Hadout #b (pp. 4-55) Dyamic Respose o Secod Order Mechaical Systems with Viscous Dissipatio orces M X + DX + K X = F t () Periodic Forced Respose to F (t) = F o si( t) ad F (t) = M u si(t) Frequecy Respose Fuctio o Secod Order Systems MEEN 617 Notes: Hadout bluis Sa Adrés 8-4
(c) Forced respose o d order mechaical system to a periodic orce excitatio Let the exteral orce be PERIODIC o requecy (period T=π/) ad cosider the system to have iitial displacemet X ad velocity V. The equatio o motio or a system with viscous dissipatio mechaism, is: d X d X M + D + K X = F si o ( t) dt dt (41) with iitial coditios V() = V ad X() = X o o The exteral orce F (t) has amplitude F o ad requecy. This type o orced excitatio is kow as PERIODIC LOADING. The solutio o the o-homogeeous ODE (41) is o the orm: ( ) ( ) st 1 st X() t X X A e A e C cos t C si t = + = + + + (4) H P 1 c s where X H is the solutio to the homogeeous orm o (1) ad such that (s 1, s ) satisy the characteristic equatio o the system: ( s ζ s ) The roots o this d order polyomial are: + + = (43) s ζ ζ ( ) 1/ 1, = 1 (44) where K D = M is the atural requecy, ζ = is the viscous Dcr dampig ratio, ad D = KM is the critical dampig coeiciet. cr MEEN 617 Notes: Hadout bluis Sa Adrés 8-41
The value o dampig ratio determies whether the system is uderdamped (ζ <1), critically damped (ζ =1), overdamped (ζ >1). Respose o d Order Mechaical System to a Periodic Loadig: ad ( ) si( ) X = C cos t + C t (45) P c s is the particular solutio due to the periodic loadig, F o si ( t). Substitutio o Eq. (4) ito Eq. (41) gives cos si ( ) { } t K M Cc DC + s + ( t) { K M} C DC = F si( t) s s o (46) sice the si() ad cos() uctios are liearly idepedet, it ollows that { } { } K M Cc+ DC s = K M C DC = F { } (47) K M D Cc = D { K M} Cs Fo i.e. a system o algebraic equatios with two ukows, C c ad C s. Divide Eq. (47) by K ad obtai: { } s s o 1 M K D K Cc = D K { 1 M K} Cs Fo K sice K D = M ; ad with ζ = the D cr (48) MEEN 617 Notes: Hadout bluis Sa Adrés 8-4
D D KM M = ζ = ζ = ζ K K D K ad X ss = F o /K is a pseudo static displacemet Deie a requecy ratio as = (49) relatig the (exteral) excitatio requecy () to the atural requecy o the system ( ); i.e. whe 1, the system operates below its atural requecy 1, the system is said to operate above its atural requecy With this deiitio, write Eq. (48) as: { } 1 ζ Cc = ζ { 1 } Cs Xss { } 1 ζ Cc = ζ { 1 } Cs Xss (5) Solve Eq. (5) usig Cramer s rule to obtai the coeiciets C s ad C c : ( ) ( ζ ) ( 1 ) ( ) ( ζ ) ζ C = X ; C = X 1 + 1 + c ss s ss (51) MEEN 617 Notes: Hadout bluis Sa Adrés 8-43
Respose o d Order Mechaical System to a Periodic Load: For a uderdamped system, < ζ < 1, the roots o the characteristic eq. have a real ad imagiary part, i.e. s 1, i 1 ( ) 1/ = ζ ζ (5) where i = 1 is the imagiary uit. The homogeeous solutio is ( cos( ) si ( )) X = e C t + C t (53) ζ t H() t 1 d d = 1 ζ is the damped atural requecy o the where ( ) 1/ system. d Thus, the total respose is Xt () = XH + XP = ( cos( ) si ( )) cos( ) si ( ) X = e C t + C t + C t + C t ζ t () t 1 d d c s where C s ad C c are give by Eq. (51). (54) At time t =, the iitial coditios are V() = Vo ad X() = Xo. The V + ζ C1 Cs C1= ( X Cc ) ad C= (55) Now, provided ζ >, the homogeeous solutio (also kow as the TRANSIENT or Free respose) will die out as time elapse. Thus, ater all trasiets have passed, the dyamic respose o the system is just the particular respose X P(t) d MEEN 617 Notes: Hadout bluis Sa Adrés 8-44
Steady State Periodic Forced Respose o Uderdamped d Order System As log as the system has some dampig (ζ > ), the trasiet respose (homogeeous solutio) will die out ad cease to iluece the behavior o the system. The, the steady-state (or quasi-statioary) respose is give by: ( ) ( ) ( ) X() t Cccos t + Cssi t = Csi t ϕ (56.a) where (C s, C c ) are give by equatio (51) as: ( ) ( ζ ) ( 1 ) ( ) ( ζ ) ζ C = X ; C = X 1 + 1 + c ss s ss ad Xss Fo K =. Deie C Ccos ( ϕ ); C Csi ( ϕ ) ta ( ϕ ) where ϕ is a phase agle, ad s = = ; the Cc ζ = = C s ( 1 ) c C = C + C = X A S C ss with A = 1 ( 1 ) + ( ζ ) ; as the amplitude ratio where = is the requecy ratio. Thus, the system respose is: X() t = Xss A si( t ϕ ) (56.b) MEEN 617 Notes: Hadout bluis Sa Adrés 8-45
Regimes o Dyamic Operatio: 1, the system operates below its atural requecy ( ) ( ζ ) 1 1; A 1 ϕ X () t X si( t) i.e. similar to the static respose ss = = 1, the system is excited at its atural requecy ( 1 ) ; A ; ϕ ( 9 ) 1 X ss π ζ π X() t si t ζ ζ <.5, the amplitude ratio A > 1 ad a resoace is i said to occur. 1, the system is said to operate above its atural requecy ( 1 ) << 1; ( ζ ) A ϕ π ( 18 ) ( π ) ( ) X () t X Asi t = X Asi t ss ss A <<< 1, i.e. very small, MEEN 617 Notes: Hadout bluis Sa Adrés 8-46
Frequecy Respose o Secod Order Mechaical System ( ϕ ) ( ) X = X Asi t or F = F si t () t ss () t o Amplitude ratio (A) 1 1 8 6 4 FRF d order system Periodic orce: Fo si( t) A dampig ratio=.5 dampig ratio=.1 dampig ratio=. dampig ratio=.5.5.5.75 1 1.5 1.5 1.75 requecy ratio () Phase agle ( ) 18 16 14 1 1 8 6 4 FRF d order system Periodic orce: Fo si( t) φ dampig ratio=.5 dampig ratio=.1 dampig ratio=. dampig ratio=.5.5.5.75 1 1.5 1.5 1.75 re que cy ra tio () MEEN 617 Notes: Hadout bluis Sa Adrés 8-47
Steady State Periodic Forced Respose o d Order system: Imbalace Load Imbalace loads are typically oud i rotatig machiery. I operatio, due to ievitable wear, material build ups or assembly aults, the ceter o mass o the rotatig machie does ot coicide with the ceter o rotatio (spi). Let the ceter o mass be located a distace (u) rom the spi ceter, ad thus, the imbalace load is a cetriugal orce o magitude F o = M u ad rotatig with the same requecy as the rotor speed (). This orce excites the system ad iduces vibratio 1. Note that the imbalace orce is proportioal to the requecy ad grows rapidly with speed. M u I practice the oset distace (u) is very small (a ew thousads o a ich). For example i the rotatig shat & disk has a small imbalace mass (m) located at a radius (r) rom the spi ceter, the it is easy to determie that the ceter o mass oset (u) is equal to (m r/m). Note that u<<r. rm u( M + m) = rm u M The equatio o motio or the d order system is d X d X M + D + K X = F si si o t = M u t dt dt ( ) ( ) 1 The curret aalysis oly describes vibratio alog directio X. I actuality, the imbalace orce iduces vibratios i two plaes (X,Y) ad the rotor whirls i a orbit aroud the ceter o rotatio. For isotropic systems, the motio i the X plae is idetical to that i the Y plae but out o phase by 9 degrees. MEEN 617 Notes: Hadout bluis Sa Adrés 8-48
the with ( F Mu o Xss = = = u = u K K = ). The system respose at steady-state is ( ϕ ) ( ) ( ϕ ) X() = X A si t = u A si t t ss X() t = u B si( t ϕ ) (59) ϕ = where ϕ is a phase agle, ta( ) ζ ( 1 ), ad B = ( 1 ) + ( ζ ) (6) is a amplitude ratio recall = is the requecy ratio. Regimes o Dyamic Operatio: 1, excitatio below its atural requecy ( ) ( ζ ) 1 1; B ad ϕ MEEN 617 Notes: Hadout bluis Sa Adrés 8-49
( ) X() t u si t i.e. little motio = = 1, the system is excited at its atural requecy ( 1 ) ; B ; ϕ ( 9 ) 1 π ζ u π u X () t si t = cos( t) ζ ζ ζ <.5, the amplitude ratio B > 1 ad a resoace is i said to occur. ( ) 1, the system operates above its atural requecy 1 ζ 1; B 1 ϕ π 18 ( π ) ( ) X () t u si t = u si t ( ) B ~ 1, at high requecy operatio, the maximum amplitude o vibratio (X max ) equals the ubalace displacemet (u) MEEN 617 Notes: Hadout bluis Sa Adrés 8-5
Frequecy Respose o Secod Order Mechaical System due to a Imbalace Load ( ϕ) ( ) X = ubsi t or F = Mu si t () t () t Amplitude ratio (B) 1 1 8 6 4 FRF d order system Imbalace orce: M u ^si( t) B dampig ratio=.5 dampig ratio=.1 dampig ratio=. dampig ratio=.5.5.5.75 1 1.5 1.5 1.75 requecy ratio () Phase agle ( ) 18 16 14 1 1 8 6 4 FRF d order system Imbalace orce: M u ^ si( t) φ dampig ratio=.5 dampig ratio=.1 dampig ratio=. dampig ratio=.5.5.5.75 1 1.5 1.5 1.75 requecy ratio () MEEN 617 Notes: Hadout bluis Sa Adrés 8-51
EXAMPLE: Z d m e M e t A catilevered steel pole supports a small wid turbie. The pole torsioal stiess is K (N.m/rad) with a rotatioal dampig coeiciet C (N.m.s/rad). X Y k m X k= Torsioal Tiess e= Torioal Dampig coeiciet c Z d M e c os t Y The our-blade turbie rotatig assembly has mass m o, ad its ceter o gravity is displaced distace e [m] rom the axis o rotatio o the assembly. I z (kg.m ) is the mass momet o iertia about the z axis o the complete turbie, icludig rotor assembly, housig pod, ad cotets. The total mass o the system is m (kg). The plae i which the blades rotate is located a distace d (m) rom the z axis as show. For a complete aalysis o the vibratio characteristics o the turbie system, determie: a) Equatio o motio o torsioal vibratio system about z axis. b) The steady-state torsioal respose θ (t) (ater all trasiets die out). c) For system parameter values o k=98,67 N.m/rad, I z =5 kg.m, C = 157 N.m.s/rad, ad m o = 8 kg, e = 1 cm, d = 3 cm, preset graphs showig the respose amplitude (i rads) ad phase agle as the turbie speed (due to wid power variatios) chages rom 1 rpm to 1, rpm. d) From the results i (c), at what turbie speed should the largest vibratio occur ad what is its magitude? e) Provide a desig recommedatio or chage so as to reduce this maximum vibratio amplitude value to hal the origial value. Neglect ay eect o the mass ad bedig o the pole o the torsioal respose, as well as ay gyroscopic eects. Note: the torque or momet iduced by the mass imbalace is T (t) = d x F u = m o e d cos(t), i.e., a uctio o requecy T( ) MEEN 617 Notes: Hadout bluis Sa Adrés 8-5
The equatio describig torsioal motios o the turbie-pole system is: I zθ + Cθ + Kθ = moe d cos t = T( ) (e.1) cos t Note that all terms i the EOM represet momets or torques. (b) Ater all trasiets die out, the periodic orced respose o the system: θss θ () t = cos φ 1/ ( 1 ) + ( ζ ) ( t ) (e.) but θ ss T m ed I m ed = = = K K Iz Iz ( ) o z o (e.3) K C with = ; = ; ζ =, ad I K I z z φ 1 ζ = ta 1 (e.4) (e.3) i (e.) leads to mo ed θ () t = cos ( t- φ) 1/ I z ( 1 ) + ( ζ ) (e.5) Let med I o θ = (e.6) z B = 1/ ( 1 ) + ( ζ ) (e.7) ad rewrite (e.5) as: θ () t = θ B cos ( t φ) (e.8) MEEN 617 Notes: Hadout bluis Sa Adrés 8-53
(c) or the give physical values o the system parameters: K rad K = 98,67 N.m/rad = = 6.8 Iz sec C I z = 5 kg m ζ = = KI.5 z C = 157. N.m.s/rad med o 8.1.3.4 θ 96 1-5 = = = = rad Iz 5 5 Ad the turbie speed varies rom 1 rpm to 1, rpm, i.e. = rpm π/3 = 1.47 rad/s to 15.66 rad/s, i.e. = =.167 to., thus idicatig the system will operate through resoace. θ ( t) = 96.4 1 rad Bcos ( t φ) Hece, the agular respose is ( 5 ) (d) Maximum amplitude o respose: sice ζ << 1, the maximum amplitude o motio will occur whe the turbie speed coicides with the atural requecy o the torsioal system, i.e. at = 1, B 1 ζ ad 1 π θ() t = θ cos t ζ θ the magitude is θ MAX = max θ = =.964 x 1 - rad, i.e. 1 times larger ξ tha θ. MEEN 617 Notes: Hadout bluis Sa Adrés 8-54
The amplitude (degrees) ad phase agle (degrees) o the polo twist are show as a uctio o the turbie rotatioal speed (RPM) amplitude o torsio (rad).1.5 4 6 8 1 1 rotor speed (RPM) Phase agle (degrees) 18 1 6 4 6 8 1 1 rotor speed (RPM) (e) Desig chage: DOUBLE DAMPING but irst BALANCE ROTOR! MEEN 617 Notes: Hadout bluis Sa Adrés 8-55