Perturbation of periodic equilibrium
|
|
- Benedict Lambert
- 5 years ago
- Views:
Transcription
1 Perturbation of periodic equilibrium by Arnaud Lazarus A spectral method to solve linear periodically time-varying systems 1
2 A few history Late 19 th century Emile Léonard Mathieu: Wave equation for an elliptical membrane moving through a fluid (1868) Achille Marie Gaston Floquet: «Sur les équations différentielles linéaires à coefficients périodiques», 1881, Comptes rendus de l académie des sciences. George William Hill [Hill, 1886]: «On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and the moon», 1886, Acta Mathematica. 2
3 A few history 20 th century Schrödinger equation with periodic potential. Parametric pendulum. A hamonic-based method for computing the stability of periodic solutions of dynamical systems. Modelling the vibration of cracked rotors in the frequency domain. 3
4 Stability of dynamical systems Fixed points: x 0 (t) = x 0 Governing equation in the state space: a. Beam under conservative compressive load 4
5 Stability of dynamical systems Fixed points: x 0 (t) = x 0 Governing equation in the state space: a. Beam under conservative compressive load 5
6 Stability of dynamical systems Fixed points: x 0 (t) = x 0 Governing equation: λ control parameter Bifurcation diagram with the ANM continuation method x 0 constant solution (static equilibrium) 6
7 Stability of dynamical systems Perturbed solution Governing equation: Perturbation: Fundamental solutions Linear system: α n : eigenvalues of J 7
8 Stability of dynamical systems Perturbed solution Governing equation: Perturbation: Fundamental solutions Linear system: α n : eigenvalues of J 8
9 Stability of dynamical systems Perturbed solution Governing equation: Perturbation: Fundamental solutions Linear system: α n : eigenvalues of J 9
10 Stability of dynamical systems Periodic equilibrium state: x 0 (t) = x 0 (t+t) Governing equation in the state space: b. Beam under compressive following force 10
11 Stability of dynamical systems Periodic equilibrium state: x 0 (t) = x 0 (t+t) Governing equation in the state space: b. Beam under compressive following force 11
12 Stability of dynamical systems Periodic equilibrium state: x 0 (t) = x 0 (t+t) Governing equation in the state space: λ control parameter x 0 (t) T-periodic solution T = 2π/ω Bifurcation diagram with to the ANM continuation method + harmonic balance 12
13 Stability of dynamical systems Perturbed solution Governing equation in the state space: λ control parameter Perturbation: x 0 (t) T-periodic solution T = 2π/ω Linear T-periodic system: J(t) T-periodic 13
14 Stability of dynamical systems Floquet theory N linearly independent solution y n (t): Floquet form: Poincaré stability p n (t) T-periodic: T = 2π/ω 14
15 Stability of dynamical systems Floquet theory N linearly independent solution y n (t): Floquet form: Poincaré stability p n (t) T-periodic: T = 2π/ω 15
16 Stability of dynamical systems Floquet theory N linearly independent solution y n (t): Floquet form: Poincaré stability p n (t) T-periodic: T = 2π/ω 16
17 Stability of dynamical systems Hill s method Fourier series expansions: Floquet form: Hill Linear T-periodic system: Hill 17
18 Stability of dynamical systems Hill s method Eigenproblem: H, infinite dimensional Hill Matrix To the infinity limit: N families of eigenvalues and eigenvectors Finite H: Most convergent members associated with l = 0 [Lazarus & Thomas, 2010] 18
19 Stability of dynamical systems The forced Guitar string Two polarizations: Modal expansion: 19
20 Stability of dynamical systems The forced Guitar string Harmonic based continuation method for quadratic dynamical systems (MANlab): Continuation of the first two modes: Dynamical system: 20
21 Stability of dynamical systems The forced Guitar string Resonance curves with H = 5: + Hill method Orbits in the p-q plane a. Amplitude of the first harmonic of p 1 and q 1 b. Phase 21
22 Stability of dynamical systems The forced Guitar string Evolution of Floquet s exponents against control parameter λ: a. Re(s) = fct(ω) for H = 5 b. Im(s) = fct(ω) for H = 3 22
23 Vibration of cracked rotors Synopsis Influence of imperfections on the global vibratory behaviour of large Rotating Machines? a. Turbo alternator b. GT-MHR project 23
24 Influence of imperfections Healthy rotor S 0 a. Simple model with 2 dofs b. Physical fundamental solutions (eigenmodes) 24
25 Influence of imperfections Healthy rotor S 0 a. Simple model with 2 dofs b. Physical fundamental solutions (eigenmodes) 25
26 Influence of imperfections Open crack S 1 T = π/ω a. Simple model with 2 dofs b. Stiffness function k(φ) in the fixed frame 26
27 Influence of imperfections Breathing crack S 2 [Andrieux & Varé, 2002] T = 2π/Ω a. Simple model with 2 dofs b. Stiffness function k(φ) in the fixed frame 27
28 Non-axisymmetrical rotating oscillators Floquet-Hill method Fourier series: + Hill matrix S 1 case: Open crack with 28
29 Non-axisymmetrical rotating oscillators Floquet-Hill method Fourier series: + Hill matrix S 2 case: Breathing crack with 29
30 Non-axisymmetrical rotating oscillators Fundamental solutions 4 independent fundamental solutions: Complex, coupling x- and y- directions. Poly-harmonic: (± ω x ±2jΩ)t or (± ω y ±2jΩ)t. a. In x- for Ω * = 0.34 and j max = 5 b. In y- for Ω * = 0.34 and j max = 5 30
31 Non-axisymmetrical rotating oscillators Stability analysis Evolution of Floquet exponents α n against rotating speed: a. Re(s) = fct(ω) for j max = 3 b. Im(s) = fct(ω) for j max = 3 31
32 Non-axisymmetrical rotating oscillators Stability analysis Evolution of Floquet exponents α n against rotating speed: a. Re(s) = fct(ω) for j max = 3 b. Im(s) = fct(ω) for j max = 3 32
33 Non-axisymmetrical rotating oscillators Strutt diagram Parametric instability of the rotating system: a. Stability map in the S 1 case a. Influence of fixed damping (S 1 case) 33
34 Non-axisymmetrical rotating oscillators Strutt diagram Parametric instability of the rotating system: a. Stability map in the S 2 case b. Influence of fixed damping (S 2 case) 34
35 Non-axisymmetrical rotating oscillators Time integration Free whirling is a linear combination of the fundamental solutions: a. Free whirling in the x-direction for Ω * = 0.3 b. Power Spectral Density 35
36 Extension to the Finite Element Method Finite Element Method (open cracks) Hypothesis: Constant spin speed Ω. Linear governing equations. Gyroscopic coupling allowed. Three-dimensional modelling for taking imperfections into account: Rotor modelled in the rotating frame R. Stator modelled in the fixed frame R (working frame). Mesh of a rotating machine 36
37 Extension to the Finite Element Method Finite Element Method Partial equilibrium equations of the rotating part in R : [1] Partial equilibrium equations of the fixed part in R: [2] Coupling between rotor and stator [3] [Combescure & Lazarus, 2008] 37
38 Extension to the Finite Element Method Finite Element Method The whole machine is a parametric oscillator with a period T: Global mesh in time domain = Relevant number of substructures in the frequency domain. 38
39 Extension to the Finite Element Method Finite Element Method Coupling between harmonic contributions considered through the rotor-stator condition [3]: Γ and Γ are chosen so that conditions could be imposed in the modal basis. Rotor-stator coupling in the time domain [3] = interface conditions between boundary nodes U j and U j. Representation of the Γ and Γ boundary in the frequency domain 39
40 Extension to the Finite Element Method Finite Element Method The modal basis Φ SS is calculated, at rest, in the frequency domain, with the component synthesis method: [Craig and Bampton, 1968] Hill s determinant for high dimensional systems Rotordynamic analysis of the whole system is performed by projected the classical equilibrium equations on the previous modal basis: 40
41 Validation on a 3D axi-symmetrical case Description Linear dynamic behaviour of «healthy» shaft: Small displacements around the equilibrium position. Balanced system rotates at a constant spin speed Ω. a. Axi-symmetrical shaft on non-isotropic bearings in the physical space b. 3D- Finite Element Modelling 41
42 Validation on a 3D axi-symmetrical case Real modes at rest Modal basis computed at rest for the four first modes: a. First beam mode in the x- direction b. First disc mode in the y-direction 42
43 Validation on a 3D axi-symmetrical case Complex modes in rotation 3D modelling: rotor in the rotating frame and stator in the fixed frame. Free whirling projected on Φ SS with j max = 0 Ω Ω a. Backward beam modes for Ω = 5 rpm Campbell diagram for flexible disc 43
44 Application to the ROTEC test bench Description Study of the free parametric oscillations of a «simple» test rig: Ω constant and negligible gyroscopic coupling. No damping consideration. The rotating beam is rectangular: Length: 0.70 m Cross-section: x m Four flexible rectangular struts: Length: 0.21 m Cross-section: x m Ω Two different configurations: Particular case: the rigid support plate is fixed. General case: the rigid support plate is free. 44
45 Application to the ROTEC test bench Description dbx dby a. Test bench general view b. Non-contacting sensors and excitation 45
46 Application to the ROTEC test bench 3D modelling (particular case) The modal basis is constituted by the two first beam modes: ω x = 20 Hz ω y = 23 Hz 2 first modes shapes at standstill 46
47 Application to the ROTEC test bench 3D modelling (particular case) 3D modelling: rotor considered in the rotating frame Free whirling projected on the Φ SS modal basis with j max = 1. Two parametric modes in the backward and forward direction. Ω First parametric mode for the spin speed Ω = 60 rpm 47
48 Application to the ROTEC test bench Experimental results (particular case) For each 1 Hz: Power Spectrum Density of the sensors responses dbx and dby in the [0-17] Hz range. ω x -2Ω ω x b. Experimental frequency spectra b. Comparison between the numerical and experimental Campbell diagrams 48
49 Application to the ROTEC test bench 3D modelling (general case) The modal basis is constituted by the four first beam modes: ω 1x = 15 Hz ω 1y = 20 Hz 2 first in-phase modes shapes at standstill 49
50 Application to the ROTEC test bench 3D modelling (general case) The modal basis is constituted by the four first beam modes: 2 first out of phase modes shapes at standstill ω 2x = 24 Hz ω 2y = 31 Hz 50
51 Application to the ROTEC test bench 3D modelling (general case) 3D modelling: rotor in the rotating frame and stator in the fixed frame Four parametric modes in the x- and y- direction: Ω j max = 2 First parametric modes in the x- direction for the spin speed Ω = 420 rpm 51
52 Application to the ROTEC test bench Experimental results (general case) For each 1 Hz: Power Spectrum Density of the sensors responses dbx and dby in the [0-14] Hz range. Ω Ω a. PSD of the db sensors for Ω = 78 Hz b. PSD of the parametric modes for Ω = 8 Hz 52
53 Taking home message Perturbation Classiclineartheory Eigenmodes Dynamical systems Constant state Periodic state Perturbation Floquet linear theory Parametric instabilities 53
54 Perturbation of periodic equilibrium Bibliography 54
Towards Rotordynamic Analysis with COMSOL Multiphysics
Towards Rotordynamic Analysis with COMSOL Multiphysics Martin Karlsson *1, and Jean-Claude Luneno 1 1 ÅF Sound & Vibration *Corresponding author: SE-169 99 Stockholm, martin.r.karlsson@afconsult.com Abstract:
More informationVibrations in Mechanical Systems
Maurice Roseau Vibrations in Mechanical Systems Analytical Methods and Applications With 112 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Contents Chapter I. Forced Vibrations
More informationVibration Dynamics and Control
Giancarlo Genta Vibration Dynamics and Control Spri ringer Contents Series Preface Preface Symbols vii ix xxi Introduction 1 I Dynamics of Linear, Time Invariant, Systems 23 1 Conservative Discrete Vibrating
More informationSAMCEF For ROTORS. Chapter 1 : Physical Aspects of rotor dynamics. This document is the property of SAMTECH S.A. MEF A, Page 1
SAMCEF For ROTORS Chapter 1 : Physical Aspects of rotor dynamics This document is the property of SAMTECH S.A. MEF 101-01-A, Page 1 Table of Contents rotor dynamics Introduction Rotating parts Gyroscopic
More informationNonlinear Rolling Element Bearings in MADYN 2000 Version 4.3
- 1 - Nonlinear Rolling Element Bearings in MADYN 2000 Version 4.3 In version 4.3 nonlinear rolling element bearings can be considered for transient analyses. The nonlinear forces are calculated with a
More informationMechanical Vibrations Chapter 6 Solution Methods for the Eigenvalue Problem
Mechanical Vibrations Chapter 6 Solution Methods for the Eigenvalue Problem Introduction Equations of dynamic equilibrium eigenvalue problem K x = ω M x The eigensolutions of this problem are written in
More informationDynamics of structures
Dynamics of structures 2.Vibrations: single degree of freedom system Arnaud Deraemaeker (aderaema@ulb.ac.be) 1 Outline of the chapter *One degree of freedom systems in real life Hypothesis Examples *Response
More informationDept.of Mechanical Engg, Defence Institute of Advanced Technology, Pune. India
Applied Mechanics and Materials Submitted: 2014-04-23 ISSN: 1662-7482, Vols. 592-594, pp 1084-1088 Revised: 2014-05-16 doi:10.4028/www.scientific.net/amm.592-594.1084 Accepted: 2014-05-19 2014 Trans Tech
More informationCOPYRIGHTED MATERIAL. Index
Index A Admissible function, 163 Amplification factor, 36 Amplitude, 1, 22 Amplitude-modulated carrier, 630 Amplitude ratio, 36 Antinodes, 612 Approximate analytical methods, 647 Assumed modes method,
More informationTable of Contents. Preface... 13
Table of Contents Preface... 13 Chapter 1. Vibrations of Continuous Elastic Solid Media... 17 1.1. Objective of the chapter... 17 1.2. Equations of motion and boundary conditions of continuous media...
More informationRefined finite element modelling for the vibration analysis of large rotating machines
Refined finite element modelling for the vibration analysis of large rotating machines Didier Combescure, Arnaud Lazarus To cite this version: Didier Combescure, Arnaud Lazarus. Refined finite element
More informationIntroduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.
Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
More informationDynamics of Rotor Systems with Clearance and Weak Pedestals in Full Contact
Paper ID No: 23 Dynamics of Rotor Systems with Clearance and Weak Pedestals in Full Contact Dr. Magnus Karlberg 1, Dr. Martin Karlsson 2, Prof. Lennart Karlsson 3 and Ass. Prof. Mats Näsström 4 1 Department
More informationUse of Full Spectrum Cascade for Rotor Rub Identification
Use of Full Spectrum Cascade for Rotor Rub Identification T. H. Patel 1, A. K. Darpe 2 Department of Mechanical Engineering, Indian Institute of Technology, Delhi 110016, India. 1 Research scholar, 2 Assistant
More informationEfficient Reduced Order Modeling of Low- to Mid-Frequency Vibration and Power Flow in Complex Structures
Efficient Reduced Order Modeling of Low- to Mid-Frequency Vibration and Power Flow in Complex Structures Yung-Chang Tan Graduate Student Research Assistant Matthew P. Castanier Assistant Research Scientist
More informationPLEASURE VESSEL VIBRATION AND NOISE FINITE ELEMENT ANALYSIS
PLEASURE VESSEL VIBRATION AND NOISE FINITE ELEMENT ANALYSIS 1 Macchiavello, Sergio *, 2 Tonelli, Angelo 1 D Appolonia S.p.A., Italy, 2 Rina Services S.p.A., Italy KEYWORDS pleasure vessel, vibration analysis,
More informationProgram System for Machine Dynamics. Abstract. Version 5.0 November 2017
Program System for Machine Dynamics Abstract Version 5.0 November 2017 Ingenieur-Büro Klement Lerchenweg 2 D 65428 Rüsselsheim Phone +49/6142/55951 hd.klement@t-online.de What is MADYN? The program system
More informationStudy of coupling between bending and torsional vibration of cracked rotor system supported by radial active magnetic bearings
Applied and Computational Mechanics 1 (2007) 427-436 Study of coupling between bending and torsional vibration of cracked rotor system supported by radial active magnetic bearings P. Ferfecki a, * a Center
More informationVIBRATION-BASED HEALTH MONITORING OF ROTATING SYSTEMS WITH GYROSCOPIC EFFECT
VIBRATION-BASED HEALTH MONITORING OF ROTATING SYSTEMS WITH GYROSCOPIC EFFECT A Thesis presented to the Faculty of California Polytechnic State University, San Luis Obispo In Partial Fulfillment of the
More informationVibro-acoustic Analysis for Noise Reduction of Electric Machines
Vibro-acoustic Analysis for Noise Reduction of Electric Machines From Flux to OptiStruct Example : Synchronous Machine Flux 2D coupling to OptiStruct Patrick LOMBARD Application Team Manager Patrick.Lombard@cedrat.com
More informationAPVC2009. Forced Vibration Analysis of the Flexible Spinning Disk-spindle System Represented by Asymmetric Finite Element Equations
Forced Vibration Analysis of the Flexible Spinning Disk-spindle System Represented by Asymmetric Finite Element Equations Kiyong Park, Gunhee Jang* and Chanhee Seo Department of Mechanical Engineering,
More informationFinite element analysis of rotating structures
Finite element analysis of rotating structures Dr. Louis Komzsik Chief Numerical Analyst Siemens PLM Software Why do rotor dynamics with FEM? Very complex structures with millions of degrees of freedom
More informationSimulation and Experimental Research on Dynamics of Low-Pressure Rotor System in Turbofan Engine
Simulation and Experimental Research on Dynamics of Low-Pressure Rotor System in Turbofan Engine Shengxiang Li 1, Chengxue Jin 2, Guang Zhao 1*, Zhiliang Xiong 1, Baopeng Xu 1 1. Collaborative Innovation
More informationSPECIAL DYNAMIC SOIL- STRUCTURE ANALYSIS PROCEDURES DEMONSTATED FOR TWO TOWER-LIKE STRUCTURES
2010/2 PAGES 1 8 RECEIVED 21. 9. 2009 ACCEPTED 20. 1. 2010 Y. KOLEKOVÁ, M. PETRONIJEVIĆ, G. SCHMID SPECIAL DYNAMIC SOIL- STRUCTURE ANALYSIS PROCEDURES DEMONSTATED FOR TWO TOWER-LIKE STRUCTURES ABSTRACT
More informationROTATING MACHINERY VIBRATION
SECOND EDITION ROTATING MACHINERY VIBRATION From Analysis to Troubleshooting MAURICE L. ADAMS, JR Case Western Reserve University Cleveland, Ohio W^ C\ CRC Press У Taylor &. Francis Group Boca Raton London
More informationAlfa-Tranzit Co., Ltd offers the new DYNAMICS R4.0 program system for analysis and design of rotor systems of high complexity
ROTORDYNAMICS OF TURBOMACHINERY Alfa-Tranzit Co., Ltd offers the new DYNAMICS R4. program system for analysis and design of rotor systems of high complexity Copyright Alfa-Tranzit Co., Ltd 2-25 e-mail
More informationDYNAMIC ANALYSIS OF ROTOR-BEARING SYSTEM FOR FLEXIBLE BEARING SUPPORT CONDITION
International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 7, July 2017, pp. 1785 1792, Article ID: IJMET_08_07_197 Available online at http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=8&itype=7
More informationANALYSIS OF HIGHRISE BUILDING STRUCTURE WITH SETBACK SUBJECT TO EARTHQUAKE GROUND MOTIONS
ANALYSIS OF HIGHRISE BUILDING SRUCURE WIH SEBACK SUBJEC O EARHQUAKE GROUND MOIONS 157 Xiaojun ZHANG 1 And John L MEEK SUMMARY he earthquake response behaviour of unframed highrise buildings with setbacks
More informationPart 1: Discrete systems
Part 1: Discrete systems Introduction Single degree of freedom oscillator Convolution integral Beat phenomenon Multiple p degree of freedom discrete systems Eigenvalue problem Modal coordinates Damping
More informationNX Nastran 10. Rotor Dynamics User s Guide
NX Nastran 10 Rotor Dynamics User s Guide Proprietary & Restricted Rights Notice 2014 Siemens Product Lifecycle Management Software Inc. All Rights Reserved. This software and related documentation are
More informationA SIMULATION STUDY OF THE ROTOR VIBRATION IN A JOURNAL BEARING
Engineering MECHANICS, Vol. 15, 2008, No. 6, p. 461 470 461 A SIMULATION STUDY OF THE ROTOR VIBRATION IN A JOURNAL BEARING JiříTůma*, Alena Bilošová**, Jiří Šimek***, Rudolf Svoboda*** The paper deals
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More informationKNIFE EDGE FLAT ROLLER
EXPERIMENT N0. 1 To Determine jumping speed of cam Equipment: Cam Analysis Machine Aim: To determine jumping speed of Cam Formulae used: Upward inertial force = Wvω 2 /g Downward force = W + Ks For good
More informationStatic & Dynamic. Analysis of Structures. Edward L.Wilson. University of California, Berkeley. Fourth Edition. Professor Emeritus of Civil Engineering
Static & Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward LWilson Professor Emeritus of Civil Engineering University of California, Berkeley Fourth Edition
More informationInternational Journal of Scientific & Engineering Research, Volume 5, Issue 7, July-2014 ISSN
ISSN 2229-5518 692 In literature, finite element formulation uses beam element or plate element for structural modelling which has a limitation on transverse displacement. Atkinson and Manrique [1] studied
More informationIntroduction to structural dynamics
Introduction to structural dynamics p n m n u n p n-1 p 3... m n-1 m 3... u n-1 u 3 k 1 c 1 u 1 u 2 k 2 m p 1 1 c 2 m2 p 2 k n c n m n u n p n m 2 p 2 u 2 m 1 p 1 u 1 Static vs dynamic analysis Static
More informationResearch Program Vibrations ENERGIFORSK Vibration Group
Vorlesungen Mechatronik im Wintersemester Research Program Vibrations ENERGIFORSK Vibration Group DIAM A Matrix Tool for Turbine and Generator Vibrations Detection, Investigation, Analysis, Mitigation
More informationRotor-dynamics Analysis Process
2001-36 Rotor-dynamics Analysis Process Mohammad A. Heidari, Ph.D. David L. Carlson Ted Yantis Technical Fellow Principal Engineer Principal Engineer The Boeing Company The Boeing Company The Boeing Company
More informationFULL SPECTRUM ANALYSIS IN JOURNAL BEARING DIAGNOSTICS
FULL SPECTRUM ANALYSIS IN JOURNAL BEARING DIAGNOSTICS Jiří Tůma and Jan Biloš VŠB-Technical University of Ostrava 17. listopadu 15, 78 33 Ostrava, Czech Republic, jiri.tuma@vsb.cz Abstract: The topic of
More informationOn The Finite Element Modeling Of Turbo Machinery Rotors In Rotor Dynamic Analysis
Proceedings of The Canadian Society for Mechanical Engineering International Congress 2018 CSME International Congress 2018 May 27-30, 2018, Toronto, On, Canada On The Finite Element Modeling Of Turbo
More informationRotor-stator contact in multiple degree of freedom systems
Rotor-stator contact in multiple degree of freedom systems A. D. Shaw 1, A. R. Champneys 2, M. I. Friswell 1 1 Swansea University, College of Engineering, UK 2 University of Bristol, Department of Engineering
More informationStochastic Dynamics of SDOF Systems (cont.).
Outline of Stochastic Dynamics of SDOF Systems (cont.). Weakly Stationary Response Processes. Equivalent White Noise Approximations. Gaussian Response Processes as Conditional Normal Distributions. Stochastic
More informationGrandstand Terraces. Experimental and Computational Modal Analysis. John N Karadelis
Grandstand Terraces. Experimental and Computational Modal Analysis. John N Karadelis INTRODUCTION Structural vibrations caused by human activities are not known to be particularly damaging or catastrophic.
More informationStructural System, Machines and Load Cases
Machine-Induced Vibrations Machine-Induced Vibrations In the following example the dynamic excitation of two rotating machines is analyzed. A time history analysis in the add-on module RF-DYNAM Pro - Forced
More informationVibration Analysis of Hollow Profiled Shafts
International Journal of Current Engineering and echnology ISSN 77-406 04 INPRESSCO. All Rights Reserved. Available at http://inpressco.com/category/ijcet Research Article Vibration Analysis of Hollow
More informationBlade Group Fatigue Life Calculation under Resonant Stresses
TEM Journal. Volume 6, Issue, Pages 73-80, ISSN 227-8309, DOI: 0.842/TEM6-25, February 207. Blade Group Fatigue Life Calculation under Resonant Stresses Zlatko Petreski, Goce Tasevski Ss. Cyril and Methodius
More informationExperimental Investigations of Whirl Speeds of a Rotor on Hydrodynamic Spiral Journal Bearings Under Flooded Lubrication
International Conference on Fluid Dynamics and Thermodynamics Technologies (FDTT ) IPCSIT vol.33() () IACSIT Press, Singapore Experimental Investigations of Whirl Speeds of a Rotor on Hydrodynamic Spiral
More informationJournal home page:
Journal home page: HTTP://WWW.SCIENCEDIRECT.COM/SCIENCE/JOURNAL/9977538 The invariant manifold approach applied to nonlinear dynamics of a rotor-bearing system European Journal of Mechanics - A/Solids,
More informationIdentification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016
Prof. Dr. Eleni Chatzi Lecture 4-09. March, 2016 Fundamentals Overview Multiple DOF Systems State-space Formulation Eigenvalue Analysis The Mode Superposition Method The effect of Damping on Structural
More informationTheory and Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory and Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 7 Instability in Rotor Systems Lecture - 2 Fluid-Film Bearings
More informationSOME INSIGHTS INTO THE SIMULTANEOUS FORWARD AND BACKWARD WHIRLING OF ROTORS
SOME INSIGHTS INTO THE SIMULTANEOUS FORWARD AND BACKWARD WHIRLING OF ROTORS Milton Dias Jr., PhD Assistant Professor of Mechanical Engineering Department of Mechanical Design State University of Campinas
More informationStability Analysis of a Hydrodynamic Journal Bearing With Rotating Herringbone Grooves
G. H. Jang e-mail: ghjang@hanyang.ac.kr J. W. Yoon PREM, Department of Mechanical Engineering, Hanyang University, Seoul, 33-79, Korea Stability Analysis of a Hydrodynamic Journal Bearing With Rotating
More informationContribution to the dynamic behaviour of bladed disks
Applied and Computational Mechanics 1 (2007) Contribution to the dynamic behaviour of bladed disks P. Lošák a, E. Malenovský a, a Institute of SolidMechanics,Mechatronics andbiomechanics, VUTBrno, Technická
More informationResearch Article Experimental and Numerical Investigations of a Dual-Shaft Test Rig with Intershaft Bearing
International Journal of Rotating Machinery Volume 7, Article ID 776, pages doi:./7/776 Research Article Experimental and Numerical Investigations of a Dual-Shaft Test Rig with Intershaft Bearing M. Guskov,,
More informationChapter 4 Analysis of a cantilever
Chapter 4 Analysis of a cantilever Before a complex structure is studied performing a seismic analysis, the behaviour of simpler ones should be fully understood. To achieve this knowledge we will start
More informationANALYSIS AND IDENTIFICATION IN ROTOR-BEARING SYSTEMS
ANALYSIS AND IDENTIFICATION IN ROTOR-BEARING SYSTEMS A Lecture Notes Developed under the Curriculum Development Scheme of Quality Improvement Programme at IIT Guwahati Sponsored by All India Council of
More informationEffects of Structural Forces on the Dynamic Performance of High Speed Rotating Impellers.
Effects of Structural Forces on the Dynamic Performance of High Speed Rotating Impellers. G Shenoy 1, B S Shenoy 1 and Raj C Thiagarajan 2 * 1 Dept. of Mechanical & Mfg. Engineering, Manipal Institute
More informationVaruvan Vadivelan. Institute of Technology LAB MANUAL. : 2013 : B.E. MECHANICAL ENGINEERING : III Year / V Semester. Regulation Branch Year & Semester
Varuvan Vadivelan Institute of Technology Dharmapuri 636 703 LAB MANUAL Regulation Branch Year & Semester : 2013 : B.E. MECHANICAL ENGINEERING : III Year / V Semester ME 6511 - DYNAMICS LABORATORY GENERAL
More informationIn this lecture you will learn the following
Module 9 : Forced Vibration with Harmonic Excitation; Undamped Systems and resonance; Viscously Damped Systems; Frequency Response Characteristics and Phase Lag; Systems with Base Excitation; Transmissibility
More informationAEROELASTICITY IN AXIAL FLOW TURBOMACHINES
von Karman Institute for Fluid Dynamics Lecture Series Programme 1998-99 AEROELASTICITY IN AXIAL FLOW TURBOMACHINES May 3-7, 1999 Rhode-Saint- Genèse Belgium STRUCTURAL DYNAMICS: BASICS OF DISK AND BLADE
More informationDynamics of Machinery
Dynamics of Machinery Two Mark Questions & Answers Varun B Page 1 Force Analysis 1. Define inertia force. Inertia force is an imaginary force, which when acts upon a rigid body, brings it to an equilibrium
More informationModal Structure Imbalance Fault Detection For Rotating Machines. Brendan Smith
Modal Structure Imbalance Fault Detection For Rotating Machines by Brendan Smith A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Control Systems Department
More informationExperimental Aerodynamics. Experimental Aerodynamics
Lecture 3: Vortex shedding and buffeting G. Dimitriadis Buffeting! All structures exposed to a wind have the tendency to vibrate.! These vibrations are normally of small amplitude and have stochastic character!
More informationDynamic behavior of turbine foundation considering full interaction among facility, structure and soil
Dynamic behavior of turbine foundation considering full interaction among facility, structure and soil Fang Ming Scholl of Civil Engineering, Harbin Institute of Technology, China Wang Tao Institute of
More informationSuppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber
Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber J.C. Ji, N. Zhang Faculty of Engineering, University of Technology, Sydney PO Box, Broadway,
More informationMULTIDIMENSIONAL DECOMPOSITION OF TIME-VARYING VIBRATION RESPONSE SIGNALS IN ROTATING MACHINERY
Mechanical Systems and Signal Processing (1997) 11(4), 577 61 MULTIDIMENSIONAL DECOMPOSITION OF TIME-VARYING VIBRATION RESPONSE SIGNALS IN ROTATING MACHINERY I. BUCHER Faculty of Mechanical Engineering,
More informationVIBRATION RESPONSE OF AN ELECTRIC GENERATOR
Research Report BVAL35-001083 Customer: TEKES/SMART VIBRATION RESPONSE OF AN ELECTRIC GENERATOR Paul Klinge, Antti Hynninen Espoo, Finland 27 December, 2001 1 (12) Title A B Work report Public research
More informationEQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION
1 EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION The course on Mechanical Vibration is an important part of the Mechanical Engineering undergraduate curriculum. It is necessary for the development
More information1872. Dynamic effect of annular flow with finite axial length on the rotor
1872. Dynamic effect of annular flow with finite axial length on the rotor De Cheng 1, Zhen-qiang Yao 2, Ya-bo Xue 3 1, 2 State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University,
More informationRobust shaft design to compensate deformation in the hub press fitting and disk clamping process of 2.5 HDDs
DOI 10.1007/s00542-016-2850-2 TECHNICAL PAPER Robust shaft design to compensate deformation in the hub press fitting and disk clamping process of 2.5 HDDs Bumcho Kim 1,2 Minho Lee 3 Gunhee Jang 3 Received:
More informationModal Analysis: What it is and is not Gerrit Visser
Modal Analysis: What it is and is not Gerrit Visser What is a Modal Analysis? What answers do we get out of it? How is it useful? What does it not tell us? In this article, we ll discuss where a modal
More informationBasics of rotordynamics 2
Basics of rotordynamics Jeffcott rotor 3 M A O a rigid rotor disk rotates at angular frequency W massless shaft acts as a spring restoring displacements disk can move only in the plane defined by axes
More informationAdvanced Vibrations. Elements of Analytical Dynamics. By: H. Ahmadian Lecture One
Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian ahmadian@iust.ac.ir Elements of Analytical Dynamics Newton's laws were formulated for a single particle Can be extended to
More informationIntroduction of Rotor Dynamics using Implicit Method in LS-DYNA
Introduction of Rotor Dynamics using Implicit Method in LS-DYNA Liping Li 1, Roger Grimes 1, Thomas Borrvall 2 1 Livermore Software Technology Corporation, Livermore, CA, USA 2 DYNAmore Nordic AB, Sweden
More informationTheory and Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory and Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 7 Instability in rotor systems Lecture - 4 Steam Whirl and
More informationAnalytic solution of simplified Cardan s shaft model
Applied and Computational Mechanics 8 (2014) 215 228 Analytic solution of simplified Cardan s shaft model M. Zají cek a,, J. Dupal a a Faculty of Applied Sciences, University of West Bohemia, Univerzitní
More informationPROJECT 2 DYNAMICS OF MACHINES 41514
PROJECT 2 DYNAMICS OF MACHINES 41514 Dynamics of Rotor-Bearing System Lateral Vibrations and Stability Threshold of Rotors Supported On Hydrodynamic Bearing and Ball Bearing. Ilmar Ferreira Santos, Prof.
More informationWORK SHEET FOR MEP311
EXPERIMENT II-1A STUDY OF PRESSURE DISTRIBUTIONS IN LUBRICATING OIL FILMS USING MICHELL TILTING PAD APPARATUS OBJECTIVE To study generation of pressure profile along and across the thick fluid film (converging,
More informationSTICK-SLIP WHIRL INTERACTION IN DRILLSTRING DYNAMICS
STICK-SLIP WHIRL INTERACTION IN DRILLSTRING DYNAMICS R. I. Leine, D. H. van Campen Department of Mechanical Engineering, Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands
More informationAA242B: MECHANICAL VIBRATIONS
AA242B: MECHANICAL VIBRATIONS 1 / 50 AA242B: MECHANICAL VIBRATIONS Undamped Vibrations of n-dof Systems These slides are based on the recommended textbook: M. Géradin and D. Rixen, Mechanical Vibrations:
More informationGT NONLINEAR DYNAMIC ANALYSIS OF A TURBOCHARGER ON FOIL-AIR BEARINGS WITH FOCUS ON STABILITY AND SELF-EXCITED VIBRATION
Proceedings of the ASME Turbo Expo 2014: Turbine Technical Conference and Exposition GT2014 June 16-20, 2014, Düsseldorf, Germany GT2014-25176 NONLINEAR DYNAMIC ANALYSIS OF A TURBOCHARGER ON FOIL-AIR BEARINGS
More informationFinite Element Vibration Analysis of a Rotating shaft System with an Open Crack by the harmonic excitation
Finite Element Vibration Analysis of a Rotating shaft System with an Open Crack by the harmonic excitation Nobuhiro NAGATA, Tsuyoshi INOUE, Yukio ISHIDA Deptartment of Mechanical Science and Engineering,
More informationFEDSM99 S-291 AXIAL ROTOR OSCILLATIONS IN CRYOGENIC FLUID MACHINERY
Proceedings of the 3 rd ASME/JSME Joint Fluids Engineering Conference 1999 ASME Fluids Engineering Division Summer Meeting July 18-23 1999, San Francisco, California FEDSM99 S-291 AXIAL ROTOR OSCILLATIONS
More informationDynamic instability of supercritical driveshafts mounted on dissipative supports effect of viscous and hysteretic internal damping
Dynamic instability of supercritical driveshafts mounted on dissipative supports effect of viscous and hysteretic internal damping Olivier Montagnier, Christian Hochard To cite this version: Olivier Montagnier,
More informationThe Effect of the Shaft Diameter and Torsional Stiffness on the Whirling Speed of the Ship Propeller Shafting System
Journal of Mechanical Design and Vibration, 2017, Vol. 5, No. 1, 37-42 Available online at http://pubs.sciepub.com/jmdv/5/1/5 Science and Education Publishing DOI:10.12691/jmdv-5-1-5 The Effect of the
More informationTheory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 2 Simpul Rotors Lecture - 2 Jeffcott Rotor Model In the
More information41514 Dynamics of Machinery
41514 Dynamics of Machinery Theory, Experiment, Phenomenology and Industrial Applications Ilmar (Iumár) Ferreira Santos 1. Course Structure 2. Objectives 3. Theoretical and Experimental Example 4. Industrial
More informationStructural Dynamics A Graduate Course in Aerospace Engineering
Structural Dynamics A Graduate Course in Aerospace Engineering By: H. Ahmadian ahmadian@iust.ac.ir The Science and Art of Structural Dynamics What do all the followings have in common? > A sport-utility
More informationPHYSICS 110A : CLASSICAL MECHANICS
PHYSICS 110A : CLASSICAL MECHANICS 1. Introduction to Dynamics motion of a mechanical system equations of motion : Newton s second law ordinary differential equations (ODEs) dynamical systems simple 2.
More informationCode_Aster. SHLL100 - Harmonic response of a bar per dynamic substructuring
Titre : SHLL100 - Réponse harmonique d'une barre par sous-[...] Date : 03/08/2011 Page : 1/5 SHLL100 - Harmonic response of a bar per dynamic substructuring Abstract: The scope of application of this test
More informationSelf-Excited Vibration
Wenjing Ding Self-Excited Vibration Theory, Paradigms, and Research Methods With 228 figures Ö Springer Contents Chapter 1 Introduction 1 1.1 Main Features of Self-Excited Vibration 1 1.1.1 Natural Vibration
More information1544. Synchronous and subsynchronous vibration under the combined effect of bearings and seals: numerical simulation and its experimental validation
1544. Synchronous and subsynchronous vibration under the combined effect of bearings and seals: numerical simulation and its experimental validation Wanfu Zhang 1, Jiangang Yang 2, Chun Li 3, Ren Dai 4,
More informationChapter 2: Rigid Bar Supported by Two Buckled Struts under Axial, Harmonic, Displacement Excitation..14
Table of Contents Chapter 1: Research Objectives and Literature Review..1 1.1 Introduction...1 1.2 Literature Review......3 1.2.1 Describing Vibration......3 1.2.2 Vibration Isolation.....6 1.2.2.1 Overview.
More informationNON-LINEAR VIBRATION. DR. Rabinarayan Sethi,
DEPT. OF MECHANICAL ENGG., IGIT Sarang, Odisha:2012 Course Material: NON-LINEAR VIBRATION PREPARED BY DR. Rabinarayan Sethi, Assistance PROFESSOR, DEPT. OF MECHANICAL ENGG., IGIT SARANG M.Tech, B.Tech
More informationRotor Dynamics. By Jaafar Alsalaet Department of Mechanical Engineering College of Engineering University of Basrah
Rotor Dynamics By Jaafar Alsalaet Department of Mechanical Engineering College of Engineering University of Basrah 1. Introduction and Definitions Rotor Dynamics is the field of science that studies the
More informationAeroelastic effects of large blade deflections for wind turbines
Aeroelastic effects of large blade deflections for wind turbines Torben J. Larsen Anders M. Hansen Risoe, National Laboratory Risoe, National Laboratory P.O. Box 49, 4 Roskilde, Denmark P.O. Box 49, 4
More informationModel of the gas journal bearing dynamics with a flexibly supported foil
Model of the gas journal bearing dynamics with a flexibly supported foil (NUM209-15) Eliza Tkacz, Zbigniew Kozanecki, Dorota Kozanecka, Jakub Lagodziski Abstract: The work is devoted to an analysis of
More informationSDLV302 Modal analysis by under-structuring: bi-- supported beam
Titre : SDLV302 Analyse modale par sous-structuration : [...] Date : 21/07/2017 age : 1/10 SDLV302 Modal analysis by under-structuring: bi-- supported beam Summary: This test validates the modal analysis
More informationThe robustness of an experimental procedure to predict the stability threshold speed in rotating machinery
The robustness of an experimental procedure to predict the stability threshold speed in rotating machinery B. Vervisch 1, D. Ghyselinck 1, M. Monte 1 K. Stockman 1, M. Loccufier 1 1 Ghent University, Department
More information