tips to review a lateral critical speed analysis document

HOW TO REVIEW? ( Simple Tips ) LATERAL CRITICAL SPEED ANALYSIS REPORT

By - Mantosh Isanchandra Bhattacharya Prepared in intention to simply a bit …………………….for Non Specialist Engineers .

Dedicated to my Parents

Lateral Rotor dynamic Analysis Report – What and How to review By – Mantosh Bhattacharya Smooth operations of Rotating Machines are one of the most primary requisites for any Industrial operating / production facility. This requisite has two fold criteria – to meet process requirement and stable operation without high vibration which will lead to short life span of equipment. There are various types of Vibrations which can lead to catastrophic failure of equipment which are- Lateral , Torsional ,Structural , Fluid induced and externally induced A lateral Rotor dynamic analysis considers the interaction between the elastic and inertia properties of the rotor and mechanical impedances bearings, seals etc. The most common problems occurring in Lateral Vibrations are

1. steady state synchronous ( multiples of operating RPM )vibration – these type of vibration can be reduced by Improving Balance Grade ( Addressed in relevant API / ISO /EN standards ), Modifying Rotor Bearing system by tuning the critical speeds , Introducing enough damping to restricting peal amplitudes when Rotor is passing through critical speed.

2. Sub harmonic Rotor instabilities .which can be avoided by raising rotor critical speed , eliminating possible fluid / lube oil induced instability by proper selection of bearing and seals .Identifying the instability onset and eliminating from operating range

These studies are carried out by the correct design of the Rotor Bearing system by optimizing stiffness and other damping characteristics. The reason of doing so lies in a basic law – Lateral Vibration Displacement (Vector Quantity) = Algebraic addition of Forces Acting on the Rotor Bearing System divided by the Rotor Bearing System Dynamic Stiffness

By knowing where these events are located in the machinery operating range, the system can be modified to remove them from the speed range or design the system to be able to tolerate the vibration level. Usually the amplitude of the lateral rotor vibration and domain of frequencies of these vibrations are the outputs from Lateral rotor dynamic analysis. They are also helpful in determining the cause of vibration concerns when compared to on line vibration signature analysis using Bode , Shaft Centerline , Orbit Plots and Full Spectrums..

Lateral rotor dynamic analysis study reports are generated by Turbo machine Vendor as normal practice with the help of various software models .It is quite difficult to check the validation of the report by end user without a similar software program unless the machine is shop tested . Sometimes a big flaw in study report is surfaced only during shop/ field test .This leads to redesign or carry out major modification of equipment and inordinate delay to tow out / start up. Hence a Lateral Rotor dynamic Report is very important document to predict machinery behavior in various operating conditions.

This article is an attempt to enable a non specialist Engineer to carry out a fruitful overview of such reports of equipments greatly used in Fertilizers, Refinery ,Oil and Gas Industry to name a few . It is assumed that Engineers are fully aware of clauses related to machinery dynamics of relevant API standards . In addition to these , API RP 684 latest edition also to be referred.

What a Lateral Rotor Dynamic Analysis Report should Contain as a Minimum. High Energy Centrifugal Pumps

A. Rotor Model i. Sketch of rotor model ii. Clear identification of bearing, shaft end and internal seals --- ( Labyrinth / honeycomb / wear ring / bush etc ) sleeves , coupling, and disc (impellers, wheels, etc) locations . An example is given below for reference Fig 1

Fig -1 Identification Table for Rotor Elements .

B. Journal Bearings Data (Most of the critical Turbo Machines use Journal Bearings )

i. Bearing type, length, pad arc length, diameter, minimum and maximum clearance, offset, number of pads, load geometry, preload and pivot type and geometry

ii. Effect of shrink fits . Note - If Foundation stiffness is less than 3.5 times the bearing stiffness , then Foundation stiffness also shall be taken into account during modeling .

iii. Oil properties and operating conditions a. Oil viscosity b. Oil flow rate and/or inlet pressure c. Inlet operating temperature range up to high lube oil trip

temperature . iv . Dynamic coefficients (plot or table) for minimum and maximum stiffness cases vs. speed

C. Bearing Pedestal Data i. Identify parameters vs. frequency (mass, stiffness and damping) . It

is a must if Rolling Element bearings are used . Vertical Pumps have much flexible rotor and casing and hence prone to structural related vibration. A long Vertical Turbine pump Rotor must be tested for validation of its mode shapes up to its 3rd order of resonance .

D. Other Forces Included in the Analysis (Machine Dependent) i. Volute fluid dynamic forces ( Volute type Pumps ) ii. Impeller Diffuser Interaction iii. Gyroscopic effect of Impeller if any . iv. Requirement and Presence of swirl brakes v. Consideration Of Lomakin Mass

E. Analysis Methods i. List of computer codes used in the analysis with a brief description

of the type of code, e.g., finite element, CFD, transfer matrix, etc.

Fig -2 Rotor Mode Shape Plot showing Nodes and Anti node If the bearings lie on Nodal points , it is impossible to correct the rotor for damping and lowering amplification factors by means of changing bearing parameters . Hence such cases must be avoided. Undamped Critical Speed (Dry Critical Speed Analysis for Centrifugal Pump )

F. Map and Mode Shapes i. Critical speed vs. support stiffness ii. Curves of the support stiffness (i.e. Kxx and Kyy for minimum and

maximum stiffness) iii. Plot, as a minimum, the first 3 critical speeds with the stiffness axis

extending to “rigid and soft support” regions

Fig -3 Example of Undamped Critical Speed Map For a Centrifugal Compressor.

iv. Duncan Hood Criterion and Frequency Ratio criterion of Stiff shaft

Rotor v. Static Analysis to evaluate Rotor Sag vis a vis annular seal element

minimum clearance for Rotor Centering . ( Refer API 610 for Flexibility Factor )

Example of such static analysis diagram is shown below in Fig -4–

Fig -4

Wet Critical Speed Analysis of a Pump is a sort of Damped Critical speed analysis where all interstage clearance wear rings , bushes etc are considered as bearings with stiffness and damping properties . This critical speed is different than Dry critical speed calculated due to “ Lomakin Effect “The accurate prediction of the rotor dynamic characteristics of centrifugal pumps is complicated by the presence of strong restoring and cross-coupling forces within annular sealing cavities such as wearing rings and thrust balancing devices. This so-called Lomakin Effect occurs when the sealing annulus shifts off-center, and can increase the first bending natural frequency of the pump rotor assembly by up to 200 percent. In addition, the lower natural frequencies may become critically damped, or in the other extreme unstable, relative to what would be expected if the rotor were run in air rather than in the pumped fluid.

G. Damped Unbalance Response Predictions

i. Identification of the frequency of each critical speed in the range from 0 to 1.5 x Machine Normal Speed .

ii. Frequency, phase and amplitude (Bode plots) at the vibration probe locations in the range 0 to 1.5 x Machine Normal Speed resulting from the unbalances . Some vendor simulate the unbalance by adding unbalance weight at coupling hub . The coupling unbalance mass shall be preferably 180 deg out of phase to Rotor imbalance to promote higher modal deformation . Such simulation calculation must be provided in unbalance calculation report.

iii. Comparison of Shop Verification test V/S Predicted Response Curve . Example shown below in fig -5–

0

5

10

15

20

25

30

35

4000 5000 6000 7000 8000 9000 10000 11000

RPM

Pea

k A

mpl

itude

(um

)

Vtest - Vectorial SubtractionVM - MRTVS - Shop Verification Test

VS VMVtest

Fig -5

iv. A plot between Modal damping factor and frequency ratio showing whether design falls under acceptable region of API criteria .

H .Campbell Diagram Example of Typical Campbell diagram pertaining to Forward and Backward whirl and associated damping is shown below – ( Red Border is API Limit of 125% of Trip speed )

Fig-6 Typical Campbell Diagram For High Energy Pumps , Campbell diagrams for Dry , API Test , New Seal and 2X Seal clearance conditions shall be provided . Note – Rotor dynamics of pump is more complicated due to lots of operating parameters which can affect Rotor behavior .which are Lomakin affect , Side radial load due to volute shapes and types , lateral and angular shaft motion might induce impeller shroud excitation force affecting the stability of high power density pump , particularly multistage pumps having flexible rotors ( operating above 1st critical speed ) . Lateral Analysis Reports for Centrifugal Compressors -

Report for Lateral and Stability Analyses Shall contain all items except D( i,ii )and Wet

Critical speed analysis shown for Centrifugal Pumps . In addition to these following information shall be furnished in Report ,

H. Interstage Seal Data i. Coefficients (when a Level 2 analysis is required) for labyrinth

seals, balance piston seal

ii. Seal type (labyrinth, honeycomb ( Straight / Tapered ) , hole pattern, etc.). Stepped Honeycomb have a distinct disadvantage over straight through Honey Comb seal.

iii. Labyrinth - Teeth on rotor, teeth on stator . iv. Seal minimum and maximum operating clearance v. Statement for requirement of shunt holes and/or swirl brakes if any .

I. Squeeze Film Dampers – Should not accepted at initial phase of design , they shall be used only when other means of stabilize the rotor is exhausted Should not used in a combination with Hole Pattern or Honeycomb seal.

J. Unbalance Response Predictions

i. Identification of the frequency of each critical speed in the range from 0 to 1.5 x Nmc

a. Frequency, phase and amplitude (Bode plots) at the vibration probe locations in the range 0 to 1.5 x Nmc resulting from the unbalances

ii. Tabulation of critical speeds, amplification factor, actual and required separation margin and scale factor

iii. Axial location, amount and phase of unbalance weights for each case

iv. Plots of amplitude and phase angle vs. speed at probe locations a. For min and max bearing stiffness cases

v. Plots of deflected rotor shape at critical speeds and Nmc – For min and max bearing stiffness cases

vi. A table of the close clearance magnitudes and locations and maximum vibration levels verifying that SP6.8.2.11.1 of API RP 684 has been met. Typical Example shown below in Fig -7

Fig -7

K. Stability Level 1 Analysis i. The calculated anticipated cross coupling, qa, (for each centrifugal

impeller or axial stage), total anticipated cross coupling, Qa, log dec and damped natural frequency at anticipated cross coupling, and Q0/Qa

ii. Kirk Donald and Fulton Plots to confirm that Compressor is safe in design .

L. Stability Level II Analysis for Centrifugal Compressors i. Description of all assumptions used in the analysis ii. Description of all dynamic effects included in the analysis. For High

Pressure Compressor, honeycomb seal performance is dependant on working temperature too . The expansion of drum and Honeycomb lowers the sealing gap thereby changing the stiffness and damping as actually calculated. Hence , it is advisable to conduct a test (as a part of acceptance test )which can enact the high pressure and high temperature scenario at Vendor works

iii. Value of log dec and frequency versus component addition for min & max bearing stiffness (Defined in SP6.8.6.2-7 of API RP 684) Example shown in Fig -8 -

Fig -8

At the end of report Vendor shall conclude the summary that confirms compliance with API requirements

Aerodynamic forces on the rotor due to volute also needed to be considered. The volute can produce asymmetric pressure gradients around the rotor. If the gas pressure in volute is quite high, then the radial load on rotor plus bearing static load can exceed the bearing radial load capability. Lateral Analysis Reports for Turbo expander Compressor with Active Magnetic Bearings In common Engineering Practice , Rotor of Turbo expander Compressor are stiff shaft Rotor. In this report apart from required conventional information listed above , Spin Inertia and Traverse Inertia of Rotor, Center of gravity of rotor , wheels with respect to common center of gravity shall be provided, type of AMB controller shall be listed for electrical stiffness of AMB (Active Magnetic Bearings ) to determine the reaction forces on bearings. There are two types of controls used – Single Input Single output SISO and Multiple Input Multiple Output MIMO . MIMO applies cross coupled electrical stiffness .

The selection of controls are based on type of machinery and configuration of modes and gyroscopic effects associated. If the equipment is to be used in very critical service , holographic technique to determine the critical speed is advised.

Fig -9 1st Mode Shape Plot of A Turbo Expander Compressor Rotor . Lateral Analysis Reports for Gas Turbine Rotors Rotor Bearing system configuration may vary between Aero derivative and Industrial type Gas Turbines. Most of the vendor information are proprietary while Gyroscopic effect and shear deformations are taken into account . A confirmation is needed whether Columb Spine effect is considered in Rotor Dynamic analysis. Due to high asymmetric bearing configuration and significantly different static loads at various bearing location ,intersections of bearing stiffness , Critical speed and operating speed range might be observed. This does not seem to pose any Rotor dynamic issues as the rotors are well damped and time tested . Rotor response due to unbalance is calculated but seldom verified with a reason of fleet of well proven design. A blade out response analysis must be performed if equipment is a propulsion engine. Steam Turbines –Special care must be taken steam injection profile and rotor bearing loading and bearing configuration . To eliminate numerical errors being a long flexible rotor bearing system having more than two bearings , two different models should be generated by altering the station numbers to compare the error margin in both calculations.

Explanation of certain aspects of Rotor dynamic Analysis. Isotropic Bearing shall have vertical and horizontal stiffness values same . Asymmetric Bearing system shall have Direct Stiffness asymmetry α= Kyy / Kxx if greater than 2.5 , Orthotropy ( Anisotropy parameter γ= ωy / ωx if greater than 1 ( Split Critical speed ) Load on Pad and Load Between Pad configuration are based on journal static load and Rotor speed criterion. Campbell Diagram with Forward and Backward Whirl - also known as "Whirl Speed Map" or a "Frequency Interference Diagram", The dotted lines show the backward whirl (BW) and forward whirl (FW) modes, respectively, which diverge as the spin speed (shown in diagonal line across the plot ) increases. When the BW frequency or the FW frequency equal the spin speed Ω, indicated by the intersections A and B with the synchronous spin speed line, the response of the rotor may show a peak and change in phase too. This is called a critical speed. In real world all disks shall have a non central feature which tend them to wobble the way a pulley wobbles if bush is not in center of shaft . This wobble creates the Gyroscopic effect and decreases the Backward whirl dominance It is known that if the rotor is supported in bearings with rotational symmetry, a rotating unbalance of the shaft cannot excite backward precessions, but if the bearings have even the slightest anisotropy, backward synchronous precessions are excited just as in the case of forward precession motions. However, internal damping forces have a marked effect in reducing the amplitudes of the backward precessions. In case of Pronounced Asymmetric Bearing system, Forward and Backward whirl can be encountered and can be excited by an unbalance response. Increasing direct damping reduces the Rotor peak response In most of the Bode Plots of Rotor Bearing response , Whirl is backward between First two critical speeds . Asymmetry in bearings are commonly used to enhance stability . The governing equations of motion are derived taking into account the effects of gyroscopic moments, and internal and external damping. Damping – A rotor bearing system has two sets of co ordinates i.e. Stator Fixed Co ordinates and Rotor Fixed Co ordinates . Rotor fixed co ordinates are based on rotor geometry and while in motion create disturbing centrifugal forces .Stator Foxed Coordinates i.e. Seal , bearing properties try to restore the rotor back to original position by damping . Damping does not shift the Rotor Critical speed away if it is inside the operating region , it lowers down the amplitude of vibration thereby absorbing the major part of energy . Damping should not be confused as reverse of stiffness . A damping ratio in negative value denotes inherent system instability.

Gyroscopic Effect – Gyroscopic effect if an impeller whirling does not lie on single plane whenever the rotor is in a slope . Based on length to diameter ratio of this effect lowers or raises the stiffness of rotor . The empherical formula to asses the gyroscopic effect on any rotor is I act = ω2 * It –υ* ω*Ip Where I act is disk effective moment of inertia , Ip is disk polar moment of inertia It is disk traverse moment of Inertia ω is rotor speed in radian / sec and υ is frequency of whirl is radian / sec . If calculated Iact is negative this means the rotor shall get stiffened and natural frequency will be raised due to this effect . If the disk are located at nodal points then gyroscopic effect are dominant because there will be a precession of disk corresponding to the changes in slope in rotor operating mode shape which will be maximum at this location . At antinodes point ,there will be no Gyroscopic effect . This study is more relevant to overhung fans , blowers and integrally geared compressors rotors and bigger diameter compressor impellers . Structural Resonance – For equipments being installed on offshore plat form , megazzine or having grout less design base frame resonance conditions must be considered. Use of sole plates beneath the base frame limits the vibrations due to structural resonance. Self Check by Engineers to have an overview of equipment integrity. Before accepting the design , always ask for a track record list ( number of units operating at near to similar operating parameters)from Vendor if number of Impellers exceeds 8 in a single casing for Centrifugal Compressor and Impellers exceeds 9 in a single casing for Centrifugal Pump. Estimation of Critical speed - Critical speed are determined by eigenvalues of the 4th order diff equation: d^4(y)/dx^4 - B^4*y=0 where B^4=rho*A/(E*I)*omega which might need complex calculations . An example of estimating First Natural Frequency for a single stage double suction pump is shown below .If the impeller mass is Mi , Shaft mass is Ms , Dia is D Young’s modulus is E and Moment of Inertia is I then first natural frequency of rotor can be calculated as Fn = (120/Π )[ (3EI) / L^3 (M+0.49Ms)]^0.5 Accordingly shaft deflection formulas also can be used based on the type of rotor arrangement to determine the first critical speed at infinite bearing stiffness . One of the hand calculated method is by using Dunkerley’s formula to determine deflection and then critical speed. There are other manual ways also to estimate Shaft Critical speed even considering Bearing stiffness .

First critical speed will be tentatively calculated using assumption made on Jeffcot rotor –for a pure translation motion the more generalized Laval- Jeffcott rotor system with flexible supports . 1/K = 1/Ks +1/2Kb ; K =2KbKs / 2Kb+Ks So if the bearing stiffness is too larger than shaft stiffness Kb>>Ks . Then this system becomes flexible rotor with rigid supports and vise versa. First natural frequency Fn1 = (1/[2*pi]*w1 W1=sqrt (k_equivalent / m_equivalent ) M_equivalent = M+0.5*m Where M is the concentrated mass at axial center of shaft and m is shaft mass K equivalent is series combination of shaft stiffness and two parallel bearing stiffness. K_equivalent is Krotor*Kbrg1*Kbrg2*Klabyrinths / (Krotor+Kbrg1+Kbrg2+ Klabyrinths) Krotor= 48*E*MI / L^3 MI= k* pi * rho*l*D^4 / 16 Modal mass ratio also to be considered to reach very close approximation . Modal mass ratio is strongly dependant on stiffness ratio of bearings and shaft . Make a Note that - The increased mass lowers the first mode frequencies and very slightly lowers the second mode frequencies. The reduced mass moment of inertia version increases the frequency of both the first and second modes, and decreases the strength of the gyroscopic effect. Take a note that bearing span of the rotor ,in any case ,should not exceed 10 times the mid shaft diameter of rotor. This criteria is applied based on numerous experiment conduct by experts . Type of Bearings being used –

Selection of bearings must follow Stribeck Curve. As a rule of thumb for Diametral Clearance of Bearing = 3.6*[√ (√N)]*D^1.25 / 100000 mm. For tilting pad radial bearing acceptable start up loads mainly depend on load direction. For load on pad type loading specific load should not exceed 1.4 Mpa and load between pads 2.2 Mpa for 5 pad bearings . For 4 pad bearings it should exceed 2.0 Mpa .If loads are exceeding then it is advisable to use bearings fitted with a jacking system . Load on pad bearing give different stiffness in different directions. Asymmetric bearing stiffness provide greater Rotor bearing system stability. For plain bearings unit load on bearings should be more than 100 psi and but less than 200 psi. unit load on bearings = Lu = Wj / L*D where Wj is journal load . For higher unit loads than 200 psi , bearing supplier must be contacted separately . Pivot stiffness shall be included in rotor dynamic analysis for spherical pivot or double radii pivot bearings . For cylindrical pivot / ball socket type pivot , it is not included as they are 2 times higher than oil film stiffness . It has been a common practice to use plain dam type , lobe sleeve bearings in gear box pinion to ensure stability particularly at super synchronous region .In this bearing arrangement L/D ratio should kept higher than 0.75 .

Confirmation that there will be no Rotor stator Rub during rotor unbalance ( RCV Criteria ) Find out the AF of the rotor using half power method from Bode Plot as shown below

Fig -10 Rotor response plot Multiply the bode amplification factor with peak to peak response at probe location the value shall be less that 75% of diametral clearance of rotoric and statoric at mid span of rotor. Similarly with a rotor mode shape clearance check with respect to amplitudes can be performed. Rotor Stability check during operation . Based on John Fulton’s extensive research , an empherical formula can be used as RSR ( Rotor Stability Ratio ) which must lie in between 0.4 to 0.7 . RSR = 1 / 3.24-0.36 ln (ρ average ) ρ average is gas density and suction and gas density at discharge . If the criterion is not met , shunt holes , swirl brakes are to be provided to ensure rotor stability . Labyrinth seal clearance do open up during a significant surge in a centrifugal compressor and can lead to higher cross coupling forces and consequently high sub synchronous vibration . Use of Rub tolerant labyrinth should be encouraged . Normally an independent design audit is necessary for a design which is at the limit of vendor proven experience or for a particularly difficult / complex application. That’s why a Rotor dynamic report should be well formatted and should carry vital information. A regular review of such reports for various types of machines backed by standard references can enhance the skill of a Non specialist Engineer to visualize Rotor Bearing System being offered by Machinery Supplier. ---------------------------------x-----------------------x---------------------------x--------------------

tips to review a lateral critical speed analysis document

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