scholarly journals How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

2021 ◽  
Author(s):  
Shobhit Jain ◽  
George Haller
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Mechanical Systems ◽  
Forced Response ◽  
High Dimensional ◽  
Nonlinear Phenomena ◽  
Finite Element Models ◽  
Engineering Structures ◽  
Backbone Curves

AbstractInvariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

Method for Analysis of Nonlinear Multiharmonic Vibrations of Mistuned Bladed Disks With Scatter of Contact Interface Characteristics

2005 ◽  
Vol 127 (1) ◽  
pp. 128-136 ◽  
Author(s):  
E. P. Petrov ◽  
D. J. Ewins
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Forced Response ◽  
Finite Element Models ◽  
Linear Vibration ◽  
Bladed Disks ◽  
Bladed Disk ◽  
Interface Characteristics ◽  
Blade Frequency

An efficient method for analysis of nonlinear vibrations of mistuned bladed disk assemblies has been developed. This development has facilitated the use of large-scale finite element models for realistic bladed disks, used hitherto in analysis of linear vibration, to be extended for the analysis of nonlinear multiharmonic vibration. The new method is based on a technique for the exact condensation of nonlinear finite element models of mistuned bladed disks. The model condensation allows the size of the nonlinear equations to be reduced to the number of degrees of freedom where nonlinear interaction forces are applied. The analysis of nonlinear forced response for simplified and realistic models of mistuned bladed disks has been performed. For a practical high-pressure bladed turbine disk, several types of nonlinear forced response have been considered, including mistuning by (i) scatter of underplatform dampers, (ii) shroud gap scatter, and (iii) blade frequency scatter in the presence of nonlinear shroud interactions.

Method for Analysis of Nonlinear Multiharmonic Vibrations of Mistuned Bladed Discs With Scatter of Contact Interface Characteristics

2004 ◽  
Author(s):  
E. P. Petrov ◽  
D. J. Ewins
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Nonlinear Vibrations ◽  
Forced Response ◽  
Finite Element Models ◽  
Contact Interface ◽  
Linear Vibration ◽  
Interface Characteristics ◽  
Blade Frequency

An efficient method for analysis of nonlinear vibrations of mistuned bladed disc assemblies has been developed. As a result, this development has facilitated the use of large-scale finite element models for realistic bladed discs, as used hitherto in analysis of linear vibration, to be extended for the analysis of nonlinear multiharmonic vibration. The new method is based on a technique for the exact condensation of nonlinear finite element models of mistuned bladed discs. The model condensation allows the size of the nonlinear equations to be reduced to the number of degrees of freedom where nonlinear interation forces are applied. The analysis of nonlinear forced response for simplified and realistic models of mistuned bladed discs has been performed. For a practical high-pressure bladed turbine disc, several types of nonlinear forced response have been considered including: (i) mistuning by scatter of underplatform dampers; (ii) mistuning by shroud gap scatter; (iii) mistuning by blade frequency scatter in the presence of nonlinear shroud interactions.

A Parallel Solution Strategy for Finite Element Models

1996 ◽  
Author(s):  
Jordan J. Cox ◽  
Jeffrey A. Talbert ◽  
Eric Mulkay
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Element Model ◽  
Decomposition Methods ◽  
Finite Element Models ◽  
Unique Contribution ◽  
Matrix Equations ◽  
Parallel Solution ◽  
The Finite Element Model ◽  
Parallel Fashion

Abstract This paper presents a method for naturally decomposing finite element models into sub-models which can be solved in a parallel fashion. The unique contribution of this paper is that the decomposition strategy comes from the geometric features used to construct the solid model that the finite element model represents. Domain composition and domain decomposition methods are used to insure global compatibility. These techniques reduce the N2 behavior of traditional matrix solving techniques, where N is the number of degrees of freedom in the global set of matrix equations, to a sum of m matrices with n2 behavior, where n represents the number of degrees of freedom in the smaller sub-model matrix equations.

Forced Response Variation of a Compressor Utilizing Blade Tip Timing, Strain Gages, and As-Manufactured Finite Element Models

2021 ◽  
Author(s):  
Daniel Gillaugh ◽  
Timothy Janczewski ◽  
Alex Kaszynski ◽  
Jeffrey Brown ◽  
Joseph Beck ◽  
...  
Keyword(s):  
Finite Element ◽  
Strain Gage ◽  
Model Updating ◽  
Forced Response ◽  
Finite Element Models ◽  
Turbine Engine ◽  
Blade Tip ◽  
Response Variation ◽  
Updating Procedure ◽  
Engine Components

Abstract The dynamic response of turbine engine components varies widely due to manufacturing deviations in the blades known as mistuning. This dynamic variation is investigated using a single stage compressor experimentally using both blade tip timing (BTT) and strain gage (SG) measurements and using as-manufactured finite element models (AMMs) on a 1st bend mode. Operational BTT and SG safety limits were generated using both averaged and AMM models via Goodman material properties. The predicted individual blade stress/deflection (S/D) ratios and strain gage ratios for this mode will be compared to the average finite element counterparts. Additionally, the correlation between BTT and SG's will be presented. This correlation will be performed using two approaches: blade maximum stress comparisons and measured response compared to the sensors safety limits. It will be shown that accounting for geometry with AMMs produce more accurate strain gage to BTT correlation compared to average models. An experimental model updating procedure is developed to increase the strain gage to BTT correlation by optimizing the location the BTT optical spot probes measure on the blade chord. Implementing this procedure using as-manufactured models are able to improve strain gage to BTT correlation.

Improving Spot Weld Models in Structural Dynamics

2003 ◽  
Author(s):  
Matteo Palmonella ◽  
Michael I. Friswell ◽  
Cristinel Mares ◽  
John E. Mottershead
Keyword(s):  
Finite Element ◽  
Structural Dynamics ◽  
Degrees Of Freedom ◽  
Dynamic Behaviour ◽  
Natural Frequencies ◽  
Spot Weld ◽  
Physical Parameters ◽  
Finite Element Models ◽  
Spot Welds ◽  
Coarse Meshes

This paper gives an overview of the finite element modelling of spot welds for the analysis of the dynamic response of structures. In particular models for dynamic analysis that use coarse meshes and equivalent parameters are considered. A major requirement for these models is their accuracy in predicting the dynamic behaviour of spot welded structures despite the low number of degrees of freedom. Three different models of spot welds are investigated [1–3] and for each model physical parameters have to be assigned based on engineering insight. The aim of the present paper is to improve the accuracy of these three models by searching for the optimum values of the parameters characterising the spot weld models using experimental data. For this purpose a benchmark structure has been analysed, consisting of a thin walled hat section beam made of two plates welded together by twenty spot welds. The predicted natural frequencies and modes of the benchmark structure have been compared to the experimental modes. Updating of the finite element models has been performed and the accuracy of the three models has been significantly improved.

Multiparametric Analysis of Resonance Peak Vibrations for Nonlinear Jointed Structures

2011 ◽  
Author(s):  
E. P. Petrov
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Frequency Domain Analysis ◽  
Resonance Peak ◽  
Test Cases ◽  
Finite Element Models ◽  
Contact Interface ◽  
Interface Elements ◽  
Practical Test ◽  
Nonlinear Contact

A methodology has been developed to analyse and to optimize the resonance peak amplitudes and frequencies for essentially nonlinear periodic vibrations of jointed structures. The frequency domain analysis of realistic finite element models of jointed structures which can contain millions of degrees of freedom is performed. The detailed description of friction, gap and other types of the nonlinear contact interfaces in jointed structures is provided by contact interface elements. The resonance peak characteristics are calculated directly as functions of several parameters of contact interfaces and excitation. The efficiency of the methodology is demonstrated on a representative set of practical test cases.

Contact Analysis of the Rotor in a Variable Capacitance Micro-Motor

2005 ◽  
Author(s):  
Wen-Ming Zhang ◽  
Guang Meng
Keyword(s):  
Finite Element ◽  
Contact Problem ◽  
Contact Stress ◽  
Numerical Solutions ◽  
Ground Plane ◽  
Contact Process ◽  
Nonlinear Phenomena ◽  
Finite Element Models ◽  
Stress Strain ◽  
Variable Capacitance

The most widely known nonlinear phenomena in Micro-electro-mechanical systems (MEMS) are probably the contact instabilities. Contact problem is an important topic in the research of micro-motors. While the micro-motor is in operation, the rotor is intended to be in electrical contact with the ground plane, and the rotor and bearing hub form a pair of contact bodies. In the paper, a mathematic model is proposed to describe the contact process and two simplified contact finite element models of the rotor, bearing and ground plane are presented to simulate the contact. The effects on the contact stress, strain and pressure are analyzed in micro-scale. The rotor-to-bearing-hub and the hemispherical-bushing-on-ground-plane configurations finite element models (FEM) are established and the implementation of the contact problem is introduced to provide the numerical solutions acted as a guide to solution of contact problems in a variable capacitance micromotor. The numerical results of the contact stress, strain and pressure and the effects of the coefficient of friction and the surface roughness of the contact pairs on contact characteristics are studied and discussed in detail. It is indicated that the nonlinear effects should not be ignored and these results must be evaluated on a relative scale to compare different design options.

scholarly journals MATHEMATICAL MODEL OF BASIC FINITE ELEMENTS IN THE FINITE-ELEMENTARY THEORY OF PORT LIFTING STRUCTURES

2018 ◽  
pp. 109-128
Author(s):  
Nikolay Nikitovich Panasenko ◽  
Alexey Vladimirovich Sinelshchikov
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Power Plants ◽  
Computational Analysis ◽  
Elementary Theory ◽  
Seismic Stability ◽  
Finite Element Models ◽  
Nuclear Facilities ◽  
Engineering Structures ◽  
Operation Rules

The problem of computational analysis of the seismic safety of lifting cranes specified by the regulatory systems (FPP "Safety Rules for dangerous production facilities using Lifting mechanisms" for standard industrial application cranes; Regulation 31.1.02-2004 "Technical operation rules for carrying and lifting equipment in the sea merchant harbors" for harbor cranes; Code of Design-031-01 "Codes of Design of antiseismic atomic power stations" and Code of Design-043-11 "Rules of Design and safe operating hoisting cranes for objects of use of atomic energy" for cranes used at the nuclear facilities) is currently under discussion, despite the emphasis on the part of scientific community. All this has led to carrying out the research which presented a vision of the problems of designing cranes in earthquake-proof design as a finite element theory of structures, and on the basis of methods of the theory of seismic stability of structures - the linear spectral method, according to the Code 14.13330.2014 "Building in earthquake areas" and the method of dynamic analysis, according to Guidelines 1.5.2.05.999.0025-2011 "Calculation and design of earthquake resistant nuclear power plants". The article highlights the trend of recent years of the increasing complexity of calculated finite element models of structures, often combining both finite element models of buildings and supporting structures, and cranes. A computational analysis of such constructions leads to a combination in the design model of finite elements of different dimensions. The article points out that both the choice of the type of finite elements and the way they are connected together within the framework of one calculation model directly affect the reliability of the results obtained. Based on the practical experience of computational analysis of complex spatial engineering structures, the article proposes stiffness and mass matrices for one-, two- and three-dimensional basic finite elements for calculating port lifting structures.

Forced Response Variation of a Compressor Utilizing Blade Tip Timing, Strain Gages, and As-Manufactured Finite Element Models

2020 ◽  
Author(s):  
Daniel L. Gillaugh ◽  
Timothy J. Janczewski ◽  
Alexander A. Kaszynski ◽  
Jeffrey M. Brown ◽  
Joseph A. Beck ◽  
...  
Keyword(s):  
Finite Element ◽  
Strain Gage ◽  
Model Updating ◽  
Forced Response ◽  
Finite Element Models ◽  
Turbine Engine ◽  
Blade Tip ◽  
Response Variation ◽  
Updating Procedure ◽  
Engine Components

Abstract The dynamic response of turbine engine components varies widely due to manufacturing deviations in the blades known as mistuning. This dynamic variation is investigated using a single stage compressor experimentally using both blade tip timing (BTT) and strain gage (SG) measurements and using as-manufactured finite element models (AMMs) on a 1st bend mode. Operational BTT and SG safety limits were generated using both averaged and AMM models via Goodman material properties. The predicted individual blade stress/deflection (S/D) ratios and strain gage ratios for this mode will be compared to the average finite element counterparts. Additionally, the correlation between BTT and SG’s will be presented. This correlation will be performed using two approaches: blade maximum stress comparisons and measured response compared to the sensors safety limits. It will be shown that accounting for geometry with AMMs produce more accurate strain gage to BTT correlation compared to average models. An experimental model updating procedure is developed to increase the strain gage to BTT correlation by optimizing the location the BTT optical spot probes measure on the blade chord. Implementing this procedure using as-manufactured models are able to improve strain gage to BTT correlation.

A Direct Two Response Approach for Updating Analytical Dynamic Models of Structures With Emphasis on Uniqueness

1990 ◽  
Vol 112 (1) ◽  
pp. 107-111 ◽  
Author(s):  
S. R. Ibrahim ◽  
C. Stavrinidis ◽  
E. Fissette ◽  
O. Brunner
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Dynamic Models ◽  
Normal Modes ◽  
Element Model ◽  
Finite Element Models ◽  
Complex Modes ◽  
Structural Responses ◽  
Stiffness And Damping ◽  
The Finite Element Model

An approach, based on utilizing only two sets of structural responses and the enforcement of the conditions for a unique solution, is presented for the updating of Finite Element Models. The responses required can be any two identified normal modes, any two identified complex modes, or two forced harmonic response vectors in the neighborhood of any two natural frequencies of the structure under test. The mass, stiffness, and damping matrices are interactively and simultaneously corrected in a direct noniterative procedure. A uniqueness factor is automatically computed in the procedure to indicate the correctability of the Finite Element Model under consideration. The number of measurement locations is assumed to be less than the number of degrees of freedom of the analytical model. Provisions for completing and smoothing the measured or identified responses are included to reduce the effects of measurement noise and identification error. Preliminary results on simple models are presented in support of the proposed technique.

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