ON THE MORTON EFFECT: SIMPLIFIED PREDICTIVE MODEL FOR A THERMALLY - PowerPoint PPT Presentation

TRC Project 40012400028 ON THE MORTON EFFECT: SIMPLIFIED PREDICTIVE MODEL FOR A THERMALLY INSTABILITY INDUCED BY DIFFERENTIAL HEATING IN A JOURNAL BEARING Lili Gu and Luis San Andres

Justification • The Morton Effect (ME) refers to a phenomenon of thermal imbalance induced instability of rotors supported by fluid film bearings. “ They keep happening … ” “ Morton Effect instabilities were like a widely-spread but undiagnosed disease. ” ----D. Childs (2015) • Rotor thermal instability (ME) was added into the rotordynamics tutorial in API 684 2015

Justification 1.Eccentricity is inevitable due to manufacturing, wear during operation, etc  eccentricity whirl yields differential heating (Fig. a)  temperature difference at the journal (Fig. b)  thermal bending  levitating vibration level. y P P P  C,2 1 H , ,1 H P  H ,2 P  P P  P C,2 ,2 H ,1 C C ,1 o x Fig. a Differential Heating [de Jongh, 2008] Fig. b Temperature Gradient

Justification • However, ME only attracts a limited attrntion in recent years. Stats from “ Web of Science ” "Morton Effect" & "Newkirk Effect" & "Spiral Vibration" &"Thermal" Publication Number Citation Number

Justification • A major reason for the lack of research is that the ME is less likely to cause catastrophe if under proper monitoring. • However, “ it did not appear immediately and did not disappear once initiated (Berot & Dourlens 2009) ” . • Lack of theoretical guidance could cause failure to eliminate ME-induced instability. • A simplified predictive tools can guarantee a continuous running and avoid a major change of rotor systems.

Objective and Executive Summary Objective: Develop a simplified & general model for the ME-induced vibrations with required accuracy. Executive Summary: 1.General excitation mechanisms for ME-alike vibrational problems. 2.Modeling of thermal evolution in ME-alike problems. 3.Develop the simplified analytical model for Morton Effect. 4.Validation of the new Morton Effect model.

ME Mechanism • Thermal bow (geometric imbalance)  ( )] T v ( ) t [ v ( ) t v ( ) ... t v t T T ,1 T ,2 T n , Thermal bow can be determined by solving heat transfer equation Thermal boundaries along • Temperature distribution rotor shaft Q  Asymmetric temperature Thermal bending

ME Mechanism • Mechanism 1: rotor bow theory          ሻ 𝐋 𝐒 𝐰 𝐔 (𝑢 ari arising from M v C +G v K K v F K v ( ) t R b R R b R T asymmetric hea as heating eff effect, , K v ( ), t excitation due to thermal bow is is nat naturally a a fun unction of of R T K , M , G rotor stifness, mass and gyroscopic matrices the the fac actors tha that ca can ca cause R R R C K , , bearing damping and stiffness mat ri ces th the ME ME-induced ins instability b b F , external forces • Mechanism 2:Equivalent mass unbalance              2 i t 𝒇 𝑼 &𝜸 are products of M v C +G v K K v F M e e R b R R b R T the thermal bow magnitude of thermal bow e v   T , i 1,2,..., n T , i 1,2,..., n  phase betwee n v and vibration vector v  , 1,2,. .., T i n

ME Mechanism Thermal bow theory Equivalent mass unbalance theory y y e g  g T g e e e um um total v T v v x x 𝐋 𝐒 𝐰 𝐔 (𝑢 ≠ 𝐍 𝐒 𝐟 𝐔 𝛻 2 𝑓 𝑗𝛻𝑢+𝛾 ሻ “ The mass unbalances will produce only small vibrations as the unbalance forces are small. However, geometric unbalances can give large vibrations even at low speed. ” -- B. Larsson (1999) Mechanism 1 is chosen for a direct coupling

Development of Thermal Bow Q  • Schmied ’ s Model (S) [Schmied, 1987] Q  y o    v       ω I v T  v p v q i 0 n T T 2 Simple, but lack of v p, heat generation factor ( 𝑅 + ) v reflection of  e q, heat dissipation factor ( 𝑅 − ) 1 dynamic properties x o v , vibration vector determined by the 𝛛 𝐨 , natural frequency system • Kellenberger Model (K) [1980] Lack of coupling            η ω I v Q with vibration 𝐰 v p q i p , T 1 n T 𝜽 𝟐 , coefficient determined by friction/shearing coefficient, dynamic properties of the system, and rotation speed. 𝐑 , normalized heat generation

Development of Thermal Bow • Schmied and Kellenberger Model (SK) Introduce equivalent dynamic coefficients to the rotor’s EOM                   a 0 v b 0 v c I v f(t)                      ω v               0 0 v 0 I v p I q i I 0 T T n T 𝐛 ′ , 𝐜 ′ , 𝐝 ′ , coefficients determined by friction/shearing coefficient and the dynamic properties of heating source ሻ 𝐠(𝐮 , external excitation vector

Development of Thermal Bow     k  Heat   Q Q f , , k , c , m , v f m f  Generation f f f y  c , lubricant friction coeffic ient f v k , c , m dynamic coe fficients of the fluid film f f f Fluid Film • Improved Model 1 (IK model) o x Journal Introduce a coefficient for heat generation to reflect dynamic properties of the system, and, normalized heat generation. • Improved Model 2 (ISK model) Introduce a coefficient for heat generation to reflect dynamic properties of the system, and, the dynamic force induced by journal whirl.

Development of Thermal Bow Under S model is better Reference K model is better significant m f than K model (Most time than S model. when p is consuming) IK has the best small prediction Indicate Instability Positive Damping Eigenvalues

Sensitive Study of Thermal Factors • p – heating factor; q – dissipation factor Frequency Frequency Damping factor Damping factor • Thermal bending frequency is mainly influenced by heating factor p • Thermal damping factor is mainly influenced by dissipation factor q • ISK model can predict the nonlinear model because it models the heating generated in the Newkirk Effect more accurately. However, the nonlinear trend is very small.

ME-Induced Thermal Bow • Identifying the heating factor and the dissipation factor   3 kA 3 R    eff q J p  2   mC 2 C c 2 1 p j , J p J , 𝐷 𝑄,𝑘 Journal specific heat capacity 𝑆 𝐾 Journal radius Shaft stiffness 𝜸 Thermal bending coefficient k 𝜑 𝑓𝑔𝑔 Effective viscosity 𝜁 Journal eccentricity ratio 𝐉𝐨𝐮𝐟𝐡𝐬𝐛𝐮𝐟 𝐪 & 𝐫 𝐣𝐨𝐮𝐩 𝐮𝐢𝐟 𝐟𝐫𝐯𝐛𝐮𝐣𝐩𝐨 𝐩𝐠 𝐮𝐢𝐟𝐬𝐧𝐛𝐦 𝐜𝐩𝐱 Model Features: Critical factors such as operational speed, bearing eccentricity, thermal and elastic properties are considered.

ME-Induced Vibration • Coupled Dynamics                v v v F M 0 D 0 K K    vib vib r vib ext                               0 0 v 0 I v p I Q v 0 T T T I  Rotor Residual 1 Rotor System O Vibration Imbalance 1   The coupled dynamics forms a  I 2 Thermo - Fluid feedback loop  Thermal Thermo - Elastic Bow Journal/Shaft • 𝐰 𝐰𝐣𝐜 , lateral vibrations . Differential Temperature O 2 • M, D, K, mass, damping & stiffness matrices. • 𝐰 𝐔 , thermal deformations (thermal bow). Using geometric constraints, this vector’s dimension can be decreased to half the dimension in 𝐰 𝐰𝐣𝐜 • 𝐋 𝑠 , shaft stiffness matrix. Its row dimension is the same as 𝐰 𝐰𝐣𝐜 and its column size corresponds to 𝐰 𝐔 . (4X2 for the Jeffcott rotor model) A critical task is to find the evolution of thermal bending 𝐰 𝐔 .

ME-Induced Vibration Journal Whirl Frequency VS Speeds Effective Temperature VS Speeds Const-visc Therm-visc Lubricant effective temperature increases with speed (almost linearly). Whirl frequencies are independent of temperature rise.

ME-Induced Vibration Influence of Temperature-Dependent Viscosity on Dynamic Coefficients Constant Speed Varying Speed More dramatic change is found at varying speeds than at a constant speed for both stiffness and damping coefficients  The rotational speed     is more dominant than pure temperature rise in the determination of dynamic coefficients.                    

ME-Induced Vibration • Model Validation Results Based on the Proposed Models Results from Reference Referenc e data

ME-Induced Vibration • Model Validation Results Based on the Proposed Models Results from Reference ≅ Important Findings: The simplified model proves reliable in predicting the Morton Effect