Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Vibration Characteristics of Hydrodynamic Fluid Film Pocket Journal Bearings

Vibration Characteristics of Hydrodynamic Fluid Film Pocket Journal Bearings Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2010, Article ID 589318, 10 pages doi:10.1155/2010/589318 Research Article Vibration Characteristics of Hydrodynamic Fluid Film Pocket Journal Bearings N. S. Feng and E.J.Hahn School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, NSW 2052, Australia Correspondence should be addressed to N. S. Feng, n.feng@unsw.edu.au Received 3 September 2010; Accepted 12 October 2010 Academic Editor: Jorge Arenas Copyright © 2010 N. S. Feng and E. J. Hahn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Theoretical analyses of hydrodynamic fluid film bearings with different bearing profiles rely on solutions of the Reynolds equation. This paper presents an approach used for analysing the so-called pocket bearings formed from a combination of offset circular bearing profiles. The results show that the variation of the dynamic bearing characteristics with different load inclinations for the pocket bearings is less than that for the elliptic bearing counterpart. It is shown that the natural frequencies as well as the critical speeds, and hence the vibrational behaviour, can also be significantly different for an industrial rotor supported by the different bearings. 1. Introduction frequently used [2]. Improvements to these techniques to achieve better solution accuracy using more complicated yet In order to increase productivity and reduce machine still manageable approaches have also been developed [3]. downtime, hydrodynamic bearings are frequently used to Alternatively, for steady-state solutions where it is possible support high-speed rotating machinery where reliability, to prepare the bearing data a priori, numerical solutions long running life, and minimum vibration levels are of to the full Reynolds equation can be tabulated or plotted primary concern. Simple circular bore journal bearings for subsequent use [4]. For different bearing types, the sometimes cause instability, which may result in catastrophic different bearing clearance profiles need to be considered failure of the machinery. To improve resistance to such fail- when obtaining the corresponding solutions [5]. Such data is ure, different bearing types with different bearing clearance also needed for the different types of pocket bearings, whose profiles have been developed and used in practice. Typical respective static and dynamic characteristics are of interest. examples are elliptic bearings and tilting pad bearings, This paper illustrates the approach used to obtain solutions the former providing stabilizing preload and the latter for these pocket bearings. minimizing the troublesome cross-coupled bearing forces. Another bearing type, formed by a combination of offset circular bearing profiles and referred to as a pocket bearing, is 2. Theory also sometimes used in machinery such as turbogenerators. 2.1. General Theory. The schematic of a simple circular The theoretical analysis of hydrodynamic bearings relies hydrodynamic fluid film bearing is shown in Figure 1.Fol- on the solutions of Reynolds equation, a partial differential equation derived from the Navier-Stokes and continuity lowing the usual assumptions of hydrodynamic lubrication equations under certain simplifying assumptions [1]. In theory, the Reynolds equation can be written as [1] order to minimise the computational effort, particularly in transient response analyses, some further simplifications ∂ ∂P ∂ ∂P ∂h 3 3 h + h =−6μU +12μV , (1) such as infinitely short or long bearing approximations are ∂X ∂X ∂Z ∂Z ∂X 2 Advances in Acoustics and Vibration where z W h = C − z sin ψ − y cos ψ,(2) ψ = θ + φ + . (3) Fluid film Upon nondimensionalisation, (1) becomes ∂ 3 ∂P D ∂ 3 ∂P h + h X O θ b ∂ψ ∂ψ L ∂Z ∂Z (4) =− 2z − y sin ψ + 2y + z cos ψ , O in which Z = , L/2 P = , 6μω(R/C) Figure 1: Schematic of a circular bearing. h y h = = 1 − z sin ψ − y cos ψ, y = ,and so forth, C C h =−z sin ψ − y cos ψ. Top lobe (5) The pressure boundary conditions are that P = 0at φ ψ Z =±1 (bearing edges) and at ψ and ψ (the boundary 1 2 coordinates at the onset and end of the fluid film, resp.). The Reynolds cavitation boundary condition is used to define the cavitation region [1]. Upon integration of the film pressure, the fluid film force Fluid film components in y and z directions are ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ψ2 L/2 F cos ψ 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ x =− P RdZdψ = . (6) F ψ −L/2 sin ψ W z Bottom lobe ψ12 The corresponding nondimensional force components are [3] ⎛ ⎞ Figure 2: Schematic of an elliptic bearing. ⎛ ⎞ ⎛ ⎞ ψ 1 f cos ψ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ =−3 P dZdψ = ⎝ ⎠ . (7) f ψ −1 sin ψ z 1 πS The above is the summary of the general theory for deter- mining the static and dynamic bearing characteristics (i.e., For small perturbations Δy, Δz, Δy ,and Δz in the y and Sommerfeld number, attitude angle, stiffness coefficients, z directions about the equilibrium position, the dynamic and damping coefficients). Different bearing configurations bearing coefficients are defined as impose different geometric relationships for the position of ⎛ ⎞ ⎡ ⎤ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞ Δ f K K Δz C C Δz the journal in the bearing. z zz zy zz zy ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ =− − . (8) Δ f K K Δy C C Δy y yz yy yz yy 2.2. Pocket Bearing Configurations. Two types of pocket Ignoring higher-order terms, the Taylor series expansion bearings are to be considered. The first type, formed in a about the equilibrium position (y , z , y = 0and z = 0) similar way to an elliptic bearing (Figure 2), is schematically yields K =−∂f /∂z,C =−∂f /∂y ,and so forth. shown in Figure 3 and is referred to as type 1. The two zz z zy z halves of a bearing block are held together, separated by Note that for more than one pad or clearance profile, shims corresponding to the ellipticity 2d . A circular hole of it is the resultant of the forces on all the pads or clearance 1 diameter D is cut, and the shims are then removed to form profiles in the y and z directions which equals zero and W, respectively, in (6); it is the perturbation in these resultant a normal elliptic bearing with the top pad centre a distance d below, and the bottom pad centre a distance d above force components which is used to evaluate the dynamic 1 1 bearing coefficients. the bearing centre. Concentric arc portions (the pockets) of Advances in Acoustics and Vibration 3 bb z β d e 1 b d O 1 bt R φ e t Figure 5: Geometric relationship between the arc centres for elliptic and type-1 pocket bearings. The two types of the pocket bearings are expected to have Figure 3: Schematic of type-1 pocket bearing. similar vibration characteristics. Normally, the bearing clearance C is used to nondi- mensionalise the film thickness and the relative journal to bearing displacements. Due to the bearing profile, each pocket bearing has two clearances: C for the pocket portion and C for the side arcs. Using different clearances will produce different nondimensional bearing coefficients, but for a given bearing, the dimensional coefficients should not be affected. Following the approach for elliptic bearings [5], C could be chosen as the nondimensionalising parameter. However, the type-2 pocket bearing does not have such a d d 2 2 corresponding clearance, though it also has two clearances: C for the pocket portion and C for the side arcs. Hence, in p s order to have comparable results, the pocket clearance C , common to both types of pocket bearings, is used as the nondimensionalising parameter. 2.3. Type-One Pocket Bearing. From (2), the nondimensional film thickness for the side arcs becomes h = − z sin ψ − y cos ψ e (9) Figure 4: Schematic of type-2 pocket bearing. and for the pocket arcs h = 1 − z sin ψ − y cos ψ. (10) a specified angular extent are then machined as shown in Note that in these equations, y and z are the nondimensional Figure 3. Such a pocket bearing has three curvature centres: journal centre coordinates from the corresponding arc oneabove onebelow thebearing centre forthe side arcs just centres. like the elliptic bearings and one coinciding with the bearing For the pocket portion: centre for the pockets. The second type of pocket bearing, referred to as type y = ε sin φ − β , 2, is shown schematically in Figure 4.Itcan be conveniently (11) machined by NC machines. It also has three arc centres, but z =−ε cos φ − β , the two for the side arcs are located at a distance d to the left and for the bottom side arcs, and right of the bearing centre, the left centre for the left arcs on both the top and bottom pads, and the right one for the y = ε sin φ − β , b b right arcs, again on both pads. The angular extent of the side (12) arcs is determined by the arc radii and the centre offset d . 2 z =−ε cos φ − β , b b b 4 Advances in Acoustics and Vibration bb α/2 O O b br α/2 Figure 6: Geometric relationship between C and C for type-1 e p pocket bearing. R where, from Figure 5, 2 2 ε = ε + δ +2εδ cos φ, (13) sin φ = sin 180 − φ . For the top side arcs, Figure 7: Geometric relationship between C and C for type-2 s p y = ε sin φ − β , t t pocket bearings. (14) z =−ε cos φ − β , t t t d d 2 2 where, from Figure 5, 2 2 ε = ε + δ − 2εδ cos φ, (15) sin 180 − φ = sin φ. O O bl br l O φ The clearances C and C are related by the pocket arc extent r e p α and the ellipticity of the side arcs d .As shown in Figure 6, at the intersection between the side and pocket arcs, one has e 2 2 2 ◦ e R = R + d − 2R d cos 180 − . (16) p 1 e p 1 Solving for R and omitting higher-order terms α α α O 2 2 R =−d cos + R − d sin ≈ R − d cos . p 1 e 1 e 1 2 2 2 Figure 8: Geometric relationship between the arc centres for type-2 (17) pocket bearing. The journal radius can be expressed as r = R − C = R − C . (18) e e p p 2.4. Type-Two Pocket Bearing. Again, following an approach Substitution of (17)into(18)gives similar to that for the type 1 bearings, one has for the side arcs C − C = R − R ≈ d cos , (19) e p e p 1 h = − z sin ψ − y cos ψ s (21) or C d cos(α/2) α e 1 ≈ 1+ = 1+ δ cos (20) 1 and for the pocket arcs C C 2 p p which is always greater than 1. h = 1 − z sin ψ − y cos ψ. (22) p 100 −20 Advances in Acoustics and Vibration 5 S and φ The clearances C and C are related by the pocket arc extent s p α and the offset of the side arc centers d .As shown in Figure 7, at the intersection between the side and pocket arcs, one has 2 2 2 ◦ R = R + d − 2R d cos 90 − . (23) p 2 s p 2 Solving for R and omitting higher-order terms, α α α 2 2 R = d sin + R − d cos ≈ R + d sin . p 2 2 s 2 2 2 2 0 0.2 0.4 0.6 0.8 1 (24) The journal radius can be expressed as S(1) S(2) φ(1) φ(2) r = R − C = R − C . (25) s s p p (a) From (24)and (25), this gives C − C ≈ d sin , (26) p s 2 Stiffness coefficients or C d sin(α/2) α s 2 ≈ 1 − = 1 − δ sin . 2 (27) C C 2 p p To relate the offset d to the ellipticity d , it is assumed 2 1 that, apart from the same pocket arc extent α, the horizontal widths of the two bearings are the same, that is, (28) R + d = R − d ≈ R . s 2 e e 1 0 0.2 0.4 0.6 0.8 1 Substitution of (24)and (17)into(28)gives K (1) K (2) yy yy α α R − d sin + d = R + d cos , (29) p 2 2 p 1 K (1) K (2) yz yz 2 2 −K (1) −K (2) zy zy or K (1) K (2) zz zz cos(α/2) (b) d = d . (30) 2 1 1 − sin(α/2) The relationships between y, z and ε, φ are the same as Damping coefficients before. However, the calculations for ε , ε , φ ,and φ need l r l r different expressions. From Figure 8, ε = ε + δ +2εδ sin φ, l 2 2 ◦ ◦ sin 90 − φ = sin 90 + φ , ε 30 (31) ε = ε + δ − 2εδ sin φ, r 2 ◦ ◦ sin 90 + φ = sin 90 − φ . 0 0.2 0.4 0.6 0.8 1 −10 C (1) C (2) yy yy 3. Solution Procedures −C (1) −C (2) yz yz −C (1) −C (2) zy zy To calculate the static and dynamic characteristics of the C (1) C (2) zz zz pocket bearings, Gauss-Seidel iteration with successive over (c) relaxation is used to solve the finite difference formulation of the Reynolds equation [5], thereby ensuring that the Reynolds condition is used as the cavitation boundary. Figure 9: Comparison of the nondimensional bearing characteris- All results here assume same zero gauge inlet, outlet, and tics for the two types of pocket bearings ((1) and (2) correspond to cavitation pressures. type 1 and 2, resp.). 6 Advances in Acoustics and Vibration 1.2 0.8 S φ 45 0.4 0 0 −90 −30 30 90 −90 −30 30 90 β β S(e) φ(e) S(p) φ(p) (a) (b) Elliptic Pocket 20 20 15 15 K 10 K 10 5 5 0 0 −90 −30 30 90 −90 −30 30 90 β β K −K K −K yy zy yy zy K K K K yz zz yz zz (c) (d) Elliptic Pocket 40 40 20 20 C C 0 0 −90 −30 30 90 −90 −30 30 90 −20 −20 C −C C −C yy yz yy yz −C C −C C yz zz yz zz (e) (f) Figure 10: Nondimensional bearing characteristics for elliptic and type-1 pocket bearings at ε = 0.2. Advances in Acoustics and Vibration 7 0.45 0.3 55 S S 0.15 −5 −90 −30 30 90 −90 −30 30 90 β β S(e) φ(e) S(p) φ(p) (a) (b) Elliptic Pocket 10 10 K 6 K 6 2 2 −90 −30 30 90 −90 −30 30 90 −2 −2 β β K −K K −K yy zy yy zy K K K K yz zz yz zz (c) (d) Elliptic Pocket 18 18 12 12 C 6 C 0 0 −90 −30 30 90 −90 −30 30 90 −6 −6 C C C C yy yz yy yz C C C C yz zz yz zz (e) (f) Figure 11: Nondimensional bearing characteristics for elliptic and type-1 pocket bearings at ε = 0.4. 8 Advances in Acoustics and Vibration 10 m TM Millhpp3 V3.2 Figure 12: Sample rotor bearing system. Elliptic Pocket 1000 2000 3000 4000 5000 1000 2000 3000 4000 5000 (rpm) (rpm) m1 m4 m1 m3 m2 syn. m2 syn. m3 (a) (b) Figure 13: Campbell diagrams for a rotor supported by (a) elliptic bearings or (b) type-1 pocket bearings. 4. Results The rotor is shown schematically in Figure 12. Table 1 lists the corresponding data used to characterise the rotor and Figure 9 compares the nondimensional static and dynamic the bearings. In house vibration analysis software was used characteristics of the two types of pocket bearings as a to determine the natural frequencies and mode shapes over function of load inclination beta (β), using the same base a speed range up to 5000 rpm. Table 2 lists these natural clearance C and the the same pocket extent of α = frequencies, and Figure 13 shows the Campbell diagrams 60 . Apart from some difference in the attitude angle, the for the elliptic bearing supports and for the type-1 pocket nondimensional characteristics are virtually identical. bearing supports. Figures 10 and 11 compare the bearing characteristics It can be seen that the system behaves quite differently of the type-1 pocket bearing to a similar elliptic bearing with the different types of bearing supports. The rotor with as functions of the load inclination angle β for given the elliptic bearing has three natural frequencies but two eccentricities ε = 0.2 and 0.4, respectively. It can be seen critical speeds in the speed range; while the rotor with the that in general there is less variation in the dynamic bearing type-1 pocket bearing has four natural frequencies but only characteristics of the pocket bearing due to the change of β. one critical speed. The second mode in both cases is a A sample vibration analysis is also performed for an backward whirl and is therefore unlikely to be excited. Using industrial rotor subjected to gravity loading and supported equivalent clearances for the different pocket bearing models, on either type of pocket bearings or similar elliptic bearings. the natural frequencies in Table 2 indicate that the two types (rad/s) (rad/s) Advances in Acoustics and Vibration 9 Table 1: Sample rotor bearing data. Rotor: Young’s modulus = 210 GPa, density = 7850 kg/m 2 2 No. Length (m) Diameter (m) Disk mass (kg) Disk Ip (kg-m)DiskId(kg-m ) 1 0.14 0.135 0 0.0358 0.0436 2 0.16 0.18 0 0.1294 0.1329 3 0.148 0.18 0 0.1193 0.1131 4 0.148 0.18 0 0.1193 0.1131 5 0.115 0.258 8.392 0.5447 0.3336 6 0.115 0.277 1.185 0.5447 0.3336 7 0.115 0.277 1.185 0.5447 0.3336 8 0.106 0.328 11.87 1.291 0.7217 9 0.106 0.353 0.921 1.291 0.7217 10 0.049 0.345 0 0.535 0.2747 11 0.076 0.44 89.29 8.643 4.408 12 0.058 0.5 47.73 6.559 3.317 13 0.15 0.556 93.86 20.57 8.875 14 0.224 0.556 139.5 30.48 14.42 15 0.2 0.556 111 25.38 12.52 16 0.219 0.556 128.4 28.57 14 17 0.217 0.556 126.6 28.19 13.88 18 0.216 0.556 122.8 27.57 13.77 19 0.215 0.556 119.1 27.05 13.66 20 0.214 0.556 119.1 26.87 13.59 21 0.221 0.547 126.7 26.83 13.6 22 0.198 0.55 117.2 24.64 12.55 23 0.159 0.704 4.15 30.62 16.34 24 0.159 0.704 4.15 30.62 16.34 25 0.16 0.497 249.4 30.81 16.46 26 0.015 0.345 0 0.1638 0.0821 27 0.168 0.353 1.467 2.056 1.335 28 0.168 0.353 1.467 2.056 1.335 29 0.17 0.353 1.484 2.081 1.359 30 0.187 0.315 0 1.419 1.043 31 0.07 0.385 38.39 3.034 1.559 32 0.238 0.315 0 1.806 1.59 33 0.238 0.315 0 1.806 1.59 34 0.067 0.42 25.1 2.904 1.489 35 0.065 0.58 67.21 12.73 6.436 36 0.07 0.58 131 22.1 11.16 Bearings: δ = 0.5, (α = 60 for pocket bearings) No. Length (m) Diameter (m) Clearance (mm) Viscosity (Ns/m)Location 1 0.092 0.18 0.1327 0.014 2 2 0.204 0.315 0.2322 0.014 32 of the pocket bearings with similar dimensions could have Compared to similar elliptic bearings, the pocket bear- quite different effects on the system vibration characteristics. ings tend to provide less fluctuation in the dynamic bearing coefficients for different load inclinations and may produce significantly different vibration behaviour in a given rotor 5. Conclusions system. The two types of pocket bearings investigated here An approach to evaluate the static and dynamic bearing (produced by different machining procedures), apart from characteristics of pocket type bearings, suited for subsequent some difference in attitude angle, have virtually identical steady-state vibration analysis of rotating machinery involv- static and dynamic bearing characteristics. ing such bearings, is presented. 10 Advances in Acoustics and Vibration Table 2: Damped natural frequencies of a rotor supported by elliptic or pocket bearings ((1) refers to type 1, etc.). Speed (rpm) 1000 2000 3000 4000 5000 Elliptic Real, Imag (rad/s) Mode 1 −19.91, 137.5 −11.17, 143.9 −7.588, 150.4 −5.047, 156.1 −3.323, 160.8 Mode 2 −4.064, 211.8 −5.900, 207.7 −6.266, 204.2 −5.913, 201.6 −5.291, 199.6 Mode 3 −129.3, 217.4 −89.73, 261.5 −84.35, 288.3 −79.77, 317.3 −70.84, 344.6 Pocket Real, Imag (rad/s) Mode 1 (1) −27.05, 192.6 −19.47, 183.2 −10.63, 184.8 −5.012, 189.2 −1.620, 193.4 (2) −30.75, 186.0 −23.41, 175.1 −15.76, 177.6 −9.796, 182.3 −6.021, 186.6 Mode 2 (1) −3.379, 212.6 −4.965, 209.7 −5.133, 207.6 −4.768, 206.3 −4.255, 205.4 (2) −3.633, 212.0 −4.647, 208.7 −4.249, 206.8 −3.419, 205.9 −2.527, 205.6 Mode 3 (1) −162.5, 75.30 −246.2, 140.1 −300.0, 189.1 −352.9, 237.0 −418.2, 293.1 (2) −201.1, 48.79 −331.0, 79.78 −390.9, 114.1 −473.7, 166.2 −566.0, 241.0 Mode 4 (1) −85.34, 57.67 −140.8, 121.7 −194.9, 206.5 −223.7, 315.3 −202.6, 425.3 (2) −111.7, 29.36 −196.7, 118.1 −258.1, 265.9 −212.5, 394.2 −155.1, 462.0 Nomenclature Superscripts (If Not Otherwise Defined): C: Radial clearance = R − r : Nondimensional C ,··· : Nondimensional bearing damping yy :Differentiation with respect to nondimensional time τ coefficients ˙: Differentiation with respect to dimensional time t D: Bearing diameter d: Ellipticity or offset Subscripts (If Not Otherwise Defined): e: Eccentricity F , F : Film force components y z o: Steady state f , f : Nondimensional force components y z b, j: Bearing, journal h:Filmthickness b, t: Bottom lobe, top lobe K ,··· : Nondimensional bearing stiffness yy e, p, s: Elliptic arc, pocket arc, side arc coefficients l, r:Leftarc,rightarc L:Bearingwidth 1, 2: Pocket type 1, pocket type 2. O: Centres P: Pressure (gauge) References R: Bearing radius r: Journal radius [1] O. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrica- S: Sommerfeld number = μωRL(R/C) /(πW) tion, McGraw-Hill, New York, USA, 1961. t:Time [2] F. W. Ocvirk, “Short bearing approximation for full journal U, V : Velocities at journal surface in X and Y bearings,” Tech. Rep. NACA TN 2808, 1952. directions, respectively [3] M. A. Rezvani and E. J. Hahn, “Limitations of the short bearing W: External load approximation in dynamically loaded narrow hydrodynamic X, Y , Z: Cartesian coordinates at bearing surface bearings,” Journal of Tribology, vol. 115, no. 3, pp. 544–549, x, y, z: Cartesian coordinates at curvature centers 1993. α:Pocketextent [4] J. W. Lund, “Rotor-bearing dynamics design technology—part β: Load inclination from vertical III: design handbook for fluid film type bearings,” Tech. Rep. AFADL-TR-65-45, Dayton, Ohio, USA, 1965. Δ: Perturbation; variation in δ: Nondimensional ellipticity/offset = d/C [5] N.S.Fengand E. J. Hahn,“Computation ofbearing charac- teristics: elliptic bearings and tilting pad bearings,” Tech. Rep. ε: Nondimensional eccentricity = e/C 1993/AM/3, UNSW, Sydney, Australia, 1993. μ: Absolute viscosity φ: Attitude angle θ: Angular coordinate from line of centres = X/R τ: Nondimensional time = ωt ω: Speed ψ: Angular coordinate from y axis International Journal of Rotating Machinery International Journal of Journal of The Scientific Journal of Distributed Engineering World Journal Sensors Sensor Networks Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Volume 2014 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Submit your manuscripts at http://www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2010 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Acoustics and Vibration Hindawi Publishing Corporation

Vibration Characteristics of Hydrodynamic Fluid Film Pocket Journal Bearings

Loading next page...
 
/lp/hindawi-publishing-corporation/vibration-characteristics-of-hydrodynamic-fluid-film-pocket-journal-0mjj0PUd0u

References

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

Publisher
Hindawi Publishing Corporation
Copyright
Copyright © 2010 N. S. Feng and E. J. Hahn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ISSN
1687-6261
eISSN
1687-627X
DOI
10.1155/2010/589318
Publisher site
See Article on Publisher Site

Abstract

Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2010, Article ID 589318, 10 pages doi:10.1155/2010/589318 Research Article Vibration Characteristics of Hydrodynamic Fluid Film Pocket Journal Bearings N. S. Feng and E.J.Hahn School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, NSW 2052, Australia Correspondence should be addressed to N. S. Feng, n.feng@unsw.edu.au Received 3 September 2010; Accepted 12 October 2010 Academic Editor: Jorge Arenas Copyright © 2010 N. S. Feng and E. J. Hahn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Theoretical analyses of hydrodynamic fluid film bearings with different bearing profiles rely on solutions of the Reynolds equation. This paper presents an approach used for analysing the so-called pocket bearings formed from a combination of offset circular bearing profiles. The results show that the variation of the dynamic bearing characteristics with different load inclinations for the pocket bearings is less than that for the elliptic bearing counterpart. It is shown that the natural frequencies as well as the critical speeds, and hence the vibrational behaviour, can also be significantly different for an industrial rotor supported by the different bearings. 1. Introduction frequently used [2]. Improvements to these techniques to achieve better solution accuracy using more complicated yet In order to increase productivity and reduce machine still manageable approaches have also been developed [3]. downtime, hydrodynamic bearings are frequently used to Alternatively, for steady-state solutions where it is possible support high-speed rotating machinery where reliability, to prepare the bearing data a priori, numerical solutions long running life, and minimum vibration levels are of to the full Reynolds equation can be tabulated or plotted primary concern. Simple circular bore journal bearings for subsequent use [4]. For different bearing types, the sometimes cause instability, which may result in catastrophic different bearing clearance profiles need to be considered failure of the machinery. To improve resistance to such fail- when obtaining the corresponding solutions [5]. Such data is ure, different bearing types with different bearing clearance also needed for the different types of pocket bearings, whose profiles have been developed and used in practice. Typical respective static and dynamic characteristics are of interest. examples are elliptic bearings and tilting pad bearings, This paper illustrates the approach used to obtain solutions the former providing stabilizing preload and the latter for these pocket bearings. minimizing the troublesome cross-coupled bearing forces. Another bearing type, formed by a combination of offset circular bearing profiles and referred to as a pocket bearing, is 2. Theory also sometimes used in machinery such as turbogenerators. 2.1. General Theory. The schematic of a simple circular The theoretical analysis of hydrodynamic bearings relies hydrodynamic fluid film bearing is shown in Figure 1.Fol- on the solutions of Reynolds equation, a partial differential equation derived from the Navier-Stokes and continuity lowing the usual assumptions of hydrodynamic lubrication equations under certain simplifying assumptions [1]. In theory, the Reynolds equation can be written as [1] order to minimise the computational effort, particularly in transient response analyses, some further simplifications ∂ ∂P ∂ ∂P ∂h 3 3 h + h =−6μU +12μV , (1) such as infinitely short or long bearing approximations are ∂X ∂X ∂Z ∂Z ∂X 2 Advances in Acoustics and Vibration where z W h = C − z sin ψ − y cos ψ,(2) ψ = θ + φ + . (3) Fluid film Upon nondimensionalisation, (1) becomes ∂ 3 ∂P D ∂ 3 ∂P h + h X O θ b ∂ψ ∂ψ L ∂Z ∂Z (4) =− 2z − y sin ψ + 2y + z cos ψ , O in which Z = , L/2 P = , 6μω(R/C) Figure 1: Schematic of a circular bearing. h y h = = 1 − z sin ψ − y cos ψ, y = ,and so forth, C C h =−z sin ψ − y cos ψ. Top lobe (5) The pressure boundary conditions are that P = 0at φ ψ Z =±1 (bearing edges) and at ψ and ψ (the boundary 1 2 coordinates at the onset and end of the fluid film, resp.). The Reynolds cavitation boundary condition is used to define the cavitation region [1]. Upon integration of the film pressure, the fluid film force Fluid film components in y and z directions are ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ψ2 L/2 F cos ψ 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ x =− P RdZdψ = . (6) F ψ −L/2 sin ψ W z Bottom lobe ψ12 The corresponding nondimensional force components are [3] ⎛ ⎞ Figure 2: Schematic of an elliptic bearing. ⎛ ⎞ ⎛ ⎞ ψ 1 f cos ψ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ =−3 P dZdψ = ⎝ ⎠ . (7) f ψ −1 sin ψ z 1 πS The above is the summary of the general theory for deter- mining the static and dynamic bearing characteristics (i.e., For small perturbations Δy, Δz, Δy ,and Δz in the y and Sommerfeld number, attitude angle, stiffness coefficients, z directions about the equilibrium position, the dynamic and damping coefficients). Different bearing configurations bearing coefficients are defined as impose different geometric relationships for the position of ⎛ ⎞ ⎡ ⎤ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞ Δ f K K Δz C C Δz the journal in the bearing. z zz zy zz zy ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ =− − . (8) Δ f K K Δy C C Δy y yz yy yz yy 2.2. Pocket Bearing Configurations. Two types of pocket Ignoring higher-order terms, the Taylor series expansion bearings are to be considered. The first type, formed in a about the equilibrium position (y , z , y = 0and z = 0) similar way to an elliptic bearing (Figure 2), is schematically yields K =−∂f /∂z,C =−∂f /∂y ,and so forth. shown in Figure 3 and is referred to as type 1. The two zz z zy z halves of a bearing block are held together, separated by Note that for more than one pad or clearance profile, shims corresponding to the ellipticity 2d . A circular hole of it is the resultant of the forces on all the pads or clearance 1 diameter D is cut, and the shims are then removed to form profiles in the y and z directions which equals zero and W, respectively, in (6); it is the perturbation in these resultant a normal elliptic bearing with the top pad centre a distance d below, and the bottom pad centre a distance d above force components which is used to evaluate the dynamic 1 1 bearing coefficients. the bearing centre. Concentric arc portions (the pockets) of Advances in Acoustics and Vibration 3 bb z β d e 1 b d O 1 bt R φ e t Figure 5: Geometric relationship between the arc centres for elliptic and type-1 pocket bearings. The two types of the pocket bearings are expected to have Figure 3: Schematic of type-1 pocket bearing. similar vibration characteristics. Normally, the bearing clearance C is used to nondi- mensionalise the film thickness and the relative journal to bearing displacements. Due to the bearing profile, each pocket bearing has two clearances: C for the pocket portion and C for the side arcs. Using different clearances will produce different nondimensional bearing coefficients, but for a given bearing, the dimensional coefficients should not be affected. Following the approach for elliptic bearings [5], C could be chosen as the nondimensionalising parameter. However, the type-2 pocket bearing does not have such a d d 2 2 corresponding clearance, though it also has two clearances: C for the pocket portion and C for the side arcs. Hence, in p s order to have comparable results, the pocket clearance C , common to both types of pocket bearings, is used as the nondimensionalising parameter. 2.3. Type-One Pocket Bearing. From (2), the nondimensional film thickness for the side arcs becomes h = − z sin ψ − y cos ψ e (9) Figure 4: Schematic of type-2 pocket bearing. and for the pocket arcs h = 1 − z sin ψ − y cos ψ. (10) a specified angular extent are then machined as shown in Note that in these equations, y and z are the nondimensional Figure 3. Such a pocket bearing has three curvature centres: journal centre coordinates from the corresponding arc oneabove onebelow thebearing centre forthe side arcs just centres. like the elliptic bearings and one coinciding with the bearing For the pocket portion: centre for the pockets. The second type of pocket bearing, referred to as type y = ε sin φ − β , 2, is shown schematically in Figure 4.Itcan be conveniently (11) machined by NC machines. It also has three arc centres, but z =−ε cos φ − β , the two for the side arcs are located at a distance d to the left and for the bottom side arcs, and right of the bearing centre, the left centre for the left arcs on both the top and bottom pads, and the right one for the y = ε sin φ − β , b b right arcs, again on both pads. The angular extent of the side (12) arcs is determined by the arc radii and the centre offset d . 2 z =−ε cos φ − β , b b b 4 Advances in Acoustics and Vibration bb α/2 O O b br α/2 Figure 6: Geometric relationship between C and C for type-1 e p pocket bearing. R where, from Figure 5, 2 2 ε = ε + δ +2εδ cos φ, (13) sin φ = sin 180 − φ . For the top side arcs, Figure 7: Geometric relationship between C and C for type-2 s p y = ε sin φ − β , t t pocket bearings. (14) z =−ε cos φ − β , t t t d d 2 2 where, from Figure 5, 2 2 ε = ε + δ − 2εδ cos φ, (15) sin 180 − φ = sin φ. O O bl br l O φ The clearances C and C are related by the pocket arc extent r e p α and the ellipticity of the side arcs d .As shown in Figure 6, at the intersection between the side and pocket arcs, one has e 2 2 2 ◦ e R = R + d − 2R d cos 180 − . (16) p 1 e p 1 Solving for R and omitting higher-order terms α α α O 2 2 R =−d cos + R − d sin ≈ R − d cos . p 1 e 1 e 1 2 2 2 Figure 8: Geometric relationship between the arc centres for type-2 (17) pocket bearing. The journal radius can be expressed as r = R − C = R − C . (18) e e p p 2.4. Type-Two Pocket Bearing. Again, following an approach Substitution of (17)into(18)gives similar to that for the type 1 bearings, one has for the side arcs C − C = R − R ≈ d cos , (19) e p e p 1 h = − z sin ψ − y cos ψ s (21) or C d cos(α/2) α e 1 ≈ 1+ = 1+ δ cos (20) 1 and for the pocket arcs C C 2 p p which is always greater than 1. h = 1 − z sin ψ − y cos ψ. (22) p 100 −20 Advances in Acoustics and Vibration 5 S and φ The clearances C and C are related by the pocket arc extent s p α and the offset of the side arc centers d .As shown in Figure 7, at the intersection between the side and pocket arcs, one has 2 2 2 ◦ R = R + d − 2R d cos 90 − . (23) p 2 s p 2 Solving for R and omitting higher-order terms, α α α 2 2 R = d sin + R − d cos ≈ R + d sin . p 2 2 s 2 2 2 2 0 0.2 0.4 0.6 0.8 1 (24) The journal radius can be expressed as S(1) S(2) φ(1) φ(2) r = R − C = R − C . (25) s s p p (a) From (24)and (25), this gives C − C ≈ d sin , (26) p s 2 Stiffness coefficients or C d sin(α/2) α s 2 ≈ 1 − = 1 − δ sin . 2 (27) C C 2 p p To relate the offset d to the ellipticity d , it is assumed 2 1 that, apart from the same pocket arc extent α, the horizontal widths of the two bearings are the same, that is, (28) R + d = R − d ≈ R . s 2 e e 1 0 0.2 0.4 0.6 0.8 1 Substitution of (24)and (17)into(28)gives K (1) K (2) yy yy α α R − d sin + d = R + d cos , (29) p 2 2 p 1 K (1) K (2) yz yz 2 2 −K (1) −K (2) zy zy or K (1) K (2) zz zz cos(α/2) (b) d = d . (30) 2 1 1 − sin(α/2) The relationships between y, z and ε, φ are the same as Damping coefficients before. However, the calculations for ε , ε , φ ,and φ need l r l r different expressions. From Figure 8, ε = ε + δ +2εδ sin φ, l 2 2 ◦ ◦ sin 90 − φ = sin 90 + φ , ε 30 (31) ε = ε + δ − 2εδ sin φ, r 2 ◦ ◦ sin 90 + φ = sin 90 − φ . 0 0.2 0.4 0.6 0.8 1 −10 C (1) C (2) yy yy 3. Solution Procedures −C (1) −C (2) yz yz −C (1) −C (2) zy zy To calculate the static and dynamic characteristics of the C (1) C (2) zz zz pocket bearings, Gauss-Seidel iteration with successive over (c) relaxation is used to solve the finite difference formulation of the Reynolds equation [5], thereby ensuring that the Reynolds condition is used as the cavitation boundary. Figure 9: Comparison of the nondimensional bearing characteris- All results here assume same zero gauge inlet, outlet, and tics for the two types of pocket bearings ((1) and (2) correspond to cavitation pressures. type 1 and 2, resp.). 6 Advances in Acoustics and Vibration 1.2 0.8 S φ 45 0.4 0 0 −90 −30 30 90 −90 −30 30 90 β β S(e) φ(e) S(p) φ(p) (a) (b) Elliptic Pocket 20 20 15 15 K 10 K 10 5 5 0 0 −90 −30 30 90 −90 −30 30 90 β β K −K K −K yy zy yy zy K K K K yz zz yz zz (c) (d) Elliptic Pocket 40 40 20 20 C C 0 0 −90 −30 30 90 −90 −30 30 90 −20 −20 C −C C −C yy yz yy yz −C C −C C yz zz yz zz (e) (f) Figure 10: Nondimensional bearing characteristics for elliptic and type-1 pocket bearings at ε = 0.2. Advances in Acoustics and Vibration 7 0.45 0.3 55 S S 0.15 −5 −90 −30 30 90 −90 −30 30 90 β β S(e) φ(e) S(p) φ(p) (a) (b) Elliptic Pocket 10 10 K 6 K 6 2 2 −90 −30 30 90 −90 −30 30 90 −2 −2 β β K −K K −K yy zy yy zy K K K K yz zz yz zz (c) (d) Elliptic Pocket 18 18 12 12 C 6 C 0 0 −90 −30 30 90 −90 −30 30 90 −6 −6 C C C C yy yz yy yz C C C C yz zz yz zz (e) (f) Figure 11: Nondimensional bearing characteristics for elliptic and type-1 pocket bearings at ε = 0.4. 8 Advances in Acoustics and Vibration 10 m TM Millhpp3 V3.2 Figure 12: Sample rotor bearing system. Elliptic Pocket 1000 2000 3000 4000 5000 1000 2000 3000 4000 5000 (rpm) (rpm) m1 m4 m1 m3 m2 syn. m2 syn. m3 (a) (b) Figure 13: Campbell diagrams for a rotor supported by (a) elliptic bearings or (b) type-1 pocket bearings. 4. Results The rotor is shown schematically in Figure 12. Table 1 lists the corresponding data used to characterise the rotor and Figure 9 compares the nondimensional static and dynamic the bearings. In house vibration analysis software was used characteristics of the two types of pocket bearings as a to determine the natural frequencies and mode shapes over function of load inclination beta (β), using the same base a speed range up to 5000 rpm. Table 2 lists these natural clearance C and the the same pocket extent of α = frequencies, and Figure 13 shows the Campbell diagrams 60 . Apart from some difference in the attitude angle, the for the elliptic bearing supports and for the type-1 pocket nondimensional characteristics are virtually identical. bearing supports. Figures 10 and 11 compare the bearing characteristics It can be seen that the system behaves quite differently of the type-1 pocket bearing to a similar elliptic bearing with the different types of bearing supports. The rotor with as functions of the load inclination angle β for given the elliptic bearing has three natural frequencies but two eccentricities ε = 0.2 and 0.4, respectively. It can be seen critical speeds in the speed range; while the rotor with the that in general there is less variation in the dynamic bearing type-1 pocket bearing has four natural frequencies but only characteristics of the pocket bearing due to the change of β. one critical speed. The second mode in both cases is a A sample vibration analysis is also performed for an backward whirl and is therefore unlikely to be excited. Using industrial rotor subjected to gravity loading and supported equivalent clearances for the different pocket bearing models, on either type of pocket bearings or similar elliptic bearings. the natural frequencies in Table 2 indicate that the two types (rad/s) (rad/s) Advances in Acoustics and Vibration 9 Table 1: Sample rotor bearing data. Rotor: Young’s modulus = 210 GPa, density = 7850 kg/m 2 2 No. Length (m) Diameter (m) Disk mass (kg) Disk Ip (kg-m)DiskId(kg-m ) 1 0.14 0.135 0 0.0358 0.0436 2 0.16 0.18 0 0.1294 0.1329 3 0.148 0.18 0 0.1193 0.1131 4 0.148 0.18 0 0.1193 0.1131 5 0.115 0.258 8.392 0.5447 0.3336 6 0.115 0.277 1.185 0.5447 0.3336 7 0.115 0.277 1.185 0.5447 0.3336 8 0.106 0.328 11.87 1.291 0.7217 9 0.106 0.353 0.921 1.291 0.7217 10 0.049 0.345 0 0.535 0.2747 11 0.076 0.44 89.29 8.643 4.408 12 0.058 0.5 47.73 6.559 3.317 13 0.15 0.556 93.86 20.57 8.875 14 0.224 0.556 139.5 30.48 14.42 15 0.2 0.556 111 25.38 12.52 16 0.219 0.556 128.4 28.57 14 17 0.217 0.556 126.6 28.19 13.88 18 0.216 0.556 122.8 27.57 13.77 19 0.215 0.556 119.1 27.05 13.66 20 0.214 0.556 119.1 26.87 13.59 21 0.221 0.547 126.7 26.83 13.6 22 0.198 0.55 117.2 24.64 12.55 23 0.159 0.704 4.15 30.62 16.34 24 0.159 0.704 4.15 30.62 16.34 25 0.16 0.497 249.4 30.81 16.46 26 0.015 0.345 0 0.1638 0.0821 27 0.168 0.353 1.467 2.056 1.335 28 0.168 0.353 1.467 2.056 1.335 29 0.17 0.353 1.484 2.081 1.359 30 0.187 0.315 0 1.419 1.043 31 0.07 0.385 38.39 3.034 1.559 32 0.238 0.315 0 1.806 1.59 33 0.238 0.315 0 1.806 1.59 34 0.067 0.42 25.1 2.904 1.489 35 0.065 0.58 67.21 12.73 6.436 36 0.07 0.58 131 22.1 11.16 Bearings: δ = 0.5, (α = 60 for pocket bearings) No. Length (m) Diameter (m) Clearance (mm) Viscosity (Ns/m)Location 1 0.092 0.18 0.1327 0.014 2 2 0.204 0.315 0.2322 0.014 32 of the pocket bearings with similar dimensions could have Compared to similar elliptic bearings, the pocket bear- quite different effects on the system vibration characteristics. ings tend to provide less fluctuation in the dynamic bearing coefficients for different load inclinations and may produce significantly different vibration behaviour in a given rotor 5. Conclusions system. The two types of pocket bearings investigated here An approach to evaluate the static and dynamic bearing (produced by different machining procedures), apart from characteristics of pocket type bearings, suited for subsequent some difference in attitude angle, have virtually identical steady-state vibration analysis of rotating machinery involv- static and dynamic bearing characteristics. ing such bearings, is presented. 10 Advances in Acoustics and Vibration Table 2: Damped natural frequencies of a rotor supported by elliptic or pocket bearings ((1) refers to type 1, etc.). Speed (rpm) 1000 2000 3000 4000 5000 Elliptic Real, Imag (rad/s) Mode 1 −19.91, 137.5 −11.17, 143.9 −7.588, 150.4 −5.047, 156.1 −3.323, 160.8 Mode 2 −4.064, 211.8 −5.900, 207.7 −6.266, 204.2 −5.913, 201.6 −5.291, 199.6 Mode 3 −129.3, 217.4 −89.73, 261.5 −84.35, 288.3 −79.77, 317.3 −70.84, 344.6 Pocket Real, Imag (rad/s) Mode 1 (1) −27.05, 192.6 −19.47, 183.2 −10.63, 184.8 −5.012, 189.2 −1.620, 193.4 (2) −30.75, 186.0 −23.41, 175.1 −15.76, 177.6 −9.796, 182.3 −6.021, 186.6 Mode 2 (1) −3.379, 212.6 −4.965, 209.7 −5.133, 207.6 −4.768, 206.3 −4.255, 205.4 (2) −3.633, 212.0 −4.647, 208.7 −4.249, 206.8 −3.419, 205.9 −2.527, 205.6 Mode 3 (1) −162.5, 75.30 −246.2, 140.1 −300.0, 189.1 −352.9, 237.0 −418.2, 293.1 (2) −201.1, 48.79 −331.0, 79.78 −390.9, 114.1 −473.7, 166.2 −566.0, 241.0 Mode 4 (1) −85.34, 57.67 −140.8, 121.7 −194.9, 206.5 −223.7, 315.3 −202.6, 425.3 (2) −111.7, 29.36 −196.7, 118.1 −258.1, 265.9 −212.5, 394.2 −155.1, 462.0 Nomenclature Superscripts (If Not Otherwise Defined): C: Radial clearance = R − r : Nondimensional C ,··· : Nondimensional bearing damping yy :Differentiation with respect to nondimensional time τ coefficients ˙: Differentiation with respect to dimensional time t D: Bearing diameter d: Ellipticity or offset Subscripts (If Not Otherwise Defined): e: Eccentricity F , F : Film force components y z o: Steady state f , f : Nondimensional force components y z b, j: Bearing, journal h:Filmthickness b, t: Bottom lobe, top lobe K ,··· : Nondimensional bearing stiffness yy e, p, s: Elliptic arc, pocket arc, side arc coefficients l, r:Leftarc,rightarc L:Bearingwidth 1, 2: Pocket type 1, pocket type 2. O: Centres P: Pressure (gauge) References R: Bearing radius r: Journal radius [1] O. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrica- S: Sommerfeld number = μωRL(R/C) /(πW) tion, McGraw-Hill, New York, USA, 1961. t:Time [2] F. W. Ocvirk, “Short bearing approximation for full journal U, V : Velocities at journal surface in X and Y bearings,” Tech. Rep. NACA TN 2808, 1952. directions, respectively [3] M. A. Rezvani and E. J. Hahn, “Limitations of the short bearing W: External load approximation in dynamically loaded narrow hydrodynamic X, Y , Z: Cartesian coordinates at bearing surface bearings,” Journal of Tribology, vol. 115, no. 3, pp. 544–549, x, y, z: Cartesian coordinates at curvature centers 1993. α:Pocketextent [4] J. W. Lund, “Rotor-bearing dynamics design technology—part β: Load inclination from vertical III: design handbook for fluid film type bearings,” Tech. Rep. AFADL-TR-65-45, Dayton, Ohio, USA, 1965. Δ: Perturbation; variation in δ: Nondimensional ellipticity/offset = d/C [5] N.S.Fengand E. J. Hahn,“Computation ofbearing charac- teristics: elliptic bearings and tilting pad bearings,” Tech. Rep. ε: Nondimensional eccentricity = e/C 1993/AM/3, UNSW, Sydney, Australia, 1993. μ: Absolute viscosity φ: Attitude angle θ: Angular coordinate from line of centres = X/R τ: Nondimensional time = ωt ω: Speed ψ: Angular coordinate from y axis International Journal of Rotating Machinery International Journal of Journal of The Scientific Journal of Distributed Engineering World Journal Sensors Sensor Networks Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Volume 2014 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Submit your manuscripts at http://www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2010 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Journal

Advances in Acoustics and VibrationHindawi Publishing Corporation

Published: Dec 27, 2010

References