Oe Degree of Freedom, Harmoically Excited Vibratios Forced Harmoic Vibratio A mechaical syem is said to udergo forced vibratio wheever exteral eergy is sulied to the syem durig vibratio Exteral eergy ca be from either a alied force or a imosed dislacemet excitatio The alied force or dislacemet excitatio may be harmoic, oharmoic but eriodic, oeriodic, or radom Harmoic excitatios are of the forms, for examle, Sice the free-vibratio resose x ( ) h 83 Mechaical Vibratios Lesso 6 t dies out with time uder each of the three coditios of damig (uderdamig, critical damig, ad overdamig), the geeral solutio evetually reduces to the articular solutio x( t ), which reresets the eady-ate vibratio Figure shows homogeeous, articular, ad geeral solutios for the uderdamed case i( ωt+φ) ( ) F e, ( ) ( ω +Φ) ( ) ( ω +Φ) F t F t F cos t, F t F si t, where F is the amlitude, ω is the frequecy, ad Φ is the hase agle usually take to be zero Uder a harmoic excitatio, the resose of the syem will also be harmoic with the same frequecy as the excitatio frequecy If the frequecy of the harmoic excitatio is close to the syem atural frequecy, the beatig heomeo will hae This coditio, kow as resoace, is to be avoided to revet failure of the syem Cosider a syem i Figure The equatio of motio is Figure : A srig-mass-damer syem mx + cx + kx F( t) Its geeral solutio is ( ) ( ) + ( ), x t x t x t h where xh( t ) is the homogeeous solutio (the solutio whe F( t ) as was udied i the free vibratio) ad x ( t ) is the articular solutio Figure : Homogeeous, articular, ad geeral solutios for the uderdamed case Coyright 7 by Withit Chatlataagulchai
Udamed Syem uder F cosω t Cosider the syem i Figure but without damer If a force F t F t acts o the mass m, the equatio of motio is give by ( ) cos where ω The solutio is ( ) ( ) mx + kx F cos ω t () ( ) ( ) + ( ), x t x t x t h x t C cosω t+ C si ω t, h x t X cos ωt Alyig iitial coditios x( ) x ad x( ) x (), ad by subitutig () ito (), the three ukows C, C, ad X ca be solved, the the solutio becomes F x F x( t) x cos si cos ωt+ ωt+ ωt k mω ω k mω Lettig / F k force F, we have δ deote the atic deflectio of the mass uder a X δ ω ω The quatity X / δ reresets the ratio of the dyamic to the atic amlitude of motio ad is called amlitude ratio (3) 83 Mechaical Vibratios Lesso 6 The total resose () ca also be writte i three cases as follows: ) For ω / ω <, we have ) For ω / ω >, we have δ x( t) Acos( ωt Φ ) + cos ωt ω ω δ x( t) Acos( ωt Φ) cos ωt ω ω 3) For, we have a beatig heomeo Lettig x x, we have ω ω ( ) ( F / m) ( cosω cosω ) x t t t ω ω ( F / m) ω+ ω ω ω si t si t ω ω Let the forcig frequecy ω be slightly less tha the atural frequecy: The ω ω we have Usig (5) ad (6), (4) becomes (4) ω ω ε (5) ω+ ω ω (6) Coyright 7 by Withit Chatlataagulchai
F m x t t t εω ( ) / siε si ω 83 Mechaical Vibratios Lesso 6 Defie eriod of beatig to be τ π ε π ( ω ω) frequecy of beatig to be ω ε ω ω b b / / Defie The lots of the total resoses of all three cases are give i Figure 3 Figure 3: Plots of total resoses of the three cases 3 Coyright 7 by Withit Chatlataagulchai
Examle : [] A recirocatig um with mass 68 kg is mouted as show below at the middle of a eel late of thickess cm, width 5 cm, ad legth 5 cm Durig oeratio, the late is subjected to a harmoic force F t t N Fid the amlitude of vibratio of the late ( ) cos683 83 Mechaical Vibratios Lesso 6 Solutio The late ca be modeled as a fixed-fixed beam with equivalet srig coat k 9EI 3 l 9 5 9 ( ) ( )( ) 48 N / m ( 5 ) 3 3 From (3), we have X F / k ω ω /48 683 48 / 68 35 m The egative sig idicates that the resose x( t ) is out of hase with the excitatio F( t ) 4 Coyright 7 by Withit Chatlataagulchai
Damed Syem uder F cosω t For the damed syem, the equatio of motio becomes 83 Mechaical Vibratios Lesso 6 mx + cx + kx F cos ω t (7) Assume the articular solutio i the form ( ) ( ω Φ ) x t X cos t, where X ad Φ are ukow coats to be determied Subitutig ito (7) ad equatig the coefficiets of cosω t ad siω t o both sides, we obtai X ad F / ( ω ) k m + c ω cω Φ k mω ta Figure 4 shows lots of forcig fuctio ad articular solutio Recall that ω k / m udamed atural frequecy, ζ c / cc c / mω, δ / F k deflectio uder the atic force F, ad r ω / ω frequecy ratio The amlitude ratio ad the hase agle is give by X δ ( r ) + ( ζ r) ω ω + ζ ω ω / (8) ad Figure 4: Reresetatio of forcig fuctio ad resose ω ζ ω ζ r φ ta ta ω r ω The lots of amlitude ratio ad hase agle versus frequecy ratio are give i Figure 5 ad Figure 6 resectively The total resose is give by ( ) ( ) + ( ) x t x t x t For a uderdamed syem, we have ζω t d h x( t) X e cos( ω t φ ) + X cos( ωt φ) 5 Coyright 7 by Withit Chatlataagulchai
where ω d ζ ω X ad Φ are ukow coats to be x x ad determied from iitial coditios For the iitial coditios ( ) x ( ) x, we have two equatios to solve for two ukows 83 Mechaical Vibratios Lesso 6 x X cosφ + X cos φ, x ζω X cosφ + ω X siφ + ωx si φ d (9) Figure 6: Variatio of Φ with r Figure 5: Variatio of X / δ with r 6 Coyright 7 by Withit Chatlataagulchai
Examle : [] Fid the total resose of a sigle degree of freedom syem with m kg, c N s / m, k 4 N / m, x, ad ω ad x uder a exteral force ( ) cos ω rad / s F t F t with F N 83 Mechaical Vibratios Lesso 6 For small values of damig, we ca take X X Q () δ δ ζ max ω ω The differece betwee the frequecies associated with the half ower ( Q / ) oits R ad R is called badwidth of the syem X / δ Q ζ Solutio From the data, we have ω k / m 4 / rad / s, δ F / k / 4 5 m, ( )( ) ζ c / c c / km / 4 5, d c ( ) ( ) ω ζ ω 5 997 rad / s, r ω / ω / 5 δ 5 X 336m / ( r ) + ( ζ r) ( 5 ) + ( 5 5) ζ r 5 5 φ ta ta 38475 r 5 Subitutig the data above ito (9), we get X 33 ad Φ 5587 Q R Badwidth Figure 7: Harmoic resose curve showig half ower oits ad badwidth To fid R ad R, we set X / Q / whose solutios are R δ i (8) to obtai 4 r r ( 4 ζ ) + ( 8 ζ ) ω / ω 7 Coyright 7 by Withit Chatlataagulchai
ω ω r R ζ, r R + ζ ω ω ω ω ω Usig the relatio + ad ( + )( ) ( R R ) 4, we have that ω ω ω ω ω ω ω ζω the badwidth is give by ω ω ω ζω Combiig the badwidth equatio with (), we obtai ω Q ζ ω ω It ca be see that Q ca be used for eimatig the equivalet viscous damig ad the atural frequecy i mechaical syems i t 3 Damed Syem uder Fe ω Let the harmoic forcig fuctio be rereseted i comlex form F t F e iωt The equatio of motio becomes as ( ) Assume the articular solutio Subitutig ito (), we have mx + cx + kx F e ω () i t ( ) x t Xe ω i t where F X ( k mω ) + icω 83 Mechaical Vibratios Lesso 6 k mω cω F i ( k mω ) c ω ( k mω ) c ω + + F iφ e, / ( k mω ) + c ω cω φ k mω ta Thus, the articular solutio (or eady-ate solutio) becomes F ( k mω ) + ( cω ) i( ωt φ ) x ( t) e / () The comlex frequecy resose of the syem is defied to be whose magitude is give by X H ( iω) F k r + i ζ r ( ω) H i kx / / F ( r ) + ( ζ r) H( iω ) ca be used i the exerimetal determiatio of the syem arameters ( m, c, ad k ) 8 Coyright 7 by Withit Chatlataagulchai
If F ( t ) F cos ωt, real art of (), which is If F ( t ) F si ωt, the corresodig articular solutio is the F x t t ( k mω ) + ( cω) ( ) cos( ω φ) / the corresodig articular solutio is the imagiary art of (), which is Suort Motio F x t t ( k mω ) + ( cω ) ( ) si( ω φ) / (3) Sometimes the base or suort of a srig-mass-damer syem udergoes harmoic motio, as show i Figure 8 The equatio of motio is give by Suosig that y( t) Y si ωt, ( ) ( ) mx + c x y + k x y 83 Mechaical Vibratios Lesso 6 the equatio of motio becomes mx + cx + kx ky+ cy ky siωt+ cωy cosωt Asi( ωt α), cω where A Y k + ( cω ) ad α ta k This is similar to havig the forcig fuctio F t F ωt actig o the syem ad the ( ) si same aalysis as the revious sectio ca be alied The articular solutio is similar to (3) ad is give by x ( t) X si( ωt φ α) Y k + ( cω) si( ωt φ / α ), ( k mω ) + ( cω) X si, ( ωt φ) cω where φ ta k mω ad 3 3 mcω ζ r φ ta ta k( k mω ) + ( ωc) + (4ζ ) r Figure 8: Base excitatio Dislacemet Trasmissibility The ratio of the amlitude of the resose x ( ) base motio ( ) y t, X Y ratio is give by t to that of the, is called the dislacemet trasmissibility The 9 Coyright 7 by Withit Chatlataagulchai
/ / X k + ( cω) + ( ζ r) Y ( k mω ) + ( cω) ( r ) + ( ζ r) 83 Mechaical Vibratios Lesso 6 The lots betwee / i Figure 9 is give X Y ad φ versus frequecy ratio r ω / ω Figure 9: The lots betwee X / Y ad φ versus frequecy ratio r ω / ω Force Trasmissibility Let F be the force trasmitted to the base or suort due to the reactios from the srig ad the dashot We have F k( x y) + c( x y ) mx The eady-ate solutio x( t ) was foud to be x( t) X ( ωt φ) Therefore, T ω si( ω φ) si( ω φ) F m X t FT t si F is called dyamic force amlitude The ratio F / ky is called the force trasmissibility ad is give by T Coyright 7 by Withit Chatlataagulchai
FT / + ( ζ r) r ky ( r ) + ( ζ r) Figure shows the force trasmissibility The force trasmissibility cocet is used i the desig of vibratio isolatio syems 83 Mechaical Vibratios Lesso 6 The eady-ate solutio is give similar to (3) by ω si( ω φ ) z t Z t ( k mω ) + ( cω ) m Y t ( ) si( ω φ / ) where mω Y r Z Y ( k mω ) + ( cω) ( r ) + ( ζ r) cω ζ r φ ta ta k mω r The ratio Z / Y is show i Figure ad lot of φ is i Figure 6,, Figure : Force trasmissibility 3 Relative Motio Let z x y deote the motio of the mass relative to the base The equatio of motio becomes + + si mz cz kz my mω Y ωt Figure : Relative Motio lot Coyright 7 by Withit Chatlataagulchai
Examle 3: [] Cosider a simle model of a motor vehicle below The vehicle has a mass of kg The srig coat is 4 kn/m ad the damig ratio of ζ 5 If the vehicle seed is km/hr, determie the dislacemet amlitude of the vehicle The road surface varies siusoidally with a amlitude of Y 5 m ad a wavelegth of 6 m 83 Mechaical Vibratios Lesso 6 Solutio From the give data, we ca comute the followig quatities: ω π f π 9 rad / s, 36 6 / 3 k 4 ω 8574 rad/s, m ω 58778 r 38653, ω 8574 X + ( ζ r) Y ( r ) + ( ζ r) + ( 5 38653), ( 38653) + ( 5 38653) X 46937Y 46937(5) 7346 m / / Coyright 7 by Withit Chatlataagulchai
Examle 4: [] A heavy machie, weighig 3 N, is suorted o a resiliet foudatio The foudatio has srig iffess k 4, N / m The machie vibrates with a amlitude of cm whe the base of the foudatio is subjected to harmoic oscillatio at the udamed atural frequecy of the syem with a amlitude of 5 cm Fid (a) the damig coat of the foudatio, (b) the dyamic force amlitude o the base, ad (c) the amlitude of the dislacemet of the machie relative to the base 3 Rotatig Ubalace 83 Mechaical Vibratios Lesso 6 Cosider a machie with rotatig ubalaced masses i Figure Solutio a) Sice, we have r Therefore, ω ω X Y + ( ζ) ( ζ) / 4 5 We the have ζ 9 The damig coat is give by c ζ c ζ km 935 N s / m c / + 4ζ b) FT Yk 4 N 4ζ Y 5 c) Z 968 m ζ 9 ( ) Figure : A machie with rotatig ubalaced masses 3 Coyright 7 by Withit Chatlataagulchai
The total mass of the machie is M, ad there are two eccetric masses m / rotatig i oosite directios with a coat agular velocity ω We cosider two equal masses m / rotatig i oosite directios i order to have the horizotal comoets of excitatio of the two masses cacel each other The equatio of motio is give by 83 Mechaical Vibratios Lesso 6 Mx + cx + kx meω si ωt The solutio of this equatio is similar to (3) ad is give by x ( t) X si( ωt φ), where meω X, / ( k Mω ) + ( cω) cω φ ta k Mω The lots betwee MX / me ad φ versus r are give i Figure ad Figure 6 Examle 5: [] The figure below deicts a Fracis water turbie Water flows from A ito the blades B ad dow ito the tail race C The rotor has a mass of 5 kg ad a ubalace (me) of 5 kg-mm The radial clearace betwee the rotor ad the ator is 5 mm The turbie is to be oerated at 6 rm The eel shaft carryig the rotor ca be assumed to be clamed at the bearigs Determie the diameter of the shaft so that the rotor is always clear of the ator Assume damig is egligible 4 Coyright 7 by Withit Chatlataagulchai
Lesso 6 Homework Problems 39, 3, 33, 339, 347, 35 83 Mechaical Vibratios Lesso 6 Homework roblems are from the required textbook (Mechaical Vibratios, by Sigiresu S Rao, Pretice Hall, 4) Refereces [] Mechaical Vibratios, by Sigiresu S Rao, Pretice Hall, 4 Solutio Settig c, we have X meω k Mω ( ) meω k r ( ) (5 ) ( π ) 5 ( π ) k 4k k 3 6 4 π N/m 4 3EI 3E π d Sice for the eel beam, k, 3 3 we have l l 64 3 4 3 64 kl (64)(4 π )( ) d 65 m 3π E 3 π (7 ) d 7 m 7 mm 4 4 4 ad 5 Coyright 7 by Withit Chatlataagulchai