ANALYSIS GUIDE FOR
`VARIABLE FREQUENCY DRIVE OPERATED CENTRIFUGAL PUMPS
`
`by
`Thomas F. Kaiser
`
`Lead Engineering Analyst
`
`Sulzer Pumps (US) Inc.
`
`Portland, Oregon
`
`Richard H. Osman
`
`Principal Product Engineer
`
`Siemens LD-A
`
`Pittsburgh, Pennsylvania
`and
`Ralph O. Dickau
`
`Senior Engineering Specialist
`
`Enbridge Pipelines
`
`Edmonton, Alberta, Canada
`
`an
`been
`has
`Thomas F. Kaiser
`Engineering Analyst with Sulzer Pumps
`(US) Inc., in Portland, Oregon since 1996,
`leading the analysis team since 2001. In
`this position he is responsible for performing
`stress,
`finite element, rotordynamic, and
`vibration analyses for new and installed
`centrifugal pump applications, calculation
`tool development, and seismic qualification
`analyses. He began his career in 1994
`with Sulzer Pumps Headquarters in Winterthur, Switzerland, as a
`development engineer working in the field of rotordynamics.
`Mr. Kaiser has B.S. and M.S. degrees (Mechanical Engineering)
`from the Swiss Federal
`Institute of Technology (ETH),
`in
`Zurich, Switzerland.
`
`is Principal
`Richard H. Osman
`Product Engineer
`for Siemens Large
`Drives-Applications (formerly Robicon), in
`Pittsburgh, Pennsylvania. He serves as
`technical advisor to the company and works
`closely with the product development group
`at Robicon. He has previously worked
`for: Westinghouse Electric Corporation,
`developing a variety of solid-state VFDs;
`Robicon Corporation, as Development
`Engineer, where he designed special purpose thyristor DC-drives;
`Manager of AC-Drives Engineering at Robicon; and Technical
`Director of Heenan Drives Ltd.. He served for five years as
`Robicon’s representative to the NEMA adjustable-speed drives
`subcommittee and two years as Chairman. He was Director of
`Drives Engineering at Halmar Robicon Group, Vice-President of
`Integrated Product Development, and Senior Vice-President of
`Technology for High Voltage Engineering.
`Mr. Osman received a BSEE degree (1965) from the Carnegie
`Institute of Technology. He has written a number of technical
`papers on VFDs, is a registered Professional Engineer, and a
`Senior Member of IEEE.
`
`Ralph O. Dickau is a Senior Engineering
`Specialist with Enbridge Pipelines,
`in Edmonton, Alberta, Canada. His
`responsibilities include pumping equipment
`specifications, selection, installation, and
`startup. He was team leader for several
`large pump and motor replacement and
`rerate programs on the pipeline system.
`Mr. Dickau is also responsible
`for
`troubleshooting pump and motor operating
`problems. He has presented seminars on pump technology for
`other pipeline companies through Enbridge Technology, has
`authored several technical papers, and has been with Enbridge for
`24 years.
`Mr. Dickau received a B.Sc. degree (Mechanical Engineering,
`1978) at the University of Alberta and is a registered Professional
`Engineer in the Province of Alberta.
`
`ABSTRACT
`
`The use of variable frequency drives (VFD) in pumping
`applications with variable-duty requirements provides the user
`with a variety of benefits, including potentially significant energy
`savings and improved reliability achieved by means of speed
`reduction and avoiding part-flow operation. Energy savings are
`primarily realized by running the equipment at high levels of
`efficiency and optimal operating speeds, matching the generated
`pump head to the exact system requirements without the use of
`energy consuming control valves. Running pumps at
`lower
`operating speeds and avoiding part-flow operation also positively
`influences component life and between maintenance intervals. The
`primary mechanical challenge of any VFD application is the wide
`continuous operating speed range. Excitation frequencies of fixed
`speed applications miss most natural frequencies of the structure,
`rotor, etc., and therefore potentially harmful resonance conditions
`often do not occur. This is no longer the case with VFD applications,
`where excitation frequencies become variable and the likelihood of
`encountering resonance conditions is greatly increased. Problems
`
`81
`
`Page 1 of 26
`
`

`

`82
`
`PROCEEDINGS OF THE TWENTY-FOURTH INTERNATIONAL PUMP USERS SYMPOSIUM • 2008
`
`and failures in pumps and associated systems that are not caused
`by resonance are generally not VFD related and are therefore
`not discussed.
`The paper gives an overview of medium-voltage VFD
`technology as well as the main categories of resonance conditions
`of concern with regard to mechanical vibrations of pump/motor
`sets. The analytical and experimental identification of resonances
`related to lateral rotor, torsional rotor, structural, and acoustic
`dynamics are discussed in detail. The applicable set of analyses
`and,
`if necessary,
`the corresponding appropriate corrective
`measures, are designed to help ensure operation free of harmful
`resonance conditions and problems caused by excessive
`mechanical vibrations.
`
`INTRODUCTION
`
`Continuous operation under resonance conditions may result in
`excessive equipment vibrations, reduced in-between maintenance
`intervals, and premature equipment failure. Resonance conditions
`with centrifugal pump applications can be divided into the four
`categories of lateral rotordynamics, torsional rotordynamics, structural
`dynamics, and acoustic resonance. Each of these categories requires
`its own specific set of analyses and checks allowing up-front
`identification of resonance conditions and corresponding corrective
`action. Resonances may then be avoided, moved (to operating
`points where the resulting mechanical vibrations are acceptable),
`or be detuned altogether. These analyses can also help eliminate the
`need for expensive factory string tests aimed at
`investigating
`vibration performance.
`A basic understanding of the most commonly applied and
`available variable frequency drive (VFD) technology and its rapid
`development over the last few decades is helpful in the assessment
`of VFD-related vibration problems and the selection of the optimal
`set of analyses and checks.
`The case studies presented in this paper give a detailed
`illustration of the analysis procedures and methods that can be
`employed in order to successfully identify resonance conditions
`of concern with VFD applications. The same methods and tools
`can also be used to study the effect of design modifications
`aimed at detuning resonances. The actual analysis work should
`be carried out by individuals specifically trained for the task. On
`a broader level, the paper indicates the type of analyses and
`checks considered necessary as well as standard analyses that
`may be omitted. The information presented may therefore be
`used as an end-user guideline for selecting/purchasing of
`analysis support from the original equipment manufacturer
`(OEM) or engineering consultants.
`The rotordynamic software tools used for the case studies are
`pump OEM in-house developments. Commercially available
`rotordynamic software may be used instead. All structural analyses
`were performed applying a general purpose finite element
`analysis software.
`
`EXCITATION SOURCES AND AMPLIFIERS
`
`This section describes the relevant excitation mechanisms and
`amplifiers of mechanical vibrations.
`
`Mechanical and Hydraulic Unbalance
`
`Mechanical unbalance occurs when the mass centerline of a
`rotating component does not coincide with the shaft centerline. A
`certain level of dissymmetry of the weight distribution is unavoidable
`in rotating equipment. For the case of two-plane balancing, the
`unbalance measured in US customary units is defined in Equation
`(1). The factor K is a balance constant, W is the mass per balance
`plane (or journal), and N is the rotor speed. The SI unit equivalent
`definition is shown in Equation (2) with the ISO Balance Quality
`Grade G, rotor-mass m, and the angular speed of rotation ␻. The
`unbalance force rotates with rotor speed (1×) and is therefore
`a sinusoidal function of time when viewed in a stationary
`
`(nonrotating) coordinate system. The corresponding definition of
`the unbalance force F in US units and SI units is given in Equations
`(3) and (4), respectively.
`
`impeller
`Geometric deviations between the individual
`channels create a nonuniform pressure distribution at
`the
`impeller outlet, which also rotates with rotor speed (1×). The
`resulting radial hydraulic force has much the same effect as
`mechanical unbalance and is therefore referred to as hydraulic
`unbalance. Hydraulic unbalance increases with increasing
`flow-rate and usually exceeds mechanical unbalance by factors.
`Unbalance affects lateral rotor and structural vibrations but not
`torsional rotor vibrations.
`
`Self-Excited Vibration
`
`Self-excited vibration, also known as rotor instability, is most
`commonly associated with radial
`journal bearings, annular
`seals, and hydraulic impeller-casing interaction. Self-excited
`vibration caused by lightly loaded cylindrical
`journal
`bearings/guide bearings in vertical pump application are the
`most common cause of instability in centrifugal pumps. The
`corresponding vibration frequency typically lies between 0.40
`and 0.50 times running speed (subsynchronous vibration),
`indicating a tangential mean fluid velocity cu inside the tight
`bearing clearance per Equation (5). The parameter R denotes the
`rotor radius at bearing location and ␻ is the angular shaft speed.
`
`Pumps with excessively worn annular seals can show the same
`phenomenon with vibration frequencies in the 0.7 to 0.9 times
`running speed range (also above 1× running speed in case of
`tangential fluid entry velocities > ␻R).
`Instabilities are caused by the nonsymmetrical pressure distribution
`of the vibrating shaft, which creates a force component acting in
`the direction of the shaft orbit. This force feeds energy to the rotor
`and thus the shaft orbital movement is accelerated. Instability
`occurs in case the energy put into the rotor exceeds the direct
`damping opposing the same vibration.
`Many vertical pump applications show a vibration component at
`or near 0.5× running speed in their amplitude spectrum (also
`referred to as oil whirl or bearing whirl). Instability usually only
`occurs in case a structural or lateral rotor natural frequency is at or
`near this 0.5× running speed frequency, changing the oil whirl into
`an oil whip condition with potentially destructive vibration levels.
`In case the operating speed is increased after the onset of instability,
`the vibration frequency will
`typically remain nearly constant,
`locked into the natural frequency of the structure or rotor as
`indicated in Figure 1.
`
`Page 2 of 26
`
`

`

`ANALYSIS GUIDE FOR
`VARIABLE FREQUENCY DRIVE OPERATED CENTRIFUGAL PUMPS
`
`83
`
`The main factors influencing the pressure pulsation magnitude
`are the radial gap between the impeller outer diameter and the
`volute/diffuser cutwater (B-gap), the percent of best efficient point
`(BEP) operation, and the impeller outlet velocity u2.
`
`B-Gap
`
`The radial distance between the impeller vane trailing edge
`and the volute cutwater or diffuser vane leading edge (B-gap)
`heavily influences the pressure pulsation amplitudes. According to
`investigations published in (Guelich and Bolleter, 1992), pressure
`pulsation amplitudes decrease on average with a power of (⫺0.77)
`on the relative radial gap as illustrated in Equation (7). For example,
`a 2 percent B-gap will produce pressure pulsation amplitudes three
`times higher than a 9 percent B-gap on an otherwise identical pump.
`
`Percent BEP Operation
`
`In Figure 3, statistical data from 36 measurements of single and
`multistage pumps are plotted as dimensionless root mean squared
`(rms) values for the frequency range of 1.25 to 20 times running
`speed, which covers vane-pass frequency. Pressure pulsations are
`normalized according to Equations (8) and (9) for US units and SI
`units, respectively. ∆PRMS is the rms value of the dimensional
`pressure pulsation measurement, ␳ is the fluid density, and u2 is
`the fluid velocity at
`impeller outlet. Flow is normalized as
`shown in Equation (10) with Q being the effective flow and QBEP
`representing the best-efficiency flow. The curves displayed in
`Figure 3 shows the strong dependency of pressure pulsations from
`the operating point with reference to percent of BEP operation
`(Guelich and Egger, 1992).
`
`Figure 1. Waterfall Vibration Plot.
`
`Measures against self-excited vibration include:
`• Reducing the mean tangential velocities in journal bearing and
`annular seal tight clearances. This may be achieved by means of
`applying rough stator surface finish, swirl breaks at annular seal
`entry, stator-side honeycomb or hole patterns, smooth rotor surface
`finish, etc.
`• Increasing bearing radial load by means of applying additional
`(intentional) misalignment in vertical pumps.
`• Avoiding critical speed situations between 0.5× running speed
`excitation and lateral or structural natural frequencies.
`• Restoring design clearances in case of excessively worn
`annular seals.
`
`Self-excited vibrations affect lateral rotordynamics and structural
`vibrations, not torsional rotordynamics.
`
`Vane-Pass Pressure Pulsations
`
`Vane-pass pressure pulsations are generated by the impingement
`of the nonuniform impeller wake flow on the volute cutwater or
`diffuser vane tips. Figure 2 depicts the nonuniform fluid velocity
`profile at the impeller outlet. These pressure pulsations travel
`through the system at the speed of sound of the pumpage. The
`frequency of the vane-pass pressure pulsations, acting on the
`stator,
`is proportional
`to the pump rotational speed N,
`the
`impeller vane count z2, and multiples (n) thereof as illustrated
`in Equation (6).
`
`Figure 3. Pressure Pulsations Versus Percent of BEP Operation.
`
`Impeller Outlet Velocity
`
`Experience indicates that pressure pulsations in geometrically
`similar pumps roughly increase with the square of the circumferential
`speed as shown in Equations (11) and (12) for US units and SI
`units, respectively:
`
`Figure 2. Wake Flow at Impeller Outlet.
`
`Page 3 of 26
`
`

`

`84
`
`PROCEEDINGS OF THE TWENTY-FOURTH INTERNATIONAL PUMP USERS SYMPOSIUM • 2008
`
`The stagnation pressure ∆Pd is a measure of the unsteady
`hydrodynamic load acting on the volute or diffuser vane.
`
`Other Influencing Parameters
`
`Parameters affecting the nonuniformity of the impeller wake
`flow and geometric parameters have an influence on the generation
`and amplitude of pressure pulsations. Among these parameters are:
`• Thickness and form of the impeller vane trailing edge.
`• Number of impeller vanes and combination of impeller vanes to
`volute vanes/diffuser vanes (e.g., odd versus even number of
`impeller vanes in double volute type pumps).
`• Impeller specific speed. Pressure pulsations generally increase
`with increasing specific speed.
`• Impeller geometry, particularly vane exit angle and shape of
`volute cutwater (blunt versus hydraulically smooth).
`• Staggering of impellers along shaft of multistage pumps and
`staggering of the two halves of double suction impellers.
`
`Vane pass pressure pulsations primarily affect structural vibrations
`(e.g., bearing housing vibrations). A certain pressure pulsation
`level will always be present in centrifugal pumps and does not need
`to represent a problem. Excessive pressure pulsations can result in
`high pump and piping vibrations, particularly in combination with
`structural resonance or acoustic resonance. This may result in
`vibration levels beyond alarm or shutdown and cause fatigue
`failures in auxiliary piping, instrumentation, etc.
`
`Other Excitation Sources
`
`Other excitation sources include broadband hydraulic forces due
`to recirculation, cavitation, and forces due to rotating stall.
`
`Amplifiers
`
`Vibration levels usually become excessive when amplified by
`resonance. A resonance condition occurs when an excitation
`frequency is within a few percent of a relevant natural frequency. In
`that condition, the excitation force is acting again once the vibrating
`component has come full cycle after the last “impact” by the force.
`The excitation force and the vibration are synchronized, and the
`vibration amplitude increases until limited by nonlinear effects.
`With regard to mechanical vibrations in VFD operated centrifugal
`pumps, resonance conditions can be divided into four categories:
`structural resonance and torsional rotor resonance are typically
`lowly damped and are likely to result in high levels of mechanical
`vibration when properly excited. Lateral rotor resonances are in
`some cases highly damped and operation on or near such a condition
`(“critical speed”) may be perfectly acceptable. Acoustic resonance
`conditions, amplifying mechanical vibrations via amplified pressure
`pulsations, are usually only lowly damped. The various resonance
`categories are discussed in detail in the following sections.
`
`LATERAL ROTORDYNAMICS
`
`General
`
`The damped lateral rotordynamic behavior of a centrifugal
`pump rotor is determined by the rotor geometry, the rotor mass and
`inertia, and the interaction forces occurring between the rotor and
`journal bearings, annular seals, and casing. Impeller wear rings,
`close-clearance bushings, and balance pistons are typical examples
`of annular seals. Casing interaction occurs at impeller location,
`between the wear rings in case of a closed impeller design, and is
`generally destabilizing. These interaction forces are nonlinear but
`may be linearized around a particular static rotor equilibrium
`position. Interaction forces vary with operating speed, pumpage
`specific gravity and viscosity, load, state of wear, etc. Solving the
`linearized homogeneous Equation of Motion (13) results in a set
`of eigenvalues.
`
`.
`
`..
`
`The linearized equation of motion consists of a stiffness-matrix
`K, damping-matrix D, mass-matrix M, and vectors of displacements
`(x), velocity (x), and acceleration (x). The complex eigenvalue λX
`is defined in Equation (14). The imaginary component ␻X represents
`the angular natural frequency. The eigenvalue determines the
`corresponding natural frequency f (Equation [15]), modal damping
`value D (Equation [16]), and mode shape as shown in Figure 4. A
`mode with negative damping D represents an instable system. The
`rotor system is laterally stable when all significant modes provide
`positive modal damping levels.
`
`Figure 4. Lateral Mode Shapes and Mechanical Model.
`
`The evaluation of the lateral rotordynamic behavior can either be
`done by solving the homogeneous equation of motion (eigenvalue
`calculation) or by specifying a set of excitation forces and subsequent
`solution of the nonhomogeneous equation of motion (forced
`response analysis).
`The eigenvalue approach and evaluation of results applying a
`combined frequency-versus-damping-ratio criterion is further
`discussed in this paper. This approach is less ambiguous compared
`to a forced response analysis because it avoids the subjective
`process of determining and applying excitation forces (typically a
`combination of mechanical and hydraulic unbalance loads).
`The results of a damped lateral rotordynamic analysis are best
`presented in the form of a Campbell diagram as illustrated in
`Figure 5, plotting the natural frequencies and modal damping
`factors versus pump operating speed. The intersection between
`a speed-dependent natural frequency line and the synchronous
`speed excitation line is called a critical speed and represents a
`resonance condition.
`
`Page 4 of 26
`
`

`

`ANALYSIS GUIDE FOR
`VARIABLE FREQUENCY DRIVE OPERATED CENTRIFUGAL PUMPS
`
`85
`
`rotor
`
`increasing lateral
`
`acceptance
`the modal-damping-versus-frequency-separation
`criterion illustrated in Figure 6 as binding. With a few exceptions
`not discussed in this paper, rotor designs in violation should be
`modified to meet this acceptance criterion. Design modifications
`aimed at improving lateral rotordynamic stability can be divided in
`two categories. A first category aims at increasing the frequency
`separation margin between lateral modes and synchronous excitation
`speed by means of increasing the rotor stiffness (Kxx) or reducing
`the rotor mass (M). A second category of modifications intends to
`increase modal damping.
`Design modifications aimed at
`natural frequencies:
`• Decreasing coupling overhung length and/or coupling weight in
`case of overhung dominated modes (Kxx, M)
`• Changing of impeller material from steel to aluminum (M)
`• Increasing shaft size (Kxx)
`• Decreasing between-bearing span (Kxx)
`• Tightening or restoring annular seal clearances (Kxx)
`• Eliminating stator-side serrations applied to reduce leakage (Kxx)
`• Applying stator-side circumferential grooves in balance pistons
`and center and throttle bushings. This reduces the adverse effect of
`piston tilting onto the direct radial annular seal stiffness (Kxx).
`• Changing from inline to back-to-back configuration, which
`reduces bearing spans and also adds damping at the center of the
`pump (Kxx)
`
`Design modifications aimed at increasing modal damping:
`• Applying stator-side swirl breaks at
`the entrance of
`impeller eye wear rings. This reduces the circumferential inlet
`swirl, which in turn reduces destabilizing annular seal
`cross-coupled stiffness.
`• Optimizing the journal bearing design. Journal bearings with
`length-over-diameter ratios above one should be avoided. The
`destabilizing effect of cross-coupled journal bearing stiffness can
`be reduced or eliminated by switching from cylindrical bearings to
`multilobe or tilting-pad designs.
`• Loading of vertical pump line-shaft bearings by means of
`applying intentional misalignment between bearings and rotor
`• Applying rough annular seal stator surface finish, stator-side
`honeycomb or hole patterns, smooth rotor surface finish
`
`In case a fixed speed centrifugal pump is converted to VFD
`operation, the continuous VFD operating-speed range may already
`be sufficiently analyzed and covered by the original fixed-speed
`lateral rotordynamic analysis. Fixed-speed lateral analyses should
`be carefully reviewed on a case-by-case basis before deciding
`whether a new lateral analysis for the VFD operated application is
`necessary or not.
`
`Case Study—Standard Lateral
`Rotordynamic Analysis Procedure
`
`The dynamic lateral stiffness and damping levels provided by
`the journal bearing fluid film depend on the stiffness of the bearing
`housing/support structure itself. With most horizontal pumps, the
`lowest bearing housing natural frequency is well above the first few
`lateral rotor bending mode natural frequencies. In these cases, the
`corresponding journal bearing support stiffness can be considered
`as near-rigid and constant over the entire speed range of
`concern. The situation is entirely different with most vertical pump
`applications. Vertical pumps are typically structurally flexible and
`significant structural modes may appear at, near, or below
`operating speed. This requires calculation of the dynamic bearing
`support stiffness by means of performing harmonic response
`
`Figure 5. Campbell Diagram.
`
`A widely used eigenvalue acceptance criterion is defined in
`Annex I of the API 610 standard (Eighth Edition, 1995; Ninth
`Edition, 2003; Tenth Edition, 2004) and the ISO 13709 standard
`(2003), respectively. The combined frequency-versus-damping
`criterion, depicted in Figure 6, is applied to each of the calculated
`lateral modes, limiting the evaluation to modes within a natural
`frequency range of zero to 2.2 times running speed. This frequency
`range covers the most typical and significant rotor lateral excitation
`forces including subsynchronous excitation, mechanical and
`hydraulic unbalance and misalignment:
`• Subsynchronous excitation (journal bearings): 0.4… 0.5× running
`speed (typical)
`• Subsynchronous excitation (annular seals): 0.7… 0.9× running
`speed (typical)
`• Rubbing: multiples of 0.5×
`• Mechanical unbalance: 1× running speed
`• Hydraulic unbalance: 1× running speed
`• Misalignment: 1× and 2× running speed
`
`Figure 6. API 610 Lateral Rotordynamic Acceptance Criterion.
`
`The API 610 (2004) lateral evaluation criterion also requires
`consideration of operating speeds outside of the defined pump
`continuous operating speed range. A speed range of 25 percent of
`minimum continuous speed to 125 percent of maximum continuous
`operating speed needs to be investigated.
`While, for cases in violation, the cited API standards allow
`proof of acceptability by means performing additional unbalance
`forced response analysis, this approach is not recommended for
`obvious reasons. An unbalance forced response analysis applies
`excitation sources at synchronous speed (1×), which cannot excite
`subsynchronous modes. It is therefore recommended to consider
`
`Page 5 of 26
`
`

`

`86
`
`PROCEEDINGS OF THE TWENTY-FOURTH INTERNATIONAL PUMP USERS SYMPOSIUM • 2008
`
`Figure 9. Amplitude Response at Upper Motor Bearing.
`
`Figure 10 shows a Bode plot of the dynamic response at the
`suction-bell bearing due to harmonic excitation at the same location.
`Amplitude responses at approximately 2.5 Hz and 20.5 Hz correspond
`to the first and third structural mode natural frequencies.
`
`analyses of the entire pump and motor structure. Alternatively a
`rotordynamic code capable of performing a combined rotor-structure
`analysis may be employed.
`The process of vertical pump lateral rotordynamic analysis is
`explained using the example of a water transport booster pump.
`The vertical turbine is VFD operated with a continuous operating
`speed range of 257 to 514 rpm and driven by a 900 horsepower
`asynchronous electric motor. The pump featuring a 42 inch
`diameter discharge nozzle and 48 inch diameter columns generates
`45 feet of differential head, moving 55,555 gpm while running at
`514 rpm rated speed.
`The forced harmonic response analyses were performed
`applying a general purpose finite element software program using
`beam elements for modeling of motor and pump structure. The
`inertia and mass effects of the rotating element were considered
`applying lumped mass elements attached to the structure at
`bearing holder and impeller locations. Internal and surrounding
`water effects were considered as additional mass effects applied
`to the beam elements. The stiffness effect of the receiver-can
`flange and gusseting was determined in separate static finite
`element (FE) analyses and considered via spring elements in the
`forced harmonic response model. Figure 7 depicts the calculated
`static receiver-can deflection due to unit-moment loading in the
`horizontal direction.
`
`Figure 7. Calculated Receiver Can Flange Deflection.
`
`Figure 8 shows the first three structural modes of the pump and
`motor structure as a result of a modal structural FE analysis. The
`modes occur at 2.6 Hz (156 cpm, first column mode), 14.2 Hz (852
`cpm, first aboveground mode), and 20.5 Hz (1230 cpm, second
`column mode), respectively. The dynamic bearing support stiffnesses
`required for the lateral rotordynamic analysis are calculated in a
`series of forced harmonic response analyses. Each of these
`analyses applied a dynamic unit-load at a single bearing location
`and varied the frequency of this harmonic load within a specified
`frequency-range. Figure 9 illustrates the dynamic displacement
`response at the upper motor bearing due to harmonic excitation
`(unit force in lateral direction acting at the same location). The
`response indicates an amplitude response peak at approximately 15
`Hz, which corresponds to the second structural mode showing large
`modal displacement in the aboveground portion of the machine.
`
`Figure 10. Bode Plot of Response at Suction-Bell.
`
`The calculated displacement curves UXX(f) are inverted to
`derive the various dynamic stiffness curves KXX(f) as indicated in
`Equation (17) and displayed in Figure 11.
`
`Figure 8. First Three Pump Structural Mode Shapes.
`
`Page 6 of 26
`
`

`

`ANALYSIS GUIDE FOR
`VARIABLE FREQUENCY DRIVE OPERATED CENTRIFUGAL PUMPS
`
`87
`
`Case Study—Rotor Instability
`
`Vertical pump bearings and bushings are typically only lightly
`loaded, which may cause rotor instability problems. The phenomenon
`is explained on the example of a vertical mixer pump intended
`for VFD operation in slurry mixing service. The single-stage,
`quad-volute pump, driven by a 200 horsepower asynchronous
`motor, underwent an endurance test at the OEM factory. The pump
`design features seven columns of 16 inch diameter, supporting the
`2 inch diameter shaft at seven line-shaft bearing locations. The
`bearing material is nickel-impregnated carbon mounted in bearing
`holders that were initially loosely positioned between column
`flanges. At startup, shaft vibration readings taken at the bottom seal
`location showed two distinct peaks of both approximately 1.2 mils
`peak-to-peak at 1× and 0.5× running speed frequency. The 50
`percent running speed frequency represents “bearing whirl.” The
`bearing whirl frequency decreased almost
`linearly over time
`starting from 18.5 Hz, indicating wearing out carbon bushings. A
`post-test inspection revealed that the diametrical bushing clearances
`increased from an initial 7 mils design clearance to as much as 400
`mils. The failed test triggered analyses and modal testing. Lowly
`damped lateral rotor modes at approximately 50 percent running
`speed and structural column modes at 0.2, 4.5, 14.0, and 29.0 Hz
`were calculated and support-structure modes at 10 and 26 Hz
`were measured by means of modal testing. Figure 14 illustrates
`the frequency and amplitude of column vibrations filtered to
`synchronous speed and bearing whirl frequency, respectively.
`The bearing whirl frequency decreases linearly with time,
`obviously because of continuous wear of the carbon bushings.
`After approximately 35 hours, the bearing whirl frequency started
`to lock into the third column natural frequency for a few hours
`(bearing whip condition), resulting in amplified vibration levels.
`
`Figure 11. Dynamic Bearing Support Stiffness KXX(f) at Suction Bell.
`
`The subsequent lateral rotordynamic analysis applied unbalance
`loads based on an ISO 1940-1 Grade G2.5 (equivalent to an API
`balance grade of 15 W/N; IRD Balancing, 2007), evaluated at rated
`operating speed. The corresponding unbalance forces in the lateral
`forced response analysis are varied proportional to the square of
`the operating speed. The significant shaft response peaks lay
`outside of the continuous VFD operating speed range and are
`therefore not an operational concern. However, the analysis clearly
`proves that shaft amplitude response peaks can occur at the natural
`frequencies of the structure (refer to Figures 12 and 13 for details).
`
`Figure 12. Lateral Shaft Response Amplitude at Upper Bearing.
`
`Figure 13. Lateral Shaft Response Amplitude at Suction Bell.
`
`Figure 14: Column Vibration and Natural Frequencies.
`
`The pump was successfully retested after implementation of
`design changes, which included fixation of the initially loose
`bearing holders, built-in misalignment between the bearing holders
`
`Page 7 of 26
`
`

`

`88
`
`PROCEEDINGS OF THE TWENTY-FOURTH INTERNATIONAL PUMP USERS SYMPOSIUM • 2008
`
`and the shaft, as well as structural modification of the pump
`support structure. The successful retest did not show any significant
`bushing wear and the bearing whirl frequency remained constant
`throughout the test.
`
`Conclusions
`
`Horizontal Pumps
`
`When converting a fixed speed application to VFD operation,
`the original lateral rotordynamic analysis may already sufficiently
`cover the new application. However, a careful review of the results
`of the original analysis and the new operating conditions is
`required on a case-by-case basis. The speed range above the
`original fixed operating speed requires special attention because of
`the increased power absorption and the increased likelihood of
`encountering critical speeds, for example critical speeds with
`coupling overhung dominated modes. The primary excitation
`mechanism of concern is synchronous speed excitation due to
`mechanical and hydraulic unbalance. API 610 Eighth through
`Tenth Editions (1995, 2003, 2004), Annex I, defines a reasonable
`lateral rotordynamic acceptance criterion.
`
`Vertical Pumps
`
`A significant coupling between the dynamic behavior of the
`structure and lateral rotordynamics is often the case. The frequency
`of lateral modes can coincide with structural natural frequencies,
`resulting in amplified shaft vibrations. The correct assessment of a
`vertical pump lateral rotordynamic behavior requires a combined
`lateral-structural analysis covering the entire continuous VFD
`operating speed range. Significant structural natural frequencies
`at 1× and 0.5× running speed should be avoided. The primary
`excitation mechanisms of concern are synchronous speed excitation
`(unbalance) and subsynchronous speed excitation around 50
`percent of running speed (bearing whirl).
`In case a resonance condition cannot be detuned, it is also
`possible to lock out/program out a specific operating speed range
`from continuous operation.
`
`TORSIONAL ROTORDYNAMICS
`
`General
`
`Torsional vibration problems usually involve resonance conditions.
`Torsional
`resonance occurs when a train torsional natural
`frequency c