Estimation of Natural Frequency of the Bearing System under Periodic Force Based on Principal of Hydrodynamic Mass of Fluid

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1 Internatonal Journal o Mechancal and Mechatroncs Engneerng Vol:, No:7, 009 Estmaton o Natural Frequency o the Bearng System under Perodc Force Based on Prncpal o Hydrodynamc Mass o Flud M. H. Pol, A. Bd, and A. V. Hosen Internatonal Scence Index, Mechancal and Mechatroncs Engneerng Vol:, No:7, 009 waset.org/publcaton/170 Abstract Estmaton o natural requency o structures s very mportant and sn t usually calculated smply and sometmes complcated. Lack o knowledge about that caused hard damage and hazardous eects. In ths paper, wth usng rom two derent models n FEM method and based on hydrodynamc mass o luds, natural requency o an especal bearng (Fg. 1) n an electrc eld (or, a perodc orce) s calculated n derent stness and derent geometrc. In nal, the results o two models and analytcal soluton are compared. Keywords Natural requency o the bearng, Hydrodynamc mass o lud method. I. INTRODUCTION STIMATION o natural requency o structures s very Emportant and sn t usually calculated smply and sometmes complcated. Lack o knowledge about that caused hard damage and hazardous eects. For example amous brdge o Sanranssco s explored because o havng the same requency as the wnd! Moreover because o not havng a smooth manner or estmatng natural requency o structures, Calculatng natural requency s very complcated usually especally when the structure subject to nteracton between a lud (ol) and a sold. In these cases, the FEM methods are powerul methods because o havng useul concept or analyzng complcated problems wth lud and sold nteracton. The FSI (Flud- Sold nteracton) s wdely nvestgated by many scentsts. When the luds move n a narrow regon, the eects o hydrodynamc mass are usually hgher than sold mass eects, although the mmersed body s hgher densty than lud [1]. Because o dsplacng o the lud when movng a body nto the lud, a gradent pressure s produced n the lud. Thereore, resultant orce on the sold body s ntroduced rom result o these orces. M. H. Pol s a member o aculty n Islamc Azad Unversty o Iran (Ahvaz branch). He s also wth Ahvaz manuacturng technology Research Center (ACECR) (Correspondng author to provde phone: ; e-mal: m_h_pol@ modares.ac.r). A. Bd, Jr., s wth Engneerng Center o IKCO, Tehran, Iran (e-mal: beh_eng@yahoo.com). A. V. Hosen s a member o aculty n Ahvaz manuacturng technology Research Center (ACECR) (phone: ; e-mal: savh@ ymal.com). In ths paper, ths s assumed that the lud s ncompressble and rctonless so the orce o lud s related to relatve acceleraton o the sold body (hydrodynamc mass) essentally. The hydrodynamc concept s descrbed by Lamb, Stokes, Batton & Brkho and others nvestgators. In ths paper, by usng rom two derent models n FEM method and based on hydrodynamc mass o luds, natural requency o an especal bearng (Fg. 1) n an electrc eld (or, a perodc orce) s calculated n derent stness and derent geometrc. The results presented by Lamb or hydrodynamc mass s used n ths paper []. Fg. 1 The bearng to be analyzed n ths paper II. THEORY For the deal stuaton, two long eccentrc cylnders are consdered wth a lud (ol) between them. The outer radus o nner cylnder and the nner radus o outer cylnder consder a and b respectvely. The potental o velocty φ s: 1 φ Vθ = ; V r r θ φ = (1) r Wth assumng ncompressble and rctonless, the lud s deal and rrgatonal low, thereore the boundary condtons are φ = x cos 1 θ at r=a r () φ = x cos θ at r=b r () Internatonal Scholarly and Scentc Research & Innovaton (7) scholar.waset.org/ /170

2 Internatonal Journal o Mechancal and Mechatroncs Engneerng Vol:, No:7, 009 Internatonal Scence Index, Mechancal and Mechatroncs Engneerng Vol:, No:7, 009 waset.org/publcaton/170 And contnuty equaton s φ 1 φ ( r ) + = 0 r r r θ A soluton or n ths condton s φ = ()cos r θ Wth solvng (4) and (5) we have: '' ' r r (4) (5) + = 0 (6) Wth solvng (6) the veloctes s ound B Vr = ( A)cosθ r B Vθ = ( + A)snθ (8) r Where constants A and B are x 1a x b A = b a (9) ba B = ( x ) 1 x b a (10) Snce the velocty o each partcle n generalzed coordnate s dened separately so Lagrange s equaton can be wrtten as F = + (7) d T T dt x x (11) Where x and T are generalzed poston and knetc energy o the lud and x s the translatonal moton o th cylnder and F s the reacton orce on that. I dsplacement o cylnder s small to compare wth lud thckness, last term n (11) can be neglected. Thereore, F d T = dt x (1) And the knetc energy s: b π 1 ρ θ( r θ ) (1) a 0 T = rldrd V + V From equatons (7), (8), (1) and (1): F = M x + ( M + M ) x 1 H 1 1 H F = ( M + M ) x ( M + M + M ) x 1 H 1 1 H (14) (15) Where F 1 and F are the reacton orce o lud on the nner and outer cylnder respectvely, and M 1 (M 1 =πa Lρ) and M (M =πb Lρ) are the mass dsplaced by nner cylnder and llng mass o outer cylnder n absence o nner cylnder respectvely. T = A x + A x A x x Or (n matrx orm) T = T x Ax In case havng two bodes T = A x + A x x + A x From (1) and (18): F = A x A x (16) (17) (18) (19) F = A 1x1 A x (0) Where F 1 and F are reacton orces on bodes 1 and. Wth assumng the body surround body, x1 = x and the lud acceleraton n each pont o the ncompressble lud equals to x and n result pressure gradent s n the lud. P = ρ. x x Ths pressure gradent produces the orces: F = ( A + A ) x = M x F = ( A + A ) x = M x 1 And A11 + A1 = M1 (1) () () (4) A1 + A = M (5) Equatons (4) and (5) are two equatons or three unknowns constant. Wth assumng x = 0 (body s xed), rom (19): F1 = A 11x1 = M Hx1 (6) The equaton (6) denes MH and wth assumng that the velocty o body 1 s known, can be calculated and then rom contnuty low, the velocty o lud can be estmated. IV. DYNAMICALLY COUPLED FLUID ELEMENT (FLUID 8) Ths element s used or showng o relaton o two ponts o a structure that are separated wth the lud. The center ponts are cylnder axs and lud lls space between them. Each element has two degrees o reedom n each node. For example moton n x and y drectons and cylnder axs les n z drecton (see Fg. 1). Ths element s used n dynamc analyzed n ths paper. III. CALCULATION OF FORCES ON TWO BODIES The same o Lamb model, when the moton o lud s ntroduced by the moton o loated body, the lud energy s a degree uncton as: Internatonal Scholarly and Scentc Research & Innovaton (7) scholar.waset.org/ /170

3 Internatonal Journal o Mechancal and Mechatroncs Engneerng Vol:, No:7, 009 Internatonal Scence Index, Mechancal and Mechatroncs Engneerng Vol:, No:7, 009 waset.org/publcaton/170 Fg. FLUID8 dynamc lud couplng element V. ASYMMETRIC HARMONIC FLUID ELEMENT (FLUID 81) Ths element s a modcaton o the PLANE5 element or modelng a lud that s surrounded n regons that don t havng net low. Ths element has 4 nodes wth DOF (n x, y and z drectons or each node) and as named s used n axal symmetrc problems. Loadng n ths element can be nonsymmetrc. (see Fg. ). Fg. FLUID81 Axsymmetrc - Harmonc contaned lud element VI. PRINCIPAL OF OPERATION: SERVO FLUID EXTERNAL PRESSURE BEARING (SFEPB) In ths bearng, ol layer s ntroduced by external pressure n system. The stness o the bearng s ntroduced by the pressure gradent n derent regons o the bearng. Thereore, when the more ree moton o the shat s mean lower stness o the bearng. As known journal bearng are two knd hydrodynamc and hydrostatc bearngs. The hydrodynamc one s workng accordng to dynamc prncples and makng lud lm between bearng and shat. Another one s workng accordng to statc prncples and s usng external pressure or workng. The SFEPB works wth two concepts assgned, but they moded the desgn. VII. CALCULATION OF STIFFNESS AND DAMPING COEFFICIENTS Stness and dampng coecents n hydrodynamc bearng are [5]: K D BD BD ηωdl = c K ε 1 ( 1 ε ) ( 1 ε ) 5/ ( N / M, Ib/ n) ηdl D1 N.sec Ib.sec =, / c m n (7) (8) And or the hydrostatc bearng P S d KBS = C0 (9) cl L a Where K BD, D BD = Radal stness and dampng o bearng K 1, D 1 = Constants c, d = Dametrc clearance and journal dameter respectvely L = Bearng length η, ε = Dynamc vscosty and eccentrcty rato Ω = Rotatonal velocty a = Kashak length C 0 = Dmensonless stness that s dened as 7.65 β (1 β ) C0 = 1 (0) β + γ(1 β) P β = 0.5 (1) PS na( l a) γ = 0.5 () π db P p = Pressure n = Number o b = crcumerental length (g. 4). Fg. 4 Hydrostatc journal parameters As shown n the equatons, the stness and dampng coecent are related together so dampng varaton aect the stness coecent. VIII. DESCRIPTION OF PROBLEM AND ASSUMPTION As shown n Fg. 1, the bearng s lled wth the lud wth envronment pressure P s =0. Because o small dsplacement o rng n z drecton, the analyss s assgned axsymmetrc and two dmensonal. The ollowng assumptons or solvng the problem are consdered. 1- Wth changng dmensons stness s not changed. - Moton s n radal drecton essentally. Internatonal Scholarly and Scentc Research & Innovaton (7) scholar.waset.org/ /170

4 Internatonal Journal o Mechancal and Mechatroncs Engneerng Vol:, No:7, Flud lows very smoothly and pressure gradent s neglgble so P S. 001Pa. 4- Usng o ormula (9), total stness o bearng s equal to 100 N/m and ol vscosty s equal to 1068 Kg / m. Internatonal Scence Index, Mechancal and Mechatroncs Engneerng Vol:, No:7, 009 waset.org/publcaton/170 IX. SOLVING OF THE PROBLEM Wth usng The ANSYS sotware and two elements FLUID8 and FLUID81, the problem s solved (see Fg. 6(a), 6(b)). FLUID8 C0MB14 FLUID8 + * /\/\//\/\ * + Fg. 6 (a) Modelng o the problem wth FLUID8& COMB14 Fg. 6 (b) Modelng o the problem wth FLUID81& COMB14 X. RESULTS The answer o numercal soluton or the two modelng shows only 1% derent (see Fg. 7). Moreover, the gradent o natural requency o system vs. changng o others parameters s shown n Fgs Fg. 7 F vs. K results or two modelng Fg. 8 The eect o changng o nner radus o outer cylnder on the natural requency o system Fg. 9 The eect o changng o outer radus o nner cylnder on the natural requency o system Fg. 10 The eect o changng o outer radus o rng n natural requency o system Fg. 11 The eect o changng o nner radus o rng on natural requency o system Internatonal Scholarly and Scentc Research & Innovaton (7) scholar.waset.org/ /170

5 Internatonal Journal o Mechancal and Mechatroncs Engneerng Vol:, No:7, 009 XI. CONCLUSION 1- The rst natural requency s 7. HZ and the second one s 11 Hz. - The natural requency s related to square root o stness as n theory. - The natural requency s related to nverse o densty. 4- As shown n Fg. 16, dampng s equal to zero then the maxmum value or natural requency s estmated. Internatonal Scence Index, Mechancal and Mechatroncs Engneerng Vol:, No:7, 009 waset.org/publcaton/170 Fg. 1 The eect o bearng o stness on the natural requency o system Fg. 1 The eect o ol densty on the natural requency o system Fg. 14 The eect o rng thckness on the natural requency o system REFERENCES [1] Klaus-Jurgen Bathe, Flud-structure nteracton, the ully coupled soluton o lud lows wth structural, Flud-structure nteracton.htm. [] Frtz,R. J., The Eect o Lquds on the Dynamc Motons o Immersed Solds, ASME,J.o Engr. or Industry,Volume 94, Feb [] Ansys Manual, secton 4 element reerences. Copyrght Elan Computer Group, Inc. [4] Ansys Manual, Mode - Frequency Analyss secton 17. Copyrght Elan Computer Group, Inc. [5] Prncple o Operaton, How the Servo lud Externally Pressurzed Bearng works, www. /Prncples O Operaton Servolud.com.htm. Fg. 15 The eect o dampng on the natural requency o system Internatonal Scholarly and Scentc Research & Innovaton (7) scholar.waset.org/ /170