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Electromagnetic force exerted by the bearing, placed on the third node, in Y direction: Influence of the air gap
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High speed machining (HSM) is a new technology used to product parts in different sectors. This technology permits time and processing cost reduction. One of the principal components of the HSM is the spindle. It is generally supported by active magnetic bearings (AMBs), in order to guarantee the greatest speed without affecting the dynamic charact...
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Citations
... where the matrix [K b ] represents the stiffness of the bearing (is reported in the work of [22][23][24]), and the matrices [M], ...
Unstable spindle rotation can result in vibrations, chatter, and increase in surface roughness, leading to productivity reduction and energy consumption rise. The combined effects of nonlinearity owing to different bearing types and spindle materials significantly impact the stability of the machine tool spindle system, which needs to be considered during the machine tool design process. This paper focuses on the stability analysis of a milling machine tool spindle. From this perspective, the dynamic behavior of the rotating shaft was investigated through analyzing the vibrations using a homogenized finite element beam model with five degrees of freedom at each nodal point. The study encompasses the Campbell diagram and stability analysis of the shaft considering three types of spindle materials (composite, steel, and functionally graded materials) as well as three types of bearings (rigid, active magnetic, and rolling bearings). The comparison between all structures stabilities conducted us to the best combination. In fact, the composite type affects the instability level. Additionally, the paper examines the influence of AMB properties, composite material characteristics and cutting parameters on natural frequencies, critical speeds, and instability thresholds of the composite spindle the findings indicated that the stability of the composite rotor is highly affected by the parameters related to the laminate composition, the parameters of the AMB and the cutting parameters. Furthermore, results indicate that increasing the control current (I0), number of turns (N), air gap (C), feed per tooth (fz), number of teeth (Nf) and axial depth (ap) enhances the stability of the composite spindle.
... Some of them were focusing on the study of shaft modeling methods. Among these studies, we can cite the finite element modeling method [1,2] and the rigid modeling method [3][4][5], which have commonly been used. In this context, Youcef-Toumi et al. [3] exploited the rigid plant dynamics method in the presence of rotor mass unbalances, gyroscopic effects and model uncertainty due to rotor speed changes. ...
... This model was then extended by the introduction of damping terms, then by the Timoshenko beam theory developed by Zorzi and Nelson [7], which takes into account the shear effects. A comparative study between elastic, rigid and coupled (rigid/elastic) movement was achieved by Hentati et al. [1] and showed that the rigid movement presents the most important vibration level followed by the coupled then the elastic ones. To ensure the bearings modeling, methods like the finite element modeling, permeance network and strength-current relation method based on the principle of virtual displacement, are usually used. ...
Background
Active magnetic bearings (AMBs) are electromechanical actuators commonly used in many industrial applications as alternatives to conventional bearings. They are generally used to ensure rotor levitation in different operating conditions and have several advantages due to its contactless and its active control properties.PurposeThe prediction of the magnetic bearing systems’ behavior is becoming a major concern in many industries. Thus, it is of paramount importance to study the system modeling and control, especially in the presence of some intrinsic defects to avoid failures. Actually, the main objective from this study was to stabilize a MIMO-coupled system in the presence of mechanical and electrical bearing’ faults.Methods
An electro-mechanical model in rigid and elastic motion describing the system dynamic behavior was proposed. Four types of linear and nonlinear controllers were developed and simulated to ensure the stabilization of the system with and without defects. The analytical simulations were elaborated to confirm the control law efficiency. A comparative study was achieved to determine the main control law leading to the highest precision and best performances.ResultsWithout control, the rotor exhibited an unstable alternating motion oscillating around the equilibrium position and its behavior showed an amplified vibratory motion in the case of the defective system especially for the eccentricity defect. The stabilization of this nonlinear and non-stable coupled model was obtained through the use of different control methods. When the two defects are simultaneously implemented for the rigid model, the PD regulator exhibit an over damped response with the longest rise time of about 0.02 s and a steady-state static error of 0.3 μm on the displacement plot while the feedback controller shows a short settling time with a tiny undershoot on the rotational movement. In contrast, backstepping revealed the best characteristics in terms of rapidity (rise time = 0.009 s) and the absence of any oscillations or undershoots. While for the direct Lyapunov controller, the equilibrium position was reached after 0.1 s. Nevertheless, it takes less than a third of the rise time needed for the backstepping method in the elastic case.Conclusion
The system has revealed such sensitivity to some intrinsic and extrinsic parameters that a little change on one of them may generate a major change in the whole system behavior. Nevertheless, the results confirm the system’s good convergence under the different controllers for the rigid plant and an adequate convergence with feedback, backstepping, and Lyapunov-based methods for the elastic motion. The nonlinear controllers demonstrated a rapid and accurate convergence for all the system states in all cases and it was proven that they require significantly less energy and showed better robustness than the linear controllers.
... This step is often expensive and requires higher computation times in order to obtain the optimal values of design parameters. In this work, we propose using interval computation (Alefeld and Mayer, 2000;Hansen and Walster, 2003;Trabelsi et al., 2015) to simulate the dynamic behavior of an electromagnetic spindle (Hentati et al., 2013;Bouaziz et al., 2016). It provides rigorous evaluation that allows designing a mechatronic system while minimizing the number of simulations and, consequently, the calculation time. ...
... The global equation of motion is formulated by applying Lagrange's formalism to the kinetic and potential energy expressions of the shaft. It is written as follows (Hentati et al., 2013;Bouaziz et al., 2016) MQ ...
The paper deals with the design approach of a subdefinite mechatronic system and focuseson the sizing stage of a gearbox of a wind turbine based on the interval computation method.Indeed, gearbox design variables are expressed by intervals to take into account the uncer-tainty in the estimation of these parameters. The application of the interval computationmethod allows minimizing the number of simulations and enables obtaining a set of solutionsinstead of a single one. The dynamic behavior of the gearbox is obtained using the finiteelement method. The challenge here is to get convergent results with intervals that reflectthe efficiency of the applied method. Thus, several mathematical formulations have beentested in static study and evaluated in the case of a truss. Then the interval computationmethod was used to simulate the behavior of the wind turbine gearbox.
... The determination of the equation of motion requires the computation of the kinetic energy of a rigid beam as well as the kinetic and potential energies when this one is considered deformable. Kinetic energy is written in the following form (Trabelsi et al. 2019;Hentati et al. 2013): ...
High Speed Machining (HSM) is a new technology used to reduce the time and cost of producing parts in several sectors. Among the main components of the HSM is the spindle. It is usually supported by Active Magnetic Bearings (AMBs), in order to achieve high speeds without affecting the dynamic characteristics of the machine. In this paper, an electromagnetic spindle supported by two AMBs is modeled and simulated using the interval calculation technique in order to assure successfully the preliminary design process. The behavior of the system is evaluated and studied with the consideration of uncertainty. The uncertainties are taken into account for the different design parameters that define the dynamic model of the spindle.
... This step is often expensive and requires higher computation times in order to obtain the optimal values of design parameters. In this work, we propose using interval computation (Alefeld and Mayer, 2000;Hansen and Walster, 2003;Trabelsi et al., 2015) to simulate the dynamic behavior of an electromagnetic spindle (Hentati et al., 2013;Bouaziz et al., 2016). It provides rigorous evaluation that allows designing a mechatronic system while minimizing the number of simulations and, consequently, the calculation time. ...
... The global equation of motion is formulated by applying Lagrange's formalism to the kinetic and potential energy expressions of the shaft. It is written as follows (Hentati et al., 2013;Bouaziz et al., 2016) MQ ...
Modeling and evaluation of uncertainties constitute indeed one of the key points when making any decision. For this, designers have to compare the measured or calculated value with a range of permissible values in order to obtain a guaranteed design process. Thus, in this
work, simulation of the dynamic behavior of an electromagnetic spindle was done based on the interval computation technique. Indeed, the use of this technique makes it possible to obtain
a set of values for different design parameters of the spindle and, consequently, to avoid making several simulations which could make the system useless, expensive or ineffective.
The proposed model is based on the combination of Matlab with ModelCenter. Matlab was used to model and simulate the system and ModelCenter to perform parametric studies to verify the influences of uncertainty on the dynamic behavior of the electromagnetic spindle and to determine the optimal design parameters.
... For example, bearingless permanent magnet motors, bearingless induction motors, bearingless switched reluctance motors. The bearingless reluctance motor has simple structure and low cost, but it has large torque fluctuation and mechanical vibration of rotor [12]- [15]. The bearingless induction motor has good high temperature resistance and can withstand large centrifugal force, but it has large rotor loss and low efficiency [16]- [18]. ...
In order to approximately obtain sinusoidal air-gap magnetic field and get larger radial levitation force of the bearingless permanent magnet synchronous motor (BPMSM), a novel BPMSM with Halbach magnetized rotor is proposed. Firstly, the suspension and rotation mechanisms of the BPMSM are introduced. The control mechanism of producing the radial levitation force in single direction and arbitrary direction is analyzed in detail. The mathematical models of the radial suspension force are given. Secondly, the comparative research on the BPMSMs with the Halbach magnetized rotor and parallel magnetized rotor is carried out. The air-gap magnetic field, back electromotive force (EMF), radial suspension force, and torque are calculated and compared. Finally, the prototype motor equipped with the Halbach magnetized rotor is constructed. Some experimental results are tested and the stable suspension operation of the proposed motor is realized.
... Six degrees of freedom are considered. Rigid displacements are also taken in account (Hentati et al., 2013). AMB are presented as spring and damper elements. ...
... The studied spindle model is presented in Fig. 1. The modeling is based on using a new approach developed by Hentati et al. (2013). This method is based on coupling both elastic and rigid spindle deformations. ...
In modern production, milling is considered the widespread cutting process in the formatting field. It remains important to study this manufacturing process as it can be subject to some parasitic phenomena that can degrade surface roughness of the machined part, increase tool wear and reduce spindle life span. In fact, the best quality work piece is obtained with a suitable choice of parameters and cutting conditions. In another hand, the study of tool vibrations and the cutting force attitude is related to the study of bearings as they present an essential part in the spindle system. In this work, a modeling of a High Speed Milling (HSM) spindle supported by two pair of Active Magnetic Bearings (AMB) is presented. The spindle is modeled by Timoshenko beam finite elements where six degrees of freedom are taken into account. The rigid displacements are also introduced in the modeling. Gyroscopic and centrifugal terms are included in the general equation. The bearings reaction forces are modeled as linear functions of journal displacement and velocity in the bearing clearance. A cutting force model for peripheral milling is proposed to estimate the tool-tip dynamic responses as well as dynamic cutting forces which are also numerically investigated. The time history of response, orbit, FFT diagram at the tool-tip center and the bearings dynamic coefficients are plotted to analyze dynamic behavior of the spindle.
Magnetic bearings’ modelling is performed using different methodologies. Generally, the linear modelling of these actuators, which is required for the use of linear controllers, gives an approximation of the nonlinear relation between the bearing load and the control current. However, this approach may have some disadvantages because the model is linearized around an equilibrium position. Thus, the performance of the linear design can decreases if there is a perturbation of the system. Although, when the variation of the rotor displacement is too small, a linearized control law can offer valid results. The main objective of this work is to study the stabilization of an electromechanical system based on the linear quadratic (LQ) control method. The stabilization of a high sped rotating shaft supported by four α-degree oriented magnetic bearings will be studied. The linearization around an equilibrium position is performed to adopt the linear control law. Some simulations results will be illustrated to evaluate the performances of the proposed controller.
The use of Active Magnetic Bearings (AMBs) in the High Speed Machining (HSM) field is increasing. In fact, they have important advantages such us: high speed which can reach 150000 rpm, high stiffness and absence of lubrication. Indeed, these bearings operate through magnetic interaction generated between the stator and the rotor. The spindle, principal component in the HSM, supported by AMBs, is considered as a source of vibration in machine tool spindles when it is affected by some defects during their functioning. These defects can affect either the rotor of the spindle like the unbalance defect, or the AMBs such us misalignment of bearings, failure of actuators, sensors, amplifier and also the eccentricity defect. This paper studies the influence of spindle or AMBs defects on the dynamic behavior of a spindle supported by two AMBs. The system modeling is based on the finite element method. Four degrees of freedom is used and the coupling between elastic and rigid displacements is taken into account.
