Modal Identification and Modeling of Bearings for Very High-Speed Rotors

Rotor-bearing systems take an important place in engineering applications and are used in many industrial systems: gas turbines, compressors, jet engines, machine-tool spindles, etc. The spindle is an essential component of manufacturing systems involving metal removal processes. Its performance largely influences the quality of machined parts. This importance has increased with the advent of High Speed Machining, High Productivity Machining and Hard Machining with low lubrication. In machining, manufacturers aim at higher cutting speeds and feed rates in order to increase productivity. These trends imply higher rotational speeds for spindles but also higher cutting forces. At the same time, manufacturers wish to improve surface roughness and tolerances of the machined parts. To meet those goals, it is essential to master the dynamic behavior of spindles. Stiffness is a fundamental parameter controlling the static and dynamic performances of the spindle. Modeling and controlling the dynamic behavior of spindles is a complex problem, because of the non-linear nature of the bearing stiffness, of its speed dependence and because of thermo mechanical effects associated with the heat dissipation in the bearings. In this dissertation, a mixed experimental-numerical method is presented for evaluating bearing stiffness of very high speed rotor-bearing systems. This method focuses on determining the stiffness properties of angular contact ball bearings used in the design of high-speed spindles for machine-tool applications. The goal of this method is to provide accurate and reliable stiffness data to improve dynamic predictive models of spindles. These models will then serve to improve and facilitate the design of high-speed spindles. A special attention is paid to the speed dependence of the bearing stiffness. The mixed identification method is based on the comparison of an experimental modal model with a numerical modal model of spindles. An optimization procedure based on a non-linear least square fit algorithm is used to estimate the bearing stiffness. The optimization criterion combines error functions based on natural frequencies and mode shapes. Based on the measurement of frequency response functions, the experimental modal parameters (natural frequencies and mode shapes) are extracted. In order to match numerical modal parameters with the experimental ones, the iterative optimization procedure updates the numerical parametric model. The model parameters are the bearing stiffness parameters to estimate. The procedure terminates once the error between the experimental and numerical parameter falls below a predefined threshold value. At this point, the bearing stiffness estimation is completed. The numerical model was developed to be easily implemented in a commercial finite element software. Moreover, the 3D finite element model allows to take into account all the surrounding structural elements which can influence the dynamic behavior of the spindle. The ball bearing is modeled as a stiffness matrix including radial, axial, tilting and coupling terms. On the other hand, a test rig using contact-free capacitive sensors to measure frequency response functions of motor-spindles was developed. Measurements can be performed either under non-rotating conditions or under rotation. Experimental modal parameters are extracted using a traditional curve fitting method. Several numerical applications were used to assess the performances of the identification method and validate it. As an example, the proposed identification procedure is applied to a mid-size (7kW) motor-spindle running at up to 70'000 rpm. Two test cases are presented: test spindle equipped with angular contact ball bearings with a contact angle of 15° and then 25°, commonly used in industrial high-speed spindles. For these two configurations, the goal is to estimate bearing stiffness and validate the method with experimental data. Several identifications were conducted at various rotational speeds. The rotational speed range was chosen to highlight the decrease in the bearing stiffness with rotational speed. In both test cases, we observed significant decrease in axial bearing stiffness with speed. With a contact angle of 15°, at 66'000 rpm, the axial stiffness can drop to 40% of its value at zero speed. With a contact angle of 25°, at 54'000 rpm, the axial stiffness can drop to 30% of its value at zero speed. The behavior of the radial stiffness is different in the two test cases. With a contact angle of 15°, the radial stiffness remains almost constant over the whole range of rotational speed. Conversely, with a contact angle of 25°, the behavior of the radial stiffness is similar to the behavior of the axial stiffness. At 54'000 rpm, its value can decrease to less than 40% of its value at zero speed. The obtained results are in good agreement with the literature and with numerically predicted stiffness values.

Experimental Contribution to the Mechanics of Herringbone Grooved Journal Air Bearings

Ivan Ognjanovic

Gas Lubricated Herringbone Grooved Journal Bearings (HGJB) have long been used in equipment such as navigation systems, optical scanners for laser printers and high speed cameras and disk spindle for flyhead testers. All such applications are characterized by a relatively small size (journal diameter

EPFL2011

Stabilized reduced basis method for parametrized advection-diffusion PDEs

Paolo Pacciarini

The aim of this master thesis is to study and develop a stabilized reduced basis method suitable for the approximation of the solution of parametrized advection-diffusion PDEs with high Péclet number, that is, roughly, the ratio between the advection coefficient and the diffusion one. Advection-diffusion equations are very important in many engineering applications, because they are used to model, for example, heat transfer phenomena or the diffusion of pollutants in the atmosphere. In such applications, we often need very fast evaluations of the approximated solution, depending on some input parameters. This happens, for example, in the case of real-time simulation. Moreover, we need rapid evaluations also if we have to perform repeated approximation of the solution, for different input parameters. An important case of this many-query situation is represented by some optimization problems, in which the objective function to optimize depends on the parameters through the solution of a PDE. The reduced basis (RB) method meets our need for rapidity and it is also able to guarantee the reliability of the solution, thanks to sharp a posteriori error bounds. We can find in literature many works about the application of the RB method to advection-diffusion problems but they mainly deal with equations in which the Péclet number is low. The need for stabilization arise from the fact that the finite element (FE) approximated solution - that the RB method aims to recover - shows strong instability problems that have to be fixed. In this work we want to go further in the study of the stabilization of the RB method for advection dominated problem in both steady and unsteady case. As regards the steady case, we first compare two possible stabilization strategies, by testing them on some test problems, in order to design an efficient stabilized reduced basis method. We will then test this method using the piecewise quadratic FE approximation as reference solution, instead of the usual piecewise linear one. We extend the method designed for the steady case to the time dependent case and we will carry out some numerical tests.

2012

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.