Rafael O. Ruiz

Asynchronous Dynamic Coefficients of a Three-Lobe Air Bearing

The study of dynamic whirl behavior of air bearings is fundamental for an adequate rotordynamic analysis and future validation of numerical predictions. This work shows the dynamic response of the air film on a three-lobe bearing under asynchronous whirl motion. One-dimensional multifrequency orbits are used to characterize the bearing rotordynamic coefficients. The test rig uses two magnetic bearing actuators to impose any given orbits to the journal. The dynamic forces are measured on the test bearing housing by three load cells. Journal whirling excitation is independent of the rotating speed, thus allowing asynchronous excitations. The multifrequency excitation is applied at each rotating speed up to 11,000 rpm, allowing the asynchronous characterization of the air film. The experimental procedure requires two linearly independent excitation sets. Thus, vertical and horizontal one-dimensional multifrequency orbits are applied as perturbations. Results show the synchronous and asynchronous dynamic coefficients of the air bearing. Asynchronous experimental results are compared to numerical estimation of the bearing force coefficients through solution of the isotropic ideal gas journal bearing Reynolds equation. Numerical dynamic coefficients are obtained as the effective coefficient values of the bearing when subject to a given orbit. A full characterization of the asynchronous rotordynamics coefficients of the bearing is presented in three-dimensional maps.

Effect of Uncertainties in the Estimation of Dynamic Coefficients on Tilting Pad Journal Bearings

It has been identified that small variations in the pad clearance and preload of a Tilting Pad Journal Bearing lead to important variations in their dynamic coefficients. Although this variation trend is already identified, a more robust statistical analysis is required in order to identify more general tendencies and quantify it. This work presents a framework that helps to identify the relation between the manufacturing tolerance of the bearing (reflected in the pad clearance and preload) and the expected variations on the dynamic coefficients. The procedure underlies the adoption of a surrogate model (based on Kriging interpolation) trained by any deterministic model available to predict dynamic coefficients. The pad clearance and preload are considered uncertain parameters defined by a proper probability density function. All statistical quantities are obtained using stochastic simulation, specifically adopting a Monte Carlo simulation employing the surrogate model. The framework is illustrated through the study of a five pad bearing.Copyright

Framework to Quantify Uncertainties in Piezoelectric Energy Harvesters

The interest of this work is to describe a framework to propagate uncertainties in piezoelectric energy harvesters (PEHs). The uncertainties are related to the random error associated to the mathematical model adopted, incomplete knowledge of the model parameters and the randomness nature of the excitation. The framework presented could be employed to conduct Prior and Posterior Robust Stochastic predictions. The prior analysis assumes a known Probability Density Function (PDF) for the uncertain variables while the posterior analysis calculates this PDF by adopting a Bayesian updating technique. The framework is particularized to evaluate the behavior of the Frequency Response Functions (FRFs) in PEHs while its implementation is illustrated by the use of a unimorph PEH. Results reveal the importance to include the model parameters uncertainties in the estimation of the FRFs. In that sense, the present framework constitutes a powerful tool in the robust design and prediction of PEH’s performance.Copyright

A View to the Energy Dissipation Mechanism of a Gas Foil Bearing’s Structure due to Dry Friction

The purpose of the present work consists on improving the understanding of the energy dissipation mechanism in the structure of a gas foil bearing. The analysis is based on an analytical model capable of predicting bumps deformation due to a load on the top coupled with dry friction forces at the top and bump ends. Models of mass-individual bump and mass-bump foil subject to a harmonic force are predicted numerically. The nonlinear behavior due to dry friction results in the possibility of stick-slip conditions over the surfaces in contact. The Variation of parameters such as excitation amplitude, mass magnitude, coefficient of friction, and bump geometry were considered. Equivalent dynamic coefficients of stiffness and damping are estimated through a least squares curve fitting, which constitutes a linearization of the system with dry friction. A computer program was developed in order to consider the effect of stick-slip. As a final product of this research the nonlinear model of the structure support was used to obtain a linear and simplified equivalent model. In most studied cases it is possible to represent the system with a linearized model of constant stiffness and viscous damping which is a variable function of the studied parameters and the frequency.Copyright

A Simple Approach to Determine Uncertainty Bounds on Bearing Rotordynamic Coefficients Identification

The majority of rotordynamic studies concerned with bearing properties identification estimate rotordynamic coefficients without addressing the issue of parameter uncertainties. Uncertainty quantification is required to establish the accuracy and therefore the robustness of the identified parameters. Accuracy on the identification methodology is hampered by measurement noise, experimental and modeling error, and numerical method. The aim of this article is to determine, by means of error analysis, the propagated uncertainty contributions in a parametric frequency-domain identification. The methodology is based on linearly independent excitations for a direct estimation of the bearing rotordynamic coefficients. Errors on measurable excitations and responses are considered in the identification strategy to evaluate uncertainties of the estimated parameters. General formulation using errors-in-variables noise model is presented for system identification, taking into account uncertainty propagation in bearing parameters estimation. Experimental measurements, obtained from a test rig, are employed to estimate rotordynamic coefficients of a three lobe air bearing and the associated uncertainties. Confidence intervals are suggested for the expected bearing coefficients. A Monte Carlo simulation is conducted to study the statistical behavior as a result of simulated stochastic uncertainty propagation for comparison purposes with the experimental evidence. Results are presented graphically to assess the influence of the uncertainty propagation on the bearing properties calculation.Copyright

Asynchronous Dynamic Coefficients of a Three Lobe Air Bearing

The study of dynamic whirl behavior of air bearings is fundamental for an adequate rotordynamic analysis and future validation of numerical predictions. This work shows the dynamic response of the air film on a three lobe bearing under non-synchronous whirl motion. One-dimensional multifrequency orbits are used to characterize the bearing rotordynamic coefficients. The test rig uses two magnetic bearing actuators to impose any given orbits to the journal. The dynamic forces are measured on the test bearing housing by three load cells. Journal whirling excitation is independent of the rotating speed, thus allowing asynchronous excitations. The multi frequency excitation is applied at each rotating speed up to 11000rpm allowing the non-synchronous characterization of the air film. The experimental procedure requires two linearly independent excitation sets. Thus, vertical and horizontal one-dimensional multi-frequency orbits are applied as perturbations. Results show the synchronous and asynchronous dynamic coefficients of the air bearing. Asynchronous experimental results are compared to numerical estimation of the bearing force coefficients through solution of the isotropic ideal gas journal bearing Reynolds equation. Numerical dynamic coefficients are obtained as the effective coefficient values of the bearing when subject to a given orbit. A full characterization of the non-synchronous rotordynamics coefficients of the bearing is presented in three dimensional maps.Copyright

Experimental Measurement of a Three Lobe Air Bearing Rotordynamic Coefficients

This work presents direct experimental measurements of air film rotordynamic coefficients on a three lobe bearing. The test rig uses two magnetic bearing actuators to impose desired test orbits to the journal. Tests are conducted at several rotating speeds up to 12,000rpm. Journal whirling excitation is independent of the rotating speed, thus allowing asynchronous excitations. One-dimensional orbits in the horizontal and vertical axes are applied as excitations at each rotating speed. The experimental results show the behavior of the rotordynamic coefficients of the air film bearing under synchronous and asynchronous excitation. The synchronous experimental results are compared to numerical estimation of the bearing force coefficients through solution of the isotropic ideal gas journal bearing Reynolds equation coupled with the pressure drop through the feeding holes. The results of this work prove the suitability of the rig to identify both the synchronous and nonsynchronous response of air fluid film bearings.© 2006 ASME

A Test Rig for Air Bearings Dynamic Characterization

This paper describes the design and operation of an experimental facility for measurement of equivalent stiffness and damping of air bearings. The rig uses two magnetic bearings to impose any given orbit to the journal, including displacement in two perpendicular directions on the rotation plane and tilting on the conical mode. Dynamic forces are measured directly on the test bearing housing. Data is gathered and processed using PC based data acquisition boards and software. Only the stiffness and damping coefficients of the fluid film are calculated as a function of the excitation frequency, being it synchronous or not. The present design allows testing air bearings up to 44 mm in external diameter and at frequencies up to 1 KHz. Preliminary testing was performed on this research that demonstrates the capability of the apparatus to measure the dynamic properties with ease and accuracy.Copyright