Optimization of Rotating Machines Vibrations Limits by the Spring - Mass System Analysis

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1 Joural of aerials Sciece ad Egieerig B 5 (7-8 (5 - doi: 765/6-6/57-8 D DAVID PUBLISHING Opimizaio of Roaig achies Vibraios Limis by he Sprig - ass Sysem Aalysis BENDJAIA Belacem sila, Algéria Absrac: The prese wor cosiss o calculae he parameers of vibraios respose of he roaig machies, i cosiss o developme a ew calculaio mehod We will calculae he respose of he sprig-mass sysem wih fiished differeces accordig o operaio parameers of he model elemes used (rigidiy, dampig ad frequecy A esimae of he absolue values of orms limis of machies vibraios will be sared Key words: vibraio, dampig, siffess, free moveme Iroducio The roaig machies prese paricular behaviors i sage rasiioal The free respose of he sysem is o be evaluaed A calculaio of hese parameers will be sared of which we will ry o deermie his respose accordig o he parameers such as he siffess ad he dampig, which represe he parameers of forces opposie o he exciaio effor odel of he achies Vibraios The roaig machies vibraios will be modeled by he sysem wih oe degree o Fig [] Roaig machie ca be compared wih his sysem: (Fig he base is replaced by he bearigs he Sprigs is replaced by he roor he mass is replaced by he body of he urbie or he pump Dampig is esured by he fluid passig i he urbie or he pump Basic Equaio The differeial equaio raslaig he behavior of Correspodig auhor: BENDJAIA Belacem, Phd sude, research fields: urbomachiery performaces ad desig, vibraios aalysis ad diagosic he sysem (displaceme of mass x ( compared o is res posiio afer release i a direcio ca be wrie: x ( + C x ( + x ( F ( ( where, is he rigidiy of he sprig which expresses ha he force of recall is proporioal o elogaio x ad he force F applied o he basis A viscous dampig (or fluid of coefficie C exers o he moveme of he mass (m, a force of dx dampig C is proporioal o he isaaeous d speed The oal respose of he moveme is: x( xl ( + xf ( ( where, x L ( is respose of he free moveme, soluio of he equaio: x ( + C x ( + x ( ( x F ( is he respose of he moveme forced, soluio of he equaio: x ( + C x ( + x ( F ( (4 The respose of he free moveme [] i he case or ξ < is: ξ x L ( e si( ξ + φ (5 C where, ξ ad C C C are criically dampig C The aural frequecy of he free moveme is: (6

2 4 Opimizaio of Roaig achies Vibraios Limis by he Sprig - ass Sysem Aalysis Fig echaical sysem wih oe degree where, / (8 Ω Ω ( ( F / + ( ξ Ω ξ g φ (9 Ω ( T π ( ε π If ε <<, he T, wih he aural pulsaio We ca have he harmoic respose forced o oe degree o Fig 4 Fig Roaig machies sysem Case of a Roaig achie (Case of exciaio by a saic ubalace, rigid roor, ha is o say a rigid roor wih a exciaio of saic ubalace (Fig 5 e is eccericiy, C is he graviy ceer ad G is he ieria ceer z ( ad y( are he co-ordiaes of he cere of graviy alog wo perpedicular direcios The equaios of he moveme is: m m d d d d ( y ( + e cos( Ω y ( c y ( ( ( z ( + e si( Ω z ( c z ( ( Fig Oe degree sysem Whe ξ, he C CC, his case is rare, ad whe ξ >, he free respose is a expoeial fucio x L ( is represeed o Fig j By holdig accou ha F( Fe, he respose of he forced moveme is: j( Ω φ x ( e (7 Fig 4 Harmoic Forced respose wih oe Degree

3 Opimizaio of Roaig achies Vibraios Limis by he Sprig - ass Sysem Aalysis 5 Wih r ( y ( + iz (, hus: jω m r ( + c r ( + r ( me Ω e We have he soluio for Ω : Ω e( j ( Ω ϕ r ( e ( Ω Ω ( ( + j (ε Tha correspods o he criical speed (Fig 6 4 Numerical Calculaio The expressios of he firs ad secod derived by usig he differeces are fiished o solve he equaio of moio of a roor: x x x ( + + x x (4, Δ i j Δ x ( x + x + x (5 Δ The respose of he free moveme is he soluio of he Eq ( Tha is o say: ( x + + ( + + x x C x+ x x (6 Δ Δ ( Δ x x x+ + CΔ + We choose he boudary codiios: ( ; x Fig 5 Fig 6 Rigid liaiso Criical speed (7 ( T; x where, T is he period, i is equivale o say ha if, he, x x+ CΔ (8 If T, he - T, ( T x + ( x + + T (9 CT + The, CT CT T ( T T + T + C his while posig ha I is he value of dampig per ui of mass We idicae he siffess per ui of mass by: ( The dampig ad he force of recalls represe opposie effors o he force of he mass These forces are represeed by he coefficies ad By aig accou of he expressios of he coefficie ε << quoed o chaper, ad ha dampig is far from ha criicizes, ad for oe period π of he moveme deadeed: T We have: π + ( 4π + If he aural frequecy is a cosa, we ca wrie: + A( B ( ( + 4A ( Wih π A ( Ce (4 The ampliude of he free vibraios is a fucio of he coefficies of siffess ad dampig quoed above I addiio, if here is o dampig, he free respose is:

4 6 Opimizaio of Roaig achies Vibraios Limis by he Sprig - ass Sysem Aalysis B ( F B ( (5 Taig accou of he equilibrium equaio, wih A, B ad B, which deped o he aural frequecy F g (6 For a raed frequecy Ω, if aig he Eq (4, F / Ω ( We will have: B / B / B B / Ω ( / Cos a e (7 (8 B / (9 Ω ( Same maer of he respose wih dampig (Eq (4 we have: Cos a e ( B B Le us evaluae his cosa: B B ( I he case of omial frequecy Ω cosa, we will hus have i geeral if here is dampig: / ( B ( Ω Ω ( ( + ( ξ The free respose varies accordig o he same maer as he raio of he coefficies of siffess ad dampig I varies accordig o he effors opposed o ha of he exciaio The force of dampig ad siffess vary accordig o he effor F Wih Eq ( ad If here is wea dampig we will have he equaio: Ω ( ( If Ω χ Ω, he: Ω χ ( (4 Thus: (5 χ ( + Same maer if here is o dampig we will have: χ Ω (6 χ ( Thus: (7 χ ( + This raio varies accordig o he oscillaio frequecy If χ, he: (8 Wih dampig: (9 Accordig o he Eq (4, if Ω, The eds owards ifiie hus he deomiaor of Eq ( eds owards zero, ha is o say: C χ (4 C 4 Case of Roaig achies The expressio of he vibraios i he case of roig machies is: Ω ( r / e (4 Ω Ω ( ( + (ε Same maer ad for omial speed of roaio Ω, we ca wrie: Ω ( r / (4 e Ω Ω ( ( + (ε I he case ha Ω χ Ω ad (ε «, we will have: χ r + χ χ (4 e χ + χ χ ( + χ If χ, he

5 Opimizaio of Roaig achies Vibraios Limis by he Sprig - ass Sysem Aalysis 7 r 5 Opimizaio of Vibraios Limis e (44 From expressio, If here is dampig: r χ ( + e (45 (5 e ad if χ, we oe: If Ω, he, eds owards ifiie hus he r τ deomiaor of Eq ( eds owards zero, ha is o (5 say: We buil Table from he sadard ISO 68- χ orm, which icludes he absolue values of roaig (46 machies vibraios or The value of coefficie τ is meioed i his able C χ We suppose ha he ampliude C is relaive o (47 equilibrium correspodig o he admissible limi The same resuls are obaied, for sprig masse vibraio accordig o orm Ad sysem or for roaig machies This value of χ correspodig o vibraio ampliude a omial speed (value sill calculaed correspods o criical speed of roaig admissible The coefficie τ aced from absolues machies, ie: vibraios, i will be he same for he measures, i C χ (48 always has a cosa value for all machie groups Ω C Wih he aural frequecy: The, ad τ τ r χ Calculae he relaive chage i he real value of Ω r e (49 over is supposed value (value o omial speed, C sill admissible accordig he orm So we have : Table Admissible vibraios i mm/s; Norm ISO 68- [] Group No No No No 4 Vibraio (i mm/s Vibraio (i mm/s Coefficie τ a ( εa ε a f f f f f a (b Fig 7 Equivalece of Specrum vibraio

6 8 Opimizaio of Roaig achies Vibraios Limis by he Sprig - ass Sysem Aalysis Δ τ ( τ (5 or Δ ε ( τ (5 The icreased off wih respec o he equilibrium codiio ( is: Δ ε (54 The differece of wo variaios is: Δ ε ε ε (55 The ew value is: + Δ ε (56 The admissible vibraio are caused by ubalace wih values i orm limis The low frequecy specrum i Fig 7a ad ha o Fig 7b of vibraios calculaio are admissible vibraios Wih Figs 7a ad 7b, we ca say ha he wo specrum are for admissible vibraios Wih ampliude of he fudameal frequecy a ( ε, we have: RS RS a ( ε a ( + ε ε (57 ( ε ε RS RS 5 Calculaios Applicaios Our machie is a machie eraied by gas urbie series GE 5P, used for elecric eergy producio wih power of W The geeraor roor mass is os ad he omial speed is, r/mi We apply he resuls which use his machie o he oher machie groups herefore; A ew form of he orm is preseed i Table I aes accou he calculaio i Table The ampliude calculaio is compared o he orm values We ca calculae wih he same maer he ew vibraio values from, he: + ε + ε + ε 4 + ε + ε + ε Thus, + ( ε + ( ε + ε + ( ε ( + ( ε + ε (58 ε ( + ( ε + ε (59 ε We should search he umber of operaios calculaio o fid he ew values ca ae, his is by solvig i equaio: >, hus: + ( ε + ε > (6 The erm sum is a geomeric progressio, so: ε ( ε + ε > (6 ε > ( ε ε (6 ( ε hus > The limi value of is for The erm i braces is equal o zero for (Table I correspods o our calculaios We oo he lower limi of his rage because a icrease i vibraio ca cause deerioraio of he equipme saus achie desig is he mai judge of he limis of admissible vibraios Our sudy ca be ae io accou i machie desig 6 Numerical Calculaio of Free Respose From equaio of he free respose wihou dampig we have: + 4A ( (6 We will have: (64 A ( π For a sysem wih several degrees (wo degrees for example were we havig o dampig: [ ] [ K ] { } { } π (65 Table Compariso vibraio calculaed i mm/s ad hose of he orm Groupe Group No Group No Group No Group No 4 Norm ISO Our calculaio

7 Opimizaio of Roaig achies Vibraios Limis by he Sprig - ass Sysem Aalysis 9 Table Calculaio o real machies achie characerisics: achie: geeraor eraied by gas urbie model GE 5P Geeraor roor mass (: abou os Nomial roaio speed:, r/mi Firs criical speed: A,5 r/mi Facor τ for all groups of machies: τ 6 Group: machie group IV Calculaios : Facor : τ Ui: ( / 58 ( 5 8 Error Calculaio : Error: ε Δ ε ( τ Error: ε Δ 8 8 ε 8 84 Global error: Δ ε ε ε % New Value of : mm / s RS a ( ε a ( ε a ( mm / s Value of : ( + ( ε + ε > ( ε ε ( ε Example: ε ( + ( ε + ε + ε ε This is he soluio of he equaio: + K Wih: [ ] [ ] { } { } j ϕ { } { Ω x e } (66 ( (67 To exrac he forced aswer we see he paricular soluio of he sysem of differeial equaio wih a secod member []: jω [ ] x( + [ C] { x( } + [ K ] { x( } { F } e (68 Wih Thus, j ϕ { } { Ω x e } ( (69 jϕ [ K Ω + jcω] { e } F For he free moveme: ϕ [ Ω + jcω] { e j } (7 K (7 7 Numerical Calculaio Applicaios I he sysem represeed o Fig 8, le us calculae he aural frequecy ad he proper

8 Opimizaio of Roaig achies Vibraios Limis by he Sprig - ass Sysem Aalysis vecors: 97 m (78 The clea vecors associaed each pulsaio are: For : (79 For : (8 8 Coclusios Fig 8 Calculaio example m x x + m x x (7 We see hese vecors ad pulsaio while respecig: ( K I (7 This expressio ca be derived from Eq (7 (For he free respose wihou dampig: ϕ [ K Ω ] { e j } (74 Thus, m de m (75 Tha is o say: 4 m 5m + (76 Thus: 48 m (77 I his sudy we deermied he expressio of he vibraios respose by aalysis of sysem sprig masse accordig o he siffess ad dampig parameers which represe he forces opposed o he effor of exciaio The vibraio respods calculaed depeds o he rigidiy, dampig ad oscillaio frequecy The res of his aricle is o revalue he limis of machie vibraio by usig a error calculaio ad jusifyig he reasoig by aalyical calculaio applied o a seleced machie We hope ha his wor will be developed i fuure ad will coribue o scieific research i his area Refereces [] Bouleger, A ad Pachaud, C 7 Aid emory Aalysis of Vibraio oiorig of achies [] Codiioal aieace by Vibraio Aalysis Thermal Power pla of sila Algeria [] Vibraio Traiig 999 SCV Swizerlad Thermal Power Pla of sila Algeria