Modal Testing of Rotors with Fluid Interaction

Modal testing of rotating structures has specific aspects, and it requires a specialized approach. Classical modal testing when applied to active (rotating) structures does not provide complete results. These aspects, and specific application of sweep frequency circular input force perturbation testing of rotors rotating in fluid environment, are discussed in this paper. Emphasis is placed on nonsynchronous perturbation of shafts rotating at a constant rotative speed. The results are presented in the direct and quadrature dynamic stiffness versus perturbation frequency formats, which permits the separation of components for easy evaluation. This perturbation technique provided new results: 1) identification of solid/fluid interaction modes; 2) identification of fluid dynamic forces in lightly loaded bearings and seals; and 3) multimode identification of rotor systems. Results of several laboratory rig experiments, the identification algorithm, and data processing techniques are discussed. A comparison with other testing methods is given.

XPERIMENTAL modal analysis has become a popular method for studying practical vibration problems of mechanical structures. Application of modal testing for parameter identification of rotating machines, which represent an important class of mechanical structures, has several specific aspects and requires a specialized approach. The results and predictions obtained by applying the classical "passive structure" modal testing to a rotating machine are usually incomplete, and not sufficiently accurate for the most important lateral modes, while providing information not significant for the rotating machine operating performance.

SPECIFIC ASPECTS OF MODAL TESTING OF ROTATING MACHINES
Most of the modal identification methods and conventional procedures of modal analysis deal with structures with assumed linear behavior. The structures are modeled by self-adjoint differential operators, and discretized by symmetric matrices. Rotating machines have an inherent nonsymmetric nature, due to rotation-related factors, such as gyroscopic effects, and fluid dynamic forces in bear-ings and seals, which provide feedback-like effects. The dynamic behavior of rotating machines can adequately be represented only by the nonself-adjoint differential operators. The discretization yields nonsymmetric matrices. The modal analysis must not only determine all classical modal parameters (for example, eigenvalues, eigenfunctions constituting the right eigenvectors and form ofeigenfunctions yielding generalized/modal masses associated with each eigenmode), but also the parameters provided by the left eigenfunctions. Decoupling of rotor precessional mode components requires the utilization of additional relations (such as biorthogonality) between left and right eigenvectors (Childs 1976], Glasgow et al. 1980], Bigret [1984], Andrews [1985], Zhang et al. [1985]).
Rotating machines can be modeled by linear equations in very limited ranges of deflections and velocities. The classical modal analysis, based on the assumption of linearity, has to be completed by taking nonlinearities into consideration. Significant nonlinear effects of geometric and physical origin in rotating machines can introduce large errors on the classical modal tests (Black et al. [1980], Fillod et al. [1985], Sullivan [1985]).
All dynamic phenomena occurring during the performance of a rotating machine are closely related to the rotative motion of the rotor (Fig. 1). The  OUTPUT (e.g., VIBRATION) FIGURE Perturbation technique for modal testing of rotating machines at their operational conditions. of rotative energy makes the system "active," containing a feedback loop. Numerous vibrational phenomena in rotating machines occur due to the transfer of energy from rotation (main performance) to vibration (undesirable side effects). Rotation of the shaft with all mechanical parts attached to it, as well as involvement in rotation of the working fluid (in fluid-flow machines, in seals, and in bearings), causes important modifications in modes and natural frequencies. In large turbomachines, additional changes can be generated by thermal effects, foundation deformations, and misalignment. All these factors cause the results of modal testing of rotating machines "at rest" ("passive structure" approach) to differ significantly from the results of testing during machine operational conditions ("active structure" approach).
Rotors, which represent the main parts of rotating machines, are similarly constrained in two lateral directions; therefore, they exhibit vibrational motion which always has two inseparably coupled, two-orthogonal lateral components (conventionally called "vertical" and "horizontal"). The lateral vibrations result in two-dimensional precessional motion of the rotor. Widely used unidirectional impulse testing, when applied to a rotating shaft, results in a response containing vertical and horizontal components, and an undetermined tangential input force components.
In practical performances of rotors, the precessional motion can contain multi-frequency, two-dimensional components, with definite relations to the direction of rotation. In the most general case, each individual component can be either forward (direction of precession the same as direction of rotation) or backward (direction of precession opposite to rotation). The direction of precessional motion is vital to the rotor integrity, as it determines the rotor stress/deformation pattern. The net deformation frequency of the rotor is equal to the difference between rotative and precessional frequencies, with their signs taken into account. During backward precession the shaft is therefore subject to high frequency deformation (sum of both frequencies). When measuring rotating ma-chine vibrations, it is important to identify each vibrational frequency component, whether it is forward or backward. Narrow band filtering, and time base/orbit analysis, are extremely helpful for this purpose. In classical modal testing, "negative" frequencies have no meaning. Applied to rotating machines, the "negative" frequency has a direct, and very significant physical interpretation related to rotor backward precession. Most important vibrational phenomena of rotating machines are associated with rotor lateral vibrations. Each mode of rotor lateral vibration contains two components (vertical and horizontal), the characteristics of which are, usually, slightly different as a result ofelasticity/mass nonsymmetry of the rotor, and supporting structure, in two orthogonal lateral directions. Modal testing of structures with closely spaced modes presents numerous difficulties.
Rotating machines belong to this category. An alleviation of the problem is offered by consideration of "pair modes" in rotating machines (for example, "first mode vertical" and "first mode horizontal"). Classical modal testing usually, though not always, deals with a large number of modes of a structure over a broad frequency range. In the performance of rotating machines, the most important are the lowest modes, and low-frequency precessional phenomena. This is because: first, the stiffness/mass characteristics of a rotor are always located in a lower range of frequencies than those of the supporting structure. The lowest modes of the rotating machine correspond, therefore, to the modes of the rotor itself. Second, the rotating machine has its own continuously active forcing function--the unbalance, which is an inseparable feature of the rotating system. The frequency of this force is equal to the rotor's actual rotative speed. The resulting motion is referred to as a "synchronous" precession. The operating speed of a single-span machine train, even if it represents dozens of thousands of rpm, seldom exceeds the third balance resonance frequency (third lateral natural frequency); therefore, main interest should be concentrated on investigating the rotor's first two or three lateral/bending modes, because the rotating machine has to survive resonant conditions ofthe lowest modes during each start-up and shutdown. The amplitudes of rotor deformations at low modes are the highest, and the low modes are usually poorly damped; therefore, they are of the greatest concern.
There is one more aspect of importance, focused on the rotor lowest modes. Almost all self-excited vibrational/precessional phenomena occurring during the performance of a rotating machine are characterized by low frequencies that are always located in the subsynchronous region (frequencies lower than the synchronous frequency). The self-excited vibrations usually occur when rotative speed is sufficiently high, and when there exists a mechanism that transfers rotative energy into selfexcited vibrations. The latter are often referred to as "rotor instabilities" (instability of the pure rotative motion after an onset of instability, with an immediately following, limit cycle of self-excited vibrations). The frequency of self-excited vibrations is either equal to a fraction of the actual rotative speed, with the same ratio to rotative speed maintained if the rotative speed varies (for example, oil whirl, partial rub) or it is rotative speed independent, and close to any rotor bending mode natural frequency (for example, oil whip, full annular rub). Most often, due to a specific role of rotor internal/structural friction, the subsynchronous vibrations of rotating machines are characterized by much higher amplitudes than supersynchronous vibrations.
When dealing with a high number ofmodes during classical modal testing, the accuracy of the phase angle readings is usually low. In rotating machines the phase angle represents an extremely important parameter. It not only gives information on the force/response relationship, but also relates the shaft lateral vibration to its rotative motion.
It also yields significant information in modal parameter identification procedures. By limiting modal testing to the lowest modes, it is possible to increase the accuracy of phase angle measurements.
Finally, the most important aspect: the results of the modal testing of rotors at their operational conditions reveal the existence of specific modes, unknown in "passive" structures. These modes are generated by solid/fluid interaction, activated by the shaft rotation, such as in fluidlubricated bearings and seals. During the rotating machine performance, these modes exhibit their activity through rotor self-excited vibrations (for example, "oil whirl" is the rotor/bearing system self-excited vibration; "oil whirl resonance," "oil whirl natural frequency," and "oil whirl mode" are modal parameters of the rotor/bearing system revealed by perturbation testing, as discussed by Bently et al. [1982bBently et al. [ , 1985c, and Muszynska [1986].
In summary, modal analysis of rotating machines provides a significant computational complexity due to the nonsymmetric nature ofrotating structure dynamic behavior. Modal testing of rotating machines should be focused on the rotor lowest bending modes, and applied to the rotor during normal operational conditions of the rotating machine.
The classical modal testing, as used in case of "passive" structures, is not the most efficient for this purpose. Better results can be obtained by applying limited frequency sweep, circular-force, perturbation testing, which will be discussed in the next section.

INPUT FUNCTIONS USED IN MODAL TESTING OF ROTATING MACHINES
Classical modal testing uses unilateral exciting forces, such as provided by hammer impacting or shaker sinusoidal excitation. Static structures exhibit symmetry (in terms of the mathematical model, all matrices are symmetric), which results in reciprocity of the cross-data: accelerance at point "p" when force is applied at point "r" is equal to the accelerance at point "r" when force is applied at point "p".
Rotors are not symmetric. Nonsymmetry in the system matrices results from rotation-generated tangential forces. The natural frequencies and lateral modes are different for each direction of rotation. They are referred to as "forward" and "backward" modes. Their corresponding natural frequencies differ in values, and these differences are functions of rotative speed. Some other operational factors may also contribute to the differences.
When the classical unilateral excitation is applied to a rotating shaft, the forced response consists of both forward and backward modes, which are difficult to separate. The best excitation for rotating shaft modal testing during machine operational conditions is a rotating circular force with distinct direction: forward (same as rotation) or backward (opposite to rotation). This type of nonsynchronous excitation allows for the easy separation of the forward and backward lateral modes, and the identification of rotation-generated terms (Stone et al. 1947]; Hull [1955]; Bently et al. [1979, 1982a,b, 1983, 1984, 1985a,b,c, 1986]). Muszynska [1986]). The term "nonsynchronous" refers to the perturbation frequency which is different from the rotative speed.
The use of a circular rotating input force perturbation system has further advantages, namely, and ease of controlling the frequency, force magnitude, and phase, by using, for instance, a controlled unbalance.
Various types of perturbation systems generating rotating forces can be applied, such as: A) an unbalanced rotating free spinner mounted on the shaft and driven by a compressed air jet flow; B) an unbalanced auxiliary shaft attached to the end of the rotating machine rotor through a pivoting bearing, and driven by a separate motor; both A) and B) allow for shaft "nonsynchronous" perturbation (Fig. 2). Two electromagnetic actuators in X Y configuration generating sinusoidal forces with 90-degree phase shift can also be applied as input circular force perturbators. Another device which provides a perturbation frequency-independent input force amplitude consists of radial spring-like elastic elements attached from one end to the rolling element bearing mounted on the rotor. On the other ends the springs are mounted in a nonsynchronously rotating and (due to its eccentricity) oscillating ring (Fig. 3). This device provides a good resolution in low frequency range testing.
In all these systems, the frequency (angular speed) ofthe perturbing force is entirely independent from the rotative speed of the main shaft: the latter rotates at a chosen constant speed, while the perturbator provides the input rotating force with sweep frequency. The shaft can be perturbed either in a forward or a reverse direction. These perturbation systems also yield very good results in "passive". In all above-mentioned methods, as well as other popular modal testing routines (such as impulse testing), the input into the system is a force; the output is a measured response in terms of mechanical displacement, velocity, or acceleration. There exists a method applied in rotating systems with fluid interactions, in which a displacement is used as an input, and the force is measured at the output (Iwatsubo et al. [1980(Iwatsubo et al. [ , 1988, Ohashi et al. [1984Ohashi et al. [ , 1988 [1986,1988], Kanki et al. [1986], Adams et al. [1988], Brennen et al. [1988]). The forced circular displacement in an orbital form is generated by a cam mechanism on the shaft rotating in a fluid environment. The output fluid dynamic force is calculated from the pressure transducer or load cell readings (Fig. 4). This method provides nonsynchronous perturbation; it is used for identification of fluid dynamic forces in seals and fluid-handling machines. A comparison ofresults yielded by this method, and the input force/output displacement method is given

RESPONSE MEASUREMENTS
Accelerometers are the most popular transducers used in classical modal testing. The testing results are presented in terms of accelerances (or "inertances") which are the ratios of response acceleration vectors to input force vectors ("vector" means amplitude and phase of a harmonic variable). Accelerometers are the most appropriate instruments for modal testing of passive structures, which deals usually with high number of modes, with nat-ural frequencies located in a relatively broad frequency range.
In a rotating machine the modes of highest interest are those of the rotor itself. Most often the rotor modes correspond to the lowest modes of the entire machine structure. The first natural frequency may occur in the range of 5 to 15 Hz. In this range of frequencies accelerometers perform very poorly. The best transducer in the low frequency range is the displacement proximity probe. When mounted in casings or bearings, the proximity probe provides relative measurements (shaft motion relative to support motion). For machines with very soft supports the proximity probe can be complemented by a seismic probe providing casing absolute measurements (for instance, a dual transducer). Results of modal testing using displacement transducers are usually presented in terms of receptances (there exists in the literature equally used names, such as "admittances," "compliances," "dynamic flexibilities"). The receptances are the ratios of the displacement response vectors to input force vectors. Receptance vectors are widely used, for instance, in rotor balancing. (They are often called "influence coefficients," but the more proper name should be "influence vectors.") The use of accelerometers, velocity pickups, or proximity transducers in measurements of mechanical structure vibrations is not only a matter of rational choice, corresponding by matching the best to the type of encountered conditions, it is also a matter of philosophical usage--although popular in modal analysis applications, accelerometers are not widely used in on-line monitoring and diagnostics of rotating machinery malfunctions. Rotor displacements, not accelerations, are the most meaningful signals for the operating personnel.
Specific changes in rotor displacements (vibration amplitudes, as well as static positions) relative to machine casing directly indicate what type of malfunctions the machine develops. Changes in the static positions indicate changes in the alignment state. These data assist in prediction of fluid whirl/whip self-excited vibrations, as well as shaft crack prevention. A specific content of vibration signals indicates presence of unbalance, misalignment, rotor-to-stator rubs, loose parts, shaft crack propagation, and other malfunctions of the rotating machine.
The most harmful vibrations for the integrity of the rotating machine are low frequency, subsynchronous vibrations, resulting usually from an instability action transfer-ring rotational energy into vibrations. Acceleration amplitudes are proportional to the square of vibration frequency. This means that when measuring rotor vibration by using accelerometers, the higher frequency components become dominant, indicating high amplitudes. The low frequency components look insignificantly small, even though they might have very high amplitudes in terms ofrotor displacements. In addition, accelerometers are most often installed outside the rotor casing; thus they measure vibrations of the outside structure, not the vibrations of the rotor, which is the main source of vibration. During transmission from the source through the structure, vibration becomes attenuated, the degree of attenuation depending on the structure transmissibility.
In summary, in the rotating machine perturbation testing, the most useful instruments to measure rotor responses are displacement transducers mounted in X Y configuration (Fig. 5), with a Keyphasor (R) once-per-turn marker for phase reference.

NONSYNCHRONOUS SWEEP FREQUENCY PERTURBATION TESTING AND DYNAMIC STIFFNESS IDENTIFICATION ALGORITHM
In the classical modal testing, identification of the structure modal parameters is usually based on curve fitting of results presented in the form of receptances (Ewins, [1984], obtained, however, when a Dynamic Stiffness approach is applied. Straight lines of dynamic stiffness components are definitely the best to fit. A dynamic stiffness algorithm for identification of modal parameters of a rotating machine rotor using nonsynchronous sweep frequency perturbation testing is outlined below.
Rotor responses to rotating force excitation bring meaningful data for identification of the system parameters. The identification procedure involves matrix inversion, thus a computer is necessary for experimental data acquisition and processing.
Consider a modal of a nonsymmetric (laterally anisotropic), anisotropically supported, flexible 2n degree of freedom (2n lateral mode) rotor rotating at a constant speed f2 (Fig. 5 which are inertia, damping, and stiffness matrices correspondingly. The matrices Dq,3_q contain "cross" damping terms, matrices [Kq,3_q] contain elements of the "cross" stiffness type. Both these matrices usually depend on the rotative speed 9. The vector [F] contains controlled perturbation (excitation) forces. For identification of the system parameters, a controlled rotating perturbation force is applied consecutively at "r" (r 1,..., n) axial locations of the rotor in either forward (s 1) or backward (s 2) direction. The "n" pairs of displacement transducers in an X Y configuration are mounted at "p" (p n) axial locations of the rotor (Fig. 5). When the force is applied at the rth location, the excitation vector used in nonsynchronous perturbation is, therefore, where co is perturbation frequency, Fsr and tsr are the perturbation force amplitude and phase, respectively. Note that the perturbation frequency w is entirely independent from the rotative speed The rotor forced response to exciting force (2) based on model (1) is as follows: [Z] col[Alslr cos(cot + 0tlslr ) Alsnr cos(cot + Otlsnr), A2slr sin(wt + Ct2slr A2snr sin(cot + Ot2snr)] ( 3) where Aqspr, Otqspr are amplitudes and phases of the responses, narrow-band filtered to the perturbation frequency. All phases are measured from the same once-perrotation marker. By substituting (3)  The nonsynchronous frequency swept rotating force input perturbation method has proved to be very efficient for identification of bearing and seal fluid dynamic forces. As a result of over ten years of testing and research, an improved model of fluid dynamic forces in lightly loaded bearings, in seals and fluid-handling machines, has been proposed (Muszynska 1988a]). The model was identified experimentally using the perturbation technique. A similar fluid force model was previously developed theoretically, and has existed in a simplified version in rotordynamic literature for at least 25 years (Bolotin [1963]; Black [1969Black [ , 1980). It has not, however, been fully exploited. The most important result of perturbation testing identification of the fluid forces was introduction of the "fluid circumferential average velocity ratio," k, as a function of shaft eccentricity. It has replaced the assumed constant "1/2" widely used in other fluid force models.
The results of perturbation testing also, yielded, conclusions regarding modal behavior of mechanical systems with fluid interactions. It was shown that the value "kf2" (fluid circumferential average angular velocity) represents (with approximation related to damping) a rotor/bearing or rotor/seal system natural frequency (Muszynska 1986]). It is associated with the specific "fluid whirl" mode of the rotor.
In the section that follows, the results of sweep frequency rotating force perturbation identification method applied to a rotor/bearing system, and the improved fluid force model in lightly loaded bearings and seals will be outlined.
The experimental rig (Fig. 7) to perform the perturbation test consisted of the main well-balanced light rotor supported in one relatively rigid bronze bushing bearing, and one oil-lubricated bearing. The main mass of the ro- tor was concentrated at the journal. An auxiliary light rotor was connected concentrically with the previous one through a pivoting rolling element bearing mounted inside the journal (Fig. 2b). The main rotor was rotating at a selected constant rotative speed. The auxiliary rotor carrying a light disk with a controlled unbalance weight was driven by a separate electric motor in the sweep frequency fashion. The unbalance provided the excitation force to the main rotor. The force phase was controlled by a Keyphasor(R), once-per-turn marker transducer. The main rotor response was measured by two displacement transducers mounted in X Y configuration at the lubricated bearing. The eccentricity of the journal inside the bearing was adjustable by using additional radial springs. In the "oil whirl resonance" experiment the journal initial eccentricity was zero, so the rotor forced X and Y responses were matching. The lubricant, T10 oil, was fed to the bearing by a supply system with controlled inlet and ambient outlet oil pressure. Four symmetrically situated radial channels, and a circumferential groove evenly distributed PERTURBATION SPEED w [rad/s 100] FIGURE 8 Typical plots of phase and amplitude of rotor response to forward and reverse nonsynchronous perturbation versus perturbation speed (0 original rotor/bearing system, S added springs, M added rotor mass, SM added springs and mass). Oil whirl resonance occurs at frequency close to f2/2, identified as .f2 (f2 constant rotative speed). Increase of the system mass causes an increase of the oil whirl resonant response amplitude, as margin of stability is reduced (Bently et al. [1982b]; Muszynska et al. [1990]). the oil inside the bearing. The oil temperature was maintained constant. Both electric motors were equipped with speed control devices. The data acquisition system consisted of the vector filter and computer. The rotor responses were continuously monitored on the oscilloscope and spectrum analyzer. The results of perturbation testing brought the following conclusions: Fluid whirl resonance frequency. The journal X and Y responses filtered to the perturbation frequency w, and displayed in the Bode format (Figs. 8 and 9) exhibit res-onances at forward perturbation frequency equal nearly half of the constant rotative speed .More precisely, the resonance occurs at o .f2 (Muszynska [1986]). At this perturbation frequency the response amplitude has a peak, and the phase angle sharply changes values through zero degree. This yields the conclusion that )g2 is the rotor/bearing system natural frequency (Muszynska 1986]). This resonance has a "quadrature" character, and became known as "oil whirl resonance" or, more generally, as "fluid whirl resonance?' Such resonance does not occur for the reverse perturbation.
Phase and amplitude of rotor response versus perturbationto-rotative speed ratio for various values of rotative speeds, 2. Increase of amplitudes with increases of 2 due to decrease of margin of stability (Bently et al. [1992a, b]; Muszynska [1986Muszynska [ , 1990).
Dynamic stiffness format. Using the dynamic stiffness algorithm (Eqs. (7) to (9)), the force/response data was then processed, and presented in the dynamic stiffness format. For the isotropic single complex degree of freedom system (x -tjy z; j Sf) the journal lat- where m, r, are the input perturbation force unbalance mass, radius, and angular orientation, respectively.
The dynamic stiffness components versus perturbation frequency are presented in Fig. 10. The direct dynamic stiffnesses are parabolas; the quadrature ones are straight lines, thus the identification of their parameters is relatively easy.
Fluid force model. The curve fitting of the dynamic stiffness components allowed identification of the fluid force model applicable for lightly loaded bearings and seals (Muszynska [1986(Muszynska [ , 1988a; Muszynska et al. (Higher rotative speeds caused an increase of oil temperature, thus a decrease of fluid radial damping reflected in quadrature dynamic stiffness graphs with lower slopes) (Muszynska [1986(Muszynska [ , 1990). K, M are rotor stiffness and mass, respectively.
[1990]). The following fluid force model F [g0 + jD(co-kf2) My(co-))2]Ae jt (11) was provided directly from the dynamic stiffness graphs; DDS Ko Mf(co-))2, QDS D(cokfa). (Note that the raw data in Fig. 10 also contain, the mechanical system parameters K and M which were subtracted, and do not appear in the fluid force model (11).) The differential form of the fluid force model in coordinates rotating with angular velocity Lfa is (Muszynska [1988a]): F KOZr + D.r + M.l',r z x a tjy zre j't In stationary coordinates z x + jy the fluid force model becomes: In Eqs. (11) (Muszynska [1986(Muszynska [ , 1990). ing force results in an appearance ofthe component known as the "cross stiffness," "Kxy." The results of perturbation testing proved that this term is directly proportional to the 8 damping, D, and to the fluid circumferential average velocity ratio, ,k. The latter provides an efficient way to control the value of this "cross stiffness" term, thus to control the stability of the rotor/bearing/seal system (Bently et al. [1988], Muszynska et al. [1989a]).
Fluid whirl mode. When the rotor is rigidly supported at one end, as it was in the case of the described experiment, the fluid whirl mode is conical. The shaft vibrates in phase with the journal, as a rigid body (Muszynska 1986(Muszynska , 1988c). Fluid force parameters as functions of various factors.
Variations of the rotative speed, including zero speed (squeeze film damper), yielded the most important conclusion regarding rotational character of the fluid dynamic force, and provided some insight into the fluid inertia effect (Figs. 9 and 10). Variations of the lubricant pressure and temperature allowed to identify the influence of these factors on the separate parameters of the fluid force model. As it is shown in Fig. 11 [1986]).
An increased lubricant temperature (decreasing viscosity) causes a decrease of the fluid radial damping (Fig. 12) where A is rotor displacement amplitude, ot is rotor response phase (also c arctan(-Ft/Fr)), w is rotor perturbation (excitation) precessional frequency (usually varying from zero to some Wmax), and Fr, Ft are radial and tangential forces, acting on the rotor. These forces are obtained by integrating the fluid pressure or measuring forces outside an elastically supported seal or bearing; F is the input force amplitude.
In both techniques the objective is identification of the transfer function of the system. More precisely, the functions sought are x(w) Complex dynamic stiffness, or /-)(w) x(-log)_= Transfer function Complex dynamic compliance orthe system.
By limiting the input to a circular periodic function, both methodologies should yield exactly the same results, provided that the system is linear, and the instrumentation allows for comparable signal-to-noise ratios. Both techniques are known in mechanics as "force excitation" and "kinematic excitation," respectively. They both serve well for identification of the lowest mode generalized (modal) parameters of the system. Figures 8, 9 velocity ratio), as discussed by Muszynska [1986Muszynska [ , 1990. In fact, instability occurs when both direct and quadrature dynamic stiffnesses are nullified at the same frequency. In comparison to the response of a classical mechanical system, the response phase is, however, ahead of the input force. This indicates the "quadrature" nature of this particular resonant phenomenon. The resonant frequency has been identified as if2. (In machinery it was sometimes called "bearing resonance?'). The occurrence of the resonance suggests that if2 represents one of the system natural frequencies. This natural frequency is generated purely by the fluid/solid interaction. The rotating shaft drags the fluid into circumferential motion, generating the fluid rotating forces which act on the shaft in a feedback loop. This fluid-related quadrature resonance was documented by Stone & Underwood in 1947, and again by Hull in 1955, but these excellent works were not immediately pursued. By inputting a constant circular displacement, and measuring the output in terms of forces, the accuracy of the results is lower (mainly due to poor phase resolution), and there is no clear physical interpretation of the results. The plot of force response amplitude versus perturbation frequency has an "anti-resonance" shape, a concave curve, like a mirror image of displacement response amplitude versus frequency (Fig. 16). The phase is the same in both techniques. It is obvious, because by definition, the phase represents the angle between the input vector and output vector, independently of the nature of the input and output functions. Note that the dip point of the response force amplitude and phase drop occur around one half of the rotative speed, i.e., the fluid circumferential average velocity ratio I is equal to about one half. The notion of the "force resonance," or rather "anti-resonance," is not known in mechanics. That is why the researchers who have used the input force methodology, and adopted direct physical interpretation of the results, have been more fortunate than those who use forces as outputs.
Note the advantage of the circular perturbation, as opposed to unilateral perturbation, in impulse testing. The results of forward perturbation (rotation and precession in the same direction), and backward perturbation (opposed direction) are significantly different (Fig. 8). The resonance occurs only for the forward perturbation, that is, the "quadrature" natural frequency Lf2 of the system has only the plus sign (classical "direct" natural frequencies governed by stiffness and mass have + and signs).
the direction opposite to shaft rotation makes dramatic changes in the fluid circumferential average velocity ratio values, considerably lowering them, and improving rotor stability features.

ANTISWIRL SEAL TESTING
The sweep.frequency rotating force nonsynchronous perturbation method was applied to identify the fluid force parameters in antiswirl seals equipped with radial and tangential inlet ports (Grant [1991]). The results provide clear effects ofthe input flow pattern on fluid force components. Water was used for series of experiments. Figure 18 illustrates an example of the data presented in terms of dynamic stiffness components for the seal input flow through radial or antiswirl ports for three rotative speeds. Based on this type of data, the identified fluid force parameters, namely, circumferential average velocity ratio ., radial stiffness, and damping reduced from approximately 70 runs covering the range from 3 to 80 psi water pressure, are presented in Figure 19. The graphs clearly show general trends of these parameters for radial and antiswirl inlets (Muszynska et al. 1992]). Figure 20 illustrates fluid circumferential average velocity ratio versus anti-swirl port pressure drop for several cases of mixed radial and tangential inlets. As seen in FIGURE 18 Direct and quadrature dynamic stiffnesses versus perturbation frequency of the seal with either radial or antiswirl inlet ports (Grant [1991]; Muszynska et al. [1992]).

ROTOR/BEARING SYSTEM IDENTIFICATION COVERING WHIRL AND WHIP MODES
In the tests for identification of the fluid whirl resonance, the main rotor was deliberately rigid and light, so that its first natural frequency of the bending mode (conventionally called here "whip") was well above the range of perturbation frequencies. The next series of tests concerned the rotor/bearing system, covering both whirl and whip modes. The main rotor was made more elastic, and a heavy disk was mounted at rotor mid-span (Fig. 21). The main rotor first natural frequency became much lower, and placed in the range of the applied perturbation frequencies. Since more modes were involved, the test required additional data. Perturbation, therefore, was applied either at the journal (as previously), or at the rotor disk (using an unbalanced, pulley-driven spinner). Each time, the response data was collected from X Y proximity transducers mounted at the bearing, and at the disk. Following the identification algorithm (7) to (9), the 2 2 matrix of complex dynamic stiffness components was identified. The results are presented in the form of Bode plot matrices from the experimental data, as well as from calculated analytical data based on the identified parameters of the system. The system model (Fig. 22) is as follows (Bently et al. [1985c]  presented in the form of Bode plot matrix exhibit two resonant ranges: the whirl resonance at lower perturbation frequency discussed on p. xxx, and the "whip" reso-nance corresponding to the rotor first bending mode. Using the dynamic stiffness algorithm (Eqs. (7) to (9)), the system parameters were identified. As it can be seen from the corresponding experimental (Fig. 23) and analytical (Fig. 24) results, the identification process provided very good agreement, and proved the model adequacy.

SQUEEZE FILM PERTURBATION TESTS, STEADY-STATE LOAD TESTS, AND IMPULSE TESTS OF ROTOR/BEARING SYSTEMS
Squeeze film tests of bearings are circular force perturbation tests of a rotor at rest, and usually performed at zero initial journal eccentricity of a rotor, so the results are isotropic. The fluid force (13) for the squeeze film tests does not contain terms depending on f2. When the dynamic stiffness components of the tested bearing are plotted versus perturbation frequency (QDS), or versus perturbation frequency squared (DDS), the parameters D, K0 and My can easily be identified from resulting straight lines. Figure 25 presents Direct Dynamic Stiffness of a bearing fluid obtained from the squeeze film test. The data exhibit very high fluid inertia effect, practically independent from fluid inlet pressure.
Steady-state load test of a rotor/bearing system is accomplished by applying a constant radial force to the rotating journal, and measuring the journal displacements in two orthogonal directions. These displacements are presented in the eccentricity ratio polar format for several values of the rotative speeds and input radial forces (from 2 to 15 lbs) (Fig. 26). The journal displacements exhibit two distinctly different patterns of behavior. In the low eccentricity range, the dominant component of the fluid force has a quadrature nature: displacement is almost 13.5"' 12.5" 7"" 7 t'l= 11.5"' e3= 9.5" l I-1 _Iqli FIGURE 21 Test rig (Bently el al. [1985c] perpendicular to the applied force, that is, the tangential (quadrature) component of the fluid force, Dkf2, is dominant. For a high rotative speed the angle between the input force and journal response is even larger than 90 which indicates a high fluid inertia effect opposing the fluid film radial stiffness* (Bently et al. 1985a]). In order to identify the fluid force parameters, the input force vector was divided by the response vector, split *The fluid dynamic force for this test is F (Ko-Dj,kf-Mf,k22)z (see Eq. (13)); the force balance is F Pe jx where P, y are amplitude and angular orientation of the input constant radial force.
By measuring "z," the fluid force can be identified.
into collinear and perpendicular to the input force components, and presented versus eccentricity ratio (Fig. 27). While the quadrature components identified as k D (rotative speed eliminated) exhibit classical, quite regular form, the fluid film dynamic stiffness in the direction of load, namely, K Mf2 2 confirms the existence of the strong fluid inertia effects, and exhibits two distinct patterns of behavior characteristic for the low and high eccentricities.
A similar steady-state load test was performed on a lowpressure oil bearing (Fig. 28). While the fluid inertia effect was smaller, similar behavior patterns in the two ranges of eccentricity were observed. In both cases (Figs. 26  the quadrature stiffness dominance ceases at lower eccentricity, when rotative speed is higher. Different dynamic behavior of the journal at low and high eccentricities corresponds to significant changes in the flow patterns: at low eccentricities the flow pattern is predominantly circumferential, due to shaft rotation, and DLf2 > K0. In this range the fluid force model (12), (13) is adequate. At high eccentricities the circumferential flow is suppressed by increasing axial flow pattern, DLf2 < K0.
The changes of flow patterns were eventually confirmed by impulse testing. The shaft (rotating at a constant speed in a fluid-lubricated bearing at one end and in a rigid bearing bushing on the other end) was impacted by a calibrated hammer. The X Y proximity probes mounted in the bearing, observed the journal transient responses. The data acquisition and processing system recorded the frequencies of the transient responses at different original journal eccentricities. These frequencies were then divided by corresponding constant rotative speeds. The results identified as fluid circumferential average velocity ratio (*) are presented in Figure 29 for two different bearings.
The circumferential pattern of flow at low eccentricity is evident, which produces the rotor/bearing system natural frequency ,f2, which was previously discovered by nonsynchronous perturbation testing. The average circumferential velocity first slightly decreases with increasing eccentricity, then collapses at higher eccentricity. Interesting enough is that when the rotative speed is high, the range of low eccentricity where circumferential flow pattern is present appears narrower than for low rotative speeds. The similar result regarding direct stiffness behavior (see Fig. 27) was obtained from the static load test.

PARAMETER IDENTIFICATION OF A ROTOR WITH STRONG GYROSCOPIC EFFECT
The overhung isotropic rotor rig schematically shown in Figure 30 was used for identification of rotor parameters by applying lateral nonsynchronous forward and backward circular force perturbation testing (Bently et al. [1986b]). The rotor natural frequencies were identified as resonant frequencies of rotor responses filtered to perturbation frequency. The sequence of polar plots of rotor vertical and axial responses for the forward and backward perturbation is shown in Figure 31. Figure 32 presents the rotor natural frequencies as functions of rotative speed, that is the well-known relationship for rotors with strong gyroscopic effect. Since the simple two-mode model of the rotor contains at least eight parameters, the lateral perturbation only was not sufficient to identify all of them, without additional assumptions. Assuming, however, the disk transverse moment of inertia is known, the remaining parameters were identified (Bently et al. [1986b]).

IDENTIFICATION OF MULTIMODE PARAMETERS OF ROTORS USING SYNCHRONOUS PERTURBATION TECHNIQUE
Practical application of nonsynchronous perturbation in rotating machines may encounter difficulties due to limited access to the rotor for instrumentation (input force actuators and response measuring transducers).
The synchronous perturbation technique is simpler, although providing useful, but limited information. This technique uses shaft rotation as sweep frequency, and controlled unbalance weights as input perturbation forces. In the easiest application, the controlled unbalances may be introduced into balancing planes of the machine.
In the nonsynchronous perturbation methodology, discussed in the previous sections, the rotating force can be applied in either forward or backward directions. The controlled unbalance can rotate only in the forward direction, FIGURE 28 Journal centerline response to a sequence of constant vertical forces P (numbers in pounds) for two values of rotative speeds. The sketch presents the axially fed oil bearing used in this test.
since it is attached to the rotor. This means that the synchronous perturbation will not provide enough data for the entire system modal parameter identification. It may bring, however, some useful information about the machine dynamics, especially when the data are taken in a time-trend sequence and, consequently, are reduced into the dynamic stiffness formats. This information may provide evidences of machine deterioration, such as wear, rotorjoint looseness, rotor-to-stator rubbing, shaft cracking, and so on.
In order to eliminate the effects of other synchronous forces that possibly exist in the system, each synchronous (1 x), whirl and "whip" natural frequencies present in the spectrum (Grant et al. 1992]); (b) Fluid circumferential average velocity ratio (,k) versus eccentricity ratio identified by impulse testing for the bearing illustrated in Fig. 28.;and (c)for the externally pressurized bearing (Muszynska 1988a] AXIAl"/-IXPROBR X/ "" '1:zaa ;= I0.6'" , LOCATION 1 FIGURE 30 Gyroscopic rotor perturbation test rig and disk angular/axial displacement relation (Bently et al. [1986b]). perturbation test is performed twice. In the first start-up test the perturbation controlled weight is introduced to the balancing plane at certain angle ;. In the second run, the same controlled weight is moved into + 180 angular location at the same radius. The x responses from these two runs are then vectorially subtracted, through the entire rotative frequency range. This eliminates the possible unknown 1 x components. The resulting differential data are then used for the system identification. An example of the synchronous perturbation application for two-mode rotor identification is outlined later in this section (Muszynska 1986(Muszynska , 1989b). Two-mode and, generally, coupled multimode approach, offer several advantages in comparison with the classical identification of separate mode parameters. The multimode identification provides parameters which more closely represent the structure parameters, and thus de-scribe the system more adequately. The identification allows locating of the physical position of specific parameters most responsible for given modes. This, for instance, makes system modifications more efficient, especially when nonlinear modifications are considered. An example offormal transformation ofmodal variables to multi-mode modal coordinates is given at the end of this section.
The two-lateral-mode model of an isotropic rotor, the subject of synchronous perturbation, is as follows (Fig. 33): where My, Dye, Kv, v, e 1, 2 are rotor multimode modal masses, damping coefficients, and stiffnesses, respectively. Note that they represent partial modal parameters of the two-mode rotor. [F] is the controlled synchronous perturbation force matrix. The perturbation force is applied at the first disk location in the first experiment, and at the second disk location in the second experiment (the forces and responses in the identification equations are actually net differential data, thus the number of necessary rotor runs is four). The matrix of the input forces resulting from two experiments is, therefore, where my, rv, ;v, v 1, 2 are unbalance weight mass, radius, and angular orientation, respectively, and the multiplier "2" results from the differential procedure of 7ectorial subtraction described above. The matrix of the system 1 net responses to controlled unbalance weights for two experiments is A21eJt21 A22eJt22 where Ave, otve, v, e 1, 2 are 1 amplitudes and phases, respectively, where v represents a number of the disk where transducers are located, e a number of experiment.
Following Eqs. (1) to (7), the dynamic stiffness matrix of the system (19) is calculated as A flexible isotropic shaft carrying two disks, and supported in relatively rigid isotropic bearings (Fig. 34) was run with low acceleration up to 6,000 rpm. The controlled unbalance of mr 0.5 x 30.5 gram x mm was installed at the inboard disk for the first run, and at the outboard disk for the second run. The rotor was equipped with one Keyphasor (R) probe and twelve X Y proximity, shaft displacement observing probes at six axial locations of the shaft. The data for modal parameter identification was taken from two vertical probes only. The horizontal probes (X) served for checking the lateral isotropy of the rotor. The data acquisition and processing system consisted of a vector tracking filter, and the specially designed computer software. An oscilloscope and a spectrum analyzer, for additional reference checking, completed the instrumentation.
The measurements were taken at the inboard and outboard vertical locations "D" and "J" (Fig. 34). The net data resulted from vectorial subtraction of 180 and 0 controlled unbalance responses taken at start-up runs are illustrated in Figure 35 in the polar plot format. Figure   36 presents the direct dynamic stiffness components.
The identified values of the rotor two-mode modal pa- Due to virtually low system damping, the quadrature dynamic stiffness components did not provide enough resolution to identify the damping matrix components. The modal damping coefficients DI, OlI were identified from the polar plots (Fig. 35) where Dm is the diagonal modal damping matrix with elements identified using the half-power bandwidth method. The additional proximity transducers observing the rotor served for evaluation of the modal correction factors, /3. Since the two-mass rotor model (Fig. 33) is a rough approximation of the rotor behavior, and since the perturbation unbalance weights, and measuring transducers practically cannot be installed at antinodal locations, the collected data carry an error. The additional data taken from other transducers allowed for correcting this error by introducing the modal correction factors/3 into the final identification algorithms (Muszynska et al. [1989b]).
In many practical cases the limited access for installing instrumentation on a rotating machine prevents one from identifying the exact modal parameters. Very often, however, there is no need for accurate values of these parameters. Their approximate values obtained from regularly collected synchronous data (without modal correction factors/3, and possible nonlinearity, and/or lateral anisotropy taken into account) represent valuable information on the machine dynamics. These values acquired in a time-trend format during the machine operation, and showing successive changes in particular dynamic stiffness components, will help in diagnosing the machine state, and in identifying the malfunction area. In addition to Bode and polar plot formats for filtered x vibration data reduction, the observed dynamic stiffness formats, which use the same raw data, provide invaluable information.
Synchronous perturbation testing is practically always done on any rotating machinery during balancing routines. The valuable dynamic stiffness data are, however, seldom appreciated, sorted out, and stored. The procedure, called here "synchronous perturbation technique," is performed during routine calibration weight balancing.
The 1 x response vectors during any transient start-up or shutdown should be stored, and then processed into the dynamic stiffness format versus f2, from which the observed dynamic stiffness components, mass, stiffness, and damping, can be evaluated. If balancing is performed only at one speed, the observed dynamic stiffness will provide discrete values only. This section will be completed with a discussion on the development of the multi-mode modal models (Muszynska, 1994]) The most often used models of mechanical structures are based on finite element grids. In rotor modeling the transfer matrix method is also used. One of the final results of these models is a set of the system natural frequencies wv, v 2n and a set of corresponding modal functions.
The uncoupled modal variables, associated with the natural frequencies, have very little immediate correlation to the physical and geometric parameters of the system. The idea of multi-mode modeling originated from the usefulness of better correspondence of modal masses and stiffnesses to the physical system masses (such as massive disks on a rotor) and stiffnesses (such as provided by relatively slim shafts). The multi-mode model is built from the modal model. An example of its formal derivation for the case of rotor/bearing system is presented below.
The modal model obtained, for instance, from the finite element method, and possibly reduced to n modes, is as follows: where Ix] is an (n 1) rotor lateral displacement vector and [Mm], [Km] are modal mass and stiffness diagonal matrices. Assume that the new multi-mode modal model of the rotor consists of n masses and n + stiffnesses connecting the masses in a chain manner. An example of such model is illustrated in Figure 33. A linear transformation x y is applied to Eq. (24), with the condition that the transformed mass matrix remains diagonal, and the stiffness matrix is symmetric tridiagonal. The first condition imposes the orthogonality of the (n n) matrix n /x, v n where tzv is the Kronecker's delta. The condition (25) results in n(n + 1)/2 equations for n : unknown elements of the matrix . The transformed Eq. (24) will have the The nonsynchronous sweep frequency lateral perturbation technique has been used to investigate the lateraltorsional cross coupling due to shaft anisotropy (Bently et al. [1991]). The rotor of the experimental rig was driven by a synchronous electric motor, connected through a laterally flexible, torsionally rigid coupling. In order to simulate anisotropy of the shaft, a part of the shaft at mid-span was machined to produce two flats. The shaft carried two disks, with 36 gear teeth on each of them. The optical pickups observed the disk gear tooth passing frequencies.
The data were then processed through a torsional signal conditioner to obtain the dynamic twist (torsional vibrations) of the shaft between two disks. The shaft lateral vibrations were observed by two sets of two proximity probes in X Y configuration. The nonsynchronous perturbation unbalanced disk driven by a separate motor was attached through a rolling element bearing at the shaft mid-span. The lateral response of the shaft to the nonsynchronous perturbation, presented in a form of the spectrum cascade plot (Fig. 37) [1992]).
The existence of these components is due to the shaft anisotropy, and can be used for its identification. The rotor torsional responses are presented in the form of overall vibration amplitude versus perturbation frequency (Fig. 38).
They exhibited high amplitudes at the excitation frequency equal to the first torsional natural frequency vt. The lateral/torsional coupling is related to the rotor residual unbalance. The other peaks of the torsional vibrations occur when the linear combinations of frequencies o9 and f2 are equal to the torsional natural frequency vt and the half of it (vt/2). These effects are due to the asymmetry of the shaft plus radial sideload, and could also be further used for identification of the shaft anisotropy, such as could be generated by a propagating transversal crack on the shaft. z=x+jy Zr Greek letters rotor lateral displacement in stationary coordinates rotor lateral displacement in rotating coordinates

FINAL REMARKS
The sweep frequency nonsynchronous circular input force perturbation testing of rotors proved to be very powerful and efficient, producing a number of results which had not been known. The most important among them are: 1) solid/fluid interaction modes of rotors rotating in fluid environment; 2) identification of the more adequate model of fluid dynamic forces in lightly loaded bearings and in seals; and 3) multimode parameter identification of rotor/bearing/seal systems.
The nonsynchronous perturbation testing should soon be used in the routing testing of rotating machines, at first, at least, on the prototype stage. The results could provide invaluable information on fluid interaction effects on the rotor, in order to assess the stability margins from fluid whirl, whip, rotating stall, surge, and evaluate other properties of rotating machinery. Less demanding synchronous perturbation testing (known as "unbalance testing"), which is now widely performed on machines, should be accompanied by appropriate data acquisition in order to extract information on "observed dynamic stiffness." This invaluable information will greatly assist in furthering machine malfunction diagnostics by using vibration monitoring. measurement axial location, p n; also: "perturbation" q displacement in "x" direction (q 1) or "y" direction (q 2) r perturbation force axial location r n; also: "radial" s perturbation forward (s 1)or backward (s 2); also: "shaft"; also: "steady state" tangential; also: "torsional" response amplitude and phase, respectively bearing diameter fluid radial damping at zero eccentricity rotor modal damping direct and quadrature dynamic stiffness, respectively force amplitude; also: fluid force transfer function rotor modal stiffnesses fluid radial stiffness at zero eccentricity bearing length unbalance mass, radius, and angular orientation, respectively rotor modal mass fluid inertia effect pressure amplitude and angular orientation of the constant radial force used in steady-state load tests time temperature rotor displacements in horizontal and vertical directions, respectively Adams, M. L., Yang, T., and Pace, S. E., 1988, May. A Seal Test Facility for the Measurement of Isotropic and Anisotropic Linear