This technical brief is to study how flexible bearings and housing affect mode localization of a nearly cyclic symmetric system with mistuning. This study is conducted via finite-element analyses and deductive reasoning. A reference system studied is a bladed disk with two groups of 24 localized modes. When bearings and housing are introduced into the reference system, their presence changes natural frequencies, mode shapes, and the number of the localized modes. Moreover, the mistuning causes bearing forces to surge for all the localized modes. A deductive reasoning based on the existing literature supports the observation from the finite-element analyses.

## Introduction

Cyclic symmetric bladed disks have been widely studied in the last several decades for their critical industrial applications, such as turbines and compressors [1,2]. Due to manufacturing tolerance, wear, or damage, the blades vary slightly from one another, and the blade variation is known as mistuning. When mistuning is present, vibration of the bladed disks may become localized at only a few blades. The phenomenon of mode localization could cause premature high-cycle fatigue and accelerate the damage of the bladed disks [3]. Moreover, when the mode localization occurs, amplitude of a few blades is largely amplified under engine order excitation of the pressure [4].

Due to its potential impact, the mode localization has been studied via different approaches, such as component-mode synthesis [5,6], the Rayleigh–Ritz method [7,8], the Taylor series [9], and the finite-element methods [10]. In many of these studies, a simplified bladed-disk model is used as a benchmark to facilitate prediction of the model localization. The benchmark model often has a single disk with multiple blades simulating a single-stage rotor. In addition, the inner rim of the bladed-disk is fixed to simplify the analysis. Although the simplified model has successfully demonstrated the phenomenon of mode localization, the model does not account for the effects of other machine components, such as the shaft supporting the bladed disk, bearings, and housing. Therefore, how these machine components would affect mode localization remains unknown. Moreover, if these components did affect mode localization, what would be the underlying mechanisms and physics? If the mode localization was affected, what would be its implication to the performance of the entire rotor assembly?

Recently, researchers started to analyze vibration of turbomachinery rotors more accurately by modeling multistage bladed disks [11–14]. A key observation is that the presence of interconnecting boundary conditions between stages affects mode localization. Specifically, some extreme localized modes present in a single-stage system are mitigated after the stages are connected together. Root causes behind the mitigation or evolution of the localized modes are not well understood. The recent studies on multistage rotor vibration echoes the three questions raised above: how would mode localization be affected, what would be the underlying mechanisms, and what would be its implications?

Motivated by the needs above, this technical brief is to study how the supporting shaft, bearings, and housing affect mode localization of a mistuned bladed disk. The study is done via two different approaches. The first approach is to observe how mode localization evolves by conducting a finite-element analysis (FEA) on a reference system. Specifically, the reference system is a mistuned, bladed disk with a fixed inner rim. To simulate the presence of bearing supports, the fixed inner rim is replaced with an elastic boundary condition. Finally, a housing is coupled with the reference system through the flexible bearings.

The second approach is via a deductive reasoning from the existing literature to obtain qualitative theoretical predictions to justify the finite-element results. For example, Chen and Shen [15] identified that mode localization in a mistuned cyclic symmetric system will only occur when the two following conditions are met: (a) there is a group of vibration modes of the tuned cyclic symmetric rotor whose natural frequencies are nearly identical and the mode shapes have similar characteristics and (b) the vibration modes in the group contain a wide range of wave numbers. Kim et al. [16] showed that the vibration modes of a rotor will evolve in a certain manner when bearings are introduced to support the rotor. Predictions of the FEA should be consistent with the conclusions drawn from these prior publications.

## Reference System

The reference system consists of two versions: a tuned system and a mistuned system. Both the systems are studied via FEA to establish a baseline. The reference system has the same geometry as used in Ref. [15], which consists of a circular disk and 24 identical blades with fixed boundary condition at the inner rim, see Fig. 1 of Ref. [15]. The disk has a density of 855 $kg/m3$ and a Young's modulus of 189 GPa. The tuned blades have a density of 855 $kg/m3$ and a Young's modulus of 85 GPa. The circular disk defines the *xy*-plane, and the out-of-plane direction is the *z* direction.

Figure 1 shows the natural frequencies of the tuned reference system, plotted with respect to the number of nodal diameters. Based on their mode shapes, these natural frequencies are grouped into several families as labeled. Rules of the labels and representative mode shape of each family can also be found in Ref. [15]. Natural frequencies are sorted in an ascending order, and corresponding mode numbers are also listed in Fig. 1 to index each natural frequency. Note that many natural frequencies have two mode numbers, because the natural frequencies are repeated. By constructing families of natural frequencies, we can easily identify the types of modes that contribute to a localized mode.

Based on the tuned reference system, mistuning is generated via random numbers with $2.7%$ standard deviation in Young's modulus of each blade. FEA shows that there are two sets of mode localization. The first set of mode localization occurs near 900 Hz, and there are 24 localized modes. The modes involved are from the B-in-0 family with blade in-plane vibration, see Fig. 7(*a*) of Ref. [15]. The second set of mode localization occurs near 1100 Hz, and there are also 24 localized modes. The modes involved are primarily from the B-out-torsional family with blade torsional vibration, see Fig. 8(*a*) of Ref. [15].

## Effects of Flexible Bearings

To study the effects of flexible bearings on mode localization, we modify the reference system with an elastic boundary condition at the inner rim. Specifically, two sets of contact elements are used to constrain upper and lower surfaces of the inner rim evenly. The movements of each set of contact elements are represented by a pilot point that is connected to a fixed point with a linear spring to simulate a flexible bearing. The two pilot points are located at $z=\xb110$ mm, respectively. The linear springs have linear stiffness of $2.16\xd7107$ N/m in the in-plane directions (i.e., *x*- and *y*-directions), linear stiffness of $5.76\xd7106$ N/m in the out-of-plane direction (i.e., *z*-direction), and rotational stiffness of 58.5 $N\xb7m/rad$ for pitch and roll (i.e., about *x*- and *y*-axes, respectively). This formulation is referred to as the “rotor–bearing system” in the sequel. The rotor–bearing system also consists of two versions: a tuned system and a mistuned system.

Figure 2 shows the natural frequencies of the tuned, bladed disk with the elastic bearings. By comparing Fig. 2 with Fig. 1, we confirm that only vibration modes with 0 or 1 nodal diameter have changed their natural frequencies when the elastic bearings are introduced. This observation is consistent with that predicted in Ref. [16]. Moreover, modes from the D-out-2 family significantly change their natural frequencies, while modes from the B-out-torsional family do not. In other words, disk-dominant modes change their natural frequencies more than the blade-dominant modes. This result is not surprising. For blade-dominant modes, there is almost no deformation at the inner rim. Therefore, the boundary condition change at the inner rim affects minimally the natural frequencies of the blade-dominant modes.

The same mistuning of 2.7% variations in the Young's modulus is now added to the tuned rotor–bearing system. FEA shows that the mode localization occurs again in the frequency range of B-in-0 and B-out-torsional families. Moreover, the number of localized modes obtained from the FEA is listed in Table 1 for comparison. For the B-in-0 family, the number of localized modes remains 24 after the bearings are introduced. In contrast, the number of localized modes in the frequency range of the B-out-torsional family is now increased to 26 from 24. The two additional modes, due to the presence of the elastic bearings, are mode 111 (at 1102.6 Hz) and mode 112 (at 1104 Hz). Their mode shapes are shown in Fig. 3, where the pink circle in the middle of each plot represents flexible bearings.

For the rest of 24 localized modes, the presence of the bearings can change their mode shapes too. For instance, Fig. 4 shows the 16th localized mode with blade torsional vibration before and after the bearings are introduced (i.e., modes 108 and 109 of the mistuned systems). The natural frequency shifts only slightly from 1102 Hz to 1101 Hz. The mode shape, however, changes dramatically. The vibration was localized at blades 11 and 12 when the inner rim is fixed. Now the vibration is localized at blade 8 when the bearing is present. From the FEA above, we reach the first conclusion—the presence of bearings may affect mode localization via the number and shape of localized modes.

The presence of mode localization is also accompanied by unbalanced bearing forces. Figure 5 shows the Euclidean norm of the modal bearing force of the tuned and mistuned rotor–bearing system. In Fig. 5, the plus markers represent bearing forces from the tuned rotor. In contrast, the circle markers represent bearing forces from the mistuned rotor, with the larger circle markers representing contribution from modes that are extremely localized. The upper plot shows the bearing force from each mode with respect to the frequency, while the two bottom plots show the bearing forces from the B-in-0 family and the B-out-torsion family in the order of ascending frequencies. When the rotor is tuned, most vibration modes except two have small bearing forces. When the mistuning is introduced, all the vibration modes have significant bearing forces.

## Theoretical Investigation

### Number of Localized Modes.

According to Ref. [15], the mode localization occurs when the following two conditions are met for a tuned cyclic symmetric system. First, there must be a group of modes that have very close natural frequencies. Second, this group of modes contains a wide range of wave numbers. When mistuning is present, vibration modes in the group can be linearly combined to form localized modes whose vibration is confined only at one or two blades. Moreover, vibration modes participating in the linear combination must have similar features in their mode shapes.

For the tuned reference system, there are two groups of modes that have very close natural frequencies, see Fig. 1. One is around 1100 Hz, and the other is around 900 Hz. Let us focus on the first group of modes around 1100 Hz. When the tuned system has no bearing, this group consists of 24 modes (i.e., modes 93–116 in Fig. 1) with very close natural frequencies. These modes encompass 0–12 nodal diameters, thus containing a wide range of wave numbers. These modes all come from the B-out-torsional family with a common feature of blade torsional vibration. Therefore, all the conditions for mode localization are met, and these 24 modes are linearly combined to form 24 localized modes shown in Table 1.

When the bearing is introduced, the group with a natural frequency around 1100 Hz grows to 26 modes (i.e., modes 94–119 in Fig. 2). In addition to the original 24 modes from the B-out-torsion family, there are two modes (114 and 115) from the D-out-3 family that have a significant blade torsion, see Fig. 6. In other words, all these 26 modes share the same feature of blade torsional vibration. Besides, these 26 modes encompass 0–12 nodal diameters, thus containing a wide range of wave numbers. Again, all the conditions for mode localization are met and these 26 modes are linearly combined to form 26 localized modes in Table 1 when the mistuning is introduced.

The analysis above explains why the number of localized modes increases from 24 to 26 when the bearing is introduced. In essence, the presence of the bearings basically introduces additional degrees-of-freedom to the reference system and thus provides additional modes that could form a group for mode localization. If the bearing coefficients are properly chosen so that the conditions for mode localization are met, these additional modes would participate in the linear combination to form additional localized modes.

The same analysis above can also explain why the group around 900 Hz does not change its number of localized modes when the bearing is introduced. When the bearing is not present, there are roughly 26 modes in this group (i.e., modes 59–84 in Fig. 1). Specifically, 24 of them are from the B-in-0 family with a common feature of blade in-plane vibration. Moreover, they encompass 0–12 nodal diameters, thus containing a wide range of wave numbers. The remaining two of the 26 modes (i.e., modes 59 and 60) are from the D-out-2 family without any blade in-plane vibration. Therefore, the 24 modes from the B-in-0 family meet the conditions of mode localization and are linearly combined to form the 24 localized modes. When the bearings are introduced, there are no additional vibration modes appearing in this frequency range. Moreover, all the vibration modes in this group around 900 Hz are almost unchanged. As a result, the model localization form by the B-in-0 family remains the same.

### Bearing Forces.

First of all, major results from Ref. [16] are summarized below to facilitate the analysis of bearing forces. Consider a tuned cyclic symmetric system consisting of *N* identical substructures with a fixed boundary (e.g., a bladed disk with a fixed inner rim). Every mode shape of the cyclic symmetric system is then characterized via a phase index *n* that defines a phase difference between the two neighboring substructures. When the bearings are introduced into the tuned cyclic symmetric system, some modes will be coupled to the bearings changing their natural frequencies. These modes are called *unbalanced modes*. Other modes will not be affected and are called *balanced modes*. Only the modes with phase indices $n=1,N\u22121,N$ may be the unbalanced modes.

The physics behind the balanced and unbalanced modes is quite straightforward. When the tuned system vibrates in a balanced mode, the inertial force (and its moments) arising from the vibration turns out to be zero. No net forces or moments are transmitted to the fixed boundary; therefore, introduction of the bearings will not affect the state of the vibration. On the contrary, when an unbalanced mode vibrates, the inertial force results in a net force or moment acting on the fixed boundary. When the bearings are present, the net force or moment deforms the bearings leading to a coupled bearing–rotor vibration affecting the natural frequency of the unbalanced modes.

For the tuned, bladed disk shown in the reference system, the phase index is related to the number of nodal diameters. The 0-nodal-diameter modes correspond to phase index *n* = *N*, while the 1-nodal-diameter modes correspond to phase indices *n* = 1 and $n=N\u22121$. Therefore, 0 - and 1-nodal-diameter modes may be affected when the bearings are introduced. The most prominent changes appear in the D-out-3 family in Figs. 1 and 2.

The presence of unbalanced modes can also be seen in the modal bearing forces plotted in Fig. 5. For the tuned system, the unbalanced modes have significantly larger bearing forces than the balanced modes. As shown in Fig. 5, the blue plus markers represent the modal bearing forces from the tuned system. Note that only two blue plus markers in the B-in-0 family (bottom left plot) and in the B-out-torsional (bottom right plot) have large bearing forces. They correspond to the 1-nodal-diameter modes, which are unbalanced. The 0-nodal-diameter modes in B-in-0 family and B-out-torsional family happen to have zero net inertia force and moment.

When the mistuning is present, all the localized modes can be expressed in terms of linear combinations of a group of modes from the tuned system. Since the group of modes must encompass a wide range of wave numbers, it will naturally include both balanced modes and unbalanced modes. Therefore, all the localized modes will have the contribution of unbalanced modes implying that all the localized modes will incur a large bearing force as shown in the red circle markers in Fig. 5.

## Effects of Housing

To study the effects of housing on mode localization, we attach the bladed disk and its bearings (cf. Fig. 3) to a flexible housing creating a new finite-element model. The housing is a square plate with a central stud, on which the bladed disk can be supported. The square plate has a density of 855 $kg/m3$ and a Young's modulus of 90 GPa. Moreover, it is fixed at its four corners. The contact elements are used to constrain the inner surface of the bladed disk and the outer surface of the stud, and their movements are represented via pilot points. A linear spring is then used to connect the two pilot points to model the flexible bearings. The bearing stiffness remains the same as that used in the rotor–bearing system. This system with the bladed disk, housing, and bearings will be referred as the “rotor–bearing–housing system” henceforth.

The housing in the form of a square plate is chosen for the sake of simplicity. The square plate certainly does not at all resemble a real housing in a turbomachine, but it does provide needed characteristics of a housing, such as natural frequencies and mode shapes. Tai and Shen [18] have shown that the effects of housing are primarily characterized through its natural frequencies and mode shapes in coupled rotor–bearing–housing vibration. Therefore, employment of a square plate as the housing is a simple way to showcase physical insights without incurring huge computational costs. At the same time, fundamental findings from the simulations will also be valid for housing of arbitrary geometry.

Figure 7 shows natural frequencies of the tuned rotor–bearing–housing system. In particular, Fig. 7 is zoomed in to focus only on the frequency ranges of B-out-torsional and B-in-0 families, for which mode localization will occur. With the addition of the housing, the number of degrees-of-freedom increases drastically and so does the number of modes. Many of these newly added modes present vibration patterns that are heavily dominated by housing deformation. The natural frequencies of these housing-dominant modes are marked as red diamonds in Fig. 7 without reference to the number of nodal diameters. According to the theoretical analysis above, more localized modes may appear when mistuning is present.

The blade mistuning of 2.7% variations in Young's modulus is then introduced into the tuned rotor–bearing–housing system. FEA results show that there are 26 localized modes in the frequency range of B-in-0 family and 26 localized modes in the B-out-torsional family, see Table 1. Compared with the mistuned reference bladed disk with bearings only, major changes occur in the frequency range of the B-in-0 family.

Regarding the two new localized modes, the most critical feature is the significant housing deformation. Figure 8 shows the two newly added localized modes, with the bladed disk on the left and the housing on the right. One is mode 104 with frequency of 870.56 Hz and the other is mode 107 with frequency 873.97 Hz. In both modes, the housing undergoes considerable out-of-plane vibration. FEA results in Fig. 8 clearly indicate that housing can affect mode localization via bearings.

The interaction between vibration of the housing and mode localization of the rotor is also observed in forced response. For example, Fig. 9 shows FRF of blade 3 as labeled in Fig. 4 from 850 Hz to 1120 Hz covering B-in-0 and B-out-torsional families. A sinusoidal force is applied in the out-of-plane direction at a point on the housing marked by the yellow dot in Fig. 8(b). The damping ratio is assumed to be 0.05%. Two reactions are calculated: (1) Euler norm of displacement at the left tip of blade 3 and (2) von Mises strain at the left root of blade 3. In Fig. 9, the highest peaks of the displacement are at 860.6 Hz, 870.56 Hz, 874.0 Hz, and 924.1 Hz. The three modes at 860.6 Hz, 874.0 Hz, and 924.1 Hz are out-of-plane housing-dominated modes. Since these three modes do not have rotor deformation, their von Mises strain is small (cf. Fig. 9). On the contrary, the mode at 870.56 Hz is one of the newly added localized modes that have rotor vibration localized on blade 3 with significant housing deformation. As a result, the von Mises strain at 870.56 Hz has the highest peak between 850 Hz and 1120 Hz. This example demonstrates that forces exerted on the housing may also excite a localized mode on the rotor via bearings.

According to the FEA above, the coupling between the housing and localized modes arises because some natural frequencies of the housing fall in the frequency ranges of localized modes. To alleviate the excitation of localized modes, the housing should be designed such that the natural frequencies of the housing are sufficiently far away from the frequency ranges of localized modes.

## Conclusions

Through finite-element simulations and deductive reasoning from existing theoretical analyses, we reach the following conclusions:

- (1)
Boundary conditions are critical to the mode localization phenomenon of a nearly cyclic symmetric system with mistuning.

- (2)
Presence of bearings and housing may not only introduce additional localized modes but may also dramatically change the shape of existing localized modes. Whether or not the presence of bearings and housing may affect mode localization depends on the stiffness of the bearings and housing.

- (3)
Presence of mistuning causes all the localized modes to become unbalanced, thus resulting in a net force or moment transmitted to the bearings for every localized mode.

- (4)
The conclusions above are well justified in the context of theoretical analyses developed in Refs. [15] and [16].

## Acknowledgment

This material was based upon the work supported by the National Science Foundation under Grant No. CMMI-0969024. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.