Analysis of the Vibrations of an Elastic Beam

Analysis of the Vibrations of an Elastic Beam - VtsjkR$R BP...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: VtsjkR$R BP Y z$As$BtR BP st ejsR${ s8 V 3BZY !I e$t%sjZ\e$tPZt{$Bt _BAj8 =R$t {BtT {}R PB8 Y YBk BP jsR${$k $ {st A RYBbt Ys Y jssj I4{$Bt jw?, 0D BP s Zt$PB8 Y$t jsR${ As8 BP jtY U R $tB %$As$Bt BAkR Y }s$sj I$&t$sj JZs$Bt ' (j (;j (1j L o{ L ) W w?, 0D, (01 (0 (?; bY W w?, 0D }RtR st s}}j$I -tsj PB{ ' $R Y 8sRR } Zt$ jtY $R Y {B^{$t BP %$R{BZR P${$Bt wj%st $P Y As8 $R %$As$t $t s 4Z$I RZ{Y sR s$D stI o{ $R Y RstIsI Rk8ABj PB Y AtI$t 8BIZjZR BP Y As8 G Iht Y I$&t$sj B}sB 6 Ak w6jDw?D ) Q; j w?D Q?; ; Bt s IB8s$t ? $t h g1 >(, UH B A PZY R}{$hI Ak $8}BR$t ABZtIsk {BtI$$BtR bY${Y b b$jj IR{$A RYBjk qB A$t Y }B{RR BP $It$Pk$t YBR ABZtIsk {BtI$$BtR b tB Ys b$Y Y RstIsI $tt }BIZ{ $ j, FX # (U jw?DFw?D Q? b Ys% PB jw?D stI Fw?D $t ? 6j, FX ) 8 U ( Q; j Fw?D Q? Q?; X ) ~ 8 U 1 Qn j Q1 j QF eU Q j Q1 F e e L Fw?D Q? Q?n Q?1 Q? ( ( Q?1 Q?1 8 U n eU Qn j Q j QF e ) Fw?De Q? n ( Q? ( Q?n Q? d X ) X ) ~ 8 U Qn j Q1 j QF Qj Q1 F Qn F eU Q; F Fw?D jw?D n e L jw?D ; Q? L e ( Q?n Q?1 Q? Q? Q?1 Q? Q? ( X ~ ? # Qn j Qj Q1 F Qn F eU Q1 j QF L Fw?D jw?D n e L j, 6F 3 e ( Q?n Q?1 Q? Q? Q?1 Q? ~ 8 U Qn j Qj Q1 F eU Qj Qn F Q1 j QF e e L Fw?D Q? Q?n Q?1 Q? Q? Q?1 ( ( Q? Q?n ) qY XtsZsj" ABZtIsk {BtI$$BtR s IhtI sR YBR bY${Y {sZR Y ABZtIsk 8R X ~ Q1 j QF Qj Q1 F Qn j Qn F eU L Fw?D jw?D n e e ( Q?n Q?1 Q? Q? Q?1 Q? B IZ{ B 9B qY s s js tZ8A BP YR {BR}BtI$t B %s$BZR bskR $t bY${Y Y As8 {st A 8BZtI B {BtRs$tI e-s8}jR $t{jZI Y {js8}I As8U jw(D ) Qj Qj w(D ) jwUD ) wUD ) ( Q? Q? Q1 j Q1 j w(D ) jwUD ) wUD ) ( Q?1 Q?1 Y R$8}jk RZ}}BI As8U jw(D ) Y P As8U Qn j Q1 j Qn j Q1 j w(D ) w(D ) wUD ) wUD ) ( Q?1 Q?n Q?1 Q?n stI Y {st$j% As8U jw(D ) Qj Q1 j Qn j w(D ) wUD ) wUD ) (3 Q? Q?1 Q?n yt Y jsR {sR Y As8 $R {js8}I s ? ) ( stI P s ? ) U yt sjj {sRR Y ABZtIsk {BtI$$BtR s s}}j$I B Fw?D sR bjj sR B jw?D sR RYBbt 1 es{Y BP YR {BR}BtIR B s I$&t Iht$$Bt BP Y IB8s$t ? qY s 8stk $tII $tht$jk 8stk BY }BRR$A$j$$R G b$jj ZR Y {st$j% As8 sR Y RstIsI -s8}j $t BZ bB, Y A{sZR $ {BR}BtIR B bYs $R 8BR sR$jk I8BtRsI $t Y jsABsBk 3B stk BP YR ABZtIsk {BtI$$BtR Y PBZY BI B}sB b$jj A RjPTsI`B$t A{sZR b Ys% 6j, FX ) j, 6FX qY B}sB $R sjRB tBtT ts$% PB stk ABZtIsk {BtI$$BtR RZ{Y Ys Y ABZtIsk 8R s$R$t sP Y hR bB $ts$BtR Ak }sR $ X ~ Qn j Q1 j QF eU e Fw?D e , Q?n Q?1 Q? ( 8 U IZ{ B 9B A{sZR $t Y$R {sR b Ys% PB Fw?D ) jw?D Q1 j Q1 j Q? (3 ( Q?1 Q?1 yt }s${Zjs Yt Y$R $8}j$R Ys sjj $t%sjZR s tBtTts$% y $R sRk B R Ys sjj PBZ BP Y R}{$h{ -s8}jR {$I sAB% Psjj $tB Y$R {sBk stI sjj -{} Y P As8 {sR s $t Ps{ }BR$$% sY Yst 6j, jX ) `ZR tBtTts$% b$Y Y {BtRJZt{ Ys sjj $t%sjZR s }BR$$% sR bjj WBb b {BtR$I Y $t%sjZ T $tPZt{$Bt JZs$Bt PB Y PBZY BI B}sBU Q; * *w?D ) (3 Q?; 7$t{ b ,tBb Y $t%sjZR 8ZR A tBtTts$% b b$ ) ; bY $R sjRB s,t B A tBtTts$% qY I$&t$sj JZs$Bt Y $R s j$ts R{BtI BI {BtRst {B^{$t I$&t$sj JZs$Bt Y {BT R}BtI$t {Yss{$R${ JZs$Bt $R ; ) ; ; ) ( n b$Y Y BBR 7 , V{{BI$tjk Y tsj RBjZ$Bt s,R Y PB8 *w?D ) d {BRY ? L 1 R$tY ? L n {BR ? L ; R$t ?3 qY ABZtIsk {BtI$$BtR s s}}j$I B IZ{ Y sA$s$tRR 4{I $t YR {B^{$tR B s R$tj sA$sk {BtRst {BR}BtI$t B Y Ps{ Ys stk tBtT9B 8Zj$}j BP st $tPZt{$Bt 8s$tR st $tPZt{$Bt isY Yst IB Y$R $t s tsj {Bt- b b$jj {sk BZ Y$R }Bs8 PB BZ R}{$h{ -s8}j BP Y {st$j% As8 bY Y ABZtIsk {BtI$$BtR s Q1 * Qn * Q* w(D ) wUD ) wUD ) (3 Q? Q?1 Q?n 7$t ? ) ( $t Y PB8Zjs PB *w?D b BAs$t d L n ) ( RB Ys n ) d *$&t$s$t Y PB8Zjs PB *w?D stI Yt R$t ? ) ( b BAs$t 1 L ; ) ( RB Ys ; ) 1 V{{BI$tjk b tBb Ys% *w(D ) *w?D ) d w{BRY ? {BR ?D L 1 wR$tY ? R$t ?D3 qYt * w?D ) d 1 w{BRY ? L {BR ?D L 1 1 wR$tY ? L R$t ?D, II * w?D ) d n wR$tY ? R$t ?D L 1 n w{BRY ? L {BR ?D3 III 7$t ? ) U Bt Y $Y YstI R$I BP ABY BP YR -}RR$BtR I$%$I$t Y hR Ak 1 stI Y R{BtI Ak n stI Yt R$t Y RZj$t -}RR$BtR JZsj B 9B b BAs$t s YB8BtBZR j$ts RkR8 BP JZs$BtR $t d stI 1 U w W w W w W {BRY U L {BR U R$tY U L R$t U d ( ) 3 R$tY U R$t U {BRY U L {BR U 1 ( 7$t{ st $tPZt{$Bt 8ZR tB A $It${sjjk 9B b tI s RBjZ$Bt b$Y d stI 1 tB ABY 9B 3B8 RstIsI j$ts sjAs Y$R {st A Y {sR Btjk $P Y I8$tst BP Y 8s$- $R 9B qYs I8$tst $R w{BRY U L {BR UD1 wR$tY U L R$t UDwR$tY U R$t UD ; ) {BRY1 U L 1 {BRY U {BR U L {BR1 U R$tY1 U L R$t1 U ) 1 {BRY U {BR U L 13 *$%$I$t Ak 1 {BRY U R$t Y RZj JZsj B 9B stI stR}BR$t b Ys% d {BR U ) 3 {BRY U qY RBjZ$BtR BP Y$R JZs$Bt PB8 st $t{sR$t RJZt{ > e > ) d, 1, 333 {sZR {BRY U tIR s}$Ijk B $tht$k sR A{B8R js Y js %sjZR BP > s %k {jBR B BII 8Zj$}jR BP 1$ U U > w1> dD$ , > 1U 3 + e e ytII PB U BP 8BIR R$9 wRsk dD Y$R s}}B-$8s$Bt $R %k {jBR PB > n 3B U ) d WbBt:R 8YBI {st A s}}j$I B Y JZs$Bt $ {BRY {BR L d ) ( b$Y Rs$t %sjZR $ , n1 BAs$t d ) d3Mxd, 1 ) 1 ;3E;d R}{$%jk 3B tsj }BR$$% %sjZR BP U b Yt Ys% d ) d3Mxd ;3E;d U , 1 ) U GYt Y I8$tst $R 9B Y bB JZs$BtR BP Y RkR8 RYBbt s JZ$%sjt ]jsjk Y hR $R Rs$RhI PB s $%t > Ak d,> ) d d , 1,> ) {BRY > U L {BR > U R$tY > U L R$t > U stI b Ys% Y wZtTtB8sj$9ID $tPZt{$BtR *> w?D ) {BRY > ? {BR > ? R$tY > ? R$t > ? 3 {BRY > U L {BR > U R$tY > U L R$t > U 7$t{ Y PBZY BI B}sB $R RjPTsI`B$t $t Y$R {sR stI Y $t%sjZR s I$R$t{ b {Bt{jZI Ys Y $tPZt{$BtR *> w?D, > ) d, 1, n, 333 sR {BtRZ{I {BtR$Z st BYBBtsj R $t Y R}s{ U1 >(, UH qYR s x U 1 sI$jk tB8sj$9I BP {BZR b$Y I$%$R$Bt Ak ( *> w?D Q? sP bY${Y b Ys% st BYBtB8sj R BP $tPZt{$BtR $t U1 >(, UH 7$t{ YR tBT 8sj$9s$Bt {BtRstR s sY {B8}j${sI B -}RR b IB tB sRRZ8 BYBtB8sj$k BP Y *> w?D $t Y I$R{ZRR$Bt B PBjjBb $ =R$t sI%st{I 8sY8s${sj {Yt$JZR AkBtI Y R{B} BP BZ {Zt }Rts$Bt $ {st A RYBbt Ys Y $tPZt{$BtR *> w?D, > ) d, 1, n, 333 `ZR {BtRZ{I s {B8}j $t Y R}s{ U1 >(, UH $t Y RtR Ys $%t stk PZt{$Bt W w?D $t Y$R R}s{ b {st }Rt Ys PZt{$Bt sR s R$R $t Y $tPZt{$BtR *> w?D yt Ps{ b Ys% R$t{ Y *> w?D s sj W w?D ) 3 >)d W, *> X d > *> w?D > ) 1 ) ,*> , ,*> ,1 8 U ( W w?D *> w?D Q?, > ) d, 1, n, 3333 qY R$R $R {Bt%t b$Y R}{ B Y tB8 $t U1 >(, UH $ $t Y RtR Ys b$Y W w?D ) b Ys% 3 >)d > *> w?D, j$8 ,W W , ) (3 WBb j ZR Zt B Y }s$sj I$&t$sj JZs$Bt b$Y bY${Y b Ast Y$R R{$Bt AZ b$Y Y R${$Bt Ys Y PB{$t 8 W w?, 0D s}}s$t Bt Y $Y YstI R$I $R $It${sjjk 9B $ ' (j (;j (1j L L o{ ) (3 (01 (0 (?; 7$t{ Y JZs$Bt $R R{BtI BI $t 0 b tI B $% st $t$$sj I$R}js{8t stI st $t$$sj %jB{$kU jw?, (D ) j( w?D, (j w?, (D ) 8( w?D3 (0 E y $R tBZY B s, 8( ; U1 >(, qH AZ PB s %s$k BP sRBtR bY${Y b b$jj tB Is$j Y Y $t$$sj I$R}js{8t j( w?D RYBZjI s Y %k jsR A R${I RB Ys $R R{BtI I$%s$% j( w?D $R IhtI stI AjBtR B U1 >(, UH stI RB Ys Y RRt$sj ABZtIsk {BtI$$BtR j( w(D ) (, j( w(D ) ( s Rs$RhI qY $t$$sj Iss PZt{$BtR s Yt -}stII $t Y $tPZt{$BtRU II I j( w?D ) 8 U ( 3 >)d j> *> w?D, 8( w?D ) 3 >)d 8> *> w?D3 8 U ( qY -}stR$Bt {B^{$tR d j> ) ,*> ,1 d j( w?D*> w?D Q?, 8> ) ,*> ,1 8( w?D*> w?D Q?, 8sk A {BtR$II ,tBbt B bYs% -t Y $t$$sj Rs j( w?D, 8( w?D $R ,tBbt y {st A RYBbt Ys RZ{Y s R}{$h{s$Bt BP Y $t$$sj Rs IT 8$tR s Zt$JZ RBjZ$Bt jw?, 0D BP Y }s$sj I$&t$sj JZs$Bt ZtIRBBI $t st s}}B}$s RtR bY${Y ss$t b IB tB Is$j Y G -}stI Y RBjZ$Bt jw?, 0D $t Y PB8 jw?, 0D ) 3 >)d j> w0D *> w?D, Y $8 %sk$t {B^{$t PZt{$BtR j> w0D A$t $t$$sjjk ZtI8$tI qB I8$t YR {B^{$t PZt{$BtR b RZAR$Z Y$R RZ8 $tB Y }s$sj I$&t$sj JZs$Bt sRRZ8$t Ys 8 Ak 8 I$&t$s$Bt b$Y R}{ B ABY ? stI 0 $R `ZR$hI qY$R RZAR$Z$Bt k$jIR Y JZs$Bt 3 >)d X Q1 j> Qj> ; ' L L o{ > j> w0D *> w?D ) (3 1 Q0 Q0 ~ 7$t{ Y *> w?D s BYBBtsj stI Yt{ $tI}tIt $t U1 >(, UH Y %stT $RY$t BP Y$R RZ8 $8}j$R Ys s{Y BP Y {B^{$tR %st$RYR js%$t ZR b$Y s R BP R{BtI BI I$&t$sj JZs$BtR $t Y j> w0DU ' Q1 j> Qj> ; L L o{ > j> w0D ) (, > ) d, 1, n, 333 3 Q01 Q0 qY s}}B}$s $t$$sj {BtI$$BtR PB YR I$&t$sj JZs$BtR s }BT %$II Ak Y -}stR$Bt {B^{$tR BP Y $t$$sj I$R}js{8t stI Y $t$$sj %jB{$k qYZR j> w(D ) j> , Qj> w(D ) 8> , > ) d, 1, n, 333 3 Q0 N:R ZR Y Ns}js{ stRPB8 {Yt$JZ B RBj% Y $t$$sj %sjZ }BAj8 }BRI Ak Y I$&t$sj JZs$Bt stI $t$$sj {BtI$$BtR `ZR R}{$T hI G Ys% ; ' X1 wRj> D wXD X j> 8> L wX wRj> D wXD j> D L o{ > wRj> D wXD ) (, p Q bY${Y $%R X j> L 8> L j> ' wX j> L 8> D L j> ' wRj> D wXD ) ) 1 3 o{ ; 1 L X L o{ ; 'X X L 'X L ' > ]B8}j$t Y RJZs $t Y ItB8$tsB stI sst$t b Ys% wRj> D wXD ) p XL ) j> p XL Q1 X j> L 8> L j> ' Q 1 1' L ; o{ > ' 1 ;'1 XL L 1' 1' ; o{ > ' X 1 ;'1 L 8> L j> p 1' XL ; o{ > ' ~ 1' Q1 d L ; o{ > ' 1 ;'1 3 WB8sjjk $R JZ$ R8sjj RB b b$jj sRRZ8 wRj> D wXD ) j> p XL C Q 1 1' 5 1 ;'1 k ( PB sjj > qYt 1 ;'1 XL L 1' ; o{ > ' L H5 o{ ; ' > h 8> L h 1' I j> n n 1 ;'1 O p XL M ; 1 o{ > ;'1 ' Q 1 o{ ; 1 L ' > ;'1 1' 3 qs,$t Y $t%R stRPB8 b Ys% j> w0D ) j> C h 8> h5 H 1' 0 CC> o{ HH: {BR ; 1 > 1 O 0O ' ;' I I L L 1' I ; o{ > ' j> n n 1 ;'1 O 1' 0 CC> o{ ; > HH: R$t ' 1 1 O 0O 3 ;' I I qYt Y RBjZ$Bt BP Y }s$sj I$&t$sj JZs$Bt b$Y W w?, 0D # ( $R $%t Ak m CC> I Iu o{ ; 3 0 1 O O{ > tj> 1' {BR HH: jw?, 0D ) 1 0 *> w?D ' ;' >)d mC L >)d 3 Nh 8> L Nh 5 tH o{ ; ' > 1' I O j> n n 1 ;'1 CC> o{ ; 0 > 1' R$t HH: ' u I I 1 O OU 0 U { ;'1 *> w?D3 yP Ww?, 0D $R tB $It${sjjk JZsj B 9B b -}stI Ys XPB{$t PZt{$Bt" $t Y PB8 3 W w?, 0D ) W> w0D *> w?D >)d stI Yt htI Ak RRt$sjjk Y Rs8 }B{RR sR ZRI sAB% Ys Y I$&T t$sj JZs$BtR PB Y j> w0D s, Y $tYB8BtBZR PB8 Qj> Q1 j> ; L L o{ > j> w0D ) W> w0D, > ) d, 1, n, 333 3 ' 1 Q0 Q0 yP Y PZt{$BtR W> w0D Ys% Ns}js{ stRPB8R bY${Y {st A -}RRI $t 8R BP j8tsk PZt{$BtR b b$jj ss$t A sAj B BAs$t {jBRI PB8 RBjZ$BtR PB Y j> w0D Ak s 8BI$h{s$Bt BP Y Ns}js{ stRPB8 }B{RR ZRI sAB% yt tsj Y RBjZ$Bt BP Y$R $tYB8BtBZR }BAj8 b$jj s, Y PB8 3 jw?, 0D ) wj> w0D L j> w0DD *> w?D, K >)d bY j> w0D $R sR $%t sAB% stI j> w0D $R Y RBjZ$Bt BP Y $tYB8BtBZR K R{BtI BI I$&t$sj JZs$Bt {BR}BtI$t B Y $t$$sj Iss j> w(D ) (, K +Z${,]Y{, e-{$RR ; d 3$tI Y JZs$Bt I8$t$t Y > PB Y $t%sjZR > ) > stI Y PB8 BP Y $tPZt{$BtR *> w?D PB Y PBZY BI B}sB w6jD w?D ) Q; j Q?; I$R{ZRRI $t Y$R R{$Bt AZ b$Y Y IBZAjk {js8}I ABZtIsk {BtI$T $BtR Q* Q* w(D ) (, *wUD ) (, wUD ) (3 *w(D ) (, Q? Q? Qj> K w(D ) (3 Q0 1 =R WbBt:R 8YBI wB stk BY tZ8${sj JZs$Bt RBj% kBZ b$RYD B BAs$t s}}B-$8s$BtR %sj$I B PBZ I{$8sj }js{R PB d stI 1 sR IR{$AI $t d sAB% s,$t U ) d n N w6jD w?D ) Q j Bt Y IB8s$t ? {BtR$R$t BP PZt{$BtR jw?D b$Y Q?; 1 PBZY I$%s$%R $t U >(, UH Rs$RPk$t ABZtIsk {BtI$$BtR jw(D ) (, Q1 j Q1 j w(D ) (, jwUD ) (, wUD ) (3 Q?1 Q?1 ; 3$tI Y PB8R BP Y $tPZt{$BtR *> w?D stI Y JZs$Bt I8$t$t ; Y > $t > ) > yt Y$R {sR Y JZs$Bt RBj%I Ak Y > {st A RBj%I -}j${$jk 3$tI st -}RR$Bt PB sjj BP Y RBjZ$BtR > stI Y {BR}BtI$t $t%sjZR > d( ...
View Full Document

This note was uploaded on 04/28/2008 for the course EMEM 641 taught by Professor None during the Spring '08 term at RIT.

Ask a homework question - tutors are online