The Extended Modal Reduction Method Applied to Rotor Dynamic Problems

1991 ◽  
Vol 113 (1) ◽  
pp. 79-84 ◽  
Author(s):  
K. Kane ◽  
B. J. Torby
Keyword(s):  
Degrees Of Freedom ◽  
Reduction Method ◽  
Mode Shapes ◽  
Dynamic Problems ◽  
Modal Reduction ◽  
Stiffness Matrices ◽  
Space Formulation ◽  
Gyroscopic Effects ◽  
Stiffness And Damping ◽  
Rotor Dynamic

In this paper, the existing Modal Reduction Method, which was developed to handle symmetric mass and stiffness matrices, is extended utilizing state-space formulation to handle nonsymmetric mass, damping, and stiffness matrices. These type of matrices typically accompany rotor dynamic problems since journal bearings supporting the rotor have nonsymmetric stiffness and damping characteristics. The purpose of modal reduction is to eliminate unimportant modes and degrees of freedom from the analytical model after they are found, so that further numerical analysis can be accelerated. The reduction described here leaves the retained eigenvalues and mode shapes unaltered from their original values. This method is demonstrated for a simple rotor problem having nonsymmetric system matrices including gyroscopic effects.

Effect of a Planetary Gearbox on the Dynamics of a Rotor System

2018 ◽  
Author(s):  
Ali Tatar ◽  
Christoph W. Schwingshackl
Keyword(s):  
Degrees Of Freedom ◽  
Dynamic Behaviour ◽  
Rotor System ◽  
Mode Shapes ◽  
Rotating System ◽  
Planetary Gearbox ◽  
Finite Element Software ◽  
Gyroscopic Effects ◽  
The Masses ◽  
Parameter Values

The dynamic analysis of rotors with bladed disks has been investigated in detail over many decades and is reasonably well understood today. In contrast, the dynamic behaviour of two rotors that are coupled via a planetary gearbox is much less well understood. The planetary gearbox adds inertia, mass, stiffness, damping and gyroscopic moments to the system and can strongly affect the modal properties and the dynamic behaviour of the global rotating system. The main objective of this paper is to create a six degrees of freedom numerical model of a rotor system with a planetary gearbox and to investigate its effect on the coupled rotor system. The analysis is based on the newly developed finite element software “GEAROT” which provides axial, torsional and lateral deflections of the two shafts at different speeds via Timoshenko beam elements and also takes gyroscopic effects into account. The disks are currently considered as rigid and the bearings are modelled with isotropic stiffness elements in the translational and rotational directions. A novel planetary gearbox model has been developed, which takes the translational and rotational stiffness and the damping of the gearbox, as well as the masses and inertias of the sun gear, ring gear, planet gears and carrier into account. A rotating system with a planetary gearbox has been investigated with GEAROT. The gearbox mass and stiffness parameters are identified as having a significant effect on the modal behaviour of the rotor system, affecting its natural frequencies and mode shapes. The higher frequency modes are found to be more sensitive to the parameter changes as well as the modes which have a higher deflection at the location of the gearbox on the rotor system. Compared with a single shaft system, the presence of a gearbox introduces new global modes to the rotor system and decouples the mode shapes of the two shafts. The introduction of a planetary gearbox may also lead to an increase or a reduction of the frequency response of the rotor system based on gear parameter values.

On Double Frequency Vibration of a Generator-Bearing System With Asymmetrical Stiffness

2002 ◽  
Author(s):  
Senlin Huang ◽  
Zhansheng Liu ◽  
Jiexian Su
Keyword(s):  
Degrees Of Freedom ◽  
Reduction Method ◽  
Integration Method ◽  
Element Model ◽  
Modal Interaction ◽  
Motion Equations ◽  
Double Frequency ◽  
Modal Reduction ◽  
Direct Integration Method ◽  
Bearing System

A finite element model for a generator-bearing system with asymmetrical stiffness is developed for investigation of the double frequency vibration. The modal reduction method is used for reducing the degrees of freedom system to improve computing efficiency, and the Newmark direct integration method is employed to solve the reduced motion equations. The two-modal interaction vibration is induced when the rotation speed is half a critical speed of the system due to asymmetry and gravity force of the generator. Such a phenomena is observed in the practical test.

Output-Only Modal Analysis of an Internal Unreachable Module Embedded in a Space Electronic Equipment

2012 ◽  
Author(s):  
Lassaad Ben Fekih ◽  
Georges Kouroussis ◽  
David Wattiaux ◽  
Olivier Verlinden ◽  
Christophe De Fruytier
Keyword(s):  
Finite Element ◽  
Modal Analysis ◽  
Degrees Of Freedom ◽  
Transverse Direction ◽  
Mode Shapes ◽  
Fe Model ◽  
Fe Method ◽  
Numeric Model ◽  
Stiffness Matrices ◽  
Output Only

An approach is proposed to identify the modal properties of a subsystem made up of an arbitrary chosen inner module of embedded space equipment. An experimental modal analysis was carried out along the equipment transverse direction with references taken onto its outer housing. In parallel, a numerical model using the finite element (FE) method was developed to correlate with the measured results. A static Guyan reduction has led to a set of master degrees of freedom in which the experimental mode shapes were expanded. An updating technique consisting in minimizing the dynamic residual induced by the FE model and the measurements has been investigated. A last verification has consisted in solving the numeric model composed of the new mass and stiffness matrices obtained by means of a minimization of the error in the constitutive equation method.

Stiffness and Damping of an Eddy Current Magnetic Bearing

1992 ◽  
Vol 114 (3) ◽  
pp. 600-605 ◽  
Author(s):  
Leehua Ting ◽  
John Tichy
Keyword(s):  
High Temperature ◽  
Perturbation Method ◽  
Eddy Current ◽  
Magnetic Bearing ◽  
Dynamic Problems ◽  
Damping Coefficients ◽  
Theoretical Predictions ◽  
Low Stiffness ◽  
Stiffness And Damping ◽  
Rotor Dynamic

The perturbation method, aided by symbolic computation software, is employed to solve Maxwell’s equations to obtain theoretical predictions of forces and stiffness and damping coefficients for a simplified eddy current magnetic bearing. The results show that this kind of bearing has low stiffness and extremely low damping compared to conventional bearings. In fact, the damping is probably negative in most cases. In addition, the cross-coupled stiffness is relatively high, further contributing to rotor-dynamic problems. Despite these drawbacks, if advanced high temperature superconducting materials become practical, eddy current magnetic bearings may be useful in many applications.

Rotor Dynamic Modeling of Gears and Geared Systems

2013 ◽  
Author(s):  
Jason A. Kaplan ◽  
Saeid Dousti ◽  
Paul E. Allaire ◽  
Bradley R. Nichols ◽  
Timothy W. Dimond ◽  
...  
Keyword(s):  
Dynamic Analysis ◽  
High Speed ◽  
Degrees Of Freedom ◽  
Forced Response ◽  
Drive Train ◽  
Mode Shapes ◽  
Element Formulation ◽  
Resonant Frequencies ◽  
Industrial Drive ◽  
Rotor Dynamic

The ability to accurately predict rotating machine resonant frequencies and to assess their stability and response to external forces is crucial from a reliability and preventive maintenance perspective. Resonant frequencies and forced response become more difficult to predict when additional complicated components such as gearboxes are present in the rotor system. Gearbox dynamics contain many complex interactions and many of the simplifying assumptions provided in the literature do not apply to most geared systems. A finite element formulation of the gearbox, which couples the axial, lateral, and torsional degrees-of-freedom of the low and high-speed shafts, is developed. It has the capability to apply to a wide variety of both spur and helical geared systems and is sufficiently robust to account for arbitrary orientation angles between the parallel shafts. This study presents a rotor dynamic analysis of an industrial drive-train consisting of a steam turbine, herringbone gearbox, and a generator using 1-D Timoshenko beam elements. The rotor dynamic analysis consists of the calculation of the damped natural frequencies, mode shapes, and provides insight into the stability of the industrial drive-train.

scholarly journals Extension of modal reduction methods to non-linear coupled structure-acoustic problems

2011 ◽  
pp. 227-245
Author(s):  
Youssef Gerges ◽  
Emeline Sadoulet-Reboul ◽  
Morvan Ouisse ◽  
Noureddine Bouhaddi
Keyword(s):  
Numerical Simulations ◽  
Linear Model ◽  
Elastic Plate ◽  
Degrees Of Freedom ◽  
Reduction Method ◽  
Static Response ◽  
Acoustic Cavity ◽  
Modal Reduction ◽  
Non Linear ◽  
Reduction Methods

This paper proposes a robust reduction method dedicated to non-linear vibroacoustic problems in the context of localized geometrical non-linearities. The method consists in enriching the truncated uncoupled modal basis of the linear model by a static response due to unit forces on the non-linear degrees of freedom and by the static response of the fluid due to the interaction with the structure. To show the effectiveness of the proposed method, numerical simulations of responses of an elastic plate closing an acoustic cavity and a hang-on exhaust are performed.

scholarly journals An analytical formulation to evaluate natural frequencies and mode shapes of high-rise buildings

2021 ◽  
Vol 8 (1) ◽  
pp. 307-318
Author(s):  
Giuseppe Nitti ◽  
Giuseppe Lacidogna ◽  
Alberto Carpinteri
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Shear Walls ◽  
Vertical Axis ◽  
Mode Shapes ◽  
Analytical Formulation ◽  
High Rise ◽  
Open Section ◽  
Stiffness Matrices ◽  
Thin Walled

Abstract In this paper, an original analytical formulation to evaluate the natural frequencies and mode shapes of high-rise buildings is proposed. The methodology is intended to be used by engineers in the preliminary design phases as it allows the evaluation of the dynamic response of high-rise buildings consisting of thin-walled closed- or open-section shear walls, frames, framed tubes, and dia-grid systems. If thin-walled open-section shear walls are present, the stiffness matrix of the element is evaluated considering Vlasov’s theory. Using the procedure called General Algorithm, which allows to assemble the stiffness matrices of the individual vertical bracing elements, it is possible to model the structure as a single equivalent cantilever beam. Furthermore, the degrees of freedom of the structural system are reduced to only three per floor: two translations in the x and y directions and a rigid rotation of the floor around the vertical axis of the building. This results in a drastic reduction in calculation times compared to those necessary to carry out the same analysis using commercial software that implements Finite Element models. The potential of the proposed method is confirmed by a numerical example, which demonstrates the benefits of this procedure.

Instability Boundaries of a Vocal Fold Modelled as a Flexibly Supported Rigid Body Vibrating in a Channel Conveying Fluid

2002 ◽  
Author(s):  
J. Hora´cˇek ◽  
J. G. Sˇvec
Keyword(s):  
Vocal Fold ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Vocal Folds ◽  
Mode Shapes ◽  
Physiological Data ◽  
Flow Induced Vibration ◽  
Aeroelastic Model ◽  
Vibration Model ◽  
Stiffness And Damping

A generalised leakage-flow induced vibration model for vibration onset of the vocal folds in the airflow is presented. Especially the influence of various physical properties of the vocal folds (e.g., their shape, adduction, mass and mass distribution) on the thresholds of instability is studied. A vibrating element with two-degrees-of-freedom (rotation and translation) placed on an elastic foundation and vibrating in the wall of a channel conveying air is used to approximate the vocal fold. The inviscid incompressible 1-D fluid flow theory is used. A generally defined shape of the vocal fold is considered for expressing the unsteady aerodynamic forces in the glottis. The parameters of the mechanical part of the model, i.e., the mass, stiffness and damping matrices are related to the known geometry, size and material density of the vocal folds as well as to the fundamental natural frequencies and damping known from the experiments. The numerical solution yields the natural frequencies, damping, mode shapes of vibration and the instability thresholds of the system. The calculated vibration and stability characteristics appear to be close to the known physiological data. The developed aeroelastic model is able to provide qualitative information, e.g., on conditions for a soft voice onset or for breathy voicing and could be helpful also in design of artificial voice prosthesis.

Modal Reduction Method for Vibrational Analysis of Structures Partially Filled With Internal Liquids

2005 ◽  
Author(s):  
Jean-Se´bastien Schotte´ ◽  
Roger Ohayon
Keyword(s):  
Degrees Of Freedom ◽  
Reduction Method ◽  
Flexible Structures ◽  
Vibrational Analysis ◽  
Galerkin Projection ◽  
The Finite Element Method ◽  
Aircraft Wings ◽  
System A ◽  
Modal Reduction ◽  
Fluid Sloshing

In the framework of the vibrational analysis of structures, we propose a method to take into account the internal liquids (like fuel) in the structure modelling. Since the “classical” added-mass hydroelastic model cannot be used for slender and flexible structures such as space launchers or civil aircraft wings (because the decoupling assumption of sloshing and hydroelasticity is not valid for such structures), we propose here a revised hydroelastic formulation which takes this coupling into account. Since this model introduces in the system a great number of degrees of freedom for the fluid, we will present a reduction method using the Ritz-Galerkin projection onto the fluid sloshing modal basis. A discretization of the reduced equations by the finite element method will be proposed and the convergence rate of this modal reduction method will be discussed on an application example.

Comportement dynamique d'une éolienne à axe vertical (développement théorique)

1988 ◽  
Vol 15 (3) ◽  
pp. 355-368
Author(s):  
Mario Veilleux ◽  
René Tinawi
Keyword(s):  
Degrees Of Freedom ◽  
Dynamic Instability ◽  
Equations Of Motion ◽  
Vertical Component ◽  
Vertical Axis ◽  
Nonlinear Static Analysis ◽  
Mode Shapes ◽  
Vertical Axis Wind Turbine ◽  
Qr Algorithm ◽  
Gyroscopic Effects

Complex frequencies and mode shapes are evaluated and presented for a guyed vertical axis wind turbine to detect any dynamic instability for a given speed of rotation. The equations of motion are developed in the rotating system of axes of the rotor to eliminate the time dependent terms. These equations take into account gyroscopic effects by evaluating the Coriolis and Circulatory (softening) matrices. The guys are replaced by nonlinear springs. The geometric stiffness matrix is also considered by performing a nonlinear static analysis that includes centrifugal effects and gravity loads, as well as compression from the vertical component of the guys. A reduction of the dynamic degrees of freedom is performed using the Rayleigh–Ritz technique. The complex frequencies and mode shapes are obtained using the QR algorithm. A program developed on a microcomputer was used to evaluate the dynamic instabilities of the ÉOLE Project. These results are described in the following paper. Key words: Vertical axis wind turbines, guys, complex frequencies, complex mode shapes, centrifugal forces, Coriolis forces, numerical software.

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