Seveth Editio CHAPTER 9 VECTOR MECHANICS FOR ENGINEERS: DYNAMICS Ferdiad P. Beer E. Russell Johsto, Jr. Mechaical Vibratios Lecture Notes: J. Walt Oler Texas Tech Uiversity 003 The McGraw-Hill Copaies, Ic. All rights reserved.
Seveth Editio Vector Mechaics or Egieers: Dyaics Cotets Itroductio Free Vibratios o Particles. Siple Haroic Motio Siple Pedulu (Approxiate Solutio) Siple Pedulu (Exact Solutio) Saple Proble 9. Free Vibratios o Rigid Bodies Saple Proble 9. Saple Proble 9.3 Priciple o Coservatio o Eergy Saple Proble 9.4 Forced Vibratios Saple Proble 9.5 Daped Free Vibratios Daped Forced Vibratios Electrical Aalogues 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9 -
Seveth Editio Vector Mechaics or Egieers: Dyaics Itroductio Mechaical vibratio is the otio o a particle or body which oscillates about a positio o equilibriu. Most vibratios i achies ad structures are udesirable due to icreased stresses ad eergy losses. Tie iterval required or a syste to coplete a ull cycle o the otio is the period o the vibratio. Nuber o cycles per uit tie deies the requecy o the vibratios. Maxiu displaceet o the syste ro the equilibriu positio is the aplitude o the vibratio. Whe the otio is aitaied by the restorig orces oly, the vibratio is described as ree vibratio. Whe a periodic orce is applied to the syste, the otio is described as orced vibratio. Whe the rictioal dissipatio o eergy is eglected, the otio is said to be udaped. Actually, all vibratios are daped to soe degree. 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-3
Seveth Editio Vector Mechaics or Egieers: Dyaics Free Vibratios o Particles. Siple Haroic Motio I a particle is displaced through a distace x ro its equilibriu positio ad released with o velocity, the particle will udergo siple haroic otio, a F W k δ + x kx x + kx 0 ( ) st Geeral solutio is the su o two particular solutios, k k x C si t + C cos t C si t + C cos t ( ) ( ) x is a periodic uctio ad is the atural circular requecy o the otio. C ad C are deteried by the iitial coditios: x ( t) C cos( t) C si + C x0 v x C ( t) C si ( t) cos C v 0 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-4
Seveth Editio Vector Mechaics or Egieers: Dyaics Free Vibratios o Particles. Siple Haroic Motio C C v 0 x 0 Displaceet is equivalet to the x copoet o the su o two vectors which rotate with costat agular velocity. ( +φ ) x x si t 003 The McGraw-Hill Copaies, Ic. All rights reserved. τ π ( v 0 ) + x x 0 aplitude period τ π ( v x ) φ ta 0 0 phase agle atural requecy C + C 9-5
Seveth Editio Vector Mechaics or Egieers: Dyaics Free Vibratios o Particles. Siple Haroic Motio Velocity-tie ad acceleratio-tie curves ca be represeted by sie curves o the sae period as the displaceet-tie curve but dieret phase agles. x v x si t x x x a x x x ( + φ ) cos si ( t + φ ) ( t + φ + π ) si si ( t + φ ) ( t + φ + π ) 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-6
Seveth Editio Vector Mechaics or Egieers: Dyaics Siple Pedulu (Approxiate Solutio) Results obtaied or the sprig-ass syste ca be applied wheever the resultat orce o a particle is proportioal to the displaceet ad directed towards the equilibriu positio. Cosider tagetial copoets o acceleratio ad orce or a siple pedulu, F t a t : W si θ l θ g θ + si θ 0 l or sall agles, + θ θ τ g l θ θ 0 si π ( t + φ ) π l g 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-7
Seveth Editio Vector Mechaics or Egieers: Dyaics Siple Pedulu (Exact Solutio) A exact solutio or leads to g θ + si θ l τ 4 l g 0 π 0 si dφ ( θ ) si φ which requires uerical solutio. K τ π π l g 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-8
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9. SOLUTION: For each sprig arrageet, deterie the sprig costat or a sigle equivalet sprig. Apply the approxiate relatios or the haroic otio o a sprig-ass syste. A 50-kg block oves betwee vertical guides as show. The block is pulled 40 dow ro its equilibriu positio ad released. For each sprig arrageet, deterie a) the period o the vibratio, b) the axiu velocity o the block, ad c) the axiu acceleratio o the block. 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-9
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9. k 4kN k 6kN SOLUTION: Sprigs i parallel: - deterie the sprig costat or equivalet sprig - apply the approxiate relatios or the haroic otio o a sprig-ass syste k 0 4 N/ 0 kg 4.4 rad s τ π τ 0.444 s P k k δ + k δ P k δ 0kN + k 0 4 N v x 0.566 s a ( 0.040 )( 4.4 rad s) xa ( 0.040 )( 4.4 rad s) v a 8.00 s 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-0
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9. k 4kN k 6kN Sprigs i series: - deterie the sprig costat or equivalet sprig - apply the approxiate relatios or the haroic otio o a sprig-ass syste k 400N/ 0 kg 6.93 rad s τ π τ 0.907 s P k k δ + k δ P k δ 0kN + k 0 4 N v x 0.77 s a ( 0.040 )( 6.93 rad s) xa ( 0.040 )( 6.93 rad s) v a.90 s 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9 -
Seveth Editio Vector Mechaics or Egieers: Dyaics Free Vibratios o Rigid Bodies I a equatio o otio takes the or x + x 0 or θ + θ 0 the correspodig otio ay be cosidered as siple haroic otio. Aalysis objective is to deterie. Cosider the oscillatios o a square plate W b si θ b θ + I but the ( ) ( ) θ [ ( ) ( ) ] b + b b W g I, 3 g 3 θ + si θ θ g + θ 5 b 5 b 3g π, τ π 5b For a equivalet siple pedulu, l 5b 3 3 0 5b 3g 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9 -
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9. k SOLUTION: Fro the kieatics o the syste, relate the liear displaceet ad acceleratio to the rotatio o the cylider. Based o a ree-body-diagra equatio or the equivalece o the exteral ad eective orces, write the equatio o otio. A cylider o weight W is suspeded as show. Deterie the period ad atural requecy o vibratios o the cylider. Substitute the kieatic relatios to arrive at a equatio ivolvig oly the agular displaceet ad acceleratio. 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-3
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9. SOLUTION: Fro the kieatics o the syste, relate the liear displaceet ad acceleratio to the rotatio o the cylider. x rθ δ x rθ α θ a rα r θ a r Based o a ree-body-diagra equatio or the equivalece o the exteral ad eective orces, write the equatio o otio. M ( ) : Wr T ( r ) ar + Iα but A M A e 0 + ( ) T T + kδ W k rθ Substitute the kieatic relatios to arrive at a equatio ivolvig oly the agular displaceet ad acceleratio. Wr W + krθ r r θ r + r θ θ + 8 3 8k 3 ( )( ) ( ) k θ 0 π τ π 3 8k θ π π 8k 3 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-4
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9.3 W τ 0 lb.3s τ.93 s The disk ad gear udergo torsioal vibratio with the periods show. Assue that the oet exerted by the wire is proportioal to the twist agle. Deterie a) the wire torsioal sprig costat, b) the cetroidal oet o iertia o the gear, ad c) the axiu agular velocity o the gear i rotated through 90 o ad released. SOLUTION: Usig the ree-body-diagra equatio or the equivalece o the exteral ad eective oets, write the equatio o otio or the disk/gear ad wire. With the atural requecy ad oet o iertia or the disk kow, calculate the torsioal sprig costat. With atural requecy ad sprig costat kow, calculate the oet o iertia or the gear. Apply the relatios or siple haroic otio to calculate the axiu gear velocity. 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-5
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9.3 W τ 0 lb.3s τ.93 s SOLUTION: Usig the ree-body-diagra equatio or the equivalece o the exteral ad eective oets, write the equatio o otio or the disk/gear ad wire. M O ( M O ) e : + Kθ I θ K θ + θ 0 I K π τ π I I K With the atural requecy ad oet o iertia or the disk kow, calculate the torsioal sprig costat. I 0 8 0.38 lb t s r 3. 0.38.3 π K 4.7 lb t rad K 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-6
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9.3 With atural requecy ad sprig costat kow, calculate the oet o iertia or the gear..93 π I 4.7 I 0.403 lb t s W τ 0 lb.3s τ.93 s Apply the relatios or siple haroic otio to calculate the axiu gear velocity. θ θ si t θ si t θ θ 90.57 rad K π τ π I K 4.7 lb t rad I K θ π π τ.93 s (.57 rad ) 5.rad s 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-7
Seveth Editio Vector Mechaics or Egieers: Dyaics Priciple o Coservatio o Eergy Resultat orce o a ass i siple haroic otio is coservative - total eergy is coserved. T + V costat x + kx costat x + x Cosider siple haroic otio o the square plate, T V Wb( cosθ ) Wb si ( θ ) 0 T [ ] Wbθ ( ) ( ) b θ + b 3 ( 5 ) b θ 3 v + I V 0 0 + T + V Wbθ T + V ( 5 ) b θ + 0 3 3g 5b 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-8
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9.4 SOLUTION: Apply the priciple o coservatio o eergy betwee the positios o axiu ad iiu potetial eergy. Solve the eergy equatio or the atural requecy o the oscillatios. Deterie the period o sall oscillatios o a cylider which rolls without slippig iside a curved surace. 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-9
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9.4 SOLUTION: Apply the priciple o coservatio o eergy betwee the positios o axiu ad iiu potetial eergy. T + + V T V T V Wh W ( R r)( cosθ ) 0 W ( )( ) R r θ T v + I V 0 3 4 ( ) ( ) R r θ + r ( R r) θ R r r θ 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-0
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9.4 Solve the eergy equatio or the atural requecy o the oscillatios. T 0 ( )( V W R r θ ) T ( R r) θ V 0 3 4 0 T + + V T V + W 4 θ ( R r) 3 ( R r) θ + 0 θ ( g)( R r) 3 ( R r) ( θ ) 4 3 g R r τ π π 3 R r g 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9 -
Seveth Editio Vector Mechaics or Egieers: Dyaics Forced Vibratios Forced vibratios - Occur whe a syste is subjected to a periodic orce or a periodic displaceet o a support. orced requecy F a : P si + x kx t + W k δ P si ( + x) x t st W k ( δ + x δ si t) x st + x kx kδ si t 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9 -
Seveth Editio Vector Mechaics or Egieers: Dyaics Forced Vibratios x x x copleetary + x particular [ C t + C cos t] x si t si + Substitutig particular solutio ito goverig equatio, x si P k t + kx P si k t P ( ) ( ) si δ t x + kx P si t x + kx kδ si t At, orcig iput is i resoace with the syste. 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-3
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9.5 SOLUTION: The resoat requecy is equal to the atural requecy o the syste. Evaluate the agitude o the periodic orce due to the otor ubalace. Deterie the vibratio aplitude ro the requecy ratio at 00 rp. A otor weighig 350 lb is supported by our sprigs, each havig a costat 750 lb/i. The ubalace o the otor is equivalet to a weight o oz located 6 i. ro the axis o rotatio. Deterie a) speed i rp at which resoace will occur, ad b) aplitude o the vibratio at 00 rp. 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-4
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9.5 SOLUTION: The resoat requecy is equal to the atural requecy o the syste. 350 0.87 lb s 3. t W 350 lb k 4(350 lb/i) k 4 ( 750) 36,000lb 3000lb t i k 36,000 0.87 57.5 rad/s 549 rp Resoace speed 549 rp 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-5
Seveth Editio Vector Mechaics or Egieers: Dyaics Saple Proble 9.5 Evaluate the agitude o the periodic orce due to the otor ubalace. Deterie the vibratio aplitude ro the requecy ratio at 00 rp. 00 rp 5.7 rad/s lb 6 oz 3.t ( oz) 0.0094 lb s t s W 350 lb k 4(350 lb/i) 57.5 rad/s P x a r ( 0.0094)( 6 )( 5.7) 5.33 lb P 5.33 3000 ( ) ( 5.7 57.5) k 0.0035 i x 0.0035 i. (out o phase) 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-6
Seveth Editio Vector Mechaics or Egieers: Dyaics Daped Free Vibratios All vibratios are daped to soe degree by orces due to dry rictio, luid rictio, or iteral rictio. With viscous dapig due to luid rictio, F a : W k( δ st + x) cx x x + cx + kx 0 Substitutig x e λt ad dividig through by e λt yields the characteristic equatio, λ + cλ + k 0 λ c ± c Deie the critical dapig coeiciet such that cc k 0 c c k k 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-7
Seveth Editio Vector Mechaics or Egieers: Dyaics Daped Free Vibratios Characteristic equatio, λ c c + cλ + k 0 λ c ± c critical dapig coeiciet k Heavy dapig: c > c c t t x λ C e λ C e + - egative roots - ovibratory otio Critical dapig: c c c x t ( C + C t) e - double roots - ovibratory otio Light dapig: c < c c x e d ( c ) t ( C t + C cos t ) si d c c c d daped requecy 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-8
Seveth Editio Vector Mechaics or Egieers: Dyaics Daped Forced Vibratios x + cx + kx P si t x x copleet ary + x particular x P k ta φ x δ [ ( ) ] [ ( )( )] + c cc ( c c )( ) c ( ) agiicatio actor phase dierece betwee orcig ad steady state respose 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-9
Seveth Editio Vector Mechaics or Egieers: Dyaics Electrical Aalogues Cosider a electrical circuit cosistig o a iductor, resistor ad capacitor with a source o alteratig voltage di q E si t L Ri 0 dt C Lq + Rq + q E si t C Oscillatios o the electrical syste are aalogous to daped orced vibratios o a echaical syste. 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-30
Seveth Editio Vector Mechaics or Egieers: Dyaics Electrical Aalogues The aalogy betwee electrical ad echaical systes also applies to trasiet as well as steady-state oscillatios. With a charge q q 0 o the capacitor, closig the switch is aalogous to releasig the ass o the echaical syste with o iitial velocity at x x 0. I the circuit icludes a battery with costat voltage E, closig the switch is aalogous to suddely applyig a orce o costat agitude P to the ass o the echaical syste. 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-3
Seveth Editio Vector Mechaics or Egieers: Dyaics Electrical Aalogues The electrical syste aalogy provides a eas o experietally deteriig the characteristics o a give echaical syste. For the echaical syste, ( x x ) + k x + k ( x ) 0 x + cx + c x ( x x ) + k ( x x ) P si t x + c For the electrical syste, q L q + R q q + C ( ) + 0 C L q q q q C ( q q ) + E si t + R The goverig equatios are equivalet. The characteristics o the vibratios o the echaical syste ay be ierred ro the oscillatios o the electrical syste. q 003 The McGraw-Hill Copaies, Ic. All rights reserved. 9-3