bearing selection with the help of samcef...

copyright LMS International 10/03/2014

Bearing selection with the help of SAMCEF Rotors

P. MORELLE – LMS 3D Division

Copyright LMS International - 2013 LMS Samtech

Bearings characteristics, the basics

2

“The rigidity (of a bearing) is determined by the type, size and operating clearance of the bearing” (Schaeffler)

Ex of a rolling element bearing : rigidity as a function of applied force and bearing type



3

How will a bearing fail ? Two different situations:

Bearing submitted to static forces :

Static load carrying capacity is based instead on the concept of “permanent plastic deformation” of the bearing material, or simply put, a “dent” in the bearing that remains after the load is removed. This “dent” could occur from a shock load, or simply applying a stationary load, such as lifting an object with a crane when the bearing is stationary.” (SKF web site)

Bearing submitted to dynamic forces :

In the dynamic load rating methods, bearings typically fail from the “inside out.” That is to say, the rolling elements repeatedly overrolling and stressing the bearing material eventually cause a “fatigue failure.” This is similar to bending a wire repeatedly until it breaks. (SKF web site)



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How will a bearing fail ? Two different situations:

Bearing submitted to static forces :

Static load carrying capacity is based instead on the concept of “permanent plastic deformation” of the bearing material, or simply put, a “dent” in the bearing that remains after the load is removed. This “dent” could occur from a shock load, or simply applying a stationary load, such as lifting an object with a crane when the bearing is stationary.” (SKF web site)

Bearing submitted to dynamic forces :

In the dynamic load rating methods, bearings typically fail from the “inside out.” That is to say, the rolling elements repeatedly overrolling and stressing the bearing material eventually cause a “fatigue failure.” This is similar to bending a wire repeatedly until it breaks. (SKF web site)



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Rotor supported by two bearings :

In order to estimate the reaction force on the bearings, it has to be given the stiffness of each of those bearings … But this stiffness is a function of the applied loads ! As a consequence, the calculation of the reaction force acting on the bearings should be an iterative process.

Classical approach : calculate static reaction under gravity. Use static load as first approximation to select bearing and then perform dynamic analysis of the selected bearing.

Consequence : many iterations may be requested


Rolling Element Bearing Modeling in Rotor Dynamics

Objective: non linear model taking into account contact stiffness, bearing kinematics, clearances New methodology : - Apply gravity, temperature, ... At t=0 so to calculate initial static non linear

equilibrium - Then apply dynamic loads (run up, additional forces, ...) so to calculate

dynamic loads acting on bearings

.



Classes: - ball bearing

- tapered roller bearing: axial + radial loading - cylindrical roller bearing: radial loading Properties: - no damping - no associated destabilizing forces - clearances -> non linear stiffness - very small cross coupling



≤>−=δδδδδδδ

j

jjHjj

sisiKQ

0)()(

23

bbi

brj N

jtRR

R πωθ

2)1(12

−+

+

−=

Jth Ball angular location: Rb Ball radius Ri Inner Race radius Jth Ball relative displacement Radial Force on jth Ball Radial Force on jth Cylinder δ radial clearance KH Hertz Constant

yx jjj ∆+∆= )sin()cos( θθδ

≤>−=δδδδδδδ

j

jjHjj

sisiKQ

0)()(

911



Misalignment Axial Preload : Axial and radial displacements: Jth Ball displacement : Force on jth Ball Force on jth Cylinder δ radial clearance

−+=−+=

δθδθδδθβθβδδ

)sin()cos())cos()sin((

jyjxrj

jyjxizzj r

02

002

000 ))cos(())sin(( AAAAA rjzjjj −+++=−= δαδαδ

≤>=

000)()(

23

j

jjHjj

sisiKQ

δδδδ

≤>=

000)()(

911

j

jjHjj

sisiKQ

δδδδ


Ball bearings : parameters

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NROL number of rolling elements RADB radius of one ball or of one cylindrical roller CL radial clearance between the rolling element L roller working length TYPE type of rolling-elements bearing

• 11 single-row radial ball bearing for radial loads • 12 cylindrical roller bearing • 13 single-row angular contact for radial and axial loads • 14 tapered roller bearing for radial and axial loads

RAD1 radius of the inner race CR1 radius of curvature of the inner race RAD2 radius of the outer race CR2 radius of curvature of the outer race MAT1 inner race material number (elastic material) MAT2 outer race material number (elastic material) MAT3 rolling element material number (elastic material) CANG initial contact angle (types 13 and 14) PRED initial axial displacement due to preload (types 13 and 14) HERT Hertz stiffness constant


Ball bearings : available results

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The following results can be accessed: styp 9524: force in the element, along the 3 element axes styp 9525: moment in the element , about the 3 element axes styp 9530: relative displacement between the two nodes, along the 3 element

axes styp 9531: relative rotation between the two nodes, about the 3 element axes MECANO/ROTORT: Components 19,20,21,22: tangent stiffness Kxx,Kyx,Kxy,Kyy in the

element system of axes

To be used as a support for better bearing selection


Analyses

Example in the Frequency Domain: Simple Rotor supported by 2 Rolling Element Bearings 1D model + Unbalance at the Disk Level



Response in the Frequency Domain: Clearances 2 and 10 microns Amplitude of displacements at the bearing level



Response in the Frequency Domain: Clearances 2 and 10 microns Amplitude of acceleration at the disk level



Time Domain: Non linear transient analyses Run-up under unbalance 0 to 12000 rpm in 10 seconds Non linear bushing to simulate the rolling bearing elements: non linear radial force



Time Domain: Non linear transient analyses Radial Displacement at Bearing Level for different Clearances 2 and 10 microns First peak at 1.5 second ( 1800 rpm ; 30 Hertz)



Time Domain: Non linear transient analyses Radial Displacement at Bearing Level for different Clearances 2 and 10 microns Second peak at 7.8 second ( 9360 rpm ; 156 Hertz)


Example : hydrodynamic bearing

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Classes : • Plain cylindrical • Partial arc • Elliptical • With lobes • With offset • Tilting-pad

Properties :

• Non linear dynamic behavior • Stiffness and damping function of the rotational speed • Cross coupling effect

Effects :

• Instability threshold (the tilting-pad bearing is highly stable) • Oil whip: unstable whirling at approximately one half the rotational speed

frequency


Hydrodynamic Bearings: Cylindrical Hydrodynamic Bearing

Hydrodynamic pivoted pad bearing

Hydrodynamic multi lobe bearing

Modelling of linking devices


Reynolds Equation

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Hydrodynamic Bearings are Interconnection Components between the rotort and the non-rotating support structure

The Hydrodynamic Pressure is solution of the Reynold’s equation Where, • h is the film thickness • p is the pressure • Ω is the rotational speed • µ is the dynamic viscosity (function of temperature, local Reynold’s number)

θµθµθ ∂∂

Ω+=

∂∂

∂∂

+

∂∂

∂∂ h

dtdh

zph

zph

R6121 33

2


Reynolds Equation

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The film thickness is function of relative displacements and angles between the rotor and the stator as well as some geometrical parameters such as clearance or preload

Examples: plain cylindrical bearing without misalignment Examples: plain cylindrical bearing with misalignment Where, • h is the film thickness • C Clerance • Δu , Δv are the relative displacements

θθθ sincos)( vuCh ∆−∆−=

θφθφθθθ sincossincos)( yx zzvuCh ∆−∆−∆−∆−=

Copyright LMS International - 2013 LMS Samtech 24

Gravity + unbalance: run-up + constant speed

Orbits for bearing #1 and # 2: Journal Equilibrium Positions

Transient Non-linear response

Copyright LMS International - 2013 LMS Samtech 25

Transient Non-linear response

FFT of the response at constant rotational speed

Harmonics in Ω, 2Ω, 3Ω, 4Ω,.. are included in the response

Linear harmonic response using the tangent stiffness and damping coefficients uses only harmonic Ω


Tilting Pads Hydrodynamic Bearings

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R : shaft radius R + C : pad radius C – C’: geometrical preload



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Cylindrical bearings : parameters

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where DIA diameter of the shaft CL nominal gap between the shaft and the stator L axial length MU viscosity CAVI cavitation flag

• 0 cavitation, if any • 1 no cavitation occurs ( This option allows to consider hydrostatic

cylindrical bearings) PSAT saturation pressure MESA

• 0 the misalignement is not taken into account (short bearing theory) • 1 the misalignment is taken into account (The 2D Reynolds equation

is solved) NPAT mesh density in the circumferential direction ( default value 720) NPAZ mesh density in the axial direction OMEG rotational speed at time 0 in rad/s


cylindrical bearings : available results

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Here also, many results are available

9524 Effort in the element, along the 3 element axes 9525 Torque in the element , around the 3 element axes

9530 Relative displacement between the two nodes, along the 3 element axes (if DISP is equal to 0, the relative position is stored)

9531 Relative rotation between the two nodes, around the 3 element axes 9901 Tangent stiffness K11 9973 Tangent stiffness K21 9903 Tangent stiffness K31 9904 Tangent stiffness K41 ... 9936 Tangent stiffness K66 9937 Tangent damping C11 9938 Tangent damping C21 ... 9972 Tangent damping C66

To be used as a support for better bearing selection



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Example Rotor on two tilting-pad bearings

• Run up from 0 to 1000 rad/s • 4 pads • Gravity • Radial bushings are put in //



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Rotor on two tilting-pad bearings • Rotor displacement within bearing



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Rotor on two tilting-pad bearings Pivots rotations (selection via the epilogue)



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Rotor on two tilting-pad bearings Radial displacements within the bearing


Example with SAMCEF Rotors

Optimization of the bearing stiffness of a steam turbine with

Boss Quattro and SAMCEF Rotors



Description of the model Length : > 60 meters 7 linking devices Blades are modeled by lumped masses Properties of the bearings (Stiffness and Damping) are known only for

Rotating speed ω=50 Hz



Position of the problem

Compute the eigen-frequencies at ω=50 Hz Due to external excitation, no frequency must be about 40 Hz

No possibility to change the geometrical data (length, radius, position of the

bearings…) We manage the stiffness of the bearings in order to avoid frequencies

close to 40 Hz



Methodology Define the model and declare the variables of the analysis

Variables are Stiffness coefficients and Damping coefficients Compute the eigenfrequencies at ω=50 Hz with Kij and Cij

Localization of the bearings that consume energy Determine which bearing can be modified

Optimization of bearing stiffness

problem is managed within Boss Quattro and Samcef Field



Step 1 : Definition of the FE model Creation of the Task 1 in Boss Quattro



Step 1 : Definition of the FE model Definition of the analysis within Samcef Field

2D Fourier Multi-harmonic modeling

Properties of the bearing can be different

Material is steel and is the same for all stages




Definition of the stiffness and damping matrix for each bearing at ω=50 Hz




Definition of the variables of the system

For each bearing :

4 Stiffness coefficients : K11, K12, K12, K21

4 damping coefficients : C11, C22, C12, C21



Step 2 : Computation of the eigenfrequencies Task 2 : Computation of the eigenfrequencies at ω=50 Hz



Step 2 : Computation of the eigenfrequencies Task 2 : Identification of the relevant bearings (Result file)

Bearing 5 and 6 are relevant to be modified



Step 2 : Computation of the eigenfrequencies Task 2 : Identification of the relevant bearings (Result file)

3D recombination of the 2D model



Step 3 : Parametric study Task 3 : Choice of a strategy

Add a parametric task to the process



Step 3 : Parametric study Task 3 : Choice of a strategy

We define three values : Kinitial, Kinitial/2 and Kinitial*2 for Bearing 5 and Bearing 6 – (Only K11 and K22)

Objective : See the influence of each properties on the response of the structure

1st value 2nd value (Initial) 3rd value K11B5 5100000000 10200000000 15300000000 K22B5 3795000000 7590000000 11385000000 K11B6 9350000000 18700000000 28050000000 K22B6 6450000000 12900000000 19350000000



Step 3 : Parametric study Task 3 : Choice of a strategy (3 possibilities)



Step 3 : Optimization study Position of the problem : Enlarge the frequency domain around 40 Hz

Definition of the optimization problem

• Minimize frequency_3 •Constraint: frequency_5 > 43

Frequency_3

Frequency_5

Starting point: frequency_3 = 36.75 frequency_5 = 41.62




Define a new task




•Selection of 8 variables




•Minimize frequency_3

•Constraint frequency_5



Step 3 : Optimization study Evolution on the 2 frequencies and convergence

Initial gap around 40 Hz

(41.62-36.75=

4.87 Hz)

Final gap around 40 Hz

(45-37.31=

7.69 Hz)



Step 3 : Optimization study Evolution on the bearing stiffness and convergence

Initial Values Final Values

K11B5 1.02e10 5333160000 K22B5 7.59e09 4055250000 K11B6 1.87e10 9991200000 K22B6 1.29e10 6892300000

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