Strojniski vestnik - Journal of Mechanical Engineering 56(2010)4, 253-259 UDC 531.2:621.822 Paper received: 23.02.2010 Paper accepted: 30.04.2010 Static Behaviour of Air Plane Pads: Comparison Between Different Feeding Solutions Guido Belforte - Federico Colombo - Terenziano Raparelli - Andrea Trivella - Vladimir Viktorov Politecnico di Torino, Department of Mechanics, Italy Air bearings are used among the others to support and guide coordinate measuring machines and instruments. There are different feeding solutions adopted to obtain the required stiffness and damping, often it is also important to minimize the air consumption. The purpose of this paper is to evaluate the effects of supply orifice number on the static characteristics of plane pad circular air bearings. Two solutions are experimentally investigated, with six and three supply inherent orifices of same dimensions. A test bench is used to obtain the static characteristics (load capacity and mass flow rate) of the pads. Then the static pressure distribution under the pads are experimentally determined in different directions. A numerical model is developed to simulate the behaviour of the pads; a comparisons between experimental and numerical results is presented. © 2010 Journal of Mechanical Engineering. All rights reserved. Keywords: gas thrust bearing, discharge coefficients, inherent orifices feeding system, circular pad 0 INTRODUCTION Gas bearings are widely used for that applications in which it is necessary to reduce friction and obtain high positioning precision. Since '60 years these components have been studied both experimentally and theoretically in order to deepen the knowledge of the phenomena that control their behaviour and develop new technical solutions. Different are the contributions to the development of the base theory on gas lubrication [1]. Concerning the gas thrust bearings in [2] and [3] the air flow under orifice compensated circular pads is analysed and in particular the pressure distribution near the supply orifices is considered. In [4] different feeding systems i.e. pocketed and annular orifices are analysed and their effects to the pad load carrying capacity, stiffness and flow rate are studied. In [5] different numerical methods to simulated the pressure distribution under the pads are discussed. In [6] a supersonic thrust bearing is studied and more recently in [7] is described an experimental activity carried on rectangular pads. With particular attention to the dynamic properties, in [8] are analysed the stiffness and the damping coefficients of circular air pads subjected to periodic forces and in [9] the pressure perturbation at the edge of the bearing pocket is analysed during air hammer. The results of numerical simulations on air pads with different orifice and porous feeding systems are shown in [10]. In [11] is presented an experimental study on the determination of the discharge coefficients with feeding systems with inherent orifices and pocketed orifices. The present paper presents a numerical and experimental work on the static behaviour of circular air pads with inherent orifices. Two pads are compared, one with six and the other with three supply orifices with the same diameter situated on the same circumference. The test bench used to measure the static properties of the pad is described. The characteristics of load capacity, stiffness and air consumption of the pads are shown, with also the pressure distributions along the radial and the circumferential directions. A numerical model based on finite difference method is used to simulate the pads. It uses experimentally identified discharge coefficients to calculate the air flow through the supply orifices. The comparison of numerical and experimental results is discussed. 1 GEOMETRY OF THE PADS The pads, both of external diameter 0e = 40 mm, are visible in Fig. 1. They are provided with a vented camera of diameter $ =10 mm concentric with the pad. The pads present N *Corr. Authors' address: Politecnico di Torino, Department of Mechanics, Corso Duca degli Abruzzi 24, Torino, Italy, andrea.trivella@polito.it supply holes (Pad 1: N = 6, Pad 2: N = 3) located on a circumference of radius rS = 12.5 mm and equally spaced. The holes are drilled on brass cylindrical inserts having diameter d0 = 4 mm. The length of the holes is l = 0.4 mm and their diameter d. Their shape and dimensions were verified with a microscope. The measured values of d for each orifice are reported in Table 1. The obtained values are the mean values measured along two orthogonal directions. For each pad is also reported the mean diameter value considering all N orifices. Table 1. Geometric parameters Pad type N Orifice position Measured orifice diameter d [mm] Mean diameter value [mm] 1 0.228 2 0.230 1 6 3 0.212 0.221 4 0.219 5 0.218 6 0.217 1 0.237 2 3 2 0.211 0.222 3 0.219 Fig. 1. Sketch of the pads The superficial roughness of the pads was verified to be approximately Ra = 0.4 ^m. In Fig. 2 are visible the pads and in Fig. 3 is shown the magnification of an orifice. Fig. 2. Photo of the pads Fig. 3. Enlargement of supply orifice 3, Pad 2 2 MATHEMATICAL MODEL The inlet mass flow rate G through a single supply port is given by ISO formula 6358 (1): G = CskTps, CSkTPS. if 0 < P- < b Ps 1 - r P P - b Ps_ 1 - b \2 where CS = 0.686 CdS P if ^ > b PS "_kL sPa (1) (2) cd is the discharge coefficient of the supply holes, Ps and Pc are the orifice upstream and downstream absolute pressures respectively. Isothermal expansion, compressible and laminar flow are assumed in the air gap. The Reynolds' equation (3) in polar coordinates is solved using the finite difference method: 1 d , 3 dP2 --1 rh - r dr [ dr 1 â(,3 dP2 -——I h - r2 dA 83 24 ßR 0T0 q = 24 ß where q = G/(r dr d3) . d(Ph ) dt The discharge coefficient cd is obtained using the empirical Eq. (4) and results as a function of h, d, and Re [11]: cd = 0.85 1 - e h\ -8.2- d (1 - e"0001Re ) (4) v y where Re = 4G / tkIs ^ . Being Re dependent on G, the determination of cd is iterative until convergence. The range of ratio h/d in which the Eq. (4) is interpolated is 0.01^0.1. The considered boundary conditions are: • P = Pa at inner and outer radius; • P = Pc downstream the supply orifices. The Reynolds equation was discretized with finite difference method with a grid of 20 nodes along the radial direction and 72 nodes along the circumferential direction between two adiacent orifices. 3 TEST BENCH AND TEST PROCEDURE DESCRIPTION The static characteristics of the pad are experimentally determined with the test bench of Fig. 4. 7 6 Fig. 4. Sketch of the test bench The set-up frame consists of a base (1), three columns (2) and a crossmember (3). The air gap h is established between the pad under test (4) and stationary bearing member (5). Pad (4) can be moved vertically by means of screw (6) and handwheel (7). The force on the pad is monitored by an estensimetric load cell (8) with full range 2 kN. The air gap is monitored by three no contact inductive transducers (9) equispaced at 120°, with resolution 0.1 p,m. A laser beam (10), visible also in Fig. 5, is used to calibrate in loco the gain of each inductive sensor. The air flow rate is measured by a flow meter situated upstream the pad. A measuring hole of diameter 0.2 mm is provided in the stationary bearing member and connected to a piezoresistive pressure transducer that measures the pressure in the air gap with a precision of 10-4 MPa. In order to read the pressure in the air gap the stationary bearing member is translated radially by a screw or it is rotated. Its radial position is read by an LVDT. In this way the pressure in correspondence of the measuring hole can be measured along the radial or the circumferential direction. Fig. 5. Photo of the test bench The pads characteristics are strongly influenced by the supply pressure value and the air clearance. An operating procedure was defined in order to obtain accurate and repeatable measures of the air clearance h. The pad, supplied with compressed air, is loaded by handwheel (7) with an increasing downwards vertical force until the air clearance is equal to zero. In this condition the displacement transducers are set to zero. In this way the reading of the displacement transducers coincides with the air clearance. Higher loads are avoided in order not to deform the pad and the stationary member. During the determination of the static characteristic the air clearance, averaged among the three transducers readings, is increased in the range 3^20 p,m and the air flow and the force are measured. All measures are relative to Ps = 0.6 MPa. Once the air clearance is imposed, the supply pressure is corrected in order to keep it constant. 9 2 1 The pressure distributions under the pad were measured with detail near the supply orifices because the pressure gradients are high. 4 EXPERIMENTAL RESULTS The static characteristics of the pads are shown in Figs. 6 to 8. Pad 1 reaches a higher load carrying capacity and expends more air flow than Pad 2. There is a maximum load capacity for low clearances, at about h = 6 ¡um. As shown in Fig. 7 for both pads the highest stiffness is obtained approx. at h = 13 | m. The optimum bearing clearance for static stiffness reduces with increasing the number of supply orifices N. 0 25 5 10 15 20 Air clearance h (¡m) Fig. 6. Load capacity F of Pad 1 and Pad 2, § 4 0) 3 £ 11.5 12 12.5 13 13.5 Radial coordinate r [mm] Fig. 11. Detail of Fig. 9 5 COMPARISON BETWEEN NUMERICAL AND EXPERIMENTAL RESULTS The experimental data are compared with numerical results obtained integrating Reynolds equation. In Figs. 12 to 15 the experimental and the numerical radial pressure distributions with h = 15 jm and PS = 0.6 MPa are compared. Figs. 12 and 13 are relative to the radial and circumferential pressure distributions under Pad 1 and Figs. 14 and 15 under Pad 2. The comparison of the results is valid for distances from the supply holes greater than 40 h because the model does not take into account pressure and velocity gradients under the supply holes. The numerical value of Pc can be compared with the experimental local maximum of pressure. x 105 ra EL 4 -0° num. — 30° num. *— 0° exp. 30° exp. 5 20 10 15 Radial coordinate r [mm] Fig. 12. Comparison of experimental and numerical radial pressure distributions, Pad 1, h=15 jm, PS =0.6MPa x 105 6*-.-■-.-■- _ 5 ra CL r 4 2 0 10 20 30 40 50 60 Circumferential coordinate 9 [deg] Fig. 13. Comparison of experimental and numerical circumferential pressure distributions, Pad 1, h=15 jm, PS =0.6MPa For Pad 1 the numerical and experimental pressure distributions are quite in good 5 3 2 3 accordance in the zone of laminar flow. In correspondence of the hole the numerical downstream pressure Pc is about 20% higher than the experimental local maximum. For Pad 2 the difference between numerical and experimental values can be due to the tolerance error in measuring the air clearance under the pad. The pressure distribution, in fact, is very sensitive to this parameter. x 105 ra SL 4 -0° num. - 60° num. -©— 0° exp. -©- 60° exp. 5 20 10 15 Radial coordinate r [mm] Fig. 14. Comparison of experimental and numerical radial pressure distributions, Pad 2, h=15 /urn, PS =0.6MPa use a lower number of supply orifices because the air flow increase predominates; • the load capacity of the pad with six supply orifices suffers less of the effect of negative stiffness at low clearance because the pressure distribution is more uniform in the pad surface; • the optimum bearing clearance for static stiffness reduces with increasing the number of supply orifices; • numerical and experimental results are quite in good agreement, considering the tolerance in the experimental measure of the air clearance and the high sensitivity of flow rate and pressure to a clearance variation. The model can be used to simulate and optimize the geometry of pads with several orifices. 6 ACKNOWLEDGEMENTS This work was supported by MIUR: PRIN project: "Study and development of gas bearings for ultra high speed electrospindle", 2006; FIRB project "Enabling technologies for multitasking high precision machining of microparts", 2007. 7 NOMENCLATURE x 10' % 50 — 4 1 0 _j_1_ 40 50 60 10 20 30 Circumferential coordinate 9 [deg] Fig. 15. Comparison of experimental and numerical circumferential pressure distributions, Pad2, h=15 /um,PS =0.6MPa 6 CONCLUSIONS The following conclusions can be stated: • the load carrying capacity and the air mass flow rate with six orifices are approximately twice than that obtained with three orifices, but at higher clearances it is convenient to b Cd d do kT l h q r rs N P Pa Ps Pc R° jO T Re S q Critical pressure ratio Discharge coefficient of supply orifices Supply orifice diameter Insert diameter Temperature coefficient = VT 0 / T Supply orifice length Air clearance Inlet mass flow rate per unit area Radial coordinate Radius of supply circumference Number of supply orifices Absolute pressure Absolute ambient pressure Absolute supply pressure Absolute supply orifice downstream pressure Gas constant = 287.6 m2/s2K Absolute temperature in normal condition = 293 K Absolute orifice upstream temperature Reynolds number Supply orifice cross section Circumferential coordinate 5 3 2 1 3 2 Pad external diameter $ Pad internal diameter U Air viscosity = 17.89 10-6 Pa s 8 REFERENCES [1] Gross, W.A. 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