Practical Approach for Solving Vibrations of Large Turbine and Generator Rotors - Reconciling the Discord between Theory and Practice

Transcription

1 SIRM th International Conference on Vibrations in Rotating Machines, Graz, Austria, February 2017 Practical Approach for Solving Vibrations of Large Turbine and Generator Rotors - Reconciling the Discord between Theory and Practice Zlatan Racic 1, Marin Racic 2 1 Owner-Director, Z-R Consulting, 34209, Bradenton, Florida, USA, zlatanraco@aol.com 2 Z-R Consulting, 53130, Milwaukee, Wisconsin, USA, mzracic@gmail.com Abstract The purpose of this paper is to illustrate a different perspective in viewing vibration problems in large rotating machines, regarding the commonly seen discord between theoretical predictions of dynamic behavior, especially the standard predicted and expected "fixes" to many vibration problems, versus observed operation in practice when unexpected vibration problems still remain or arise anew. The paper will also discuss the key root causes behind this discord, and behind unexpected or unexplainable machine vibration in operation, usually after a major outage. In short, the primary cause in a substantial portion of such cases is the presence of significant, axially distributed mass eccentricities, which have a profound effect on the dynamic behavior of large flexible turbine-generator rotors. These cases require a different approach versus the methodology traditionally utilized for identifying and resolving unbalance in general on a variety of rotating machines of different sizes and operating speeds. The paper also presents and describes an improved rotor balancing approach when dealing with such cases. In particular, the focus must be on properly compensating the asymmetry of the mass axis of the rotor (its natural centroidal axis, considering eccentricity) relative to its designed or intended geometric center axis, without bending the rotor at any speed. When balancing significantly eccentric rotors, it is necessary to follow a new balancing method using 2N+1 balancing planes, where "N" is the highest mode reached in operation. 1 Introduction and Overview It s tempting to write off the special cases where expected turbine-generator rotor behavior diverges from expectations, and where solutions to vibration problems fail, by ascribing the problem to unknowable variables or nonlinearity. However, experience has suggested a somewhat consistent framework around these nonstandard problems and their true root causes, which are usually rooted in the presence of significant rotor mass eccentricities or bows. In a wider sense, these causes ultimately and indirectly originate in the application of rotor condition assumptions used within theoretical rotordynamics for predictive modelling purposes, and carrying and applying these same assumptions to practical shop and field practice. These assumptions are primarily related to identifying rotor eccentricities and runouts, and to balancing methods of large flexible turbine and generator rotors on low speed or high speed balancing machines. This is not to point out a problem with modern rotordynamics theory, but rather with the practical means of application of the theory to oftenunverified real-life rotor conditions, where a rotor does not adhere to the boundary conditions and assumptions required within the theory for its predictions to be applicable. Practicing technologists who base shop balancing procedures on traditional theory often neglect the particular significance of existing distributed mass eccentricities within the rotor, assuming them as equivalent to "unbalance mass", and thereby focus balancing efforts on resolving dynamic modal displacement responses observed at critical speeds. We consider "unbalance mass" here as localized mass(es) that can produce centrifugal force or elastic bending of the accelerated, spinning rotor, but that don't otherwise statically shift or alter the position or skew of the axial mean mass axis of the rotor away from its designed geometric axis (that is, the ideally straight line connecting all journal and coupling centers). However, the presence of axially distributed mass eccentricities, such as a permanent bow for example, does statically shift or skew the position of the mean mass axis. This has a profound effect on the dynamic behavior of flexible turbine and generator rotors in operation, and requires a different perspective than that traditionally utilized for identifying and resolving unbalance. The primary focus needs to be on the often-overlooked overall static asymmetry of the mass axis of the rotor relative to its geometric or design axis, and on rigid-mode behavior, which is usually the source of unexpected dynamic behavior, especially at speeds at and above the fundamental system resonance velocity (first critical speed).

2 To better conceptually define the idea meant by rigid-mode behavior in this paper, consider the motion of a bowed flexible rotor at higher speed which is undergoing some amount of induced elastic bending while in some orbital lateral translational or pivotal rocking motion. We can conceptually dissociate this total rotor motion into two parts, a rigid component and a flexible component. The rigid-mode component of the motion is representative of the dynamic motion of the rotor in its innate bowed shape. This is generally limited to motion of the journals within the oil film of hydrodynamic bearings. The flexible or elastic-bending component is representative of any superimposed bending deviation of the rotor from its innate shape in any radial plane. The rigid-mode rotor motion is tied purely to any distributed mass eccentricity on the rotor and to its effect on the position and skew of the rotor s mean mass axis relative to its intended inertial axis. The elastic-bending component is tied to the centrifugal force generated at any given speed by any ponts of unbalance on the rotor. This can additionally include the centrifugal force generated by distributed mass eccentricity, and also includes any amplified modal responses from resonant excitation. There is some conceptual overlap in this visualization, in that mass eccentricity both affects the static symmetry of the rotor, and can act as unbalance to induce bending deflection, while what we re labelling as unbalance mass can induce bending deflection, but is too small or localized to alter the rotor s static symmetry and mass axis position. In diagnosing the cause of observed vibration and in identifying an ideal balancing solution, the rigid-mode behavior should be viewed and treated as fundamental, and the elastic-bending component should be secondary. Rotors in a "free" state, not coupled to other rotors and with a moment-free drive system, as on a balancing machine, naturally rotate about their true mean mass axis above the first system critical speed (within the limits of any constraints, that is). Therefore, standard balancing approaches utilizing measurements at speeds above the first critical can inadvertently balance an "unbalanced" rotor with significant distributed mass eccentricities around a centroidal rotation axis (the mean mass axis) that is different than the operating axis to which it will be constrained when coupled in the field (its geometric axis, as defined by the line connecting the coupling centers). This can frequently lead to unexpected or stubborn vibration issues upon the restart of a unit in a power plant following a major outage. This recognition of rotational axis change also explains why the rigid-mode eccentricities of a rotor body (any permanent bow or distributed asymmetry on the rotor between the constraints/bearings) need to be fully corrected (statically compensated, restoring the symmetry) in a balancing facility at rotational speeds at or below the fundamental critical speed of the rotor-bearing system, and why certain defects (static runouts of couplings and journals) must be resolved first by machining, prior to balancing 1. Lastly, when balancing significantly eccentric rotors, it is necessary to follow a new balancing method using 2N+1 balancing planes (similar to a method applied in practice by GE [2]), where "N" is the highest mode reached in operation. During the balancing process when utilizing the 2N+1 balancing method, correction masses are simultaneously placed first (axially) symmetrically, and then redistributed axially if needed, prior to proceeding with the more widely applied standard method of modal balancing using influence coefficients. 2 Common Points of Discord between Theory and Practice Many engineers likely recognize that at any number of power plants, turbine-generator rotor vibration problems go on sometimes for years without a proper resolution, despite the best efforts of OEMs and plant engineers and consultants. In many such cases, the vibration behavior observed and measured in the field does not conveniently correspond to the expected or predicted behavior of standard rotordynamic theory. Likewise, the suspected root causes and proposed solutions predicted by standard theory do not always reliably resolve the turbine-generator vibration problems at hand. It is important to state up front that there is no fundamental problem or any gross error within classical mechanics or associated theory of rotordynamics. It has been very well proven in its predictive ability in machine design over many decades of development, and has been developed to take into account the existence of rotor defects such as bows and mass eccentricities, and effects of rotor train and bearing misalignment. The problem leading to the discord between theoretical predictions and real-life observations of vibration problems in power plants is based in greatest part on the assumptions used when developing a theoretical dynamic model of the system. When attempting to solve a real vibration problem based on theoretical expectations, it is crucial to understand the framework and boundaries within which the system model was constructed. Theoretical simulations based on closed system models by necessity must view and represent real-life complex open systems under a set of closed-system assumptions. It would be mathematically impossible to accurately model a complete, open system with multitudes of parametric variables and perpetual inputs and outputs of an open system model [1]. The combined turbine-generator rotor-bearing system can be thought of as comprising two interconnected equilibriums, one being the non-rotating static equilibrium between the shaft and the bearing, and the other being the rotating quasi-dynamic equilibrium of the forces in the spinning rotor itself. Each of these 1 Flexible rotor overhangs with couplings and their associated dynamic effects, either from mass eccentricities or from gyroscopic forces are a more complex and separate topic not considered in this paper.

3 equilibriums affects the other, and instability in one can produce instability in the other. The non-rotating, static equilibrium of the rotor position in the bearings remains generally stable (only following the shaft centerline path), held by gravity load which is constant, and oil hydrodynamic forces which govern the rotor s elevation in the bearings. The rotating dynamic equilibrium remains referenced to the geometric center of the shaft or its neutral centerline. The dynamic forces originating from rotor rotation/orbit are vectorially summed with the static forces from the static equilibrium in the bearings, with the net summation then governing the position and orientation of the rotor in its bearings. In an eccentric rotor, the internal moments and forces dynamically generated by eccentric mass will at some sufficient speed overcome the gravity load forces acting on the rotor, and will alter and govern the rotor s orientation in its static bearing equilibrium. In traditional rotordynamics theory, one key assumption is the general static radial symmetry of a rotor about its geometric axis. This is the straight line connecting its coupling and journal centers, or the intendedcentroidal axis, about which a rotor is constrained to spin while in operation if coupled in the field. In this view, any mass asymmetry present is considered as an equivalent mass unbalance, and it is referenced to and superimposed upon this geometric axis. All mass eccentricity and unbalance is furthermore considered and assumed as additive under the principle of superposition, as equivalent to a single mass unbalance of some total mass and at some single axial position and phase angle reference. Additionally, the geometric axis is maintained as a consistent and singular absolute frame of reference to define the predicted motion (oscillation or deflection) of the rotor body within the modeled speed range. Standard models present the resulting deflection or bending response of the rotor as the motion of an equivalent point mass in an analogous linear spring-mass-damper system driven in harmonic or cyclic oscillation. Any mass asymmetry is calculated and represented as an equivalent, rotating centrifugal force (m*e*ω 2 ), pulling outward on the rotor with a line of action or point of origination at this geometric axis, which then is translated into a sinusoidal forcing function driving a linear mass-spring-damper system. To approximate real behavior, two such orthogonal motions in a common radial plane can then be vectorially added and taken to represent lateral orbital motion, and create a two-dimensional representation of a shaft displacement orbit (at any radial plane(s) represented, either with discrete axial steps or axially continuous). This representation works very well at modeling critical speed responses, and through the use of damping in the model, simulates the phase shift through a linear-motion analog. The real generated vibration of a rotor is not a single simple motion and is not truly as modeled by a linear mass-spring-damper analog, since the vibration of the rotor itself is really a predominantly synchronous translational motion in an orbit, without any real cyclic oscillation of the shaft itself (if the orbit is circular). That said, there still is a standard oscillatory component within the total rotor-bearing system response. Although the vibrating rotor is really synchronously rotating while in a translating orbit (which we refer to as "precession" here), its bearing support (oil film primarily, but also the shell, housing and pedestal) can be viewed as a linear, non-rotating spring, with an effective line of action matching the attitude angle of the rotor in the bearing. However, the representation of rotor motion using a mass-spring-damper analog neglects the full effect of the offset or skewed mass axis of the rotor, created by the presence of significant distributed mass eccentricities or permanent bows, which produces notable dynamic effects during rotor acceleration under input torque. Similarly, a key limitation of this linearized representation is its persistent global coordinate reference (assumed also to be an inertial reference frame) to the geometric axis of the rotor as the point of origination of all motion and forces through the full speed range. All rotation is represented as being centered about the geometric axis of the rotor, and all centrifugal force always acts in a line originating from this geometric axis as well. In such a model, all rotation and all motion is assumed to naturally occur about this single rotor axis. This fact alone provides balancers in service shops and in the field incorrect justification to ignore the true effects of eccentricities on dynamic rotor behavior in practice. By extension, this representation suggests that any reduction of vibration displacement amplitudes by balancing is to be achieved solely through countering the centrifugal driving force about this same geometric axis, under the assumption that this is a consistent representation of rotor motion at any speed while balancing a rotor in free-free mode in a balancing facility (not coupled to other rotors, driven by a moment-free connection), or as assembled in the field. If dealing with localized unbalances on an otherwise effectively concentric rotor, this is an accurate representation and works well, as long as the correction mass is placed at the same axial location as the unbalance mass, not creating any internal bending moment. The primary point of discord between this modeled representation and practical application comes about in the presence of significant, distributed mass eccentricity, particularly in more flexible rotors. (An applicable threshold of significant, by the author's experience is about 0.050mm (0.002, 0-peak) of 1x radial rotor body eccentricity.) In a significantly eccentric rotor, the natural tendency of rotation is not about the geometric axis, as is modeled, but is about the mean mass axis of the rotor, rotating in a state of natural dynamic symmetry by the conservation of angular momentum. In operating machines, rotors are often held and constrained to rotate about a specific static designed axis when attached or fixed in some manner, and this position may or may not correspond in practice with the rotor s true centroidal center-of-mass axis, given the potential for defects such as bows and mass eccentricities or static-bending eccentricity induced by the misalignment of multiple assembled rotors (when off-perpendicular couplings are forced together, for instance).

4 The most common, central cause in practice of persistent vibration among rotors in horizontal machines operating in a gravitational environment is distributed mass asymmetry, such that the true mass axis of the rotor is offset and/or skewed by some distance from the designed geometric axis of the rotor. When this situation is present in flexible rotor/bearing systems, it creates a unique challenge with regard to identifying an ideal balance solution, many times leading to a less than ideal result in practice once the rotor is installed in the field, compared to what was expected. The divergence between the expected post-balancing rotor behavior and the actual observed behavior is rooted in tracking the applicable rotation axis and reference frame from which the unbalance is acting, as the rotor accelerates through its speed range in the balancing facility. Vibration monitoring sensors referenced to a fixed, absolute frame, which only track the motion of the rotor surface, cannot directly discern any change in rotational axis. This change is really the transition or switching of the axis that is nearest to the physical center of the lateral translation orbit. Initially, the rotor s geometric axis is at the center, with mass axis or unbalance precessing around it (being within the rotor body). In a flexible rotor, while passing the peak of the first critical speed, the actual mass axis transitions to the center, and the geometric axis (which nonetheless still remains the axis about which drive torque is applied) effectively orbits around the mass axis. In significantly eccentric rotors, when above the first critical speed, and upon reaching a phase angle of 180 degrees, this process of switching rotation axes will create a natural measured vibration amplitude of twice the measure of the actual inherent eccentricity (runout) present in the rotor (or rotor train), as seen by sensors referenced to stationary globalized coordinates. The practical evidence of the change in axis can be seen upon studying the measured dynamic data presented in Bode plots and Nyquist (polar) plots. The static DC signal presented in shaft centerline plots can provide an indication of a corresponding shift in the quasi static position and orientation of the rotor, relative to a fixed global reference. Shaft centerline plots will also indicate the existence of a rotor bow or significant eccentric mass as a hysteresis in the path of the geometric center of the journal between rotor acceleration under torque from standstill, and deceleration without torque back to standstill again. If an eccentric rotor is coupled in a rotor train, where the geometric line of the coupling centers does not line up with the actual center-of-mass line along the rotor body, then when the rotor is accelerated, the induced mass eccentricity will produce forces, and these forces will be expressed as a vibration response, either as rotor displacement amplitudes (kinetic energy) or as seismic vibration (potential energy) at the bearings. These forces represent the energy required to maintain and hold the rotor in a state of forced, non-centroidal rotation within its constraints. Such a constraint can be either a bearing or a neighboring coupled rotor within a rotor train. These constraints do not themselves provide the originating force to hold and constrain the rotation of the rotor. The originating force is gravity, and this is applicable to any turbine and generator rotor operating horizontally and in a gravity environment. The bearing(s) provide only a reaction force against the weight of the rotor, and against any dynamic forces generated by the rotor while under rotation. If an eccentric rotor is initially accelerated with applied torque, the rotor will immediately tend to rotate about its actual center of mass, but cannot, because this motion is constrained by the force of gravity, keeping the concentric journals continually pinned against the bearings. The rotor is therefore held in forced, non-centroidal rotation. This condition effectively maintains an eccentricly precessing centroidal mass-axis within the rotorbearing system. Every point of eccentric mass will generate a net-asymmetric tangential force while undergoing rotation. Likewise, any static asymmetric mass that is maintained under rotation will produce asymmetric reactive centrifugal force radially outward from to the forced axis of rotation, equal and opposite to an asymmetric centripetal force (which results from circular motion from torque), following Newton's 3rd law of action and reaction. This force (or more accurately, asymmetric inertia) pulls the mass of the constrained flexible rotor radially outward, acting against the points of constraint (the bearings, or couplings of neighboring rotors). This induced bending deflection progresses until the peak of the first critical speed (peak measured vibration displacement amplitude at ~90 degrees phase lag). At this point, the rotor overcomes its forced noncentroidal manner of rotation, and nearly instantaneously transitions to a natural, centroidal rotation about its mean mass axis, while any dynamically induced bending deflection reduces, as the rotor approaches the end of the fundamental system critical speed range at a phase angle of 180 degrees. (A much more detailed explanation of the mechanisms of this process is found in the references [7, 8] ). To reduce or eliminate such vibration motion, rather than treat the observed response from this force as from mass unbalance, the more effective goal or viewpoint should be to first compensate for or remove the underlying static asymmetry, and restore and permit the natural tendency of the rotor to rotate about its geometric axis at all speeds, to create a condition where the designed geometric axis and the mass axis coincide. This prevents all rigid-mode motion from occurring, and in so doing, can nearly eliminate the responses at critical speeds from resonant modal excitation. This can only be accomplished if the rotor is not bent or distorted in any way by the compensating forces generated by the correction weights at any speed. Current engineering methods and predictive models in rotordynamics do not focus on the natural tendency of a rotor to rotate about its center of mass axis (as a natural state of least action ), and balancing engineers inadvertently carry the assumption that a rotor would always rotate about its designed axis, at all speeds, including across all critical speeds, with or without having defects or eccentricities. With this assumption, the standard focus on treating vibration falls simply on methods of suppressing and handling the resulting forces or

5 vibration responses observed at measuring points, by forcefully bending and pushing the rotor s geometric axis to go where the engineer wants it to go (and inadvertently creating internal bending moments), rather than to focus on how to create a symmetry that brings the rotor to that desired condition under its own natural tendency, maintaining its inherent shape. This simple but often subtly overlooked difference in concept and mindset further affects the diagnosis of rotor vibration problems and recommended engineering solutions, commonly leading to incorrectly identified root causes and in many cases leading to improperly applied rotor balancing. Figure 1: Natural behavior tendency of a permanently bowed rotor, below and above the first critical speed. 2 If the mass axis is angularly skewed, meaning the mass eccentricity is axially asymmetric, the measured amplitude will subsequently increase proportionally to speed once the rotor is further accelerated above the first system critical frequency range (supercritical), creating moments as well as generating additional radially asymmetric forces. This type of increasing amplitude trend (which is based within the rigid-mode behavior of the rotor) is commonly misconstrued as building to the rotor s second critical speed response. This is based on the observation of out-of-phase motion of the ends of the rotor, referenced to the vector angles of the responses relative to a common keyphasor mark as the rotor passes by the timing trigger sensor, referenced to globalized coordinates. This behavior occurs as a rigid-mode motion of the rotor, and can also be taken as representative of the orbiting motion of the straight-line mean mass axis of the rotor. This occurs as supercritical rotation under input torque creates centrifugal force on the residual eccentric mass present on each half of the rotor individually. This centrifugal force is a reaction against the points of constraint on each half of the rotor (each rotor half is essentially an individual modal element), which include the bearing at one end of the rotor and the axially central nodal/pivot point of the rotor s pivotal orbital motion. If the rotor is sufficiently flexible, then there will likely be additional elastic bending from the modal response at the second harmonic, which is superimposed on top of the rigid-mode pivotal rocking motion. The assumed traditional solution to this behavior is to treat the problem by balancing by using a couple shot. In a flexible rotor, this weight distribution bends back the out-of-phase ends of the rotor (against the "inertial" nodal pivot constraint at the axial mass center of the rotor). However, this does not leave the rotor balanced about its geometric axis, and the rotor retains internal bending moments and induced elastic bending deflection elsewhere within the rotor body. Meanwhile, because the real root cause of this behavior is axial asymmetry of the distributed mass eccentricity driving the rotor s rigid pivotal mode, the ideal solution instead requires an axial redistribution of any existing balancing weights while at the same time controlling their radial angle. While determining a balance solution, the second-critical resonant modal response should not be the primary focus, nor, should there even be an initial attempt to balance the second critical. Rather, the problem needs to be viewed as an extension of rigid-mode behavior and as an axial redistribution of the correction masses used to solve the fundamental critical speed response, with proper axial redistribution of correction weights. 2 Note: When a rotor is rotating at speeds below its system critical speed range, it behaves as in the upper image in the figure, since it is constrained by gravity in its bearings. Above the peak of the first critical speed range, the rotor's natural tendency is that in the lower image in the figure. However, if the rotor were unconstrained entirely, rotated in free space or rotated vertically, the natural tendency of rotation would resemble the lower figure at any speed.

6 Attempts to resolve this kind of vibration, which originates in significant eccentricity, either by field balancing, with adjacent rotors constrained by each other, or by applying traditional balancing methods to the rotor in a "free state" in a balancing facility, will not bring about the desired or expected results when the residual allowable eccentricities exceed about mm, (0.002" 0-pk), for ISO-1940 class G2.5 rotors. Or, if the measured vibration displacement amplitudes are reduced, as seen by vibration sensors at the points of constraint (the bearings), the "lower vibration" is achieved only at the expense of transforming potentially damaging forces at the reaction points at the bearings into internal cyclic bending moments elsewhere within the rotor or rotor train. The effects of these created internal moments can commonly cause certain invisible cumulative damage to the rotor over time, such as micro fretting of contact surfaces (especially coupling faces, which become offsquare), shrunk on component contacts, weakening of generator rotor insulation, axial excitation of the resonance frequency of turbine free-standing blades, etc. Without restoring true mass symmetry to the rotor, the vibration problems caused by distributed eccentricity are not truly resolved. The presence of unresolved mass eccentricities in a rotor train can also be an indirect cause of subsynchronous whirling, simply because it leads to operating misalignment or improper change of bearing gravity loading. This commonly appears at the uncoupled outboard bearings in a rotor train, or at bearings with a light gravity load, and its root cause is often incorrectly attributed to bearing instabilities or insufficient dynamic parameters of the bearing such as stiffness or damping. While it is true that in some cases the resulting response motion and forces may be better absorbed and suppressed by passive control methods, such as with a modified bearing and support design, the real root cause and driving force comes from distributed mass eccentricities, usually either from a permanently bowed rotor, or induced from rotor train misalignment affecting journal gravity loading [1]. 3 Rotor Service Shop Procedures In summary, the single most important point of discord between theory and practice is in the recognition and treatment of significant, residual distributed mass eccentricities present on flexible turbine and generator rotors. Similarly, this extends to mass eccentricities that arise within a coupled rotor train due to horizontal and vertical misalignment of the bearings, as a result of off-square or off-center couplings being used for reference during the field alignment process. The presence of unidentified and unresolved eccentricities creates numerous unique dynamic effects that can misguide diagnosticians and trouble-shooters, and create rotordynamic behavior that sometimes even appears to not follow the laws of physics, as has been personally overheard exclaimed in the field. Very often, the existence of these kinds of problems really originates in the treatment of the rotor in a service shop during an outage. Many service shops, in addition to being under economic pressure, operate under the assumption that the shop outage practices with regard to their effect on a rotor s dynamic behavior are already adequate and optimized, given that they have been performed the same way and refined and streamlined for the past 50 years or more. Additionally, these methods have been researched, developed and applied by OEMs, and are modeled and supported by standard general rotordynamics theory in academia. One key element affecting practical rotordynamics is that the shop outage practices and balancing methods, and field alignment methods and tolerances used in the industry were all created by OEMs, following the best theory, based on the installation of new machines coming from the factory with properly machined concentric rotors and square couplings, all within design tolerances, i.e. with assured static symmetry. These corresponding shop and balancing and field alignment procedures require that kind of rotor condition in order for the methods and theoretical assumptions to fully apply in practice. For these new, factory rotor conditions, the standard theoretical methods and assumptions do work well. The problem comes from applying these same methods and assumptions to used rotors in the service industry, where those assumptions on rotor coupling and eccentricity condition are not necessarily true, and not absolutely verified in practice (or corrected). In particular, perhaps the largest problematic assumption is assuming that any found static defects in symmetry can always be balanced later in a balancing facility or in the field via the applied dynamic forces from correction weights. Such an assumption is not applicable in many situations, and it is crucial to distinguish between vibration caused by permanent distributed mass eccentricity or bows (which creates a skewed and/or a permanent offset between the mass axis and geometric axis), and vibration caused by excitation from unbalance (which only trivially alters the mass axis, such as a lost blade). These two distinct sources of vibration each require their own appropriate set of assumptions and appropriate balancing approach. Models of rotordynamic vibration in standard theory focus primarily on unbalance, along with certain associated assumptions. In particular, theory assumes that unbalance will create a predictably measurable phase lag angle to point to the net radial location of the unbalance in its entirety, under the assumption that the full measured vibration response arises only from resonant excitation and induced deflection from this unbalance. In eccentric, flexible rotors, this assumption neglects the rigid-mode contribution from the constrained natural tendency to rotate about the rotor s true mass axis in the subcritical speed range. This also neglects the measured effect on phase angle from the built-in runout seen under the sensors due to the inherent nonconcentric shape of the rotor itself, even though the slow-roll amplitude is commonly considered and removed by runout subtraction in the measurement signal during the balancing process. (In most cases, using runout

7 subtraction actually removes the best indication of the presence and effect of distributed eccentricity, which would otherwise assist in recognizing the true balance condition of the rotor.) Furthermore, the model of viewing rotor vibration displacement through the critical speed range as caused solely by unbalance assumes a constant and consistent reference frame, centered about the rotor s geometric axis through the entire rotor speed range. This assumption is accurate, but only as long as the rotor is otherwise radially symmetric at each axial plane between the bearing constraints, aside from the particular point(s) of unbalance. If significant eccentricity is present, the measured vibration amplitude seen at any one instant in time is composed of the vectorial sum of the temporary bending deflections created by unbalance (which follow the standard model of phase lag behavior), plus the temporary additional bending deflections created by distributed mass eccentricity, plus the permanent rotor runout at a constant phase angle. These sources when in combination often create a distorted phase lag angle compared to what is expected. In practice, sometimes the entire first critical speed region is passed through with a total measured phase lag of only 40 degrees, as witnessed by the author for example. Balancing this response by placing weights using traditional balancing phase lag assumptions might improve the resonant response, but actually worsen the rigid-mode behavior. In such a situation, standard balancing of the first critical response may worsen the second critical response, and then balancing the second critical response again disturbs and worsens the first critical response, and so on in an endless cycle. These kinds of difficulties in practice are a strong indication of dealing with combined unbalance and distributed inherent bow or eccentricity, and prove that the fixed mass asymmetry created by the eccentricity is not being properly compensated. Furthermore, the reference frame identifying the center of orbital precession is not constant for an eccentric rotor through the full speed range, and the primary precession center changes from the rotor s geometric axis to the mean mass axis when passing through the first critical speed peak, and this transition defines the new shaft centerline position of the journals. Balancing weights calculated based on responses at speeds above the first critical will inadvertently be placed in reference to the rotor s actual mass axis (which at supercritical speeds acts as the primary axis of rotor orbital translation/precession), effectively balancing the rotor about this mass axis, and not about the geometric axis, as is assumed within the theory. In a balancing facility, when the rotor is in a free state (moment free), the balance condition might appear good. These effects likely will not be noticeable or measured by sensors tracking solely a dynamic signal (AC voltage signal), although it can be observed as a change in the static signal (DC voltage signal) showing the static position of the rotor (orbit center), thereby indicating a shift in position in space of its axis of rotation. However, if the rotor is coupled to adjacent rotors in the field, and thereby constrained to its geometric axis through the entire speed range, the rotor will experience high vibration and appear to be unbalanced. Ultimately, this type of situation occurs when the incorrect set of assumptions is used in assessing the measured behavior, treating a rotor with permanent distributed mass eccentricity as if it were an otherwise concentric rotor exhibiting a forced or resonant response from unbalance. This often leads to ineffective attempts to further diagnose the rotor vibration or to further attempt standard balancing methods, when distributed eccentricity is the true root cause of the vibration problem. 4 Rotor Balancing - Modified Approach for Bowed and Eccentric Rotors The great majority of rotor vibration problems can be effectively resolved by appropriately restoring mass symmetry to the rotor about the intended-centroidal operating axis; that is, restoring symmetry about the designed geometric axis connecting the coupling and journal centers. The only criteria are to do so without bending or distorting the rotor, which requires sufficient and proper axial distribution of balancing weights, and to perform most of the balancing procedure in practice at speeds near or below the first critical speed peak of the rotor/bearing system [2,7]. Most traditional balancing methods calculate weight placement based on a premise of bringing all displacement amplitude vectors to zero, which is not a suitable approach when balancing flexible rotors with significant eccentricity. This carries the incorrect implication that that all measured displacement is the result of rotor bending deflection, and wrongly suggests that an ideal balancing solution would lead to straightening the rotor while eliminating all bending deflection responses. In rotors with mass eccentricity, the real situation is more complex, and the rigid-mode motion and flexible resonant responses must be considered separately. Up to the first critical speed peak (rotor system resonance frequency), the measured motion seen by a sensor is a combination of the rigid-mode lateral translational motion, plus the inherent built-in eccentricity (runout) passing under the measuring probe, plus upon entering the resonant speed region includes the elastic bending from centrifugal force from residual mass unbalance and distributed mass eccentricity, which pulls outward from the geometric axis of the rotor. Above the first critical peak, the measured motion is a combination of the rigidmode pivotal rocking of the rotor due to axial asymmetry of mass eccentricity, plus inherent runout, plus elastic bending from centrifugal force as before, but now pulling outward from the actual mass axis of the rotor, not from the geometric axis. Since standard balancing approaches consider the measured displacement as a singular response from a singular cause, it becomes very difficult to properly dissociate a measured displacement amplitude measurement into its component causes and determine the ideal radial and axial weight distribution. The ultimate goal is not to un-bow, or straighten the rotor with the balancing weights, but rather balancing weights need to be distributed

8 axially across a sufficient number of balancing planes (2N+1), in essence mirroring the eccentricity distribution with the dynamic forces created from balancing weights. This weight placement should in aggregate create a virtual mass axis that coincides with inertial axis and the designed geometric axis of the rotor. When dealing with fixed eccentricity or a rotor bow, it is necessary to first fully restore radial mass symmetry to the rotor about its geometric axis at lower speeds (subcritical frequencies), in what we could consider rigid mode balancing, to ensure that all rotation remains about the geometric axis at all higher speeds (supercritical frequencies), ensuring it effectively acts as the natural centroidal axis of rotation. This must be completed first, while the rotor is operating in the lower-speed environment where gravitational force dominates its motion in the bearings, before balancing any higher critical speed resonant responses [9]. Because so much of this modified balancing process, and very often nearly all of it, is performed at lower speeds, using this balancing method can save a substantial amount of time and runs and cost in the balancing facility, which helps the bottom line and outage schedule, and at the same time gives a better result in the field.[ 3, 6, 7]. It is also important to comment on the traditional rotordynamics concepts of static unbalance" (parallel offset of the mass axis away from the geometric axis), and dynamic unbalance" (static angular skew of the mass axis), both referenced to globalized coordinates. Dynamic unbalance is really a misnomer, because it is also a static unbalance integral to the rotor. The distinguishing feature of dynamic unbalance is that it has a static axially asymmetric mass distribution. (In some unique cases, along with the axial distribution, it is also necessary to give consideration to radial angular placement, such as when axially distributed eccentricities form a "spiral".) Typically in theory, these two types of unbalance are both assumed to exist simultaneously but are treated independently, and are distinguished based on whether they excite the first critical response (in-phase lateral motion) or the second critical response (out-of-phase pivotal motion) at a particular speed. As good practice, only after the condition is achieved where the geometric axis and mass axis are effectively coincident, (within allowable limits by ISO, or OEM standards) should any further balancing at higher speeds above the first critical be performed. Typically, responses at higher speeds above the first critical are of a pivoting nature, with the rotor ends out of phase with each other, and this is traditionally balanced with a weight couple, as an independent aspect of the total balancing solution. However, this is properly effective only if the rotor is otherwise concentric, or if it is truly rigid. If dealing with a flexible rotor with axially asymmetric distributed eccentricity, placing weights in a couple can actually worsen the true balance condition, even if the displacement amplitudes at the journal measuring points may appear improved. This happens because the weight couple in many such cases acts to bend or distort the rotor about its points of constraint at the bearings or about the nodal point (inertial constraint) of the rigid pivoting motion at the rotor's axial center-of mass. At the same time, a weight couple does not actually restore mass symmetry about the rotor s geometric axis, and the placed weights may even worsen the net mass asymmetry. As described earlier, this leads to residual vibration problems once the rotor is constrained to its geometric axis in the field. In practice, if an out-of-phase, rocking motion is seen on an eccentric rotor above the first critical speed, the necessary solution is instead an axial redistribution of the correction weights used to resolve the static response of the first critical (in-phase ends), without using additional correction weights. This can be done correctly only in a balancing facility, not in the field with limited access of balancing planes. This approach restores the original symmetry of the rotor about its geometric axis, leading to a far more effective balancing solution. As previously mentioned, on large turbine-generator rotors, the threshold of distributed eccentricity that requires a modified balancing approach and alternate set of assumptions on expected behavior is approximately mm (0.002 ; 0-pk). First, though, we have to distinguish if rotor eccentricity is on the main rotor body between the journals or if it is outboard of the journals. Any eccentricity beyond about mm (0.001 ; 0-pk) that is outboard of the journals can only be resolved by machining, and cannot be resolved by balancing [4], but it can greatly influence the overall balance condition of the assembled turbine-generator rotor train. However, nearly any amount of rotor body eccentricity, meaning mass eccentricity in between the two journals, can be successfully balanced and resolved in a balancing facility, or on a low speed balancing machine with hard supports, as long as certain requirements are met and a specific unique balancing procedure utilizing 2N+1 balancing planes is applied. [2, 7] To guarantee reliable long-term operation in the field of an eccentric flexible rotor running above its first critical speed, it is also crucial to not introduce any bending or distortion into the rotor during balancing, which results when using too concentrated of weight placement (often due to an incorrect assumption of taking distributed mass eccentricity as equivalent to some unbalance in a single plane). Many balancers attempt to balance rotors with inherent permanent bows or eccentricities by placing excessive amounts of weights in an effort to push and contort the bowed or eccentric rotor back into a straight line about its geometric axis. In the process, the rotor is not truly balanced, but only eventually the nodal points of the mode shape are shifted under the measurement sensors at the journals, creating the appearance of a balanced rotor, while the weights create internal bending moments and leave the rotor in a distorted shape in operation. These induced cyclic bending moments are the cause of many unexplainable damages found on turbine rotors in the form of turbine blade cracking, and even more commonly on generator rotors, in the form of premature electrical faults and various stress cracks. All of these faults are usually found on machines during inspections in a subsequent major outage. Other common examples of related faults include the development of off-square coupling faces on rigid couplings, and damage and seizing of coupling bolts.

9 5 Applying Rigid Mode Balancing" to Flexible Rotors The key requirement of the rotor itself for balancing when any significant eccentricity is present is that the rotor must have at least three balancing planes, two endplanes and a midplane. If a rotor has more "significant" body runout (> 0.050mm, or > 0.002", 0-pk), but only has two balancing planes on the ends, then a third plane, either a machined grove or bolt-holes must be added at the midplane, or it can be practically guaranteed that the rotor will cause problems in the field after shop balancing, even if the vibration amplitudes in a balancing facility when using only two planes might look successful. If a third, center balancing plane cannot be added due to material conditions or thermal stresses, then the only chance of good operation in the field is to remove the eccentricity in the shop either by machining the full rotor to a new centerline, by truing the centers, or by thermal straightening, to bring the rotor body runout under the threshold of 0.050mm (2 mils), as referenced to the journals [4]. Historically, there have been numerous proposals and experiments of balancing in three planes, but they have not yet been raised to a level of general acceptance within established theories to become scientifically approved as necessary, at least as far as the rotor service industry is concerned. To have a proper and effective way of balancing and resolving the system responses of a flexible, bowed or eccentric rotor, there was a need to develop a new balancing method [7]. This balancing method is a hybrid of rigid-mode balancing combined with the standard influence coefficient method, and is based on displacement readings, though it could also be applied based on measured bearing reaction forces [2]. To account for rotor flexibility, this balancing method is based on the inverse of a particular Finite Element modeling requirement. In the FE modeling of rotors, the minimum number of nodes of modal elements required in the model to ensure that no rotor modes are missed in calculation, based on the highest mode in the rotor s operating range, works out to the formula 2N+1, where N is the highest mode reached in rotor operation [1]. This correlates to the minimum number of axial divisions (or balancing planes) required to divide a rotor by, to ensure that each resulting division or element will then behave sufficiently as a rigid beam through the full speed range. A well known idea in balancing is that any purely rigid rotor can be fully balanced in any two balancing planes. By taking a flexible rotor and dividing it using 2N+1 divisions (nodal points) as in the FE model, each resulting division or element can be assured to behave as a fully rigid beam, with a defined number of constraints and degrees of freedom. As a result, each element can be properly balanced as a rigid beam in two planes. The inner planes shared by two neighboring elements are essentially used twice. This means that any correction weights for a flexible rotor passing through the first critical speed (frequency region of the of the fundamental harmonic resonance of the rotor system), or solving the so-called static component of the rotor response effectively compensating the translational rigid mode, must initially be placed simultaneously in three balancing planes, symmetrically axially distributed. When dealing with a more flexible rotor operating above its second mode (N=2), a minimum of five planes may be used to compensate for the mode seen at the second harmonic. This second mode is a result of an axially asymmetric distribution of the rotor mass eccentricity. The second mode response (second harmonic of the rotor, superimposed over the rocking rigid-mode) can be potentially resolved in three planes by axially redistributing the initial static correction used for the first rigid-mode response without adding any additional correction masses. In cases of a second mode for more slender, flexible rotors, the proper axial redistribution may have to be spread among five balancing planes, and determined with additional trial runs with pairs of weights to obtain additional influence coefficients for proper redistribution. The weights given by a single influence coefficient calculation (from a pair of equivalent forces) are actually placed simultaneously as a pair within each end of a single modal element. This axial redistribution is optimized through trial runs as necessary until all dynamic cyclic reaction forces in the bearings are vanished or displacements are minimized [2]. This latter procedure is not performed at the Figure 2: Representation of rotor element divisions and weight placements in the 2N+1 plane balancing method.

10 actual speed of the second critical resonant response, but rather at a speed not more than up to 50% above the first critical [7]. If a very flexible rotor (such as newer types of generator rotors) operates at or above its first flexural mode, or its so-called third critical, then the final trim balancing to achieve desired amplitude limit values in operation is determined using modal influence coefficients, but in this stage is run and measured at speed. It is even better to balance at rotor overspeed to assure a well-balanced condition when placed in the field, where different bearings and support stiffness may alter the relative response and speed of the third critical. The trial weight sets must be placed following the well known modal balancing weight distributions, where the sum of the forces and sum of the moments of the weight distribution is zero, so as not to disturb the already corrected rigid-mode solution by inadvertently altering the corrected net mass axis of the rotor [5]. Utilizing the 2N+1 plane balancing method is beneficial for balancing any rotor, but it is absolutely essential when balancing any large flexible rotor with "significant" residual eccentricity. Overall, it is essential that the rigid mode forces are resolved first at lower speeds, before even considering balancing at higher critical speeds [9]. Because this balancing procedure is done mostly at lower speeds, the author of the original iteration of this approach [2] named it Pseudo-High Speed Balancing. The modified balancing method described in this paper has therefore been named similarly as the Quasi-High Speed Balancing Method [7]. As a result of using this axial distribution of balancing weights, any rotor will maintain its inherent shape, even if it is bowed, without any distortion during operation, which can be termed as running dynamically straight i.e. with minimum residual error [3]. This prevents any internal cyclic bending moments, which can be a problem with generator rotors especially, and prevents high cyclic forces being transmitted into the bearings, which often happens when standard balancing methods are applied to flexible and eccentric rotors in the balancing facility, or when balancing an assembled rotor train in the field. The rotor also remains balanced about and spinning and orbiting about its geometric axis at all speeds, and behaves as if it were concentric, since now the geometric axis and mass axis are coincident. The most important factor is that this method restores symmetry to the rotor about its geometric axis connecting the coupling and journal centers, which is the line that the rotor is constrained to run about in the field when coupled to adjacent rotors. The final balance weight distribution (and the generated force vectors) will approximately mirror the distribution of mass eccentricity on the rotor. Figure 3: Comparison of post-balancing condition between balancing in N and 2N+1 planes. 6 Conclusion It is important to review the assumptions used to create the common theoretical understanding and predictions of general rotordynamics behavior, and to recognize the rotor conditions and areas in practical rotor behavior and balancing that diverge from those assumptions, in order to more accurately identify root causes and effective solutions to vibration problems with rotors in operation. Of course, the effects of resonances and damping and stiffness of the system are still of great importance when dealing with external forces acting on rotors in operation, and these are generally well-studied and optimized by designers and engineers. However, many unexpected vibration problems have little to do with the common areas of focus of resonances and typical unbalance, but rather are centered on the presence of mass eccentricities and the rigid-mode behavior of