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Actively compensated aerostatic thrust bearing: design, modelling and experimental validation

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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.

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Notes

  1. Subscripts written in capital letters refer to physical points of the described mechanical structure.

  2. Laplace domain variables are indicated by capital letters.

  3. 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

  4. 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

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Correspondence to Luigi Lentini.

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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

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