Nonlinear vibration analysis of steam turbine bladed disk with friction contact between adjacent blades

Transcription

1 Nonlinear vibration analysis of steam turbine bladed disk with friction contact between adjacent blades Josef Voldřich Ukraine, November 2018 Проект Развитие международного сотрудничества с украинскими ВУЗами в областях качества, энергетики и транспорта г. Харьков, 11/2018

2 Степан Прокопович Тимошенко A grand native of Ukraine, the father of modern engineering mechanics. - His portrait, as the only one, has displayed with reverence in my workroom for 20 years. - He was a giant for mathematical modelling of strength problems in mechanical engineering. Encouragement: - Let s not be afraid to use MATHEMATICS for solution of our actual problems. - And it is not inevitable to look up only to commercial FEM software.

3 Content Motivation and introduction Methodology Calculations of nonlinear vibration for bladed disks Contact stiffnesses Conclusions

4 Motivation and introduction Pilsen ( inhabitants): Steam turbines produced by Doosan Škoda Power Ltd. Nuclear reactors WWER for NPPs produced by Škoda JS, Ltd. (25 so far). Well-known for its brewing The world s first-ever pilsner type blond lager, making it the inspiration for much of the beer produced in the world today, many of which are named pils, pilsner and pilsener. The Škoda company (established 1859) was one of the biggest European arms factories up to the WW II. Pilsen

5 Motivation and introduction shroud tie-boss Demand from Doosan Škoda Power Ltd. : Vibration analysis of bladed disk with 66 blades of type LSB48 (1220 mm) 277,269 DOFs

6 Motivation and introduction renovation of steam turbines low-pressure stages in NPP Temelín by Doosan Škoda Power NPP Temelín in south Bohemia Reactor heat power [%] Before modernization LP [MWe] Power rise garanted [MWe] Power rise measured [MWe] , ,8 Turbine room

7 Methodology Modal analysis Eigenvalue problem for the whole bladed disk ω 2 M + K w = 0 K = K 0 K 1 0 K N 1 K 0 K 1 0 K N 1 K K K N 1 K 0 K 1 K N 1 K 0 It is sufficient to consider only a sector with 1 blade the reference sector M = M 0 0 M M Eigenvalue problém for the reference sector: ω 2 M + K 0 + K 1 e ikσ + K N 1 e ikσ v = 0 turbin room M, K 0 : mass and stiffness matrices K 297k DOFs 1, K N 1 : sector-to-sector coupling N : number of blades, σ = 2π N, k : harmoni index (nodal diameter)

8 Methodology Vibration of the reference sector M d2 u t dt 2 + B du t dt + Ku t + f L u t t, u t + f R u t, u t + t = p t Linear part Nonlinear forces between blades (friction contact) Steady-state vibration responce can be represented by Fourier series n n Substituing into the equation we obtain n Excitation forces p t = P c k cos kωt + P s k sin kωt u t = U 0 + U c k cos kωt + U s k sin kωt = U 0 + U k e ikωt + U k e ikωt k=1 k=1 k=1 Z k ω U k + F k U P k = 0, Z k ω = K + ikωb kω 2 M, k = 0,, n Matrix of dynamic stiffnesses for the system without couplings

9 Methodology Nonlinear vibration, Multiharmonic Balance Method 4 important steps for the solution of equation 1) Calculate a multiharmonic FRF matrix of the linear part of the system Z k ω 2) Separate linear and nonlinear degrees of freedom U k = U k ln, U k nln 1 = A k A k = A l n ln k nl n ln A k l n nln A k nl n nln A k 3) Split the equation into linear and nonlinear part m r= iη k,r Ω 2 k,r kω φ H 2 k,r φ k,r Z k ω U k + F k U P k = 0, k = 0,, n U ln l n k + A nln k F nln k U nln A k P k ln = 0 ln U nln nl n k + A nln k F nln k U nln 4) Accomplish effective calculation U nln A k P k nln = 0 nln F k nln U nln using F k U = 0, F k nln U nln

10 Methodology Mathematical model of contacts Dry Coulomb friction model for contactforces is used μ N 0 x, y k t, k n friction coefficient normal preloading force of the contact relative displacement of nodes of the nonlinear contact element in the tangential and the normal directions tečné a normálové kontaktní tuhosti Calculation of harmonic coefficients for nonlinear forces U nln nln F k [Petrov E.P. and Ewins D: Analytical formulation of friction interface elements for analysis of nonlinear multiharmonic vibrations of bladed disks. ASME Journal of Turbomachinery 125: , 2003]

11 Methodology mathematical model of contacts Concept of contact stiffnesses Finite number m of mode shapes A k m r= iη k,r Ω 2 k,r kω 2 φ H k,r φ k,r Notion of contact stiffness is misleading, because it does not label only physical properties of contact. Contact stiffnesses compensate higher mode shapes that are not included in the approximation of A k. Calculation of contact stiffnesses: Linearized contact stick or μ=0 Model of nonlinear systém resonant frequency Measurement resonant frequencies FEM calculations natural frequencies

12 Methodology Computational diagram Modal anallysis of the sector without binding ANSYS Calculation of contacts localization and preload ANSYS Operating FEM mesh Quiescent FEM mesh NPP Temelín in souh Bohe Modal analysis of the system with stuck contacts ANSYS NONVIBCS In-house software Input file with description of analysis Results of modal analysis Input file specification of excitation Input file specification of binding Analysis of nonlinear vibration for cyclically symmetric system with contacts NONVIBTW_V4 In-house software Input files NONVIB_VIEW In-house software Pictures Graphs Output files Stress field ANSYS Numbers

13 Calculation of nonlinear vibration LSB mm, steel Low x z φ High y Low LSB mm, titanium alloy Each blade has two integral coupling: Tie-boss middle part Shroud (bandage) top Shr oud TieB oss High Blade geometries of LSB48 and LSB54 are moderately different. High High Contact surfaces are plane.

14 Calculation of nonlinear vibration Normalizovaná výchylka kmitání špičky lopatky Normalized max vibration amplitude, shroud contact LSB 48, excitation by traveling waves with 2 ND k T = 0, N/mm = 0,35 = 0,2 k T = N/mm s1 = 0,35 = 0,2 St-St systém Normalizovaná budící frekvence Limited number (10) of mode shapes for approximation of blade dynamic behavior necessity to fit contact stiffnesses (influenceing resonant frequencies of computational model) s2 s3 Normalized excitation frequency

15 Normalized max vibration amplitude, shroud contact Normalized tangential force, shroud Calculation of nonlinear vibration Normalizovaná výchylka kmitání špičky lopatky k T = 0, N/mm k T = N/mm s1 St-St systém = 0,35 = 0,35 s2 = 0,2 = 0,2 s Normalizovaná budící frekvence Normalized excitation frequency Normalizovaná smyková síla f T, shroud = 0,2 = 0, Normalizovaný relativní prokluz, shroud Normalized relative displacement, shroud - Higher friction coeficient, µ more friction energy dissipated in contact. - Higher energy dissipated in contacts does not lead to drop of displacement aplitudes. friction damper??

16 Normalized contact force, shroud Normalized contact force, shroud Calculation of nonlinear vibration Normalizované kontatní síly, shroud f N f T - f N Fáze kmitu [rad] Phase of vibration [rad] Normalizované kontaktní síly, shroud f T f N - f N Fáze kmitu [rad] Phase of vibration [rad] Normalizovaná výchylka kmitání špičky lopatky k T = 0, N/mm k T = N/mm s1 St-St systém = 0,35 = 0,35 s2 = 0,2 = 0,2 s Frictional sliding occurs in contact. - Excitation by travelling wave of 2ND two cycles within the period (0,2π).

17 Normalized max vibration amplitude, shroud contact Calculation of nonlinear vibration LSB 48, excitation by harmonic variation of rotor torque moment Normalizovaná výchylka kmitání špičky lopatky Sl-Sl system systém Sl-St system systém St-St system systém = 0.2 0, Normalizovaná budící frekvence Normalized excitation frequency - Linearized problems: Contacts in durable ideal sliding ( Slide ) or sticking ( Stick ). - Sl-Sl system = sliding without friction for Shroud and Tie-Boss contact. - Dangerous situation: the excitation frequency of variation of torque moment is close to the natural frequency of Sl-Sl system. - Contact couplings could activate a sleeping resonance regime.

18 Normalized max vibration amplitude, shroud contact Calculation of nonlinear vibration LSB 54, excitation by traveling waves with 8 ND Normalized excitation frequency - Analogous character as in the case of LSB 48 excited with 2ND: higher values of µ higher amplitudes of displacements of forced vibrations - loss of numerical konvergency for µ > cca 0.6.

19 Tangential force ft, shroud [kn] Dissipated power, shroud [W] Calculation of nonlinear vibration µ Relative displacement of coupled nodes, shroud [mm] Normalized excitation frequency Analysis of nonlinear vibration results: - Dissipated power in level tens W for each blade, if slipping. - Higher displacement amplitude rise for higher friction coefficient.

20 Normalized contact forces, shroud Calculation of nonlinear vibration LSB 54, excitation by traveling waves with 8 ND Phase of vibration [rad] Phase of vibration [rad] Phase of vibration [rad] - Excitation by traveling wave with 8 ND 8 cycles within the period (0,2π). - Higher friction coefficient higher amplitude of tangential and normal contact forces

21 Calculation of nonlinear vibration 0,988 ω 8,1,St 0,994 ω 8,1,St Normalizované kontaktní síly, shroud 1 f N f 0.2 T f -1 N Fáze kmitu [rad] Normalizované kontaktní síly, shroud f T f N -1 - f N Fáze kmitu [rad] µ = 0.6 : big amplitude of normal forces for excitat. frequencies close to resonance intersection of curves for µf N and -µf N We can interpreted the intersection as contact separation.

22 Contact stiffnesses The compliance of interface contact elements NPP Temelín insouth Bohemia Related to the deformations of the asperities of the contacting surfaces. May be omitted (an order of magnitude higher). 1 = = k t k x k t,comp k n k y k n,comp Related to the compensation of omitted higher modes. k t k t,comp k n k n,comp turbineroom To our knowledge, up to now, no general method for finding the values of contact stiffnesses has been introduced. We developed a (computational) method that fits the contact stiffnesses effectively, at least in our case of bladed disks (mentioned also in this lecture).

23 Conclusions Harmonic Balance Method (HBM) - It is possible to use to calculations of steady state of nonlinear vibration for bladed - It is possible to perform the calculations on PC Contact stiffnesses - They compensate higher mode shapes omitted in the approximation of dynamic compliance of a calculated system - It is necessary to fitted these contact stiffnesses Ne vždy lze třecí vazby považovat pouze za třecí tlumiče: - Větší disipovaný výkon nevede pokaždé k většímu poklesu amplitud výchylek. - Prokluz ve třecí vazbě může být příčinou nárůstu amplitud výchylek. - Separation of contact surfaces can occur Notion of sleeping resonant regime

24 Thank You very much