Influence of Blade Flexibility on the Dynamic Response Simulation of a Turbomolecular Pump on Magnetic Bearings

2019 ◽  
Author(s):  
Alysson Bruno Barbosa Moreira ◽  
Fabrice Thouverez
Keyword(s):  
Mechanical Model ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Coupled System ◽  
Forced Response ◽  
Second Order ◽  
Magnetic Bearings ◽  
Flexible System ◽  
Turbomolecular Pump ◽  
Rigid Body Modes

Abstract This paper proposes the simulation of a complete mechanical model of a turbomolecular pump rotor, including rotor and blades flexibility, suspended by controlled active magnetic bearings. The mechanical model is composed of an eight stage blisk, attached to a shaft. Magnetic forces are linearized by first order Taylor expansion around a given point. Including blades and rotor flexibility makes the mechanical system asymmetric, so the equations of motion for the coupled system have periodic terms. A modal controller was designed to control rigid body modes, since the number of sensors is limited and no state observer is implemented. PID controllers are used for low frequency modes combined with second order filters to damp high frequency modes. First of all, stability analysis was carried out for the axisymmetric case. Secondly, blades flexibility was included. Forced response of the whole system to an impulsive force was studied. Divergent responses for the system in rotation were obtained as a second order filter pole possibly interacts with blades modes. Taking second order filters off the control loop allowed the system to be stable. These results show that the analysis method developed here is efficient to evaluate the performance of a controller in closed loop with the complete flexible system. This method may be used in industrial design processes as computation times for the complete system are very short.

Influence of Blade Flexibility on the Dynamic Response Simulation of a Turbomolecular Pump on Magnetic Bearings

2020 ◽  
Vol 142 (4) ◽  
Author(s):  
Alysson Bruno Barbosa Moreira ◽  
Fabrice Thouverez
Keyword(s):  
Mechanical Model ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Coupled System ◽  
Forced Response ◽  
Second Order ◽  
Magnetic Bearings ◽  
Flexible System ◽  
Turbomolecular Pump ◽  
Rigid Body Modes

Abstract This paper proposes the simulation of a complete mechanical model of a turbomolecular pump rotor, including rotor and blades flexibility, suspended by controlled active magnetic bearings. The mechanical model is composed of an eight stage blisk, attached to a shaft. Magnetic forces are linearized by the first-order Taylor expansion around a given point. Including blades and rotor flexibility makes the mechanical system asymmetric, so the equations of motion for the coupled system have periodic terms. A modal controller was designed to control rigid body modes, since the number of sensors is limited and no state observer is implemented. PID controllers are used for low frequency modes combined with the second order filters to damp high frequency modes. First of all, stability analysis was carried out for the axisymmetric case. Second, blades flexibility was included. Forced response of the whole system to an impulsive force was studied. Divergent responses for the system in rotation were obtained as a second order filter pole possibly interacting with blades modes. Taking the second order filters off the control loop allowed the system to be stable. These results show that the analysis method developed here is efficient to evaluate the performance of a controller in closed loop with the complete flexible system. This method may be used in industrial design processes as computation times for the complete system are very short.

Singular Mode Characteristics of Fluid-Structure Interaction Modal Formulations

2012 ◽  
Author(s):  
Jeffrey L. Cipolla
Keyword(s):  
Low Frequency ◽  
Order Reduction ◽  
Coupled System ◽  
Forced Response ◽  
Coupled Modes ◽  
Model Order ◽  
Linear Elastic ◽  
Frequency Domains ◽  
Rigid Body Modes ◽  
Mode Characteristics

Modal formulations for linear acoustic and vibration problems are important for model order reduction as well as physical interpretation and insight. In the case of structural acoustic systems, a number of formulations exist for the computation of the modes of the coupled system: these may be referred to as ‘coupled modes’, ‘in-water modes’, etc. These modes have the desirable property that they diagonalize the undamped structural-acoustic problem, making forced-response computations in the time- and frequency-domains trivial. In this paper, we review a number of alternative formulations for the undamped FSI mode problem, and concentrate on a particular aspect: the existence and nature of the singular modes of the systems, i.e. the modes at zero frequency. Corresponding to rigid-body modes in linear elastic systems, these modes are essential for accurate low-frequency performance of reduced-order models. It is found that the original, nonsymmetric system of Zienkiewicz and Newton [53] maintains physically reasonable singular mode properties, while many other formulations do not.

Nonlinear Analysis of a Rigid Rotor on Magnetic Bearings

1995 ◽  
Author(s):  
C. Nataraj
Keyword(s):  
Nonlinear Analysis ◽  
Equations Of Motion ◽  
Forced Response ◽  
Magnetic Bearings ◽  
Rigid Rotor ◽  
Stability Criteria ◽  
Safe Operation ◽  
Qualitative And Quantitative ◽  
Quantitative Results ◽  
Nonlinear Equations Of Motion

A simple model of a rigid rotor supported on magnetic bearings is considered. A proportional control architecture is assumed, the nonlinear equations of motion are derived and some essential nondimensional parameters are identified. The free and forced response of the system is analyzed using techniques of nonlinear analysis. Both qualitative and quantitative results are obtained and stability criteria are derived for safe operation of the system.

scholarly journals Moduli and modes in the Mikado model

Soft Matter ◽  
2021 ◽  
Author(s):  
Brian Tighe ◽  
Karsten Baumgarten
Keyword(s):  
Shear Modulus ◽  
Mechanical Model ◽  
Low Frequency ◽  
Vibrational Modes ◽  
Prior Work ◽  
Fiber Networks ◽  
Elastic Shear Modulus ◽  
Flexible Fiber ◽  
Elastic Shear

We determine how low frequency vibrational modes control the elastic shear modulus of Mikado networks, a minimal mechanical model for semi-flexible fiber networks. From prior work it is known that...

scholarly journals Higher order Hamiltonian Monte Carlo sampling for cosmological large-scale structure analysis

2021 ◽  
Vol 502 (3) ◽  
pp. 3976-3992
Author(s):  
Mónica Hernández-Sánchez ◽  
Francisco-Shu Kitaura ◽  
Metin Ata ◽  
Claudio Dalla Vecchia
Keyword(s):  
Fourth Order ◽  
Large Scale ◽  
Equations Of Motion ◽  
Large Scale Structure ◽  
Real Space ◽  
Higher Order ◽  
Second Order ◽  
Scale Structure ◽  
Integration Schemes ◽  
Reconstruction Methods

ABSTRACT We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretization of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case for high-dimensional cases. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 h−1 Mpc side and 2563 cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth order in the leap-frog scheme shortens the burn-in phase by a factor of at least ∼30. This implies that 75–90 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3.0 fewer gradient computations for meshes of 2563 cells. In the considered cosmological scenario, the traditional leap-frog scheme turns out to outperform higher order integration schemes only when considering lower dimensional problems, e.g. meshes with 643 cells. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.

Analytical Solution for Rotational Rub-Impact Plate Under Thermal Shock

2016 ◽  
Vol 32 (3) ◽  
pp. 297-311
Author(s):  
T.-Y. Zhao ◽  
H.-Q. Yuan ◽  
B.-B. Li ◽  
Z.-J. Li ◽  
L.-M. Liu
Keyword(s):  
Thermal Shock ◽  
High Frequency ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Cantilever Plate ◽  
Analysis Method ◽  
Multiple Frequency ◽  
Width Direction ◽  
Blade Tip ◽  
High Frequency Vibrations

AbstractThe analysis method is developed to obtain dynamic characteristics of the rotating cantilever plate with thermal shock and tip-rub. Based on the variational principle, equations of motion are derived considering the differences between rubbing forces in the width direction of the plate. The transverse deformation is decomposed into quasi-static deformation of the cantilever plate with thermal shock and dynamic deformation of the rubbing plate under thermal shock. Then deformations are obtained through the calculation of modal characteristics of rotating cantilever plate and temperature distribution function. Special attention is paid to the influence of tip-rub and thermal shock on the plate. The results show that tip-rub has the characteristics of multiple frequency vibrations, and high frequency vibrations are significant. On the contrary, thermal shock shows the low frequency vibrations. The thermal shock makes the rubbing plate gradually change into low frequency vibrations. Because rub-induced vibrations are more complicated than those caused by thermal shock, tip-rub is easier to result in the destruction of the blade. The increasing friction coefficient intensifies vibrations of the rubbing plate. Minimizing friction coefficients can be an effective way to reduce rub-induced damage through reducing the surface roughness between the blade tip and the inner surface of the casing.

A Second Order Lagrangian Model for Irregular Ocean Waves

2005 ◽  
Vol 128 (3) ◽  
pp. 177-183 ◽  
Author(s):  
Sébastien Fouques ◽  
Harald E. Krogstad ◽  
Dag Myrhaug
Keyword(s):  
Specular Reflection ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Wave Theory ◽  
Ocean Waves ◽  
Second Order ◽  
Lagrangian Model ◽  
Lagrangian Description ◽  
Linear Wave ◽  
Irregular Solution

Synthetic aperture radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, slope, and mean curvature are studied.

Simulation of Engine Vibration on Nonlinear Hydraulic Engine Mounts

2005 ◽  
Author(s):  
A. R. Ohadi ◽  
G. Maghsoodi
Keyword(s):  
Equations Of Motion ◽  
Low Frequency ◽  
Governing Equations ◽  
Engine Mounts ◽  
Engine Vibration ◽  
Engine Mount ◽  
Rotational Motions ◽  
The Time Domain ◽  
Nonlinear Equations Of Motion ◽  
Four Cylinders

In this paper, vibration behavior of engine on nonlinear hydraulic engine mount including inertia track and decoupler is studied. In this regard, after introducing the nonlinear factors of this mount (i.e. inertia and decoupler resistances in turbulent region), the vibration governing equations of engine on one hydraulic engine mount are solved and the effect of nonlinearity is investigated. In order to have a comparison between rubber and hydraulic engine mounts, a 6 degree of freedom four cylinders V-shaped engine under inertia and balancing masses forces and torques is considered. By solving the time domain nonlinear equations of motion of engine on three inclined mounts, translational and rotational motions of engines body are obtained for different engine speeds. Transmitted base forces are also determined for both types of engine mount. Comparison of rubber and hydraulic mounts indicates the efficiency of hydraulic one in low frequency region.

Towards a Technique for Nonlinear Modal Analysis

2012 ◽  
Author(s):  
Simon A. Neild ◽  
Andrea Cammarano ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Transformation Process ◽  
Second Order ◽  
Identity Transformation ◽  
Modal Testing ◽  
System Response ◽  
Weakly Nonlinear ◽  
Nonlinear Modal Analysis

In this paper we discuss a theoretical technique for decomposing multi-degree-of-freedom weakly nonlinear systems into a simpler form — an approach which has parallels with the well know method for linear modal analysis. The key outcome is that the system resonances, both linear and nonlinear are revealed by the transformation process. For each resonance, parameters can be obtained which characterise the backbone curves, and higher harmonic components of the response. The underlying mathematical technique is based on a near identity normal form transformation. This is an established technique for analysing weakly nonlinear vibrating systems, but in this approach we use a variation of the method for systems of equations written in second-order form. This is a much more natural approach for structural dynamics where the governing equations of motion are written in this form as standard practice. In fact the first step in the method is to carry out a linear modal transformation using linear modes as would typically done for a linear system. The near identity transform is then applied as a second step in the process and one which identifies the nonlinear resonances in the system being considered. For an example system with cubic nonlinearities, we show how the resulting transformed equations can be used to obtain a time independent representation of the system response. We will discuss how the analysis can be carried out with applied forcing, and how the approximations about response frequencies, made during the near-identity transformation, affect the accuracy of the technique. In fact we show that the second-order normal form approach can actually improve the predictions of sub- and super-harmonic responses. Finally we comment on how this theoretical technique could be used as part of a modal testing approach in future work.

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