TAN Activity2 Section5.1
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Transcript of TAN Activity2 Section5.1
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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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Name: Ceferino Kevin A. Tan Course and Year: BSECE - 4EE 179.1 Section: 4! "a#orator$ Sc%edu&e: onda$ 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'-(
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 $#stem and its +ree&,od#
Diaram
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Name: Ceferino Kevin A. Tan Course and Year: BSECE - 4EE 179.1 Section: 4! "a#orator$ Sc%edu&e: onda$ 1'-(
• 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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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'-(
//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#$"5
v = ode(v0, t0, t, f) //read references for ode tool
clf
%lot&d(t, v)
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Name: Ceferino Kevin A. Tan Course and Year: BSECE - 4EE 179.1 Section: 4! "a#orator$ Sc%edu&e: onda$ 1'-(
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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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'-(
"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#$"5
0 = 0 //set initial %osition
%rie0 = 0 //set initial velocit
= ode(.0 %rie0, t0, t, f)
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8/18/2019 TAN Activity2 Section5.1
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Name: Ceferino Kevin A. Tan Course and Year: BSECE - 4EE 179.1 Section: 4! "a#orator$ Sc%edu&e: onda$ 1'-(
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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8/18/2019 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'-(
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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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'-(
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
$ample code 1
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Name: Ceferino Kevin A. Tan Course and Year: BSECE - 4EE 179.1 Section: 4! "a#orator$ Sc%edu&e: onda$ 1'-(
$ample code 2
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8/18/2019 TAN Activity2 Section5.1
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Name: Ceferino Kevin A. Tan Course and Year: BSECE - 4EE 179.1 Section: 4! "a#orator$ Sc%edu&e: onda$ 1'-(
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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Name: Ceferino Kevin A. Tan Course and Year: BSECE - 4EE 179.1 Section: 4! "a#orator$ Sc%edu&e: onda$ 1'-(
(. 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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Name: Ceferino Kevin A. Tan Course and Year: BSECE - 4EE 179.1 Section: 4! "a#orator$ Sc%edu&e: onda$ 1'-(
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$stem 2%en b is c%an)edF
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Name: Ceferino Kevin A. Tan Course and Year: BSECE - 4EE 179.1 Section: 4! "a#orator$ Sc%edu&e: onda$ 1'-(
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(