Abstract
Active compensation is an effective method for increasing air bearing static and dynamic performance. This paper describes the design, modelling and experimental validation of an actively compensated externally pressurized gas bearing. The active compensation is obtained through the support compensation strategy. With this strategy, the system’s initial working position is restored by compensating for air gap variations through adjustments to the bearing vertical dimension. The described bearing consists in a conventional thrust bearing which is integrated with a multilayer piezoelectric actuator, a compliant mechanism and a digital controller. Nevertheless the non-linear nature of the air system, a simple linear model results to be an effective choice for neighbour of equilibrium conditions. Results demonstrate the good accuracy of the model and the system’s good capacity of rejecting external force disturbances.
Similar content being viewed by others
Notes
Subscripts written in capital letters refer to physical points of the described mechanical structure.
Laplace domain variables are indicated by capital letters.
This is the maximum voltage of the actuator. The actuator, especially in the presence of high dynamics, has to be supplied up to its maximum nominal voltage (100 V) in order to preserve its performance
The curves designated by continuous lines are a polynomial interpolation of the measured experimental points.
Abbreviations
- b :
-
Air pad width (m)
- C(s):
-
Controller transfer function
- \(c_a\) :
-
Air gap damping coefficient (Ns/m)
- E :
-
Young’s Modulus (Pa)
- F :
-
Load applied on the ATB (N)
- \(F_{eff}\) :
-
PZT preloading force (N)
- \(F_0\) :
-
Piezo actuator blocking force (N)
- \(F_{pzt}\) :
-
Load applied on PZT (N)
- \(f_{pzt}\) :
-
PZT generated force (N)
- \(G_1\) :
-
Air pad passive transfer function (\(\upmu\)m/N)
- \(G_2\) :
-
Air pad passive active function (\(\upmu\)m/V)
- g :
-
Gravitational acceleration (m/\({\mathrm{s}}^2\))
- h :
-
Air gap height (m)
- H :
-
Controlled (or working) height (m)
- \(K_{Amp.}\) :
-
Piezo amplifier static gain (–)
- \(K_{t,b}\) :
-
Bending concentration factor (–)
- \(k_{a}\) :
-
Air gap stiffness (N/m)
- \(k_{l}\) :
-
Load stiffness (N/m)
- \(k_{pzt}\) :
-
Piezo actuator axial stiffness (N/m)
- L :
-
Maximum bearing pad length (m)
- \({\varDelta }L\) :
-
PZT stroke (m)
- m :
-
Supported mass (kg)
- \(m_s\) :
-
Mass of the calibrated weights (kg)
- \(m_v\) :
-
Mass of the movable structure (kg)
- \(P_s\) :
-
Absolute supply pressure measured in the air reservoir (Pa)
- Q :
-
Air flow rate (l/min-ANR)
- r :
-
Flexure radius (m)
- t :
-
Flexure hinge thickness (m)
- \(u_{pzt}\) :
-
Deformation due to the piezo shrinking/stretching action (m)
- \(\dot{u}_{pzt}\) :
-
Deformation velocity due to the piezo shrinking/stretching action (m/s)
- \(V_{pzt}\) :
-
Control voltage to the piezo actuator (V)
- \(V_{s}\) :
-
Control voltage to the piezo amplifier (V)
- w :
-
Flexure hinge depth (m)
- \(W_{Z_{G_{Ref.}}, Z_{G_2}}\) :
-
Closed loop transfer function (–)
- Z :
-
Air pad height (m)
- \(z_i\) :
-
Displacement of the i-th point (m)
- \(\dot{z}_i\) :
-
Velocity of the i-th point (m/s)
- \(\ddot{z}_i\) :
-
Acceleration of the i-th point (m/s)
- \(\theta _{Z,max}\) :
-
Maximum flexure rotation (rad)
- \(\omega\) :
-
Excitation frequency (rad/s)
- \(\nu\) :
-
Poisson’s coefficient (–)
- AIR:
-
Active inherent restrictor
- ATB:
-
Aerostatic thrust bearing
- ECR:
-
Exhaust control restrictor
- FEM:
-
Finite element method
- LVDT:
-
Linear variable displacement transducer
- PI:
-
Proportional–integrative controller
- PRBM:
-
Pseudo Rigid Body Model
- PZT:
-
Piezoelectric actuator
References
Raparelli T, Viktorov V, Colombo F, Lentini L (2015) Aerostatic thrust bearings active compensation: critical review. Precis Eng. ISSN 0141-6359. doi:10.1016/j.precisioneng.2015.11.002
Al-Bender F, Van Brussel H (1994) Active dynamic stiffness control of aerostatic bearings. In: Proceedings of 19th international seminar on modal analysis and structural dynamics, pp 187–197. Leuven
Snoeys R, Al-Bender F (1987) Development of improved externally pressurized gas bearings. KSME J 1(1):81–88. ISSN 1738-494X. doi:10.1007/BF02953383
Talukder HM, Stowell TB (2003) Pneumatic hammer in an externally pressurized orifice-compensated air journal bearing. Tribol Int 36(8): 585–591, ISSN 0301-679X. doi:10.1016/S0301-679X(02)00247-5
Mizumoto H, Arii S, Kami Y, Goto K, Yamamoto T, Kawamoto M (1996) Active inherent restrictor for air-bearing spindles. Precis Eng 19(23):141–147. ISSN 0141-6359. doi:10.1016/S0141-6359(96)00041-4
Mizumoto H, Shimizu T (1993) An infinite-stiffness aerostatic spindle with active restrictors. J Jpn Soc Precis Eng 59(4):607–612. doi:10.2493/jjspe.59.607
Mizumoto H, Matsubara T, Yamamoto H, Okuno K, Yabuya M (1991) An infinite-stiffness aerostatic bearing with an exhaust-control restrictor. In: Seyfried P, Kunzmann H, McKeown P, Weck M (eds) Prog Precis Eng. Springer, Heidelberg, pp 315–316. doi:10.1007/978-3-642-84494-2_35
Aguirre G, Al-Bender F, Van Brussel H (2010) A multiphysics model for optimizing the design of active aerostatic thrust bearings. Precis Eng 34(3):507–515. ISSN 01416359. doi:10.1016/j.precisioneng.2010.01.004
Al-Bender F (2009) On the modelling of the dynamic characteristics of aerostatic bearing films: from stability analysis to active compensation. Precis Eng 33(2):117–126. ISSN 0141-6359. doi:10.1016/j.precisioneng.2008.06.003
Aguirre G , Al-Bender F, Van Brussel H (2008) A multiphysics coupled model for active aerostatic thrust bearings. In: Advanced intelligent mechatronics, 2008. IEEE/ASME international conference on AIM 2008, pp 710–715. doi:10.1109/AIM.2008.4601747
Rejado Gorka A, Al-Bender F, Van Brussel H (2008) Dynamic stiffness compensation with active aerostatic thurst bearings. In: Proceedings of the international conference on noise and vibration engineering, pp 105–117
Aoyama H, Watanabe I, Akutsu K, Shimokohbe A (1988) An ultra precision straight motion system (1st report). J Jpn Soc Precis Eng 54(3):558–563. doi:10.2493/jjspe.54.558
Matsumoto H, Yamaguchi J, Aoyama H, Shimokohbe A (1988) An ultra precision straight motion system (2nd report). J Jpn Soc Precis Eng 54(10):1945–1950. ISSN 0912-0289. doi:10.2493/jjspe.54.1945. http://ci.nii.ac.jp/naid/110001369376/en/
Horikawa O, Shimokhobe A (1990) An active air bearing : control of radial axis motion and stiffness. JSME Int J Ser 3 Vib Control Eng Eng Ind 33(1):55–60. doi:10.1299/jsmec1988.33.55
Belforte G, Raparelli T, Viktorov V, Trivella A (2007) Discharge coefficients of orifice-type restrictor for aerostatic bearings. Tribol Int 40(3):512–521. doi:10.1016/j.triboint.2006.05.003
Belforte G, Colombo F, Raparelli T, Trivella A, Viktorov V (2011) Comparison between grooved and plane aerostatic thrust bearings: static performance. Meccanica 46(3):547–555. ISSN 1572-9648. doi:10.1007/s11012-010-9307-y
Colombo F, Raparelli T, Viktorov V (2009) Externally pressurized gas bearings: a comparison between two supply holes configurations. Tribol Int 42(2):303–310, ISSN 0301679X. doi:10.1016/j.triboint.2008.06.014
Belforte G, Colombo F, Raparelli T, Trivella A, Viktorov V (2008) High-speed electrospindle running on air bearings: design and experimental verification. Meccanica 43(6):591–600. ISSN 1572-9648. doi:10.1007/s11012-008-9135-5
Colombo F, Lentini L, Raparelli T, Viktorov V (2017) Experimental identification of an aerostatic thrust bearing. In: Advances in Italian mechanism science, pp 441–448. Springer, Berlin
Yong YK, Lu T-F, Handley DC (2008) Review of circular flexure hinge design equations and derivation of empirical formulations. Precis Eng 32(2):63–70, ISSN 0141-6359. doi:10.1016/j.precisioneng.2007.05.002
Lobontiu N (2014) Compliance-based matrix method for modeling the quasi-static response of planar serial flexure-hinge mechanisms. Precis Eng 38(3):639–650. ISSN 01416359. doi:10.1016/j.precisioneng.2014.02.014
Niezrecki C, Brei D, Balakrishnan S, Moskalik A (2001) Piezoelectric actuation: state of the art. Shock Vib Digest 33(4):269–280. ISSN 0583-1024. doi:10.1177/058310240103300401
ANSI/IEEE Std 176-1987 (1988) IEEE standard on piezoelectricity. 10.1109/IEEESTD.1988.79638
Lin C-J, Yang S-R (2006) Precise positioning of piezo-actuated stages using hysteresis-observer based control. Mechatronics 16(7):417–426. ISSN 0957-4158. doi:10.1016/j.mechatronics.2006.03.005
© Physik Instrumente (PI) GmbH & Co. KG. Piezoelectric actuators, 2014
Paros JM, Weisbord L (1965) How to design Flexure Hinges. Mach Des 37:151–156
Howell LL (2001) Compliant mechanisms. A Wiley-Interscience publication, Wiley. ISBN 9780471384786. https://books.google.it/books?id=tiiSOuhsIfgC
Lobontiu N, Paine JSN, Garcia E, Goldfarb M (2002) Design of symmetric conic-section flexure hinges based on closed-form compliance equations. Mech Mach Theory 37(5):477–498. ISSN 0094114X. doi:10.1016/S0094-114X(02)00002-2
Lobontiu N, Garcia E, Hardau M, Bal N (2004) Stiffness characterization of corner-filleted flexure hinges. Rev Sci Instrum 75(11):4896–4905. doi:10.1063/1.1806999. http://scitation.aip.org/content/aip/journal/rsi/75/11/10.1063/1.1806999
Lobontiu N, JSN Paine, Garcia E, and Goldfarb M (2000) Corner-filleted flexure hinges. J Mech Des 123(3):346–352. ISSN 1050-0472. doi:10.1115/1.1372190
Pilkey WD, Pilkey DF (2008) Peterson’s stress concentration factors. ISBN 9780470048245. doi:10.1002/9780470211106
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Rights and permissions
About this article
Cite this article
Colombo, F., Lentini, L., Raparelli, T. et al. Actively compensated aerostatic thrust bearing: design, modelling and experimental validation. Meccanica 52, 3645–3660 (2017). https://doi.org/10.1007/s11012-017-0689-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-017-0689-y