ied force:

eferrin) to t%e free-#od$ dia)ram, 2e )et

M d

2 x (t )

d t 2 +f v

dx (t )dt

+ Kx(t )=f ( t ) . )1*

T%e second order &inear di0erentia& e5uation ;1< descri#es t%e re&ations%i

#et2een t%e dis&acement and t%e a&ied force. T%e di0erentia& e5uation

can t%en #e used to stud$ t%e time #e%avior of ()t* under various c%an)es

of t%e a&ied force.

T%e o#=ectives #e%ind mode&in) t%e mass-damer s$stem can #e man$ and

ma$ inc&ude:

• /nderstandin) t%e d$namics of suc% s$stem.

+iure 1% Mass&$prin $#stem and its +ree&,od#

Diaram


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• Stud$in) t%e e0ect of eac% arameter on t%e s$stem suc% as mass M,

t%e friction coe3cient ,, and t%e e&astic c%aracteristic '()t*.• >esi)nin) a ne2 comonent suc% as damer or srin).

• eroducin) a ro#&em in order to su))est a so&ution.

4%2 -sin $cilab in $olvin Ordinar# Di.erential quations

Sci&a# can %e& so&ve &inear or non&inear ordinar$ di0erentia& e5uations ;+>E<

usin) ode too&. To s%o2 %o2 $ou can so&ve +>E usin) Sci&a#, 2e 2i&& roceed

in t2o 2a$s. e rst see %o2 2e can so&ve a rst order +>E and t%en see

%o2 2e can so&ve a second order +>E.

4% 3 +irst Order OD0 $peed ruise ontrol (ample

Assume a ?ero srin) force 2%ic% means t%at ' . E5uation ;1< #ecomes

M d

2 x ( t )

d t 2 +f v

dx ( t )dt =f (t ) )2*

or

M dv (t )

dt +f v v (t )=f (t ) )3*

since

a(t )=dv (t )

dt =

d2 x (t )

d t 2 ,

and v (t )=dx (t )

dt .

E5uation ;(< is a rst order &inear +>E and 2e can use Sci&a# to so&ve for t%is

di0erentia& e5uation.

e can mode& and so&ve for e5uation ;(< #$ 2ritin) t%e fo&&o2in) scrit in

Sci&a#. You can a&so inut t%e codes direct&$ into t%e Sci&a# Conso&e.

Code:


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//declare constant values

M = 750

B = 30

Fa = 300 //the force f(t)

//differential equation definition

function dvdt = f(t,v)

dvdt = (Fa-B*v)/M

endfunction

//initial conditions ! t=0, v=0

t0 = 0

v0 = 0

t = 0"0#$"5

v = ode(v0, t0, t, f) //read references for ode tool

clf

%lot&d(t, v)


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4% 4 $econd Order OD0 Mass&$prin $#stem (ample

6n rea&it$, t%e srin) force andor friction force can %ave a more com&icated

e8ression or cou&d #e reresented #$ a )ra% or data ta#&e. or instance, a

non&inear srin) can #e desi)ned suc% t%at t%e e&astic c%aracteristic is

K xr (t )

2%ere r @ 1. i)ure ' is an e8am&e of a non&inear srin).

6n suc% case, e5uation ;1<

#ecomes

M d

2 x (t )

d t 2 +f v

dx (t )dt

+ K x r( t )=f (t ) )4*

E5uation ;(< reresents anot%er ossi#&e mode& t%at descri#es t%e d$namic

#e%avior of t%e mass-damer s$stem under e8terna& force. /n&ie e5uation;1


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"et x (t )= X 1 , sod X

1

dt = X

2 anddx( t )

dt = X

2 ,

sod X

2

dt =

−f v M

X 2−

K

M X

1

r+f (t ) M .

6n vector form, &et X = X

1

X 2

=[ x x ' ] 7 dX dt =[ d X

1

dt

d X 2

dt ]=[ x ' x ' ' ]

so 2e )et

dX

dt

=

[ X

2

−f v M X 2−

K

M X 1r

+f ( t ) M

] .

T%e Sci&a# ode too& can no2 #e used:

'ode"

//declare constant values

M = 750

B = 30

Fa = 300 //the force f(t) = $5

r = $

//differential equation definition

function %rie = f(t,)

%rie($) = (&)

%rie(&) = -(B/M)*(&)-(/M)*((($))+r)(Fa/M)

endfunction

t0 = 0

t = 0"0#$"5

0 = 0 //set initial %osition

%rie0 = 0 //set initial velocit

= ode(.0 %rie0, t0, t, f)


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clf

%lot&d(t, ($,")) // ($,")1 %lots the %osition vs tie

// (n, ")1 to %lot a solution in atri

for

// 2ith diension n


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8 Assessment

8%1 Assinment

1. S%o2 and discuss t%e )ra%s o#tained from t%e sam&e simu&ations.

T%e rst )ra% s%o2s

t%e ve&ocit$-time

re&ation of an o#=ect,

%avin) a mass of 7!

), #ein) inuenced

#$ a force of a (N movin) at one

direction, a&& 2%i&e

under t%e coe3cient

of friction of

ma)nitude ( Nm. T%e o#=ectDs )iven

initia& condition is vo ms at to s.

At rst )&ance, t%e function aears to#e &inear, 2%en in fact, it is not.e used t%e ode function ;ordinar$

di0erentia& e5uation


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y (t 0)= y0

T%e second sam&e code $ie&ds t%e )ra%s%o2n to t%e ri)%t.

T%is )ra% &ots t%e osition of an$ ointin t%e srin) 2it% resect to time.

Notice t%at t%e rate #$ 2%ic% t%e ositionc%an)es over time increases, %ence t%e

u2ard ara#o&ic c%aracteristic of t%e)ra%.

$creens"ots

$ample code 1


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$ample code 2


12/16!osition vs% 9ime


8%2 (ercise 1

eferrin) to t%e mass-srin) s$stem e8am&e:

1. *&ot t%e osition and t%e seed of t%e s$stem #ot% 2it% resect to time

in searate )ra%s.

:rap" of position vs time

:rap" of velocit# vs time

'. C%an)e t%e va&ue of r to ' and (. *&ot t%e osition and seed #ot% 2it%

resect to time. >iscuss t%e resu&ts.

;"en r2


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(. it% r 1, var$ t%e va&ue of ' ;mu&ti&$ #$ ! and 1< and discuss t%e

resu&ts.;it" ' >8

;it" ' 18

A "i"er sprin constant will result in a ?bouncier@ sprin% 9"at is a

"i"er sprin constant results in an increased frequenc# of

oscillation of t"e sprin over a period of time% =t is also noticeable

t"at t"e rane of motion "as decreased for increasin values of t"e

sprin constant '%


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8%3 (ercise 2

Consider t%e mec%anica& s$stem deicted in t%e )ure.

T%e inut is )iven #$ f(t), and t%e outut is

)iven #$ y(t). >etermine t%e di0erentia&

e5uation )overnin) t%e s$stem and usin) Sci&a#,

2rite a scrit and &ot t%e s$stem resonse suc%

t%at forcin) function f(t) = 1. "et m = 10, k = 1, and b

= 0.5.

a. T%e ea dis&acement of t%e #&oc is a#out .#. At time t , t%e outut reac%es ea dis&acement.c. 6f k = 10, t%e ea va&ue of $ is .d. %at %aens to t%e s$stem 2%en b is c%an)edF


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B Ceferences

C6SE (' "inear Contro& S$stems "a#orator$ anua&. ;'11, +cto#ereartment, Kin) a%d /niversit$ of *etro&eum Ginera&s.

Sci&a# Enterrises . ;'1(

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TAN Activity2 Section5.1