Finite element analysis of rotating structures

Size: px
Start display at page:

Download "Finite element analysis of rotating structures"

Transcription

1 Finite element analysis of rotating structures Dr. Louis Komzsik Chief Numerical Analyst Siemens PLM Software

2 Why do rotor dynamics with FEM? Very complex structures with millions of degrees of freedom Wide spread industrial applications Aviation industry: Auto industry: Propellers Aero engines Starters Drive trains Energy industry: Power generator turbines Windmills

3 Tutorial topics Theoretical foundation of rotor dynamics Critical speeds and stability analysis Dynamic response solutions A demonstration example An industrial case study Application recommendations

4 A rotating mass point z z e 3 k e 2 Ω j y y r r x m r 3 = i= 1 3 = i= 1 r e i r r ( t) e ( t) i i Ω = Ω xi + Ω y j + Ω z k i i e 1 x r ( t) = r ( Ωt) i i e ( t) = e ( Ωt) i i

5 Velocities and accelerations Velocity dr dr de de = e + r = v e + r = v dt dt dt dt i i i i i i i i i= 1 i= 1 i= 1 i= 1 Acceleration Special derivatives Acceleration dv dv de de d e = dt dt dt dt dt i i i i ei vi vi ri 2 i = 1 i = 1 i = 1 i = 1 de a = a + v + r dei ( t) = Ω e dt i i 2 i i 2 i= 1 dt i= 1 dt i 2 d ei 2 d e dω de = ei + Ω dt dt dt dω a = a + 2 Ω v + r + Ω ( Ω r) dt i

6 Equilibrium of forces Inertial force Fin = ma Force in rotating system F = ma Centrifugal force F = mω ( Ω r) cf Coriolis and Eotvos force Euler force Force equilibrium in rotating system F = 2mΩ v C dω FE = m r dt F = Fin Fcf FC FE

7 Rotating mass particle of a flexible structure Ω = 0i + 0 j + Ωk z = z i j e 1 y y r r Ωt x x u m ux u = uy u z cos Ωt sin Ωt 0 H = sin t cos t 0 Ω Ω r = [ H] r + [ H] u i j k Ω e1 = 0 0 Ω = cos Ωt sin Ωt 0 deɺ 1 dt

8 Analysis of flexible rotating structure Fundamental problem of calculus of variations Euler-Lagrange differential equation solution t 1 I = f ( uɺ, u, t) dt = extremum t 0 d f f = 0 dt uɺ u Hamilton s principle of conservative systems t t 1 0 Ldt = extremum The Lagrangian Lagrange s equation of motion L = E ( uɺ, u, t) E ( u) k d E E dt u ɺ u u E k k p + = p 0

9 Kinetic energy of mass particle Kinetic energy Ek 1 T = mrɺ rɺ 2 Mass particle velocity Rotation matrix derivative Auxiliary matrices Mass particle kinetic energy rɺ = Hɺ ( r + u) + Huɺ sin Ωt cos Ωt 0 Hɺ = Ω cos t sin t 0 Ω Ω = ΩH T T J = H H = 0 1 0, P = H H = m Ek = Ω r Jr + Ω r Ju + Ω u Ju + Ω uɺ P r + Ω u Puɺ + uɺ Iuɺ 2 2 T 2 T 2 T T T T T ( )

10 Kinetic energy based equation of motion Displacement derivative Velocity and time derivative Lagrange s equation u d dt E k uɺ = m Ω Jr + Ω Ju + ΩPuɺ E k 2 2 T ( ) T = m( Ω P uɺ + Iuɺɺ ) d Ek Ek dt u = ɺ u m Ω P uɺ + Iuɺɺ Ω Jr Ω Ju + ΩP uɺ T 2 2 T ( ) Equation of motion miuɺɺ + mωp uɺ Ω mju Ω mjr = T [ ] ɺɺ [ ] ɺ [ ] m u + 2 Ω c u Ω z u = { f c }

11 Non-conservative systems Full Lagrange s equation d E E dd dw dt uɺ u u duɺ du E k k p + + = 0 Elastic potential energy Dissipative energy E p D D W m u E p Work potential W

12 Elastic structure equilibrium equation Full Lagrange s equation Potential energy Stiffness matrix Geometric stiffness Damping matrices External forces d E k E E k p dd dw + + = ; i = 1,2,... n dt uɺ i ui ui duɺ i dui de p, E T { } [ B] [ E][ B]{ u} dv [ K ]{ u} d u = = de { } T ( σ ε ) E = 1 2 ( dv + σ ε dv ) = E + E T p 0 0 p, G p, E [ B '][ S ][ B' ]{ u} dv [ K ]{ u} p, G 0 d u = = dd d u = + { ɺ} dw d u = { } [ D]{ u ɺ } [ K ]{ u} { F} B G

13 Equation of motion in rotating system Second order, full, ordinary, non-homogeneous system M uɺɺ t D C uɺ t K Z K K u t F t 2 ( ) + ( + 2 Ω ) ( ) + ( Ω ( G ) + Ω B ) ( ) = ( ) Standard structural matrices (symmetric, pos. def.) M, D, K Geometric stiffness (centrifugal stiffening, spd) Centrifugal softening (symmetric, positive definite) Circulatory matrix (skew-symmetric) Gyroscopic (Coriolis) matrix (skew-symmetric) Active load vector K G Z K B C F

14 Centrifugal softening concept Particle at the end of flexible rod (only axial displacement) K r m u F c Small displacement (linear) solution Ku = Fc = mrω 2 Large displacement (nonlinear) solution 2 Ku = m( r + u) Ω Large displacement adjustment ( K Ω m) u = mrω 2 2 Static equilibrium with centrifugal softening 2 ( K Ω Z) u = Fc

15 Centrifugal stiffening concept F b Particle at the end of flexible beam (bending deformation) r u K b F c Moment equilibrium Force equilibrium Linear formulation Centrifugal stiffening matrix Fb r Fcu = K bur F r Fc ( K b + ) u = F r c Fb u = K bu F r c 2 mrω = = mω = Ω r b 2 2 K G

16 Equation of motion in fixed system Equations of motion M uɺɺ ( t) + ( D + Ω C) uɺ ( t) + ( K + Ω K ) u( t) = F ( t) No centrifugal (softening or stiffening) effect present Change in coefficient and content of gyroscopic matrix No Coriolis, only gyroscopic terms related to rotational DOF B Rotating system solution may be converted to fixed for visualization

17 θ z σ Coupling flexible rotating and zɶ = Bβ z stationary structures yɶ y r y ψ r x ϕ Ωt xɶ x θ w v ψ u ϕ ρ Aα r m u ϕ { ρ} = v ;{ α} = ψ w θ ϕ σ x β = ψ σ = σ y θ σ z { } ;{ } { u} σ β = ρ α cos Ωt sin Ωt 0 H = sin Ωt cos Ωt { } { } { } r = σ + [ B]{ β} + [ H]{ r} + [ H] ρ + [ H][ A]{ α}

18 Time dependent coupled matrices Mass matrix M( Ω t) = M + M sin( Ω t) + M cos( Ω t) + M sin(2 Ω t) + M cos(2 Ωt ) Coriolis matrix C( Ω t) = C + C sin( Ω t) + C cos( Ω t) + C sin(2 Ω t) + C cos(2 Ωt) Centrifugal matrix Z( Ω t) = Z + Z sin( Ω t) + Z cos( Ω t) + Z sin(2 Ω t) + Z cos(2 Ωt)

19 Frequency domain solutions Harmonic solution iωt u( t) = e u( ω) ( ) i ω t F t = e F ( ω ) Frequency domain problem 2 ( ω M + iωd + K) u( ω) = F( ω) Time dependent matrices M ( Ω t ) = M + M ( Ω t) D ( Ω t ) = D + 2 Ω C ( Ω t ) K Ω t = K Ω Z Ω t K + Ω K 2 ( ) ( ( ) G ) B Free vibrations problem λ M + λd + K ϕ = λ = α + iω 2 ( ) 0;

20 Next topic Theoretical foundation of rotor dynamics Critical speeds and stability analysis Dynamic response solutions A demonstration example An industrial case study Application recommendations

21 Critical speed analysis Frequency of a mode (in RPM) f ( Ω ) = ω( Ω) / (2 π ) The critical speeds are found by intersecting the mode lines with the np lines where frequency is the nth multiple of rotor speed f ( Ω ) = np In rotating system intersection with - 0P indicates forward whirl - 2P indicates backward whirl 0 P : f = 0 2 P : f = 2* Ω In fixed system intersection with - 1P line indicates both whirls = negative slope - backward = positive slope forward 1 P : f = Ω Zero slope is indicative of a linear motion in both systems

22 Whirl directions Complex eigenvector ϕ( Ω ) = ϕ + ϕ i Re Im The mode shape whirl is w = ϕ ϕ = w i + w j + w k Re Im x y z Forward Backward Linear motion w z > w z < 0 0 w < ε z

23 Stability analysis Complex eigenvalue real part α ( Ω) The damping ratio Modal convention by 2 g ( Ω ) = 2 α ω The mode is in the Stable region Unstable region g ( Ω ) < 0 g ( Ω ) > 0

24 Undamped rotating mass particle Equation of motion in time domain problem m 0 0 m k 0 m 0 0 m m 0 0 k 0 m 2 { uɺɺ } + 2Ω { uɺ } + Ω { u} = { p} y Frequency domain free vibrations k m x m 0 0 m k 0 m m + Ω m 0 + Ω = 0 k 0 m λ 2 λ 2 ϕ { } { } z k Eigenvalue solution is purely imaginary λ ( Ω ) = 0 ± ( Ω ± 1)i

25 Critical speeds of undamped rotating mass particle Rotating system Campbell diagram Critical speeds - 0P at 1 Hz : forward whirl - 2P at 1 Hz: backward whirl Stability: Unconditional since g = 0

26 Damped rotating mass particle problem Equation of motion in time domain m 0 d 0 0 m k 0 m 0 0 m 0 d m 0 0 k 0 m 2 { uɺɺ } + + 2Ω { uɺ } + Ω { u} = { p} y Frequency domain free vibrations + + = 0 m 2Ωm d 0 k Ω m 2 2 m 0 d 2Ωm k Ω m 0 λ λ ϕ 2 Eigenvalue solution has real part λ ( Ω ) = ± α ( Ω ) ± ω ( Ω )i { } { 0} z d m k x

27 Damped rotating mass particle stability Damping ratio plot Mode shapes Forward whirl: 1 unstable above 1 Hz Backward whirl: 2 stable throughout

28 Next topic Theoretical foundation of rotor dynamics Critical speeds and stability analysis Dynamic response solutions A demonstration example An industrial case study Application recommendations

29 Dynamic response analyses Frequency response Synchronous analysis: excitation frequency is a function of rotor speed load is scaling with speed: simulate mass unbalance Asynchronous analysis: constant rotor speed, excitation frequency is changing load does not scale with speed: simulate gravity load Transient response Synchronous analysis: excitation time function is in synchrony with rotor speed simulate the start-up or wind-down process Asynchronous analysis: excitation time function is changing for constant rotor speed simulate behavior at the operational point of the rotor

30 Damped rotating mass particle frequency response Forward rotating force load exciting forward whirl: mode 1 Asynchronous analysis: dotted Synchronous analysis: solid Peaks increase due to the decreasing damping of mode 1 Fixed reference system result

31 Damped rotating mass particle frequency response Backward rotating force load exciting backward whirl: mode 2 Asynchronous and synchronous analysis both dotted Peaks decrease due to the increasing damping of mode 2 Fixed reference system result

32 Damped rotating mass particle transient response Response to a sweep load Sine excitation in x-direction Cosine excitation in y-direction Synchronous analysis Resonance at critical speed: 1 Hz Fixed reference system results

33 Damped rotating mass particle transient response Response to a sweep load Sine excitation in x-direction Cosine excitation in y-direction Asynchronous analysis Resonance at critical speed: 1 Hz Fixed reference system results

34 Next topic Theoretical foundation of rotor dynamics Critical speeds and stability analysis Dynamic response solutions A demonstration example An industrial case study Application recommendations

35 A demonstration example Long cylindrical shaft Heavy central hub Spring and damper supports Ω Rotor speed: Ω = RPM Solutions sought: Critical speeds and mode shapes Mass unbalance and start-up

36 Critical speed analysis Modes 1,2 Symmetric bending forward-backward whirl Mode 3 - Torsion mode Mode 4 Asymmetric bending backward whirl Mode 5 - Asymmetric bending forward whirl Mode 6 Axial mode Mode 5 Forward Whirl Mode 4 Backward Whirl 1P Mode 6 Axial Critical speeds: Independent: 1,460 RPM Backward: 8,420 RPM Forward: 17,800 RPM Mode 3 Torsion Mode 1, 2 Forward and backward 1460 RPM Critical Speed Backward & Forward 8420 RPM Critical Speed Backward RPM Critical Speed Forward

37 0 RPM Symmetric Hz

38 24,000 RPM Symmetric forward whirl

39 24,000 RPM Symmetric backward whirl

40 0 RPM Asymmetric Hz

41 24,000 RPM Asymmetric forward whirl

42 24,000 RPM Asymmetric backward whirl

43 Mass unbalance simulation Force load on central hub Lateral and vertical direction Phased with positive rotation F me π π t 2 2 x( Ω ) = 4 Ω sin(2 Ω ) F me π π t π 2 2 y ( Ω ) = 4 Ω sin(2 Ω + / 2) Unbalance: mass = m kg eccentricity = e mm Critical Speed 1460 RPM (24 Hz) Critical Speed RPM (296 Hz) Frequency response analysis RPM Result: Forward whirl modes excited Due to force positive rotation Rotating Rotor Non-Rotating Rotor

44 Rotor start-up simulation Force load and eccentricity as before t=.3 sec Critical Speed at 24 Hz t = 3.0 sec Critical Speed at 296 Hz Transient response analysis 0-4 sec Result: Forward whirl modes excited Short dwell time

45 Next topic Theoretical foundation of rotor dynamics Critical speeds and stability analysis Dynamic response solutions A demonstration example An industrial case study Application recommendations

46 Industrial case study Turbine wheel with 30 blades Range: 0 25,000 RPM Number of steps: 100 Number of nodes: 45,350 Number of elements: 24,199 Degrees of freedom: 135,035

47 Critical speed analysis 16,969 RPM Critical Speed Backward whirl 1,389 RPM Critical Speed Backward whirl 4,664 RPM Critical Speed Backward whirl 4,085 RPM Critical Speed Forward whirl

48 Rotor disk tilting modes Mode shape at 20,000 RPM Real Imaginary

49 Synchronous circumferential bending Real Imaginary Mode shape at 20,000 RPM

50 Asynchronous circumferential bending Real Mode shape at 20,000 RPM Imaginary

51 Shaft symmetric bending Real Mode shape at 20,000 RPM Imaginary

52 Next topic Theoretical foundation of rotor dynamics Critical speeds and stability analysis Dynamic response solutions A demonstration example An industrial case study Application recommendations

53 Coupled solution strategies The fully coupled equations of motion between the support and rotor contain periodic terms Hence evaluation of the solution at multiple azimuth angles is required and makes the solution very expensive Coupled solution strategies: execute at a discrete azimuth angle for a rotor speed sweep with specified speed discretization execute a full circle azimuth sweep with specified angular discretization at a given fixed rotor speed

54 Reference system selection The fixed (non-rotating) reference system is an inertial system while the rotating coordinate system is not The equations of motion are different in the two reference systems, resulting in different but compatible solutions for certain models The content of some of the participating matrices is also different between the two reference systems Rotating reference frame is the preferred analysis solution for general finite element models with large radius components It is also possible to solve the equation of motion in one reference frame and convert the solution to the other one Presenting results in the fixed reference system is more intuitive for the engineer

55 Modeling approach Models built from solid elements must be analyzed in the rotating system, since the nodal rotations are not present in the model, but the rotational matrices in the fixed system contain only terms related to the nodal rotations This can be overcome by covering solid model surfaces with thin shells, but this approach has challenges in establishing the proper characteristics Models built from shell elements may be analyzed in both systems, but rotational system is still preferred, because the shell normal DOFs are either not present or approximated

56 The effect of rotor symmetry Symmetric rotors Turbine engines I x = I y Unsymmetric rotors Automobile crankshaft I x I y Fixed reference system formulation Rotating reference system formulation C = f ( I ); Z = 0 z C = f ( I, I, I ); Z = f ( I, I, I ) z x y z x y Symmetric rotor may be analyzed with fixed zero azimuth angle based time independent coupling terms Unsymmetric rotor must be analyzed with azimuth angle based time dependent coupling terms

57 The effect of support symmetry Symmetric support Classical bearings via spring and dampers k = k ; b = b xy yx xy yx Example: tower of a vertical axis wind turbine Unsymmetric support Fluid film (journal) bearings k k ; b b xy yx xy yx Example: support of horizontal axis turbines Symmetric support may be analyzed with fixed zero azimuth angle based time independent coupling terms Unsymmetric support must be analyzed with azimuth angle based time dependent coupling terms

58 Advanced applications of rotordynamics Simulate the rotational loads occurring during aircraft maneuvers Simulate the blade loss and rubbing phenomena in aero engines with nonlinear rotor dynamic solutions Simulate the dynamic behavior of wind turbines by real-time coupling with unsteady airflow solution Simulate the operation of fossil turbines by co-simulating with heat-fluid flow solution

59 References

60 Conclusion Thank you for your interest in the topic and attention! Solution results were produced by NX Nastran NX is a registered trademark of Siemens PLM Software Inc NASTRAN is a registered trademark of NASA

SAMCEF 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 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 information

NX Nastran 10. Rotor Dynamics User s Guide

NX 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 information

Towards Rotordynamic Analysis with COMSOL Multiphysics

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 information

Structural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.

Structural 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 information

Structural Dynamics Lecture 2. Outline of Lecture 2. Single-Degree-of-Freedom Systems (cont.)

Structural Dynamics Lecture 2. Outline of Lecture 2. Single-Degree-of-Freedom Systems (cont.) Outline of Single-Degree-of-Freedom Systems (cont.) Linear Viscous Damped Eigenvibrations. Logarithmic decrement. Response to Harmonic and Periodic Loads. 1 Single-Degreee-of-Freedom Systems (cont.). Linear

More information

Nonlinear Rolling Element Bearings in MADYN 2000 Version 4.3

Nonlinear 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 information

Effects 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. 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 information

Advanced Vibrations. Elements of Analytical Dynamics. By: H. Ahmadian Lecture One

Advanced 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 information

Lifecycle simulation in the automotive industry

Lifecycle simulation in the automotive industry Lifecycle simulation in the automotive industry White Paper Advances in dynamic response with NX Nastran Accelerated product development cycles are just one symptom of an extremely competitive automotive

More information

COPYRIGHTED MATERIAL. Index

COPYRIGHTED 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 information

Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I

Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix

More information

Dynamic Analysis of An 1150 MW Turbine Generator

Dynamic Analysis of An 1150 MW Turbine Generator Dyrobes Rotordynamics Software https://dyrobes.com 1 PWR2005-50142 Abract Dynamic Analysis of An 1150 MW Turbine Generator Edgar J. Gunter Fellow ASME RODYN Vibration Inc Charlottesville, Va. 22903 DrGunter@aol.com

More information

Dynamics of Rotor Systems with Clearance and Weak Pedestals in Full Contact

Dynamics 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 information

Lecture 27: Structural Dynamics - Beams.

Lecture 27: Structural Dynamics - Beams. Chapter #16: Structural Dynamics and Time Dependent Heat Transfer. Lectures #1-6 have discussed only steady systems. There has been no time dependence in any problems. We will investigate beam dynamics

More information

Finite element method based analysis and modeling in rotordynamics

Finite element method based analysis and modeling in rotordynamics Finite element method based analysis and modeling in rotordynamics A thesis submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of

More information

Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.

Introduction 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 information

Part 1: Discrete systems

Part 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 information

Dynamic Model of a Badminton Stroke

Dynamic Model of a Badminton Stroke ISEA 28 CONFERENCE Dynamic Model of a Badminton Stroke M. Kwan* and J. Rasmussen Department of Mechanical Engineering, Aalborg University, 922 Aalborg East, Denmark Phone: +45 994 9317 / Fax: +45 9815

More information

Vibration Dynamics and Control

Vibration 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 information

Dept.of Mechanical Engg, Defence Institute of Advanced Technology, Pune. India

Dept.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 information

Theory 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 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 information

Introduction to Vibration. Mike Brennan UNESP, Ilha Solteira São Paulo Brazil

Introduction to Vibration. Mike Brennan UNESP, Ilha Solteira São Paulo Brazil Introduction to Vibration Mike Brennan UNESP, Ilha Solteira São Paulo Brazil Vibration Most vibrations are undesirable, but there are many instances where vibrations are useful Ultrasonic (very high

More information

VIBRATION ANALYSIS OF TIE-ROD/TIE-BOLT ROTORS USING FEM

VIBRATION ANALYSIS OF TIE-ROD/TIE-BOLT ROTORS USING FEM VIBRATION ANALYSIS OF TIE-ROD/TIE-BOLT ROTORS USING FEM J. E. Jam, F. Meisami Composite Materials and Technology Center Tehran, IRAN jejaam@gmail.com N. G. Nia Iran Polymer & Petrochemical Institute, Tehran,

More information

Theory 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 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 information

Modeling and Performance Analysis of a Flywheel Energy Storage System Prince Owusu-Ansah, 1, Hu Yefa, 1, Philip Agyeman, 1 Adam Misbawu 2

Modeling and Performance Analysis of a Flywheel Energy Storage System Prince Owusu-Ansah, 1, Hu Yefa, 1, Philip Agyeman, 1 Adam Misbawu 2 International Conference on Electromechanical Control Technology and Transportation (ICECTT 2015) Modeling and Performance Analysis of a Flywheel Energy Storage System Prince Owusu-Ansah, 1, Hu Yefa, 1,

More information

Dynamics of Machinery

Dynamics 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 information

Basics of rotordynamics 2

Basics 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 information

Rotor 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 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 information

Robust shaft design to compensate deformation in the hub press fitting and disk clamping process of 2.5 HDDs

Robust 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 information

A Study on the Tube of Integral Propeller Shaft for the Rear-wheel Drive Automobile Using Carbon Composite Fiber

A Study on the Tube of Integral Propeller Shaft for the Rear-wheel Drive Automobile Using Carbon Composite Fiber A Study on the Tube of Integral Propeller Shaft for the Rear-wheel Drive Automobile Using Carbon Composite Fiber Kibong Han Mechatronics Department, Jungwon University, 85 Munmu-ro, Goesan-gun, South Korea.

More information

ANALYSIS AND IDENTIFICATION IN ROTOR-BEARING SYSTEMS

ANALYSIS 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 information

PLEASURE VESSEL VIBRATION AND NOISE FINITE ELEMENT ANALYSIS

PLEASURE 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 information

Introduction of Rotor Dynamics using Implicit Method in LS-DYNA

Introduction 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 information

Dr.Vinod Hosur, Professor, Civil Engg.Dept., Gogte Institute of Technology, Belgaum

Dr.Vinod Hosur, Professor, Civil Engg.Dept., Gogte Institute of Technology, Belgaum STRUCTURAL DYNAMICS Dr.Vinod Hosur, Professor, Civil Engg.Dept., Gogte Institute of Technology, Belgaum Overview of Structural Dynamics Structure Members, joints, strength, stiffness, ductility Structure

More information

3D Finite Element Modeling and Vibration Analysis of Gas Turbine Structural Elements

3D Finite Element Modeling and Vibration Analysis of Gas Turbine Structural Elements 3D Finite Element Modeling and Vibration Analysis of Gas Turbine Structural Elements Alexey I. Borovkov Igor A. Artamonov Computational Mechanics Laboratory, St.Petersburg State Polytechnical University,

More information

202 Index. failure, 26 field equation, 122 force, 1

202 Index. failure, 26 field equation, 122 force, 1 Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic

More information

Name: Fall 2014 CLOSED BOOK

Name: Fall 2014 CLOSED BOOK Name: Fall 2014 1. Rod AB with weight W = 40 lb is pinned at A to a vertical axle which rotates with constant angular velocity ω =15 rad/s. The rod position is maintained by a horizontal wire BC. Determine

More information

a) Find the equation of motion of the system and write it in matrix form.

a) Find the equation of motion of the system and write it in matrix form. .003 Engineering Dynamics Problem Set Problem : Torsional Oscillator Two disks of radius r and r and mass m and m are mounted in series with steel shafts. The shaft between the base and m has length L

More information

Dynamic Analysis of Pelton Turbine and Assembly

Dynamic Analysis of Pelton Turbine and Assembly Dynamic Analysis of Pelton Turbine and Assembly Aman Rajak, Prateek Shrestha, Manoj Rijal, Bishal Pudasaini, Mahesh Chandra Luintel Department of Mechanical Engineering, Central Campus, Pulchowk, Institute

More information

Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I

Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix

More information

Control of Earthquake Induced Vibrations in Asymmetric Buildings Using Passive Damping

Control of Earthquake Induced Vibrations in Asymmetric Buildings Using Passive Damping Control of Earthquake Induced Vibrations in Asymmetric Buildings Using Passive Damping Rakesh K. Goel, California Polytechnic State University, San Luis Obispo Abstract This paper summarizes the results

More information

Engineering Science OUTCOME 2 - TUTORIAL 3 FREE VIBRATIONS

Engineering Science OUTCOME 2 - TUTORIAL 3 FREE VIBRATIONS Unit 2: Unit code: QCF Level: 4 Credit value: 5 Engineering Science L/60/404 OUTCOME 2 - TUTORIAL 3 FREE VIBRATIONS UNIT CONTENT OUTCOME 2 Be able to determine the behavioural characteristics of elements

More information

AA242B: MECHANICAL VIBRATIONS

AA242B: 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 information

Some effects of large blade deflections on aeroelastic stability

Some effects of large blade deflections on aeroelastic stability 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 5-8 January 29, Orlando, Florida AIAA 29-839 Some effects of large blade deflections on aeroelastic stability

More information

On The Finite Element Modeling Of Turbo Machinery Rotors In Rotor Dynamic Analysis

On 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 information

Program System for Machine Dynamics. Abstract. Version 5.0 November 2017

Program 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 information

This equation of motion may be solved either by differential equation method or by graphical method as discussed below:

This equation of motion may be solved either by differential equation method or by graphical method as discussed below: 2.15. Frequency of Under Damped Forced Vibrations Consider a system consisting of spring, mass and damper as shown in Fig. 22. Let the system is acted upon by an external periodic (i.e. simple harmonic)

More information

Introduction to Vibration. Professor Mike Brennan

Introduction to Vibration. Professor Mike Brennan Introduction to Vibration Professor Mie Brennan Introduction to Vibration Nature of vibration of mechanical systems Free and forced vibrations Frequency response functions Fundamentals For free vibration

More information

Theory & Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati

Theory & Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Theory & Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 8 Balancing Lecture - 1 Introduce To Rigid Rotor Balancing Till

More information

JEPPIAAR ENGINEERING COLLEGE

JEPPIAAR ENGINEERING COLLEGE JEPPIAAR ENGINEERING COLLEGE Jeppiaar Nagar, Rajiv Gandhi Salai 600 119 DEPARTMENT OFMECHANICAL ENGINEERING QUESTION BANK VI SEMESTER ME6603 FINITE ELEMENT ANALYSIS Regulation 013 SUBJECT YEAR /SEM: III

More information

EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION

EQUIVALENT 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 information

2C9 Design for seismic and climate changes. Jiří Máca

2C9 Design for seismic and climate changes. Jiří Máca 2C9 Design for seismic and climate changes Jiří Máca List of lectures 1. Elements of seismology and seismicity I 2. Elements of seismology and seismicity II 3. Dynamic analysis of single-degree-of-freedom

More information

Complex modes analysis for powertrain and driveline applications

Complex modes analysis for powertrain and driveline applications Complex modes analysis for powertrain and driveline applications T. Parikyan 1 1 AVL List GmbH, Advanced Simulation Technologies Hans-List-Platz 1, A-8020, Graz, Austria e-mail: tigran.parikyan@avl.com

More information

Study of coupling between bending and torsional vibration of cracked rotor system supported by radial active magnetic bearings

Study 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 information

Theory 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 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 information

Use of Full Spectrum Cascade for Rotor Rub Identification

Use 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 information

Simulation 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 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 information

Critical Speed Analysis of Offset Jeffcott Rotor Using English and Metric Units

Critical Speed Analysis of Offset Jeffcott Rotor Using English and Metric Units Dyrobes Rotordynamics Software https://dyrobes.com Critical Speed Analysis of Offset Jeffcott Rotor Using English and Metric Units E. J. Gunter,PhD Fellow ASME February,2004 RODYN Vibration Inc. 1932 Arlington

More information

Lecture 9: Harmonic Loads (Con t)

Lecture 9: Harmonic Loads (Con t) Lecture 9: Harmonic Loads (Con t) Reading materials: Sections 3.4, 3.5, 3.6 and 3.7 1. Resonance The dynamic load magnification factor (DLF) The peak dynamic magnification occurs near r=1 for small damping

More information

ROTATING MACHINERY VIBRATION

ROTATING 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 information

Introduction to Mechanical Vibration

Introduction to Mechanical Vibration 2103433 Introduction to Mechanical Vibration Nopdanai Ajavakom (NAV) 1 Course Topics Introduction to Vibration What is vibration? Basic concepts of vibration Modeling Linearization Single-Degree-of-Freedom

More information

Finite Element Analysis Lecture 1. Dr./ Ahmed Nagib

Finite Element Analysis Lecture 1. Dr./ Ahmed Nagib Finite Element Analysis Lecture 1 Dr./ Ahmed Nagib April 30, 2016 Research and Development Mathematical Model Mathematical Model Mathematical Model Finite Element Analysis The linear equation of motion

More information

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016

Identification 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 information

Structural Dynamics A Graduate Course in Aerospace Engineering

Structural 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 information

WORK SHEET FOR MEP311

WORK 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 information

Research Program Vibrations ENERGIFORSK Vibration Group

Research 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 information

DYNAMICS OF MACHINERY 41514

DYNAMICS OF MACHINERY 41514 DYNAMICS OF MACHINERY 454 PROJECT : Theoretical and Experimental Modal Analysis and Validation of Mathematical Models in Multibody Dynamics Holistic Overview of the Project Steps & Their Conceptual Links

More information

Rotor-dynamics Analysis Process

Rotor-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 information

Alfa-Tranzit Co., Ltd offers the new DYNAMICS R4.0 program system for analysis and design of rotor systems of high complexity

Alfa-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 information

PROJECT 1 DYNAMICS OF MACHINES 41514

PROJECT 1 DYNAMICS OF MACHINES 41514 PROJECT DYNAMICS OF MACHINES 454 Theoretical and Experimental Modal Analysis and Validation of Mathematical Models in Multibody Dynamics Ilmar Ferreira Santos, Professor Dr.-Ing., Dr.Techn., Livre-Docente

More information

Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter.

Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter. Chapter 3 Mechanical Systems A. Bazoune 3.1 INRODUCION Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter. 3. MECHANICAL ELEMENS Any mechanical system consists

More information

Methodology of modelling the flexural-torsional vibrations in transient states of the rotating power transmission systems

Methodology of modelling the flexural-torsional vibrations in transient states of the rotating power transmission systems Methodology of modelling the flexural-torsional vibrations in transient states of the rotating power transmission systems Tomasz Matyja Faculty of Transport Department of Logistics and Aviation Technologies

More information

28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)

28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) 28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.

More information

EFFECT OF HYDRODYNAMIC THRUST BEARINGS ON ROTORDYNAMICS

EFFECT OF HYDRODYNAMIC THRUST BEARINGS ON ROTORDYNAMICS The 12th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery Honolulu, Hawaii, February 17-22, 2008 ISROMAC12-2008-20076 EFFECT OF HYDRODYNAMIC THRUST BEARINGS ON ROTORDYNAMICS

More information

VIBRATION-BASED HEALTH MONITORING OF ROTATING SYSTEMS WITH GYROSCOPIC EFFECT

VIBRATION-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 information

Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6.2, 6.3

Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6.2, 6.3 M9 Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6., 6.3 A shaft is a structural member which is long and slender and subject to a torque (moment) acting about its long axis. We

More information

Nonlinear Dynamic Analysis of a Hydrodynamic Journal Bearing Considering the Effect of a Rotating or Stationary Herringbone Groove

Nonlinear Dynamic Analysis of a Hydrodynamic Journal Bearing Considering the Effect of a Rotating or Stationary Herringbone Groove G. H. Jang e-mail: ghjang@hanyang.ac.kr J. W. Yoon PREM, Department of Mechanical Engineering, Hanyang University, Seoul, 133-791, Korea Nonlinear Dynamic Analysis of a Hydrodynamic Journal Bearing Considering

More information

D && 9.0 DYNAMIC ANALYSIS

D && 9.0 DYNAMIC ANALYSIS 9.0 DYNAMIC ANALYSIS Introduction When a structure has a loading which varies with time, it is reasonable to assume its response will also vary with time. In such cases, a dynamic analysis may have to

More information

NON-LINEAR ROTORDYNAMICS: COMPUTATIONAL STRATEGIES

NON-LINEAR ROTORDYNAMICS: COMPUTATIONAL STRATEGIES The 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery Honolulu, Hawaii, February 1-14, NON-LINEAR ROTORDNAMICS: COMPUTATIONAL STRATEGIES Tom J. Chalko Head of Rotordynamic

More information

Investigation of Coupled Lateral and Torsional Vibrations of a Cracked Rotor Under Radial Load

Investigation of Coupled Lateral and Torsional Vibrations of a Cracked Rotor Under Radial Load NOMENCLATURE Investigation of Coupled Lateral and Torsional Vibrations of a Cracked Rotor Under Radial Load Xi Wu, Assistant Professor Jim Meagher, Professor Clinton Judd, Graduate Student Department of

More information

Dynamics of assembled structures of rotor systems of aviation gas turbine engines of type two-rotor

Dynamics of assembled structures of rotor systems of aviation gas turbine engines of type two-rotor Dynamics of assembled structures of rotor systems of aviation gas turbine engines of type two-rotor Anatoly А. Pykhalov 1, Mikhail А. Dudaev 2, Mikhail Ye. Kolotnikov 3, Paul V. Makarov 4 1 Irkutsk State

More information

EFFECTIVE MODAL MASS & MODAL PARTICIPATION FACTORS Revision F

EFFECTIVE MODAL MASS & MODAL PARTICIPATION FACTORS Revision F EFFECTIVE MODA MASS & MODA PARTICIPATION FACTORS Revision F By Tom Irvine Email: tomirvine@aol.com March 9, 1 Introduction The effective modal mass provides a method for judging the significance of a vibration

More information

Structural 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 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 information

EN40: Dynamics and Vibrations. Final Examination Wed May : 2pm-5pm

EN40: Dynamics and Vibrations. Final Examination Wed May : 2pm-5pm EN40: Dynamics and Vibrations Final Examination Wed May 10 017: pm-5pm School of Engineering Brown University NAME: General Instructions No collaboration of any kind is permitted on this examination. You

More information

Back Matter Index The McGraw Hill Companies, 2004

Back Matter Index The McGraw Hill Companies, 2004 INDEX A Absolute viscosity, 294 Active zone, 468 Adjoint, 452 Admissible functions, 132 Air, 294 ALGOR, 12 Amplitude, 389, 391 Amplitude ratio, 396 ANSYS, 12 Applications fluid mechanics, 293 326. See

More information

Structural Matrices in MDOF Systems

Structural Matrices in MDOF Systems in MDOF Systems http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 9, 2016 Outline Additional Static Condensation

More information

Modal Analysis: What it is and is not Gerrit Visser

Modal 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 information

Table of Contents. Preface... 13

Table 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 information

PROJECT 2 DYNAMICS OF MACHINES 41514

PROJECT 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 information

A Guide to linear dynamic analysis with Damping

A Guide to linear dynamic analysis with Damping A Guide to linear dynamic analysis with Damping This guide starts from the applications of linear dynamic response and its role in FEA simulation. Fundamental concepts and principles will be introduced

More information

Structural System, Machines and Load Cases

Structural 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 information

Vortex-induced vibration of a slender single-span cylinder

Vortex-induced vibration of a slender single-span cylinder Vortex-induced vibration of a slender single-span cylinder N. Oikou Delft University of Technology, the Netherlands The goal of this paper is to study the vortex-induced vibration of slender cylindrical

More information

Aeroelastic effects of large blade deflections for wind turbines

Aeroelastic 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 information

Static and Dynamic Analysis of mm Steel Last Stage Blade for Steam Turbine

Static and Dynamic Analysis of mm Steel Last Stage Blade for Steam Turbine Applied and Computational Mechanics 3 (2009) 133 140 Static and Dynamic Analysis of 1 220 mm Steel Last Stage Blade for Steam Turbine T. Míšek a,,z.kubín a aškoda POWER a. s., Tylova 57, 316 00 Plzeň,

More information

Cardan s Coupling Shaft as a Dynamic Evolutionary System

Cardan s Coupling Shaft as a Dynamic Evolutionary System American Journal of Modern Physics and Application 2017; 4(2): 6-11 http://www.openscienceonline.com/journal/ajmpa Cardan s Coupling Shaft as a Dynamic Evolutionary System Petr Hrubý 1, Zdeněk Hlaváč 2,

More information

Influence of friction coefficient on rubbing behavior of oil bearing rotor system

Influence of friction coefficient on rubbing behavior of oil bearing rotor system Influence of friction coefficient on rubbing behavior of oil bearing rotor system Changliang Tang 1, Jinfu ang 2, Dongjiang Han 3, Huan Lei 4, Long Hao 5, Tianyu Zhang 6 1, 2, 3, 4, 5 Institute of Engineering

More information

Introduction to structural dynamics

Introduction 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 information

41514 Dynamics of Machinery

41514 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 information

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods

More information

Vibrations Qualifying Exam Study Material

Vibrations Qualifying Exam Study Material Vibrations Qualifying Exam Study Material The candidate is expected to have a thorough understanding of engineering vibrations topics. These topics are listed below for clarification. Not all instructors

More information