Rotordynamic Energy Expressions for General Anisotropic Finite Element Systems

2017 ◽  
Author(s):  
Manoj Settipalli ◽  
Venkatarao Ganji ◽  
Theodore Brockett
Keyword(s):  
Finite Element ◽  
Mode Shape ◽  
Degrees Of Freedom ◽  
Element Formulation ◽  
Energy Distributions ◽  
Displacement Vectors ◽  
Critical Components ◽  
Time Invariant ◽  
Alternative Set ◽  
Orbit Characteristics

It is often desirable to identify the critical components that are active in a particular mode shape or an operational deflected shape (ODS) in a complex rotordynamic system with multiple rotating groups and bearings. The energy distributions can help identify the critical components of a rotor bearing system that may be modified to match the design requirements. Although the energy expressions have been studied by researchers in the past under specific limited conditions, these expressions require computing the displacements and velocities of all degrees of freedom over one full cycle. They do not address the overall time-dependency of the energies and energy distributions, and their effect on the interpretation of a mode shape or an ODS. Moreover, a detailed finite element formulation of these energy expressions including the effects of anisotropy, skew-symmetric stiffness, viscous and structural damping have not been identified by the authors in the open literature. In this article, a detailed account of orbit characteristics and planarity for isotropic and anisotropic systems is presented. The effect of orbit characteristics on the energy expressions is then discussed. An elegant approach to obtaining time-dependent kinetic and strain energies of a mode shape or an ODS directly from the structural matrices and complex eigenvectors/displacement vectors is presented. The expressions for energy contributed per cycle by various types of damping and the destabilizing skew-symmetric stiffness that can be obtained in a similar way are also shown. The conditions under which the energies and energy distributions are time-invariant are discussed. An alternative set of energy expressions for isotropic systems with the degrees of freedom reduced by half is also presented.

Rotordynamic Energy Expressions for General Anisotropic Finite Element Systems

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Manoj Settipalli ◽  
Venkatarao Ganji ◽  
Theodore Brockett
Keyword(s):  
Finite Element ◽  
Mode Shape ◽  
Degrees Of Freedom ◽  
Element Formulation ◽  
Energy Distributions ◽  
Displacement Vectors ◽  
Critical Components ◽  
Time Invariant ◽  
Alternative Set ◽  
Orbit Characteristics

It is often desirable to identify the critical components that are active in a particular mode shape or an operational deflected shape (ODS) in a complex rotordynamic system with multiple rotating groups and bearings. The energy distributions can help identify the critical components of a rotor bearing system that may be modified to match the design requirements. Although the energy expressions have been studied by researchers in the past under specific limited conditions, these expressions require computing the displacements and velocities of all degrees-of-freedom (DOFs) over one full cycle. They do not address the overall time dependency of the energies and energy distributions, and their effect on the interpretation of a mode shape or an ODS. Moreover, a detailed finite element formulation of these energy expressions including the effects of anisotropy, skew-symmetric stiffness, viscous and structural damping have not been identified by the authors in the open literature. In this article, a detailed account of orbit characteristics and planarity for isotropic and anisotropic systems is presented. The effect of orbit characteristics on the energy expressions is then discussed. An elegant approach to obtaining time-dependent kinetic and strain energies of a mode shape or an ODS directly from the structural matrices and complex eigenvectors/displacement vectors is presented. The expressions for energy contributed per cycle by various types of damping and the destabilizing skew-symmetric stiffness that can be obtained in a similar way are also shown. The conditions under which the energies and energy distributions are time-invariant are discussed. An alternative set of energy expressions for isotropic systems with the DOFs reduced by half is also presented.

scholarly journals Generalized finite element formulation for efficient first-order plastic hinge analysis

2019 ◽  
Vol 11 (3) ◽  
pp. 168781401983636
Author(s):  
Dae-Jin Kim ◽  
Hong-Jun Son ◽  
Yousun Yi ◽  
Sung-Gul Hong
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Plastic Hinge ◽  
Computational Cost ◽  
Finite Element Formulation ◽  
Solution Technique ◽  
Element Formulation ◽  
Plastic Hinges ◽  
First Order ◽  
Generalized Finite Element

This article presents generalized finite element formulation for plastic hinge modeling based on lumped plasticity in the classical Euler–Bernoulli beam. In this approach, the plastic hinges are modeled using a special enrichment function, which can describe the weak discontinuity of the solution at the location of the plastic hinge. Furthermore, it is also possible to insert a plastic hinge at an arbitrary location of the element without modifying its connectivity or adding more elements. Instead, the formations of the plastic hinges are achieved by hierarchically adding more degrees of freedom to existing elements. Due to these features, the proposed methodology can efficiently perform the first-order plastic hinge analysis of large-frame structures. A generalized finite element solution technique based on the static condensation scheme is also proposed in order to reduce the computational cost of a series of linear elastic problems, which is in general the most time-consuming portion of the first-order plastic hinge analysis. The effectiveness and accuracy of the proposed method are verified by analyzing several representative numerical examples.

A New Hierarchical Technique for the Multiscale Modeling of Carbon Nanostructures

2005 ◽  
Author(s):  
Arash Mahdavi ◽  
Eric Mockensturm
Keyword(s):  
Finite Element ◽  
Atomic Structure ◽  
Carbon Nanostructures ◽  
Degrees Of Freedom ◽  
Finite Size ◽  
Multiscale Methods ◽  
Atomic Scale ◽  
Modeling Technique ◽  
Element Formulation ◽  
Kinematic Constraints

We present a new hierarchical modeling technique called the Consistent Atomic-scale Finite Element (CAFE´) method [1]. Unlike traditional approaches for linking the atomic structure to its equivalent continuum [2-7], this method directly connects the atomic degrees of freedom to a reduced set of finite element degrees of freedom without passing through an intermediate homogenized continuum. As a result, there is no need to introduce stress and strain measures at the atomic level. This technique partitions atoms to masters and salves and reduces the total number of degrees of freedom by establishing kinematic constraints between them [5-6]. The Tersoff-Brenner interatomic potential [8] is used to calculate the consistent tangent stiffness matrix of the structure. In this finite element formulation, all local and non-local interactions between carbon atoms are taken into account using overlapping finite elements (Figure 1b). In addition, a consistent hierarchical finite element modeling technique is developed for adaptively coarsening and refining the mesh over different parts of the model (Figure 2a, 2b). The stiffness of higher-rank elements is approximated using the stiffness of lower-rank elements and kinematic constraints. This process is consistent with the underlying atomic structure and, by refining the mesh, molecular dynamic results will be recovered. This method is valid across the scales and can be used to concurrently model atomistic and continuum phenomena so, in contrast with most other multiscale methods [4-7], there is no need to introduce artificial boundaries for coupling atomistic and continuum regions. Effect of the length scale of the nanostructure is also included in the model by building the hierarchy of elements from bottom up using a finite size atom cluster as the building block (Figures 2a, 2b). In this method by introducing two independent field variables, the so-called inner displacement is taken into account (Fig. 3b). Applicability of the method is shown with several examples of deformation of carbon nanostructures such as graphene sheet, nanotube, and nanocone, subjected to different loads and boundary conditions.

scholarly journals Dynamics of on-board rotors on finite-length journal bearings subject to multi-axial and multi-frequency excitations: numerical and experimental investigations

2021 ◽  
Vol 22 ◽  
pp. 35
Author(s):  
Yvon Briend ◽  
Eric Chatelet ◽  
Régis Dufour ◽  
Marie-Ange Andrianoely ◽  
Franck Legrand ◽  
...  
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Experimental Validation ◽  
Parametric Instability ◽  
Experimental Investigations ◽  
Element Formulation ◽  
Hydrodynamic Bearing ◽  
Mass Unbalance ◽  
Moving Base ◽  
Technology Applications

On-board rotating machinery subject to multi-axial excitations is encountered in a wide variety of high-technology applications. Such excitations combined with mass unbalance forces play a considerable role in their integrity because they can cause parametric instability and rotor–stator interactions. Consequently, predicting the rotordynamics of such machines is crucial to avoid triggering undesirable phenomena or at least limiting their impacts. In this context, the present paper proposes an experimental validation of a numerical model of a rotor-shaft-hydrodynamic bearings system mounted on a moving base. The model is based on a finite element approach with Timoshenko beam elements having six degrees of freedom (DOF) per node to account for the bending, torsion and axial motions. Classical 2D rectangular finite elements are also employed to obtain the pressure field acting inside the hydrodynamic bearing. The finite element formulation is based on a variational inequality approach leading to the Reynolds boundary conditions. The experimental validation of the model is carried out with a rotor test rig, designed, built, instrumented and mounted on a 6-DOF hydraulic shaker. The rotor’s dynamic behavior in bending, torsion and axial motions is assessed with base motions consisting of mono- and multi-axial translations and rotations with harmonic, random and chirp sine profiles. The comparison of the predicted and measured results achieved in terms of shaft orbits, full spectrums, transient history responses and power spectral densities is very satisfactory, permitting the experimental validation of the model proposed.

Quench Process Modeling and Optimization

2000 ◽  
Author(s):  
Ravi S. Bellur-Ramaswamy ◽  
Nahil A. Sobh ◽  
Robert B. Haber ◽  
Daniel A. Tortorelli
Keyword(s):  
Particle Size ◽  
Finite Element ◽  
Process Parameters ◽  
Degrees Of Freedom ◽  
Programming Algorithm ◽  
Precipitate Size ◽  
Rate Theory ◽  
Element Formulation ◽  
Balancing Energy ◽  
Finite Element Node

Abstract We optimize continuous quench process parameters to produce a desired precipitate distribution in aluminum alloy extrudates. To perform this task, an optimization problem is defined and solved using a standard nonlinear programming algorithm. Ingredients of this algorithm include a cost function, constraint functions and their sensitivities with respect to the process parameters. These functions are dependent on the temperature and precipitate size which are obtained by balancing energy to determine the temperature distribution and by using a reaction-rate theory to determine a discrete precipitate particle size distribution. Both the temperature and the precipitate models are solved via the finite element method. Since we use a discrete particle size model, there are as many as 105 degrees-of-freedom per finite element node. After we compute the temperature and precipitate size distributions, we must also compute their sensitivities. This seemingly intractable computational task is resolved by using an element-by-element discontinuous Galerkin finite element formulation and a direct differentiation sensitivity analysis which allows us to perform all of the computations on a PC.

USE OF MODAL FLEXIBILITY AND NORMALIZED MODAL DIFFERENCE FOR VIBRATION MODE SHAPE EXPANSION

2009 ◽  
Vol 09 (04) ◽  
pp. 765-775 ◽  
Author(s):  
WEI-XIN REN ◽  
BIJAYA JAISHI
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mode Shape ◽  
Degrees Of Freedom ◽  
Vibration Mode ◽  
Mode Shapes ◽  
Modal Expansion ◽  
Simply Supported ◽  
Modal Flexibility ◽  
Actual Test

Proposed herein are two possible ways for mode shape expansion for future use. The first method minimizes the modal flexibility error between the experimental and analytical mode shapes corresponding to the measured degrees of freedom (DOFs) to determine the multiplication matrix. In the second method, Normalized Modal Difference (NMD) is used to calculate the multiplication matrix using the analytical DOFs corresponding to the measured DOFs. This matrix is then used to expand the measured mode shape to unmeasured DOFs. A simulated simply supported beam is used to demonstrate the performance of the methods. These methods are then compared with two most promising existing methods, namely the Kidder dynamic expansion and the modal expansion methods. It is observed that the performance of the modal flexibility method is comparable with existing methods. NMD also have the potential to expand the mode shapes though it is seen to be more sensitive to the distribution of error between finite element method and actual test data.

Substructuring Developments in the Energy Finite Element Formulation

2006 ◽  
Author(s):  
Geng Zhang ◽  
Nickolas Vlahopoulos ◽  
Jiulong Sun
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Simulation Models ◽  
Element Formulation ◽  
System Of Equations ◽  
Element Analysis ◽  
Global System ◽  
Computational Capability ◽  
New Development ◽  
Energy Finite Element Analysis

In Naval applications of the Energy Finite Element Analysis (EFEA) there is an increasing need for developing comprehensive models with a large number of elements which include both structural and interior fluid elements, while certain parts of the structure are considered to be exposed to an external heavy fluid loading. In order to accommodate efficient computations when using simulation models with a large number of elements, joints, and domains, a substructuring computational capability has been developed. The new algorithm is based on dividing the EFEA model into substructures with internal and interface degrees of freedom. The system of equations for each substructure is assembled and solved separately and the information is condensed to the interface degrees of freedom. The condensed systems of equations from each substructure are assembled in a reduced global system of equations. Once the global system of equations has been solved the solution for each substructure is pursued. Important issues which have been considered in the new development originate from the necessity to define substructure interfaces along joint locations. The discontinuity of the energy density variables and the proper formulation of the joints across substructure interfaces have been considered in the new algorithm. In order to demonstrate the validity of the developments and the computational savings a set of previous applications where simulation results were compared to test data is repeated using the substructuring algorithm.

Beam Theories for Torsional-Bending Response of Ship Hulls

1991 ◽  
Vol 35 (03) ◽  
pp. 254-265 ◽  
Author(s):  
P. Terndrup Pedersen
Keyword(s):  
Mathematical Model ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
Higher Order ◽  
Response Analysis ◽  
Element Formulation ◽  
Transition Matrices ◽  
Ship Hulls ◽  
The Mathematical Model ◽  
Bending Response

A consistent one-dimensional finite-element procedure for analysis of the coupled torsional-bending response of thin-walled beam structures such as ship hulls is presented. At each element end there are three translations, three rotations and one classical Vlasov warping degree of freedom plus possibly N degrees of freedom associated with higher order generalized warping deformation modes. These higher order warping modes are generated from an eigenvalue problem associated with the homogeneous plane stress equilibrium condition for the individual beam cross sections. The assembly of the beam elements to the global model is performed by use of transition matrices which assure compatibility between the elements in the sense of least squares. Numerical examples are included which demonstrate the accuracy of the mathematical model and the applicability of the proposed analysis procedure for calculation of torsion-horizontal bending response of a containership hull. Even if the higher order warping modes are not included in the finite element formulation it is found that the mathematical model is quite accurate for overall response analysis of hull structures.

Finite Element Formulation of Multiphasic Shell Elements for Cell Mechanics Analyses in FEBio

2018 ◽  
Vol 140 (12) ◽  
Author(s):  
Jay C. Hou ◽  
Steve A. Maas ◽  
Jeffrey A. Weiss ◽  
Gerard A. Ateshian
Keyword(s):  
Finite Element ◽  
Cell Membrane ◽  
Degrees Of Freedom ◽  
Fluid Pressure ◽  
3D Models ◽  
Solid Matrix ◽  
Cell Physiology ◽  
Element Formulation ◽  
Isolated Cells ◽  
Shell Elements

With the recent implementation of multiphasic materials in the open-source finite element (FE) software FEBio, three-dimensional (3D) models of cells embedded within the tissue may now be analyzed, accounting for porous solid matrix deformation, transport of interstitial fluid and solutes, membrane potential, and reactions. The cell membrane is a critical component in cell models, which selectively regulates the transport of fluid and solutes in the presence of large concentration and electric potential gradients, while also facilitating the transport of various proteins. The cell membrane is much thinner than the cell; therefore, in an FE environment, shell elements formulated as two-dimensional (2D) surfaces in 3D space would be preferred for modeling the cell membrane, for the convenience of mesh generation from image-based data, especially for convoluted membranes. However, multiphasic shell elements are yet to be developed in the FE literature and commercial FE software. This study presents a novel formulation of multiphasic shell elements and its implementation in FEBio. The shell model includes front- and back-face nodal degrees-of-freedom for the solid displacement, effective fluid pressure and effective solute concentrations, and a linear interpolation of these variables across the shell thickness. This formulation was verified against classical models of cell physiology and validated against reported experimental measurements in chondrocytes. This implementation of passive transport of fluid and solutes across multiphasic membranes makes it possible to model the biomechanics of isolated cells or cells embedded in their extracellular matrix (ECM), accounting for solvent and solute transport.

Stability of Non-Axisymmetric Rotor and Bearing Systems Modeled With Three-Dimensional-Solid Finite Elements

2019 ◽  
Vol 142 (1) ◽  
Author(s):  
Joseph Oh ◽  
Alan Palazzolo ◽  
Lingnan Hu
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Computational Efficiency ◽  
Degrees Of Freedom ◽  
Monodromy Matrix ◽  
Three Dimensional ◽  
Element Formulation ◽  
System Matrix ◽  
Rotor Bearing ◽  
Solid Finite Elements

Abstract Although rotors are simplified to be axisymmetric in rotordynamic models, many rotors in the industry are actually non-axisymmetric. Several authors have proposed methods using 3D finite element, rotordynamic models, but more efficient approaches for handling a large number of degrees-of-freedom (DOF) are needed. This task becomes particularly acute when considering parametric excitation that results from asymmetry in the rotating frame. This paper presents an efficient rotordynamic stability approach for non-axisymmetric rotor-bearing systems with complex shapes using three-dimensional solid finite elements. The 10-node quadratic tetrahedron element is used for the finite element formulation of the rotor. A rotor-bearing system, matrix differential equation is derived in the rotor-fixed coordinate system. The system matrices are reduced by using Guyan reduction. The current study utilizes the Floquet theory to determine the stability of solutions for parametrically excited rotor-bearing systems. Computational efficiency is improved by discretization and parallelization, taking advantage of the discretized monodromy matrix of Hsu's method. The method is verified by an analytical model with the Routh–Hurwitz stability criteria, and by direct time-transient, numerical integration for large order models. The proposed and Hill's methods are compared with respect to accuracy and computational efficiency, and the results indicate the limitations of Hill's method when applied to 3D solid rotor-bearing systems. A parametric investigation is performed for an asymmetric Root's blower type shaft, varying bearing asymmetry and bearing damping.

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