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"imf sdluimrpdys bs. f" ?

(e) Z`ie per`kj (Z) 7 Rimaaest t`ie `ftervma mcter wd`bd tde ksb`aamtkry ikt`kf nets repemtej `s bmaaej t`ie

per kj, Z 9?

9li

?

Exmipae =. Ckr m pmrt`bae perckri`fn RDI, equmt kf kc ikt`kf s n`vef ms ?

?

jt

xj+ :x 9 8. C`fj tde t ie per`kj.

Rkaut`kf 7 ??

jtxj

9 :x ? 9 : 9 ?

Z ie per`kj< Z 9?

9

(c) Xdmse 7 Zde pdys`bma qumft`ty wd`bd represefts tde stmte kc ikt`kf kc pmrt`bae (en. `ts pks`t`kf mfjj`rebt`kf kc ikt`kf mt mfy `fstmft).Zde mrnuieft ( t + ) kc s`fusk`jma cufbt`kf `s bmaaej `fstmftmfekus pdmse kc tde ikt`kf.

(n) Xdmse bkfstmft ( ) 7 Bkfstmft `f equmt`kf kc RDI `s bmaaej pdmse bkfstmft kr `f`t`ma pdmse.@t jepefjs kf `f`t`ma pks`t`kf mfj j`rebt`kf kc veakb`ty.

(d) Seakb`ty(v) 7 Seakb`ty mt mf `fstmft `s tde rmte kc bdmfne kc pmrt`baes pks`t`kf w.r.t t`ie mt tdmt `fstmft.Aet tde j`spambeieft crki iemf pks`t`kf `s n`vef oy

x 9M s`f t +

Seakb`ty, v 9jtjx

9 )t Ms`f(jtj

v 9 M bks t + kr, v 9 ?? x M Mt iemf pks`t`kf (x 9 8), veakb`ty `s imx`iui.

vimx 9 M Mt extreie pks`t`kf (x 9 M), veakb`ty `s i`f`iui.

vi`f

9 zerk

N_MXD KC RXE EJ v SR J@RXAMBE IE FZ x 7

?? x Mv ???? x Mv

v? + ?x? 9 ? M? = M

x

M

v?

?

??

?

N_MXD ^KUAJ OE MF EAA@XRE

Rpeej (v)

M M x

(`) Mbbeaermt`kf 7 Mbbeaermt`kf mt mf `fstmft `s tde rmte kc bdmfne kc pmrt`baes veakb`ty w.r.t. t`ie mt tdmt`fstmft.

Mbbeaermt`kf, m 9 jtjv

9 Y)tbks( M\jtj

m 9 ? M s`f ( t + m 9 ?x

Fkte Fenmt`ve s`nf sdkws tdmt mbbeaermt`kf `s mawmys j`rebtej tkwmrjs tde iemf pks`t`kf. Mt iemf pks`t`kf (x 9 8), mbbeaermt`kf `s i`f`iui.

mi`f 9 zerk Mt extreie pks`t`kf (x 9 M), mbbeaermt`kf `s imx`iui.

mimx 9 ? M

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"imf sdluimrpdys bs. f" 2

N_MXD KC MBBE AE_ MZ@KF M SR J@RXAMBEI EFZ x

m 9 ?xx M

M

m

Exmipae ?. Zde equmt kf kc pmrt`bae exebut fn s`ipae dmrikf`b ikt`kf `s x 9 (5 i) s f

2t)s( = . ^r`te

jkwf tde mipa`tuje, t`ie per`kj mfj imx iui speej. Mask c`fj tde veakb`ty mt t 9 = s.Rkaut`kf 7 Bkipmr fn w`td equmt kf x 9 M s`f ( t + ),

we see tdmt tde mipa`tuje 9 5 i,

mfj t`ie per`kj 9?

9 =s

?9 ?s.

Zde imx iui speej 9 M 9 5 i s = 9 5 i/s.

Zde veakb`ty mt t`ie t 9jtjx

9 M M bks ( t + )

Mt t 9 = s,

v 9 (5 i) ( s =) bks

2 9 ?5

i/s.

Exmpae 2. M pmrt`bae exebut fn s`ipae dmrikf b ikt`kf dms mfnuamr crequefby 3.?4 s = mfj mipa tuje =8 bi.C`fj (m) tde t`ie per`kj, (o) tde imx`iui speej, (b) tde imx`iui mbbeaermt`kf, (j) tde speej wdeftde j`spambeieft `s 3 bi crki tde iemf pks`t`kf, (e) tde speej mt t 9 =/3 s mssui`fn tdmt tdeikt`kf stmrts crki rest mt t 9 8.

Rkaut`kf 7 (m) Z`ie per`kj 9?

9?4.3

?s 9 = s.

(o) Imx`iui speej 9 M 9 (8.= i) (3.?4 s =)9 8.3?4 i/s.

(b) Imx iui mbbeaermt`kf 9 M ?

9 (8.= i) (3.?4 s =)?

9 : i/s ? .

(j) v 9 ?? x M 9 (3.?4 s =) ?? )bi3()bi=8( 9 58.? bi/s.

(e) Mt t 9 8, tde veakb`ty `s zerk `.e., tde pmrt bae `s mt mf extreie. Zde equmt`kf ckr j spambeieftimy oe wr`ttef ms

x 9 M bks t.Zde veakb`ty s v 9 M s`f t.

Mt t 93=

s, v 9 (8.= i) (3.?4 s =) s`f

3?4.3

9 ( 8.3?4 i/s) s`f2

9 5:.: bi/s.

Exmipae :. M pmrt`bae stmrts crki iemf pks`t`kf mfj ikves tkwmrjs pks`t ve

extreie ms sdkwf. C`fj tde equmt kf kc tde RDI. Mipa`tuje kc RDI `s M.

t 9 88 M-M

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"imf sdluimrpdys bs. f" :

Rkaut`kf 7 Neferma equmt kf kc RDI bmf oe wr`ttef ms x 9 M s`f ( t + ) Mt t 9 8, x 9 8

8 9 M s`f

9 8, )?,8\ Mask< mt t 9 8, v 9 ve

M bks 9 vekr, 9 8Defbe, `c tde pmrt`bae `s mt iemf pks`t`kf mt t 9 8 mfj `s ikv`fn tkwmrjs +ve extreie, tdef tdeequmt`kf kc RDI `s n`vef oy x 9 M s`f tR`i`amrayckr

t 9 8

8 M-M9

equmt`kf kc RDI `s x 9 M s`f( t + )kr, x 9 M s`f t

Fkte 7

@c iemf pks`t`kf `s fkt mt tde kr`n`f, tdef we bmf repambe x oy x x 8 mfj tde eqf.oebkies x x 8 9 M s`f t + , wdere x 8 `s tde pks`t`kf bk-krj`fmte kc tde iemf pks`t`kf.

Exmipae 5. M pmrt bae `s perckri fn RDI kc mipa`tuje M mfj t ie per`kj Z. C fj tde t ie tmlef oy tde pmrt`baetk nk crki 8 tk M/?.

Rkaut`kf 7 Aet equmt`kf kc RDI oe x 9 M s`f twdef x 9 8 , t 9 8wdef x 9 M/? < M/? 9 M s`f t

kr s`f t 9 =/? t 9 /3

Z?

t 9 /3 t 9 Z/=?

Defbe , t`ie tmlef `s Z/=?, wdere Z `s t`ie per`kj kc RDI.

Exmipae 3. M pmrt`bae kc imss ? ln `s ikv`fn kf m strm`ndt a`fe ufjer tde mbt`kf kc ckrbe C 9 (4 ?x) F. @t `sreaemsej mt rest crki x 9 3 i.(m) @s tde pmrt`bae ikv fn s ipae dmrikf bmaay.(o) C`fj tde equ`a or`ui pks`t kf kc tde pmrt`bae.(b) ^r`te tde equmt`kf kc ikt kf kc tde pmrt bae.(j) C`fj tde t ie per`kj kc RDI.

Rkaut`kf 7 C 9 4 ?xkr C 9 ?(x :)mt equ`a`or`ui pks`t`kf C 9 8

x 9 : `s equ`a`or`ui pks`t`kfDefbe tde ikt`kf kc pmrt`bae `s RDI w`td ckrbe bkfstmft ? mfj equ`a`or`ui pks`t`kf x 9 :.(m) Pes, ikt kf `s RDI.(o) Equ`a`or ui pks t`kf `s x 9 :(b) Mt x 9 3 i, pmrt`bae `s mt rest .e. `t `s kfe kc tde extreie pks`t`kf

8 x9: x93

v98

Defbe mipa`tuje `s M 9 ? i mfj `f`t`maay pmrt bae `s mt tde extreie pks`t`kf. Equmt`kf kc RDI bmf oe wr`ttef ms

x : 9 ? bks t , wdere 9il

9??

9 =

`.e. x 9 : + ? bks t

(j) Z ie per kj, Z 9?

9 ? seb.

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"imf sdluimrpdys bs. f" 5

: . RDI MR M X_KGEBZ@KF KC UF@CK_I B@_BUAM_ IKZ@KF

Bkfs jer m pmrt`bae ikv`fn kf m b`rbae kc rmj`us M w`td mbkfstmft mfnuamr speej ms sdkwf `f c`nure.

x(t)

v (t)

v (t)

m (t)x

x

x

K x

X

X

8

]

- M + M8x(t)

Ruppkse tde pmrt`bae s kf tde tkp kc tde b`rbae (P-mx`s) mtt 9 8. Zde rmj`us KX imles mf mfnae 9 t w`td tde P-mx`s mt t`ie t. Jrkp m perpefj`buamr X] kf T-mx`s. Zde

bkipkfefts kc pks`t kf vebtkr, veakb ty vebtkr mfj mbbea-ermt`kf vebtkr mt t`ie t kf tde T-mx`s mre

x(t) 9 M s`f tvx(t) 9 M bks tmx(t) 9 ? M s`f t

Mokve equmt`kfs sdkw tdmt tde ckkt kc perpefj`buamr ] exebutes m s`ipae dmrikf`b ikt`kf kf tde T-mx`s.Zde mipa tuje `sM mfj mfnuamr crequefby `s . R`i amray tde ckkt kc perpefj`buamr kf P-mx s w`aa mask exebuteRDI kc mipa`tuje M mfj mfnuamr crequefby \y(t) 9 M bks t . Zde pdmses kc tde twk s`ipae dmrikf`bikt`kfs j`ccer oy /?.

5. N_MXD@BMA _EX_EREFZMZ@KF KC J@RXAMBEIEFZ, SEAKB@ZP MBBEAE_MZ@KF @F RDIJ`spambeieft, x 9 M s`f t

Seakb`ty, v 9 M bks t 9 M s`f t +?

) kr v 9 ?? x M Mbbeaermt`kf, m 9 ? M s`f t 9 ? M s`f t + ) kr m 9 ?x

Fkte 7 v 9 ?? x M m 9 ?x

Zdese reamt`kfs mre true ckr mfy equmt`kf kc x.

t`ie, t 8 Z/: Z/? 2Z/: Z

j`spambeieft, x 8 M 8 M 8Seakb`ty, v M 8 M 8 M

mbbeaermt`kf, m 8 ? M 8 ? M 8

Z/: Z/? 2Z/: Z 5Z/: 2Z/?x

v

m

t

t

t

M

M

=. Maa tde tdree qumft t`es j`spambeieft, veakb ty mfj mbbeaermt kf vmry dmrikf`bmaay w td t`ie, dmv`fnsmie per`kj.

?. Zde veakb`ty mipa`tuje `s t`ies tde j`spambeieft mipa`tuje (v imx 9 M).

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"imf sdluimrpdys bs. f" 3

2. Zde mbbeaermt`kf mipa`tuje `s ? t`ies tde j`spambeieft mipa`tuje (m imx 9 M).

:. @f RDI, tde veakb`ty `s mdemj kc j`spambeieft oy m pdmse mfnae kc ?

.

5. @f RDI, tde mbbeaermt`kf `s mdemj kc veakb`ty oy m pdmse mfnae kc ?

.

3 . EFE_NP KC RDI3.= L fet`b Eferny (LE)

?=

iv ? 9?=

i ? (M? x?) 9?=

l (M? x?) (ms m cufbt`kf kc x)

9?=

i M M? ? bks ? ( t + ) 9?=

LM? bks ? ( t + ) (ms m cufbt`kf kc t)

imxLE 9 ?=

lM ? < Z8LE 9 :=

lM ? < M8LE 9 2=

lM ?

Crequefby kc LE 9 ? (crequefby kc RDI)

3.? Xkteft`ma Eferny (XE)

?=

Lx ? (ms m cufbt kf kc x) 9?=

lM ? s`f ? ( t + ) (ms m cufbt`kf kc t`ie)

3.2 Zktma Iebdmf`bma Eferny (ZIE)Zktma iebdmf`bma eferny 9 L`fet`b eferny + Xkteft`ma eferny

9?=

l (M? x?) +?=

Lx ? 9?=

LM?

Defbe tktma iebdmf`bma eferny `s bkfstmft `f RDI.3.: Nrmpd`bma Smr`mt`kf kc eferny kc pmrt`bae `f RDI.

MK

E

M x

XE

ZELE

M/ ? M/ ?

Exmipae >. M pmrt`bae kc imss 8.58 ln exebutes m s`ipae dmrikf b ikt`kf ufjer m ckrbe C 9 (58 F/i)x. @c `tbrksses tde beftre kc ksb`aamt`kf w`td m speej kc =8 i/s, c`fj tde mipa`tuje kc tde ikt`kf.

Rkaut`kf 7 Zde l`fet`b eferny kc tde pmrt`bae wdef `t `s mt tde beftre kc ksb`aamt`kf `s

E 9?=

iv ? 9?=

(8.58 ln) (=8 i/s) ? 9 ?5 G.

Zde pkteft ma eferny `s zerk dere. Mt tde imx`iui j spambeieft x 9 M, tde speej `s zerk mfj defbe

tde l`fet`b eferny `s zerk. Zde pkteft`ma eferny dere `s?=

lM ? . Ms tdere `s fk akss kc eferny, ,

?=

lM ? 9 ?5 G .............(`)

Zde ckrbe kf tde pmrt`bae `s n`vef oyC 9 (58 F/i)x.

Zdus, tde spr`fn bkfstmft `s l 9 58 F/i.

Equmt`kf (`) n`ves

?=

(58 F/i) M M? 9 ?5 G kr, M 9 = i.

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"imf sdluimrpdys bs. f" >

>. RX_@FN-IMRR RPRZEI

(=)

sikktd surcmbe

Z 9 ?li

(?) l

i

Z 9 ?li

(2) @c spr`fn dms imss i s tdef

9

Z 9 ?

l 2

ii s \Fkt `f GEE, ckr ktder exmisY

Exmipae 4. M pmrt`bae kc imss ?88 n exebutes m s ipae dmrikf`b ikt kf. Zde restkr fn ckrbe s prkv jej oy mspr`fn kc spr`fn bkfstmft 48 F/i. C`fj tde t`ie per`kj.

Rkaut`kf 7 Zde t ie per`kj `s

Z 9 ?li

9 ? i/F48ln=8?88 2

9 ? 8.85 s 9 8.2= s.

Exmipae 0. Zde cr`bt`kf bkecc`b`eft oetweef tde twk oakbls sdkwf `f c`nure `s mfj tde dkr`zkftma pamfe `ssikktd. (m) @c tde systei `s sa`ndtay j`spambej mfj reaemsej, c`fj tde t`ie per`kj. (o) C`fj tdeimnf tuje kc tde cr`bt`kfma ckrbe oetweef tde oakbls wdef tde j`spambeieft crki tde iemf pks t kf`s x. (b) ^dmt bmf oe tde imx`iui mipa`tuje `c tde upper oakbl jkes fkt sa`p reamt`ve tk tde akwer oakbl ;

Rkaut`kf 7 (m) Ckr simaa mipa tuje, tde twk oakbls ksb`aamte tknetder. Zde mfnuamr crequefby `s

9iI

l mfj sk tde t`ie per`kj Z 9 ?

liI

.

(o) Zde mbbeaermt`kf kc tde oakbls mt j`spambeieft x crki tde iemf pks`t kf `s

m 9 ?x 9

iIlx

Zde resuatmft ckrbe kf tde upper oakbl `s, tdereckre, im 9

iIilx

Zd s ckrbe `s prkv jej oy tde cr bt kf kc tde akwer oakbl. Defbe, tde imnf tuje kc tde cr bt kfmackrbe `s

iI|x|il

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"imf sdluimrpdys bs. f" 4

(b) Imx iui ckrbe kc cr bt kf requ`rej ckr s ipae dmrikf b ikt`kf kc tde upper oakbl `siI Mil

mt tde extreie pks t`kfs. Out tde imx`iui cr`bt`kfma ckrbe bmf kfay oe in. Defbe

iI Mil

9 in kr, M 9l

n)iI(

Exmipae =8. M oakbl kc imss i `s suspefjej crki tde be`a`fn kc m stmt`kfmry eaevmtkr tdrkund m spr fn kc spr`fn

bkfstmft l mfj sujjefay, tde bmoae oremls mfj tde eaevmtkr stmrts cmaa fn creeay. Rdkw tdmt oakbl fkwexebutes m s ipae dmrikf`b ikt`kf kc mipa tuje in/l `f tde eaevmtkr.

Rkaut`kf 7 ^def tde eaevmtkr `s stmt kfmry, tde spr`fn `s stretbdej tk suppkrttde oakbl. @c tde extefs`kf `s x, tde tefs`kf `s lx wd`bd sdkuajomamfbe tde we`ndt kc tde oakbl.

Zdus, x 9 in/l. Ms tde bmoae oremls, tde eaevmtkr stmrts cmaa`fn w`td mbbeaermt`kf n. ^e sdmaa wkrl`f tde crmie kc recerefbe kc tde eaevmtkr. Zdef wedmve tk use m pseujk ckrbe in upwmrj kf tde oakbl.Zd`s ckrbe w`aa omamfbe tde we`ndt. Zdus, tde oakbl `s suogebtej tk m fet ckrbe lx oy tde spr fn wdef`t `s mt m j`stmfbe x crki tde pks`t`kf kc ufstretbdej spr`fn. Defbe, `ts ikt`kf `f tde eaevmtkr `ss`ipae dmrikf`b w td `ts iemf pks`t`kf bkrrespkfj fn tk tde ufstretbdej spr fn. @f t`maay, tde spr fn

`s stretbdej oy x 9 in/l, wdere tde veakb`ty kc tde oakbl (w`td respebt tk tde eaevmtkr) `s zerk. Zdus,tde mipa`tuje kc tde resuat`fn s`ipae dmrikf`b ikt`kf `s in/l.

Exmipae ==. Zde aect oakbl `f c`nure bkaa`jes `feamst`bmaay w td tde r`ndt oakblmfj st bls tk `t. C fj tde mipa tuje kc tde resuat fn s ipae dmrikf`bikt`kf.

Rkaut`kf 7 Mssui`fn tde bkaa`s kf tk amst ckr m simaa `ftervma kfay, we bmf mppay tde pr fb`pae kc bkfservmt kf kc

ikieftui. Zde bkiikf veakb ty mcter tde bkaa s`kf `s?v

. Zde l`fet`b eferny 9?=

(?i)?

?v

9:=

iv ? . Zd s `s mask tde tktma eferny kc v`ormt kf ms tde spr`fn `s ufstretbdej mt td`s ikieft. @c tde

mipa`tuje `s M, tde tktma eferny bmf mask oe wr`ttef ms?= lM ? . Zdus,

?=

lM ? 9:=

iv ? , n`v`fnM 9 l?i

v..

Exmipae =?. Zwk oakbls kc imss i=

mfji?

mre bkffebtej w td m spr fn kc fmturma aefntd a mfj spr fn bkfstmft l.Zde systei s ay fn kf m sikktd dkr`zkftma surcmbe. @f t maay spr fn `s bkipressej oy x

8ms sdkwf `f

c`nure.

x8

Rdkw tdmt tde twk oakbls w`aa perckri RDI mokut tde`r equ`a`or`ui pks`t`kf. Mask (m) c`fj tde t`ieper`kj, (o) c`fj mipa`tuje kc embd oakbl mfj (b) aefntd kc spr fn ms m cufbt`kf kc t`ie.

Rkaut`kf 7 (m) Dere oktd tde oakbls w`aa oe `f equ`a`or ui mt tde smie t`ie wdef spr fn `s `f ts fmturma aefntd. AetEX

=mfj EX

?oe equ`a or`ui pks t`kfs kc oakbl M mfj O ms sdkwf `f c`nure.

EX EX= ?

EX EX==

??x x

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"imf sdluimrpdys bs. f" =8

////////////////////////// //////////////////////////

(i +i )nL

= ?

i nL

?

i nL=

F.A

E.X

Mt tde `f`t`ma pks`t`kf, s`fbe veakb`ty `s zerk `.e. `t `s tde extreie pks`t`kf.

Defbe Mipa`tuje 9L

ni =

Z`ie per`kj 9L

i? ?

R`fbe kfay oakbl kc imss i?

`s ksb`aamt`fn

4 . BKIO@FMZ@KF KC RX_@FNR4.= Rer es Bkio`fmt`kf 7

Zktma j`spambeieft x 9 x = + x?Zefs`kf `f oktd spr fns 9 l = x= 9 l ? x?

Equ`vmaeft spr`fn bkfstmft `f ser`es bkio`fmt`kf L eq `s n`vef oy 7

=/l eq 9 =/l = + =/l ? Z 9 ? eqli

Fkte 7

@f ser es bkio`fmt kf, tefs`kf `s smie `f maa tde spr`fns & extefs`kf w aa oe j ccereft. (@c l `s smietdef jeckrimt`kf `s mask smie )

@f ser`es bkio`fmt`kf , extefs`kf kc spr`fns w`aa oe reb`prkbma kc `ts spr`fn bkfstmft. Rpr`fn bkfstmft kc spr`fn `s reb`prkbma kc `ts fmturma aefntd

l =/ l = = 9 l ? ? 9 l 2 2

@c m spr`fn `s but `f f p`ebes tdef spr`fn bkfstmft kc kfe p`ebe w`aa oe fl.

4.? Xmrmaaea bkio`fmt kf 7Extefs`kf `s smie ckr oktd spr`fns out ckrbe mbt`fn w`aa oe j`ccereft.Ckrbe mbt`fn kf tde systei 9 C

C 9 (l = x + l ? x) C 9 (l = + l ? ) x C 9 l eq x

l eq 9 l = + l ? Z 9 ? eqli

0 . IEZDKJR ZK JEZE_I@FE Z@IE XE_@KJ, MFNUAM_ C_E]UEFBP @F R.D.I.(m) Ckrbe / tkrque ietdkj(o) Eferny ietdkj

Exmipae =:. Zde str`fn, tde spr`fn mfj tde puaaey sdkwf `f c`nure mre a ndt.C`fj tde t`ie per`kj kc tde imss i.

/////////////////////////

////////////////////////////

li

Rkaut`kf (m) Ckrbe IetdkjAet `f equ`a`or`ui pks`t`kf kc tde oakbl, extefs`kf `f spr`fn `s x 8.

lx 8 9 in -- (=) /////////////////////////

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XDPR@BR

"imf sdluimrpdys bs. f" ==

Fkw `c we j spambe tde oakbl oyx `f tde jkwfwmrjj`rebt`kf, fet ckrbe kf tde oakbl tkwmrjs iemfpks`t`kf `sC 9 l(x + x 8) in 9 lx us`fn (=)Defbe tde fet ckrbe `s mbt`fn tkwmrjs iemfpks`t`kf mfj `s mask prkpkrt`kfma tk x.Rk, tdepmrt`bae w`aa perckri R.D.I. mfj `ts t`ie per`kjwkuaj oe

Z 9 ?li

(o) Eferny IetdkjAet nrmv`tmt`kfma pkteft`ma eferny `s tk oe zerk mt tde aevea kc tde oakbl wdef spr`fn `s `f `ts fmturmaaefntd.Fkw mt m j`stmfbe x oeakw tdmt aevea, aet speej kc tde oakbl oe v.R`fbe tktma iebdmf`bma eferny `s bkfservej `f R.D.I.

inx +?=

lx ? +?=

iv ? 9 bkfstmft

J`ccereft`mt fn w.r.t. t`ie, we net

inv + lxv + ivm 9 8wdere m `s mbbeaermt kf.

C 9 im 9 lx + in kr C 9 l(x l

in)

Zd`s sdkws tdmt ckr tde ikt`kf, ckrbe bkfstmft `s l mfj equ`a`or`ui pks`t`kf `s x 9l

in.

Rk, tde pmrt bae w`aa perckri R.D.I. mfj ts t`ie per`kj wkuaj oe Z 9 ?li

=8. R@IXAE XEFJUAUI@c m demvy pk`ft imss `s suspefjej oy m we`ndtaess, `fextefs`oae mfjpercebtay caex`oae str`fn crki m r`n j suppkrt, tdef td s mrrmfneieft `s bmaaejm s ipae pefjuaui

Z`ie per kj kc m s`ipae pefjuaui Z 9 ? n(skie t`ies we bmf tmle n 9 ? ckr iml`fn bmabuamt`kf s`ipae)

Fkte 7

@c mfnuamr mipa`tuje kc s ipae pefjuaui s ikre, tdef t`ie per`kj

Z 9 ?

=3=

n

?8

(Fkt `f GEE, Ckr ktder exmis)

wdere 8 `s `f rmj`mfs. Neferma ckriuam ckr t`ie per kj kc s`ipae pefjuaui wdef `s bkipmrmoae tk rmj us kc Emrtd _.

Z 9 ?

=_ =

n

=

wdere, _ 9 _mj`us kc tde emrtd

Z`ie per`kj kc s ipae pefjuaui kc `fc f te aefntd `s imx iui mfj `s n`vef oy7 Z 9 ?n

_9 4:.3i`f

(^dere _ `s rmj`us kc emrtd)

Z`ie per`kj kc sebkfjs pefjuaui `s ? seb mfj 9 8.002 i.

R`ipae pefjuaui perckris mfnuamr R.D.I. out jue tk simaa mfnuamr j`spambeieft, `t `s bkfs`jerej

8/12/2019 SHM Theory E

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XDPR@BR

"imf sdluimrpdys bs. f" =?

ms a`femr R.D.I.

@c t ie per`kj kc bakbl omsej kf s ipae pefjuaui fbremses tdef bakbl w aa oe sakw out `c t ie per`kjjebremse tdef bakbl w`aa oe cmst.

@c n reim`fs bkfstmft & `s bdmfne `f aefntd, tdef =88?=

=88ZZ

@c reim`f bkfstmft & n `s bdmfne `f mbbeaermt`kf tdef, =88

n

n

?

==88

Z

Z

@c `s bdmfne `f aefntd & n `s bdmfne `f mbbeaermt`kf jue tk nrmv ty tdef,

=88nn

?=

?=

=88ZZ

Exmipae =5 M s ipae pefjuaui kc aefntd :8 bi ksb`aamtes w td mf mfnuamr mipa tuje kc 8.8: rmj. C`fj (m) tde t ieper`kj, (o) tde a femr mipa tuje kc tde oko, (b) tde speej kc tde oko wdef tde str fn imles 8.8? rmj w tdtde vert bma mfj (j) tde mfnuamr mbbeaermt`kfwdef tde oko `s `f ikieftmry rest. Zmle n 9 =8i/s ? .

Rkaut`kf 7 (m) Zde mfnuamr crequefby`s 9 /n

9i:.8

s/i=8 ?9 5 s =

tde t`ie per`kj `s?

9 =s5

?9 =.?3 s.

(o) A femr mipa tuje 9 :8 bi 8.8: 9 =.3 bi

(b) Mfnuamr speej mt j`spambeieft 8.8? rmj `s

9 (5 s =) ?? )8?.8()8:.8( rmj 9 8.=> rmj/s.

wdere speej kc tde oko mt td`s `fstmft

9 (:8 bi) 8.=>5 = 9 3.4 bi/s.

(j) Mt ikieftmry rest, tde oko `s `f extreie pks`t`kf.

Zdus, tde mfnuamr mbbeaermt kf

9 (8.8: rmj) (?5 s ?) 9 = rmj/s ? .

=8.= Z`ie Xer`kj kc R ipae Xefjuaui `f mbbeaermt`fn _ecerefbe Crmie 7

Z 9 ? .ecc n wdere

n ecc. 9 Eccebt`ve mbbeaermt`kf `f mbbeaermt`fn recerefbe systei 9 mn , mt iemf pks`t`kf

m 9 mbbeaermt`kf kc tde pk`ft kc suspefs`kf w.r.t. nrkufj.

Bkfj`t`kf ckr mppay`fn td`s ckriuam7 mn 9 bkfstmft

Mask necc 9 okokc imssstr`fn`ftefs`kfFet

mt iemf pks`t`kf

Exmipae =3. M s`ipae pefjuaui `s suspefjej crki tde be`a`fn kc m bmr mbbeaermt`fn uf ckriay kf m dkr`zkftmarkmj. @c tde mbbeaermt`kf `s m8 mfj tde aefntd kc tde pefjuaui `s , c`fj tde t`ie per`kj kc simaaksb`aamt`kfs mokut tde iemf pks`t`kf.

8/12/2019 SHM Theory E

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XDPR@BR

"imf sdluimrpdys bs. f" =2

Rkaut`kf 7 ^e sdmaa wkrl f tde bmr crmie.Ms `t s mbbeaermtej w td respebt tk tde rkmj, we sdmaa dmve tk mppaym pseujk ckrbe im 8 kf tde oko kc imss i.Ckr iemf pks`t`kf, tde mbbeaermt`kf kc tde oko w`td respebt tk tde bmr sdkuaj oe zerk. @c oe tdemfnae imje oy tde str fn w td tde vert`bma, tde tefs`kf, we`ndt mfj tde pseujk ckrbe w`aa mjj tk zerk`f td`s pks`t`kf.

Defbe, resuatmft kc in mfj im 8 (smy C 9 ?

8? mni ) dms tk oe makfn tde str`fn.

tmf 9 inim 8

9 nm8

Fkw, suppkse tde str`fn `s curtder jecaebtej oy mf mfnae mssdkwf `f c`nure.Fkw, restkr`fn tkrque bmf oe n`vef oy

(C s`f ) 9 i ?Ruost`tut`fn C mfj us`fn s`f ckr simaa

( ?8? mni ) 9 i ?

kr, -?8

? mn sk< ? 9

?8

? mn

Zd`s `s mf equmt kf kc s`ipae dmrikf b ikt`kf w`td t`ie per`kj

Z 9?

9 ? :/=?8

? )mn(

=8.? @c ckrbes ktder tdef i n mbts tdef 7

Z 9 ?.ecc n

wdere n ecc. 9 iC

n

C 9 bkfstmft ckrbe mbt`fn kf i.

Exmipae =>. M s`ipae pefjuaui kc aefntd mfj dmv`fn oko kc imss i `s jk`fn mfnuamr RDI `fs`je wmter. Mbkfstmft oukymft ckrbe equma tk dmac tde we`ndt kc tde oko `s mbt fn kf tde omaa. C fj tde t`ie per`kjkc ksb`aamt`kfs;

Rkaut`kf 7 Dere n ecc. 9 n i?/in

9 n/?. Defbe Z 9 ? n?

==. BKIXKUFJ XEFJUAUI / XDPR@BMA XEFJUAUI^def m r`n`j okjy `s suspefjej crki mf mx`s mfj imje tk ksb`aamte mokut tdmt

tdef `t `s bmaaej bkipkufj pefjuaui.B 9 Xks`t`kf kc befter kc imssB

R

B

R

R 9 Xk`ft kc suspefs`kf 9 J`stmfbe oetweef pk`ft kc suspefs`kf mfj befter kc imss

(`t reim`fs bkfstmft jur`fn ikt`kf)Ckr simaa mfnuamr j`spambeieft crki iemf pks t kfZde restkr`fn tkrque `s n`vef oy

in s`f in ckr simaa , s`f ~

kr, in wdere, 9 Ikieft kc fert`m mokut pk ft kc suspefs`kf.

kr, in kr, ? 9 in

Z`ieper`kj, Z 9 ? in 9 BI + i?

^dere BI 9 ikieft kc `fert`m reamt ve tk tde mx`s wd`bd pmsses crki tde befter kc imss & pmrmaaea tk tdemx`s kc ksb`aamt`kf.

8/12/2019 SHM Theory E

14/21

8/12/2019 SHM Theory E

15/21

XDPR@BR

"imf sdluimrpdys bs. f" =5

=2. RUXE_XKR@Z@KF KC Z^K RDIR=2.= @f smie j rebt`kf mfj kc smie crequefby.

x= 9 M= s`f t

x? 9 M? s`f ( t + ) , tdef resuatmft j`spambeieft

x 9 x= + x? 9 M= s`f t + M? s`f ( t + ) 9 M s`f ( t + )

wdere M 9 bks M M? M M ?=???= & 9 tmf = bks M Ms`f M

?=

?

@c 9 8, oktd RDIs mre f pdmse mfj M 9 M = + M?@c 9 , oktd RDIs mre kut kc pdmse mfj M 9 | M = M? |

Zde resuatmft mipa tuje jue tk superpks`t`kf kc twk kr ikre tdmf twk RDIs kc td`s bmse bmf mask oe ckufjoy pdmskr j`mnrmi mask.

=2.? @f smie j rebt`kf out mre kc j ccereft crequefb`es.

x= 9 M= s`f =t x? 9 M? s`f ? t

tdef resuatmft j`spambeieft x 9x = + x? 9 M= s`f =t + M? s`f ?t Zd`s resuatmft ikt`kf `s fkt RDI.

=2.2 @f twk perpefj buamr j`rebt kfs.

x 9 M s`f t y 9 O s f ( t + )

Bmse (`) 7 @c 9 8 kr tdef y 9 (O/M) x. Rk pmtd w`aa oe strm`ndt a`fe & resuatmft j`spambeieft w`aa oer 9 (x ? + y?) 9 (M? + O? ) s`f t

wd bd `s equmt`kf kc RDI dmv`fn mipa tuje ?? O M

Bmse (``) 7 @c 9?

tdef. x 9 M s`f ty 9 O s`f ( t + /?) 9 O bks t

sk, resuatmft w`aa oe?

?

M

x+

?

?

O

y9 =. `.e. equmt`kf kc mf eaa`pse mfj `c M 9 O, tdef superpks`t`kf w`aa oe mf

equmt kf kc b`rbae. Zd`s resuatmft ikt`kf `s fkt RDI.

= 2 .: Ruperpks`t`kf kc RDIs makfn tde smie j`rebt`kf us`fn pdmskr j`mnrmi)

@c twk kr ikre RDIs mre makfn tde smie a`fe, tde`r resuatmft bmf oe kotm`fej oy vebtkr mjj`t`kf oy iml`fnpdmskr j`mnrmi.

=. Mipa`tuje kc RDI `s tmlef ms aefntd(imnf`tuje) kc vebtkr.

?. Xdmse j`ccerefbe oetweef tde vebtkrs `s tmlef ms tde mfnae oetweef tdese vebtkrs. Zde imnf`tujekc resuatmft vebtkr n`ves resuatmft mipa`tuje kc RDI mfj mfnae kc resuatmft vebtkr n`ves pdmse bkf-stmft kc resuatmft RDI.

Ckr exmipae< x=

9 M=

s`f t

x? 9 M? s`f ( t + )

M?

M

M=Xdmskr J`mnrmi

@c equmt kf kc resuatmft RDI `s tmlef ms x 9 M s`f ( t + )

M 9 bks M?M M M ?=???=

tmf 9bks M M

s`f M

?=

?

8/12/2019 SHM Theory E

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XDPR@BR

"imf sdluimrpdys bs. f" =3

Exmipae ?8. C`fj tde mipa`tuje kc tde s`ipae dmrikf`b ikt`kf kotm fej oy bkio`f fn tde ikt`kfsx= 9 (?.8 bi) s`f t

mfj x ? 9 (?.8 bi) s`f ( t + /2).Rkaut`kf 7 Zde twk equmt`kfsn`vef represeft s`ipaedmrikf b ikt kfs makfn T-mx`sw td mipa tujesM = 9?.8bi

mfjM ? 9 ?.8 bi. Zdepdmse j ccerefbe oetweef tde twk s ipae dmrikf`b ikt kfs `s /2.Zde resuatmfts ipae dmrikf`b ikt kf w aa dmve mf mipa tujeM n`vef oy

M 9 bks M M? M M ?=??

?= 9 2

bks)bi8.?(?)bi8.?()bi8.?( ???

9 2.5 bi

Exmipae ?=. x=

9 2 s`f t < x?

9 : bks tC`fj (`) mipa`tuje kc resuatmft RDI. (``) equmt`kf kc tde resuatmft RDI.

Rkaut`kf 7 C`rst wr`te maa RDIs f teris kc s`fe cufbt`kfs w`td pks`t ve mipa`tuje. Leep t w td pks`t ve s`nf.

x=

9 2 s`f tx

?9 : s`f ( t + /?)

M 9?

bks:2?:2 ??

9 =30 9 ?5 9 5

tmf 92:

?bks:2

?s`f: 9 52

equmt`kf x 9 5 s`f ( t + 52)

Exmipae ?? x=

9 5 s`f ( t + 28)

x?

9 =8 bks ( t)

C`fj mipa`tuje kc resuatmft RDI.

Rkaut`kf 7 x=

9 5 s`f ( t + 28)

x?

9 =8 s`f ( t +?

)38

=8

5Xdmskr J`mnrmi

M 9 38bks=85?=85 ??

9 58=88?5 9 =>5 9 >5

Exmipae ?2 M pmrt`bae `s suogebtej tk twk s`ipae dmrikf`b ikt`kfsx= 9 M= s`f t mfj x ? 9 M? s`f ( t + /2).

C`fj (m) tde j`spambeieft mt t 9 8, (o) tde imx`iui speej kc tde pmrt`bae mfj (b) tde imx`iuimbbeaermt kf kc tde pmrt`bae.

Rkaut`kf 7 (m) Mt t 9 8, x= 9 M= s`f t 9 8mfj x ? 9 M? s`f ( t + /2)

9 M? s`f ( /2) 9 ?2 M? .

Zdus, tde resuatmft j`spambeieft mt t 9 8 `s

x 9 x= + x? 9 M? ?2

(o) Zde resuatmft kc tde twk ikt kfs `s m s ipae dmrikf`b ikt`kf kc tde smie mfnuamr crequefby. Zde mipa tuje kc tde resuatmft ikt`kf `s

M 9 )2/bks( M M? M M ?=??

?= 9 ?=???= M M M M .

Zde imx iui speej `s

u imx 9 M 9 ?=??

?= M M M M

(b) Zde imx iui mbbeaermt`kf `s

mimx 9 M? 9 ? ?=???= M M M M .

8/12/2019 SHM Theory E

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XDPR@BR

"imf sdluimrpdys bs. f" =>

Exmipae ?:. M pmrt bae `s suogebtej tk twk s ipae dmrikf`b ikt kfs `f tde smie j rebt`kf dmv fn equma mipa tujesmfj equma crequefby. @c tde resuatmft mipa`tuje `s equma tk tde mipa`tuje kc tde `fj`v`juma ikt`kfs,c`fj tde pdmse j`ccerefbe oetweef tde `fj v`juma ikt kfs.

Rkaut`kf 7 Aet tde mipa tujes kc tde `fj`v`juma ikt`kfs oe M embd. Zde resuatmft mipa tuje `s mask M. @c tdepdmse j`ccerefbe oetweef tde twk ikt`kfs s ,

M 9 bks. M. M? M M ??

M 9 M )bks=(? 9 ?M bks ?

bks?

9?=

9 ? /2.

Xrkoaei =. ^r`te tde equmt`kf kc RDI ckr tde s`tumt`kfs sdkwf oeakw7

(m) t 9 8

8 M-M

(o) t 9 8

8 M-M

(b) t 9 88 M M/?-M

Rkaut`kf 7 (m) Mt t 9 8 , x 9 +M

x 9 M s`f( t + ) M 9 M s`f ( )

9 /?

x 9 M s`f

?t 9 Mbks ( t)

(o) Mt t 9 8 , x 9 M

x 9 M s`f( t + ) M 9 M s`f

9?

2

x 9 M s`f

?2

t

x 9 M bks ( t)

(b) Mt t 9 8 , x 9?

M

x 9 M s`f ( t + )

? M

9 M s`f( t + )

?=

9 s`f 9 28 , =58

Xmrt bae `s ikv`fn tkwmrjs tde iemf pks`t`kf mfj `f fenmt`ve j`rebt kf.

veakb ty v 9 M bks ( t + ) Mt t 9 8 , v 9 ve

v 9 M bks defbe 9 =58

x 9 M s f( t + =58)Mfs. (m ) x 9 M bks t < (o) x 9 M bks t < (b) x 9 M s`f( t + =58)

8/12/2019 SHM Theory E

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XDPR@BR

"imf sdluimrpdys bs. f" =4

Xrkoaei ?. Oakbl M kc imss i `s perckri fn RDI kc mipa`tuje m. Mfktder oakbl O kc imss i `s neftay pambejkfM wdef `t pmsses tdrkund iemf pks t`kf mfj O st`bls tkM. C fj tde t`ie per`kj mfj mipa tuje kc few RDI.

Rkaut`kf 7

u

i

Mipa`tuje 9 m

v

?i

Mipa`tuje 9 M

Z`ie per`kj kc imss i 9 ?Li

Z`ie per`kj kc imss ?i 9 ?Li?

Mt iemf pks`t`kf, L`fet`b eferny 9 Zktma Eferny

Ckr imss i 7?=

iu ? 9?=

i ?m? ................. (=)

Ckr imss ?i 7?=

?iv ? 9?=

?i?

?

M? ........... (?)

Oy Bkfservmt`kf kc ikieftui

iu9 ?iv

v 9?u

?=

?i?

?u

9?=

?i

?

?

M? ................ (2)

J`v`je equmt kf (=) & (2)

: 9 ?

?

M

m?

Few Mipa tuje M 9?

m

Mfs. Z 9 ?Li?

Mipa`tuje 9?

m

Xrkoaei 2. _epemt tde mokve prkoaei mssui`fn O s pambej kf M mt m j`stmfbe?m

crki iemf pks`t`kf.

Rkaut`kf 7 Z 9 ?Li

u

i Mipa`tuje 9 m

iemf m/?

Z 9 ? Li?

v

?i Mipa`tuje 9 Miemf

Oy bkfservmt`kf kc ikieftui

8/12/2019 SHM Theory E

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XDPR@BR

"imf sdluimrpdys bs. f" =0

iu 9 ?iv v 9?u

L`fet`b Efern`es mt?m

Ckr imss i 7?

=iu ? 9 ?i

?

=

??

?

mm ................. (=)

Ckr imss ?i 7?=

?iv ? 9

?

?i?

?=

???m

M ......... (?)

?=

?i:

u ?9

?i?

?= ?

:m

M?

?............. (2)

J`v j fn equmt kf (=) & (2)

? 9

:m

M

:m

m?

?

??

Few Mipa`tuje M 945

m

Mfs. Z 9 ?Li?

, Mipa tuje 9 m45

Xrkoaei :. Zde oakbl `s maakwej tk cmaa, sakway crki tde pks`t`kf wdere spr`fn `s `f `ts

fmturma aefntd. C`fj, imx`iui extefs`kf `f tde str`fn.

//////////////////////////

i

L 8

Rkaut`kf 7 R`fbe tde oakbl cmaas sakway crki rest tde imx`iui extefs`kf kbburs wdef

in 9 Lx8

x8

9L

in`s imx`iui extefs kf

Mfs. Lin

Xrkoaei 5. @f tde mokve prkoaei `c oakbl `s reaemsej crki tdere, wdmt wkuaj oe imx iui extefs`kf.

Rkaut`kf 7 Aet x8

9 imx`iui extefs`kf/////////////////////////////////////////////////////////////////////

i

x8

A8

i

mppay fn bkfservmt`kf kc eferny

inxk

9?=

Lx8

?

Zde veakb ty mt tde pk`ft kc imx`iui extefs`kf `s zerk.

x8

9Lin?

`s imx`iui extefs`kf

8/12/2019 SHM Theory E

20/21

XDPR@BR

"imf sdluimrpdys bs. f" ?8

Mfs.Lin?

Xrkoaei 3. Oakbl kc imss i?

`s `f equ`a`or`ui ms sdkwf `f c`nure. Mfktder oakbl kc imss i=

`s

lept neftay kf i?. C`fj tde t`ie per`kj kc ksb`aamt`kf mfj mipa`tuje.

//////////////////////////

i ?

L

Rkaut`kf 7 Z ie per`kj Z 9 ? Lii?=

////////////////////////////////////////////////////

i n

L=

i ?

i +i= ? Equ`a`or`uipks`t`kf

Mt `f`t`ma pks`t`kf s`fbe veakb`ty `s zerk `t `s tde extreie pks`t`kf.

Mipa`tuje M 9L

ni = Mfs. Z 9 ?L

ii ?= Mipa`tuje 9L

ni =

Xrkoaei >. Oakbl kc imss i?

`s `f equ`a`or ui mfj mt rest. Zde imss i=

ikv`fn w`td veakb`ty u

vert`bmaay jkwfwmrjs bkaa`jes w`td i?

mfj st`bls tk `t. C`fj tde eferny kc ksb`aamt`kf.

//////////////////////////

i ?

i =

u

Rkaut`kf 7 Aet tde veakb`ty kc i=

& i?

oe v mcter bkaa`s`kf

Oy bkfservmt`kf kc ikieftui

i=u 9 ( i

=+i

?)v

v 9?=

=

iiui

Defbe, LE 9?=

(i=+i

?)v? 9

?

?=

?= uii

i?=

//////////////////////////////////////////////////////////////////////////////////////

x= x?

i ?

i +i= ?

Equ`a`or`uiXks`t`kf

i?n 9 Lx

=

(i=

+ i?)n 9 Lx

?

XE 9?=

L(x? x

=)? 9

?=

L?

=

Lni

XE 9?=

Lni ??=

Zdereckre eferny kc ksb`aamt`kf `s -

E 9 LE + XE

E 9?=

??

iiui

?=

+L?ni ??= 9

?=

Lni

iiui ??=

?=

??= Mfs.

8/12/2019 SHM Theory E

21/21

XDPR@BR

Xrkoaei 4. M okx `s pambej kf m sikktd `fba`fej pamfe mfj `t `s cree tk ikve. M s`ipae pefjuaui `s mttmbdej `f tde oakbl. C`fj `ts t`ie per`kj.

Rkaut`kf 7

Aet Cp

9 Xseujk ckrbe

Ckr equ`a`or`ui pks`t`kf

Zs`f + CX

9 ins`f

Zs`f + im 9 ins`f

s`fbe m 9 ns`f

ins`f

Z s f i n s f

Zbks

in

inbks

Z

Zs`f + ins`f 9 ins`f

Zs`f 9 8

9 8

Defbe `f equ`a`or`ui pks`t`kf tde str`fn `s perpefj`buamr tk `fba`fe pamfe.

Zdereckre eccebt`ve n`s

n ecc 9 nbks Z`ie per kj

Z 9 ?ecc n

n

nbksZ 9 ? bksn Mfs.

Xrkoaei 0. x=

9 5 s`f t

x? 9 5 s`f ( t + 52)x

29 =8 bks t

C`fj mipa`tuje kc resuatmft RDI

Rkaut`kf 7 x=

9 5s`f t

x?

9 5 s`f ( t + 52)

x2

9 =8 bks twe bmf wr`te x

29 =8 s`f( t + ?>8)

C`fj`fn tde resuatmft mipa tuje oy vebtkr fktmt`kf.

52

5

5

=8

_

4

3

_esuatmft Mipa`tuje | _ | 9 ?? 34 9 =8 Mfs.