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

Résumé

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.

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