# Patent

## Title
Method for predicting vibrational characteristics of rotating structures

## Abstract

A method is disclosed for analyzing the vibrational characteristics of rotating devices, such as hard disk drives and jet engines, that are coupled through bearings to flexible supports. In the method, the rotating device is discretized, for example, using a mesh suitable for finite element analysis. The support is also discretized. The natural frequency of elastic vibration modes for the rotating device is calculated and the natural frequency of vibration modes for the support is calculated. The mode shapes are then calculated and a set of modal basis corresponding to the mode shapes is utilized. Bearing stiffness and damping matrices are input, and the Lagrangian equations of motion are integrated numerically in the modal space, to calculate the vibrational response of the rotating device and support. The modal space results are then transformed into ordinary space. The vibrational analysis may be used to design devices having desired vibrational characteristics.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims the benefit of Provisional Applications No. 60/473,748, filed May 27, 2003, and No. 60/517,684, filed Nov. 6, 2003, the benefit of which is hereby claimed under 35 U.S.C. § 119.


STATEMENT REGARDING FEDERALLY-SPONSORED RESEARCH OR DEVELOPMENT
The present invention was made in the course of research that was supported by the National Science Foundation under grants CMS-0244116 and CMS-9820745. The United States Government may have certain rights to the invention herein.


FIELD OF THE INVENTION
This invention relates to methods and systems for calculating the vibrational characteristics of devices having rapidly rotating components, such as computer hard disk drives, jet engines, and the like.
BACKGROUND OF THE INVENTION
Rotating machines appear everywhere in our daily life, from high-tech applications, such as jet engines and computer hard disk drives (“HDD”) to ordinary household appliances, such as washing machines and refrigerator compressors. Rotating machines typically include a rotating part (rotor), a stationary part (stator or housing), and bearings. The rotors will have very different geometry for various applications. In jet engines, for example, the rotors are generally slender rods carrying relatively heavy disks with turbine blades. For an HDD, the rotor is generally a plurality of flexible disks attached in a spaced stack to a hub. Similarly, the stator or housing may have wide-varying geometry. The bearings connect or interface between the rotating and stationary parts. The bearings may include radial and thrust bearings to take the various loads during use. Typical bearing types include rolling element bearings, such as ball bearings, and hydrodynamic (fluid) bearings.
Traditional vibration analysis of rotating machines rests on several major assumptions. First, the rotors often have simplified geometry. For example, turbine engines are often modeled as rotating flexible rods with rigid or flexible disks. Conversely, HDDs are frequently modeled as rotating flexible disks on rigid hubs. As a result, vibration analysis models developed for one application often cannot be applied to other applications. Moreover, real rotors usually have much more complicated geometry than that utilized in existing models. Therefore, vibration analysis based on the simplified geometry often cannot predict responses accurately. Generally, the modeling of the housing and/or stator is minimal, to simplify the analysis. For example, the housing is often assumed to be rigid in modeling HDD systems. For jet engines, the housing is often modeled as a simple lumped-parameter system with a few degrees of freedom. Forced responses predicted from these simplified models may lose accuracy, even when the excitation frequency is low.
For the past decade, rapid technology advances in various industries call for more powerful vibration analysis tools for rotating machines. For example, high-speed, high-density HDDs require accurate geometric design of housing and rotors to meet stringent vibration specifications. For modem high-thrust turbine engines, housing flexibility can affect rotor performance considerably. All these new developments require greater vibrational analysis accuracy and challenge the assumptions made in traditional vibration analysis of rotating machines. Here is a specific example encountered in the HDD industry.
FIG. 1 shows a simplified model of an HDD spindle motor. The rotor consists of a rigid hub, two elastic disks (modeled by the classical plate theory), and a cantilever shaft (modeled as a Euler-Bernoulli beam). The stator is assumed to be rigid. The ball bearings are modeled as linear springs and dampers. In addition, the rotor spins with constant speed ω3. With these assumptions, the equations of motion have been derived using Lagrange's equation. FIG. 2 compares the theoretical natural frequencies of the rotor with experimental measurements as a function of the spin speed. Note that there is a significant discrepancy (about 15%) between the theoretical and experimental results for the first pair of modes denoted by (0,1), which are critical modes controlling read/write accuracy in hard disk drives.
The primary option to such vibrational analyses of an HDD spindle motor in spindle design is trial-and-error methods that are time-consuming and expensive. The time and expense limit the number of different rotor system designs that can be studied. As a result, the dynamic performance cannot be optimized with such few samples. Moreover, experimental evidence suggests that the vibrational response of the rotor may be significantly impacted by the flexible response of the fixed components, such as the base casting and the top cover. This implies that, with proper design of the fixed components, the rotor performance can be improved. However, existing disk/spindle vibration models do not include the flexibility of the fixed components and, therefore, are not effective for optimizing the dynamic response of the whole hard disk drive.
SUMMARY OF THE INVENTION
Motivated by the industrial needs and academic challenges, the present invention is directed to a unified approach to analyze vibration of rotating machines. The unified approach will be valid for rotors and stators with arbitrary geometry and complexity. There are three complementary elements that make the unified approach possible. First, a complete set of basis must be available to discretize vibration (e.g., infinitesimal displacement field) of the spinning rotor. One such set of basis is normal vibration modes of the corresponding stationary rotor. To accommodate complex rotor geometry, the normal vibration modes can be obtained through finite element analyses (“FEA”), for example. Similarly, a complete set of basis exists for the stationary housing. Second, use of Lagrange's equation and discretization will lead to a set of gyroscopic differential equations of motion. Finally, the equations of motion may be integrated in modal space using a finite element analysis. A derivation and application of the method for analyzing an apparatus with an axisymmetric rotor and flexible stationary parts are presented in the related U.S. Provisional Application No. 60/473,748, which is hereby incorporated by reference. The method is extended to asymmetric rotors and periodic rotors in U.S. Provisional Application No. 60/517,684, which also is hereby incorporated by reference.
For the present invention, the equations of motion are derived by considering the rotating part, stationary part, and the bearings, individually. For the rotating and stationary parts, the kinetic and potential energies are derived and discretized in terms of a complete set of normal modes. The gyroscopic equations of motion are derived through Lagrange's equation.
A method is disclosed for improving the vibrational characteristics of an apparatus having a rotating component, for example and not by way of limitation, hard disk drives, jet engine and/or jet engine components, motors, and the like. In particular, a method is disclosed wherein a model of the device, including the rotating component, is analyzed to determine some or all of the vibrational characteristics of the device, including accounting for the centrifugal and gyroscopic effects, by solving the equations of motion for the model, and the resulting information is used to improve the design of the apparatus.
More particularly, in an embodiment of the present invention, the a computer model of a first version of the device is generated. The model may include, for example, a finite element grid representing the first version of the device. A set of independent vibration shape functions for the rotating element are selected, and the stationary natural frequencies of the rotating element corresponding to the independent vibration shape functions are calculated. The vibration mode shape functions may be, for example, the vibration mode shape. The gyroscopic coefficients of the rotating element are calculated from the independent vibration shape functions, and the gyroscopic equations of motion are solved to obtain some of the vibrational characteristics of the rotating element through generalized coordinates corresponding to the independent vibration shape functions. A convenient method of obtaining the gyroscopic equations of motion is through use of the Lagrangian equation. The obtained vibrational characteristics are then transformed from modal space into three-dimensional physical space, and are used to modify the first version of the device.
In another embodiment of the invention a computer model of a first version of a device having a rotating element, a stationary element, and at least one bearing disposed between the rotating element and the non-rotating element is generated. A first set of independent vibration shape functions for the rotating element are selected, and the stationary natural frequencies of the rotating element corresponding to the first set of independent vibration shape functions are calculated. The gyroscopic coefficients of the rotating element from the first set of independent vibration shape functions are calculated. A second set of independent vibration shape functions for the non-rotating element are selected, and the natural frequencies of the non-rotating element corresponding to the second set of independent vibration shape functions are calculated. The set of gyroscopic equations of motion are then solved to obtain some of the vibrational characteristics of the rotating element through generalized coordinates corresponding to the first and second sets of independent vibration shape functions. The obtained vibrational characteristics are transformed from modal space into three-dimensional physical space, and are used in modifying the first version of the device using the vibrational characteristics to generate a second version of the device.



BRIEF DESCRIPTION OF THE DRAWINGS
The foregoing aspects and many of the attendant advantages of this invention will become more readily appreciated as the same become better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings, wherein:
FIG. 1 shows a simplified model of a hard disk drive spindle motor that may be analyzed using the method of the present invention;
FIG. 2 is a Campbell plot comparing typical calculated natural frequencies of a hard drive rotor, such as that shown in FIG. 1, calculated using prior art methods, with typical experimental measurements of natural frequencies as a function of the spin speed;
FIG. 3 illustrates prescribed rotational base excitation (γx, γy, γz) and linear base excitation s(t);
FIG. 4 illustrates a coordinate systems for the rotating part;
FIG. 5 shows the kinematics to determine bearing deformation;
FIG. 6 shows a spinning circular disk with evenly-spaced radial slots;
FIGS. 7A-7B show a spinning fork, indicating (a) redial vibration and (b) torsional vibration; and
FIG. 8 is a flow chart illustrating a method of using the present algorithm in the design of a device having a rotating component.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
A method for calculating the vibrational characteristics of devices having a rapidly rotating component, such as a rotor, will be described with reference to the figures, wherein like numbers indicate like elements. Vibrational characteristics, as used herein, includes the natural frequencies and mode shapes of the device, and may also include the mechanical responses of the device to externally applied forces and/or prescribed motions applied to a device having a flexible rotating component. The disclosed method is suitable for analyzing apparatuses such as hard disk drives, jet engines, motors, spin-stabilized satellites, and the like. As will be readily apparent to persons of skill in the art, the accurate analysis and prediction of the vibrational response of such apparatus—such as the elastic deformation and relative motion between the various elements, the frequency response, time-domain response, natural frequency, vibration mode shapes and damping factors—are important to the component design process.
The present method is developed for the general case of analyzing an apparatus having an asymmetric rotor supported within a flexible, stationary housing. The method is then applied to the simplified special case of a rotating asymmetric rotor with simplified boundary conditions, and wherein the rotor is rotationally periodic, such as a compressor or turbine stage for a jet engine. In this special case, the computational requirements are significantly reduced because the coefficients in the resulting equations of motion reduce to constants.
Finally, the application of the methods disclosed herein to the design of rotating apparatus will be discussed.
1. Formulation of the Stationary Part
An apparatus, such as a disk drive, having a rotating part and a stationary part, is considered. The stationary part is modeled as an elastic structure with arbitrary geometry, wherein the stationary part may be subject to externally applied motion, which may be thought of, for example, as mounting the apparatus onto a rigid shaker table that provides linear and angular base excitations. Let O be a convenient reference point on the stationary part. With O being the origin, one can define an inertia frame XYZ with XY plane being parallel to the shaker table, initially. In addition, the XYZ frame has unit vectors I, J, and K.
The stationary part is subjected to a rigid-body base excitation in the form of prescribed three-dimensional infinitesimal rotations γx, γy, and γz, as shown in FIG. 3, which shows a first rotation about the X axis, followed by a rotation about the new Y′ axis, and then a rotation about the new Z′′ axis. Due to the rigid-body rotations, the stationary part rotates to a new set of coordinates {circumflex over (X)}Ŷ{circumflex over (Z)} with unit vectors Î, Ĵ, and {circumflex over (K)}. For convenience, the coordinates {circumflex over (X)}Ŷ{circumflex over (Z)} will be termed “shaker frame” herein. For infinitesimal rotation, the coordinate systems XYZ and {circumflex over (X)}Ŷ{circumflex over (Z)} are related through




  
    
      
        
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The angular velocity and acceleration of the shaker frame, respectively, are

ω{circumflex over (X)}Ŷ{circumflex over (Z)}≈{dot over (γ)}xÎ+{dot over (γ)}yĴ+{dot over (γ)}z{circumflex over (K)}  (2)

and

{dot over (ω)}{circumflex over (X)}Ŷ{circumflex over (Z)}≈{umlaut over (γ)}xÎ+{umlaut over (γ)}yĴ+{umlaut over (γ)}z{circumflex over (K)}  (3)

In addition to the angular base excitations, the stationary part is also, in general, subjected to a linear base excitation s(t) defined as

s(t)≡sx(t)I+sy(t)J+sz(t)K  (4)

As a result of the linear base excitation, the origin O moves to an new position O′, as indicated in FIG. 3.
Now consider an arbitrary point P on the stationary part. The motion of point P consists of two parts. One is the rigid-body motion consistent with the base excitations. The other is an elastic deformation superimposed on the rigid-body motion. When the base excitation is absent, the location of P is defined through the position vector

r≡rxI+ryJ+rzK  (5)

With the angular base excitations, the rigid-body rotation causes point P to move to its new position {circumflex over (r)} defined as

{circumflex over (r)}≡rxÎ+ryĴ+rz{circumflex over (K)}  (6)

Therefore, the rigid-body displacement for point P is

RP′≡s(t)+({circumflex over (r)}−r)  (7)

which consists of infinitesimal translation and rotation.

Let W(B)({circumflex over (r)},t) be the elastic deformation of the stationary part at point P, where the superscript (B) refers to the stationary base. The elastic deformation W(B)({circumflex over (r)},t) can be approximated in terms of nb vibration modes of the stationary part as




  
    
      
        
          
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where

Wn(B)({circumflex over (r)})≡Wxn(B)({circumflex over (r)})Î+Wyn(B)({circumflex over (r)})Ĵ+Wzn(B)({circumflex over (r)}){circumflex over (K)}  (9)

is the n-th vibration mode shape of the stationary part and qn(B)(t) is the corresponding generalized coordinate. In addition, the mode shapes satisfy the orthonormality conditions

∫Wm(B)({circumflex over (r)})·Wn(B)({circumflex over (r)})dm=δmn  (10)

and

∫Vb[Wm(B)({circumflex over (r)}),Wn(B)({circumflex over (r)})]dV=[ωn(B)]2δmn  (11)

where δmn is the Kronecker delta, ωn(B) is the natural frequency of the n-th vibration mode of the stationary part, and Vb[●] is the potential energy operator for the stationary part.

With the rigid-body motion RP′ and elastic deformation W(B)({circumflex over (r)}, t), the displacement of P is

RP≡s(t)+({circumflex over (r)}−r)+W(B)({circumflex over (r)}, t)  (12)

As a result, the velocity of point P is





  
    
      
        
          
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where (8) has been used. If higher order terms are neglected, the kinetic energy of the stationary part becomes





  
    
      
        
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where mB is the mass of the stationary part, and IB is the mass moment of inertia tensor of the stationary part relative to the origin O. Also,

Js1(B)≡∫{circumflex over (r)}dm  (15)

Jan(B)≡∫Wn(B)({circumflex over (r)})dm  (16)

and

Jbn(B)≡∫{circumflex over (r)}×Wn(B)({circumflex over (r)})dm  (17)

Finally, the potential energy of the stationary part is





  
    
      
        
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2. Formulation of the Rotating Part

The rotating part is modeled as an asymmetric elastic solid. The rotating part has arbitrary geometry with centroid G located on the spin axis. In other words, the rotor is perfectly balanced. Moreover, the rotating part can undergo rigid-body motion as well as elastic deformation. To formulate the motion of the rotating part, first define a virgin state for reference. In the virgin state, the system (consisting of the rotor, the housing, and the bearings) experiences no elastic deformation and no rigid-body motion except the spin of the rotor. Moreover, let G′ denote the location of the centroid in the virgin state (see FIG. 4). (In other words, G and G′ coincide in the virgin state, but will correspond to two different points during an actual motion.)
With G′ being the origin, one can define an inertia Cartesian coordinate system {tilde over (X)}{tilde over (Y)}{tilde over (Z)} for the rotating part with {tilde over (Z)} being the spin axis in the virgin state. (In actual motion, the bearings can deform and the rotor might wobble; therefore, {tilde over (Z)} may not be the spin axis.) Also, one can define a rotating coordinate system xyz with constant angular velocity ω3. To be convenient, also assume that xyz coincides with the principal axes of the rotating part in the virgin state. Note that the xyz axes are not attached to the rotating part, because the rotating part can rock and translate but the xyz axes do not. Nevertheless, the xyz axes are the best coordinate system to describe the motion of the rotating part. Let i, j, and k be the unit vectors of the xyz coordinate system. Then




  
    
      
        
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Let us now consider the motion of a point P on the rotating part. In the virgin state, the position of P is defined by vector {tilde over (r)} (FIG. 4), where

{tilde over (r)}={tilde over (r)}xi+{tilde over (r)}yj+{tilde over (r)}zk  (20)

In the deformed state, the position of and arbitrary point P with respect to the inertia frame XYZ becomes

RP(S)({tilde over (r)},t)={right arrow over (OG′)}+{tilde over (r)}+W(S)({tilde over (r)},t)  (21)

where W(S)({tilde over (r)},t) is the displacement of P that includes rigid-body translation, rigid-body rocking, and elastic deformation.

Under the coordinate system xyz, there exists several sets of complete mode shapes. For example, when the rotor is stationary, undamped, and subjected to free boundary conditions, the rotor will have a set of complete mode shapes

Wn(S)({tilde over (r)})≡Wxn(S)({tilde over (r)})i+Wyn(S)({tilde over (r)})j+Wzn(S)({tilde over (r)})K  (22)

with natural frequencies ωn(S), n=0, 1, 2, . . . . For rotors with complicated geometry, Wn(S)({tilde over (r)}) and ωn(S) can be calculated through finite element analyses, for example. Moreover, these mode shapes are orthogonal satisfying

∫Wm(S)({tilde over (r)})·Wn(S)({tilde over (r)})dms=δmn  (23)

and

∫VS[Wm(S)({tilde over (r)}),Wn(S)({tilde over (r)})]dVs=[ωn(S)]2δmn  (24)

where δmn is the Kronecker Delta and Vs[●] is the potential energy operator for the rotor.

It will be appreciated by persons of skill in the art that other methods could alternatively be used to generate a complete set of mode shapes. For example, a fixed boundary condition for the rotating part at the bearings could be used. Furthermore, the present method would work even using assumed vibration shapes (independent vibration shape functions) for the rotating part. MEIROVITCH, L., Elements of Vibration Analysis, McGraw Hill, Inc., New York, 1975, Chap. 7.4, “Assumed-Modes Method,” pp. 266-269. Moreover, the independent vibration shape functions do not need to satisfy all the actual boundary conditions of the rotating part. These and other similar options for generating a suitable set of mode shapes are known in the art.
Note that the mode shapes Wn(S)({tilde over (r)}) also include six infinitesimal rigid-body modes with zero natural frequencies. Let us label the zero-th mode as the infinitesimal rigid-body spin of the rotor, i.e.,




  
    
      
        
          
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                3
              
            
          
          ⁢
          
            (
            
              
                
                  -
                  
                    y
                    ~
                  
                
                ⁢
                i
              
              +
              
                
                  x
                  ~
                
                ⁢
                j
              
            
            )
          
        
      
    
    
      
        (
        25
        )
      
    
  




where Ī3 is the centroidal mass moment of inertia about the spin axis. Because the vibration mode shapes are complete, any infinitesimal vibration can be expanded in terms of Wn(S)({tilde over (r)}). Therefore, the vibration RP(S)({tilde over (r)},t) of the rotor in (21) can be approximated as





  
    
      
        
          
            R
            P
            
              (
              S
              )
            
          
          ⁡
          
            (
            
              
                r
                ~
              
              ,
              t
            
            )
          
        
        ≈
        
          
            
              OG
              ′
            
            _
          
          +
          
            r
            ~
          
          +
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
            
            ⁢
            
              
                
                  W
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  
                    r
                    ~
                  
                  )
                
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        26
        )
      
    
  




where qn(S)(t) is the generalized coordinates whose response is to be determined, and ns is the number of modes retained in the series for approximation. Note that the rigid-body spin of the rotor W0(S)({tilde over (r)}) does not appear in RP(S)({tilde over (r)},t), because the spin of the rotor is redundant and has been described by the rotation of the coordinate system xyz.

With (26), the velocity of point P is




  
    
      
        
          
            
              R
              .
            
            P
            
              (
              S
              )
            
          
          ⁡
          
            (
            
              
                r
                ~
              
              ,
              t
            
            )
          
        
        =
        
          
            ω
            ×
            
              r
              ~
            
          
          +
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
            
            ⁢
            
              
                
                  W
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  
                    r
                    ~
                  
                  )
                
              
              ⁢
              
                
                  
                    q
                    .
                  
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
          +
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
            
            ⁢
            
              
                [
                
                  ω
                  ×
                  
                    
                      W
                      n
                      
                        (
                        S
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          ~
                        
                        .
                      
                      )
                    
                  
                
                ]
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        27
        )
      
    
  




where ω=ω3k. With the velocity field (27), the kinetic energy of the rotor is





  
    
      
        
          T
          S
        
        =
        
          
            
              1
              2
            
            ⁢
            
              
                I
                _
              
              3
            
            ⁢
            
              ω
              3
              2
            
          
          +
          
            
              1
              2
            
            ⁢
            
              
                ∑
                
                  n
                  =
                  1
                
                
                  n
                  s
                
              
              ⁢
              
                
              
              ⁢
              
                
                  [
                  
                    
                      q
                      .
                    
                    n
                    
                      (
                      S
                      )
                    
                  
                  ]
                
                2
              
            
          
          +
          
            
              1
              2
            
            ⁢
            
              ω
              3
              2
            
            ⁢
            
              
                ∑
                
                  n
                  =
                  1
                
                
                  n
                  s
                
              
              ⁢
              
                
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  
                    n
                    s
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    λ
                    mn
                  
                  ⁢
                  
                    
                      q
                      m
                      
                        (
                        S
                        )
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    
                      q
                      n
                      
                        (
                        S
                        )
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
          
          +
          
            
              ω
              3
            
            ⁢
            
              
                ∑
                
                  n
                  =
                  1
                
                
                  n
                  s
                
              
              ⁢
              
                
              
              ⁢
              
                
                  J
                  bn
                  
                    (
                    S
                    )
                  
                
                ⁢
                
                  
                    
                      q
                      .
                    
                    n
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
          
          +
          
            
              ω
              3
              2
            
            ⁢
            
              
                ∑
                
                  n
                  =
                  1
                
                
                  n
                  s
                
              
              ⁢
              
                
              
              ⁢
              
                
                  J
                  cn
                  
                    (
                    S
                    )
                  
                
                ⁢
                
                  
                    q
                    n
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
          
          +
          
            
              ω
              3
            
            ⁢
            
              
                ∑
                
                  n
                  =
                  1
                
                
                  n
                  s
                
              
              ⁢
              
                
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  
                    n
                    s
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    g
                    nm
                  
                  ⁢
                  
                    
                      q
                      m
                      
                        (
                        S
                        )
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    
                      
                        q
                        .
                      
                      n
                      
                        (
                        S
                        )
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




where

λmn=λnm≡δmn−∫Wmz(S)({tilde over (r)})Wnz(S)({tilde over (r)})dm  (29)

Jbn(S)≡k·∫{tilde over (r)}×Wn(S)({tilde over (r)})dm  (30)

Jcn(S)≡∫[{tilde over (r)}·Wn(S)({tilde over (r)})−{tilde over (r)}zWnz(S)({tilde over (r)})]dm  (31)

gmn=−gmn≡k·∫[Wm(S)({tilde over (r)})×Wn(S)({tilde over (r)})]dm  (32)

In addition, λmn are the centrifugal stiffening/softening coefficients, and gnm are the gyroscopic coefficients for the rotating part. For (29) to (32), the integration is carried out with respect to the rotating part. Also, δmn in (29) is the Kronecker delta. Finally, the potential energy of the rotating part is




  
    
      
        
          V
          S
        
        =
        
          
            
              1
              2
            
            ⁢
            
              ∫
              
                
                  
                    V
                    s
                  
                  ⁡
                  
                    [
                    
                      
                        W
                        
                          (
                          S
                          )
                        
                      
                      ,
                      
                        W
                        
                          (
                          S
                          )
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  
                
                ⁢
                
                  d
                  V
                
              
            
          
          =
          
            
              1
              2
            
            ⁢
            
              
                ∑
                
                  n
                  =
                  1
                
                
                  n
                  s
                
              
              ⁢
              
                
                  [
                  
                    
                      ω
                      n
                      
                        (
                        S
                        )
                      
                    
                    ⁢
                    
                      
                        q
                        n
                        
                          (
                          S
                          )
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  ]
                
                2
              
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




3. Bearing Deformation

The purpose of this section is to derive bearing deformations in terms of qn(B)(t) and qn(S)(t). Let A and A′ be two mating surfaces of a bearing (e.g., inner race and outer race), where A is on the rotating part and A′ is on the stationary part. When the spindle vibrates, the two bearing surfaces move relative to each other. In addition, the relative motion could be linear or angular.
To determine the bearing deformation, we define the location and orientation of the bearing surfaces first. In the virgin state, the bearing A is undeformed and its position is

{right arrow over (OA)}≡pxI+pyJ+pzK  (34)

where px, py, and pz are the Cartesian coordinates of bearing A with respect to the origin O (see FIG. 5). In the virgin state, also define the relative position of A to G′ as

{tilde over (r)}A={right arrow over (G′A)}≡lxI+lyJ+lzK  (35)

where lx=ly=0 for a realistic bearing. Moreover, define the relative position of G′ to O as

{right arrow over (OG′)}≡txI+tyJ+tzK  (36)

Note that {right arrow over (OG′)}+{right arrow over (G′A)}={right arrow over (OA)}; therefore,

ti+li=pi,i=x,y,z  (37)

To determine the linear bearing deformation, let rA and rA′ be the position vectors of A and A′, respectively. Then rA−rA′ defines the linear bearing deformation. According to FIG. 5 and (26),




  
    
      
        
          r
          A
        
        =
        
          
            (
            
              
                
                  t
                  x
                
                ⁢
                I
              
              +
              
                
                  t
                  y
                
                ⁢
                J
              
              +
              
                
                  t
                  z
                
                ⁢
                K
              
            
            )
          
          +
          
            
              l
              z
            
            ⁢
            K
          
          +
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
                
                  W
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  
                    
                      r
                      ~
                    
                    A
                  
                  )
                
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        38
        )
      
    
  




According to (12),





  
    
      
        
          r
          
            A
            ′
          
        
        =
        
          
            s
            ⁡
            
              (
              t
              )
            
          
          +
          
            (
            
              
                
                  p
                  x
                
                ⁢
                
                  I
                  ^
                
              
              +
              
                
                  p
                  y
                
                ⁢
                
                  J
                  ^
                
              
              +
              
                
                  p
                  z
                
                ⁢
                
                  K
                  ^
                
              
            
            )
          
          +
          
            
              ∑
              
                n
                =
                1
              
              
                n
                b
              
            
            ⁢
            
              
                
                  W
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  
                    
                      r
                      ^
                    
                    A
                  
                  )
                
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        39
        )
      
    
  




Based on (38) and (39), the linear bearing deformation is

rA−rA′≡ΔxI+ΔyJ+ΔzK  (40)

where





  
    
      
        
          
            
              
                Δ
                x
              
              ≡
              
              ⁢
              
                
                  -
                  
                    s
                    x
                  
                
                -
                
                  
                    p
                    z
                  
                  ⁢
                  
                    γ
                    y
                  
                
                +
                
                  
                    p
                    y
                  
                  ⁢
                  
                    γ
                    z
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      n
                      =
                      1
                    
                    
                      n
                      s
                    
                  
                  ⁢
                  
                    
                      [
                      
                        
                          
                            
                              W
                              xn
                              
                                (
                                S
                                )
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  r
                                  ~
                                
                                A
                              
                              )
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            ω
                            3
                          
                          ⁢
                          t
                        
                        -
                        
                          
                            
                              W
                              yn
                              
                                (
                                S
                                )
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  r
                                  ~
                                
                                A
                              
                              )
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            ω
                            3
                          
                          ⁢
                          t
                        
                      
                      ]
                    
                    ⁢
                    
                      
                        q
                        n
                        
                          (
                          S
                          )
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    1
                  
                  
                    n
                    b
                  
                
                ⁢
                
                  
                    
                      W
                      xn
                      
                        (
                        B
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          ^
                        
                        A
                      
                      )
                    
                  
                  ⁢
                  
                    
                      q
                      n
                      
                        (
                        B
                        )
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        41
        )
      
    
  
  
    
      
        
          
            
              
                
                  Δ
                  y
                
                ≡
                
                ⁢
                
                  
                    -
                    
                      s
                      y
                    
                  
                  -
                  
                    
                      p
                      x
                    
                    ⁢
                    
                      γ
                      z
                    
                  
                  +
                  
                    
                      p
                      z
                    
                    ⁢
                    
                      γ
                      x
                    
                  
                  +
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    
                      ∑
                      
                        n
                        =
                        1
                      
                      
                        n
                        s
                      
                    
                    ⁢
                    
                      
                        [
                        
                          
                            
                              
                                W
                                xn
                                
                                  (
                                  S
                                  )
                                
                              
                              ⁡
                              
                                (
                                
                                  
                                    r
                                    ~
                                  
                                  A
                                
                                )
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              ω
                              3
                            
                            ⁢
                            t
                          
                          -
                          
                            
                              
                                W
                                yn
                                
                                  (
                                  S
                                  )
                                
                              
                              ⁡
                              
                                (
                                
                                  
                                    r
                                    ~
                                  
                                  A
                                
                                )
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              ω
                              3
                            
                            ⁢
                            t
                          
                        
                        ]
                      
                      ⁢
                      
                        
                          q
                          n
                          
                            (
                            S
                            )
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                  -
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    ∑
                    
                      n
                      =
                      1
                    
                    
                      n
                      b
                    
                  
                  ⁢
                  
                    
                      
                        W
                        yn
                        
                          (
                          B
                          )
                        
                      
                      ⁡
                      
                        (
                        
                          
                            r
                            ^
                          
                          A
                        
                        )
                      
                    
                    ⁢
                    
                      
                        q
                        n
                        
                          (
                          B
                          )
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        and
      
    
    
      
        (
        42
        )
      
    
  
  
    
      
        
          
            
              
                Δ
                z
              
              ≡
              
              ⁢
              
                
                  -
                  
                    s
                    z
                  
                
                -
                
                  
                    p
                    y
                  
                  ⁢
                  
                    γ
                    x
                  
                
                +
                
                  
                    p
                    x
                  
                  ⁢
                  
                    γ
                    y
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      n
                      =
                      1
                    
                    
                      n
                      s
                    
                  
                  ⁢
                  
                    
                      
                        W
                        zn
                        
                          (
                          S
                          )
                        
                      
                      ⁡
                      
                        (
                        
                          
                            r
                            ~
                          
                          A
                        
                        )
                      
                    
                    ⁢
                    
                      
                        q
                        n
                        
                          (
                          S
                          )
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      n
                      =
                      1
                    
                    
                      n
                      b
                    
                  
                  ⁢
                  
                    
                      
                        W
                        zn
                        
                          (
                          B
                          )
                        
                      
                      ⁡
                      
                        (
                        
                          
                            r
                            ^
                          
                          A
                        
                        )
                      
                    
                    ⁢
                    
                      
                        q
                        n
                        
                          (
                          B
                          )
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        43
        )
      
    
  




Note that the coordinate transformation (19) is used, because Wn(S)({tilde over (r)}A) in (38) is represented in coordinate systems xyz and has to be transformed into the inertia frame XYZ.
For the angular deformation of the bearing, the shaker frame {circumflex over (X)}Ŷ{circumflex over (Z)} attached to the stationary part undergoes an infinitesimal rotation from shaker excitation

γ≈γxÎ+γyĴ+γz{circumflex over (K)}  (44)

Moreover, the elastic deformation of the stationary part will induce an infinitesimal rigid-body rotation




  
    
      
        
          α
          ≡
          
            
              
                α
                x
              
              ⁢
              
                I
                ^
              
            
            +
            
              
                α
                y
              
              ⁢
              
                J
                ^
              
            
            +
            
              
                α
                z
              
              ⁢
              
                K
                ^
              
            
          
        
        =
        
          
            1
            2
          
          ⁢
          
            ∇
            
              ×
              
                
                  ∑
                  
                    n
                    =
                    1
                  
                  
                    n
                    b
                  
                
                ⁢
                
                  
                    
                      W
                      n
                      
                        (
                        B
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          ^
                        
                        A
                      
                      )
                    
                  
                  ⁢
                  
                    
                      q
                      n
                      
                        (
                        B
                        )
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        45
        )
      
    
  




where {circumflex over (r)}A is the position vector of A in the shaker frame. Or more explicitly,





  
    
      
        
          α
          x
        
        =
        
          
            
              ∑
              
                n
                =
                1
              
              
                n
                b
              
            
            ⁢
            
              
                
                  
                    1
                    2
                  
                  ⁡
                  
                    [
                    
                      
                        
                          ∂
                          
                            W
                            zn
                            
                              (
                              B
                              )
                            
                          
                        
                        
                          ∂
                          
                            Y
                            ^
                          
                        
                      
                      -
                      
                        
                          ∂
                          
                            W
                            yn
                            
                              (
                              B
                              )
                            
                          
                        
                        
                          ∂
                          
                            Z
                            ^
                          
                        
                      
                    
                    ]
                  
                
                
                  
                    r
                    ^
                  
                  =
                  
                    
                      r
                      ^
                    
                    A
                  
                
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
          ≡
          
            
              ∑
              
                n
                =
                1
              
              
                n
                b
              
            
            ⁢
            
              
                α
                xn
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        46
        )
      
    
  
  
    
      
        
          α
          y
        
        =
        
          
            
              ∑
              
                n
                =
                1
              
              
                n
                b
              
            
            ⁢
            
              
                
                  
                    1
                    2
                  
                  ⁡
                  
                    [
                    
                      
                        
                          ∂
                          
                            W
                            xn
                            
                              (
                              B
                              )
                            
                          
                        
                        
                          ∂
                          
                            Z
                            ^
                          
                        
                      
                      -
                      
                        
                          ∂
                          
                            W
                            zn
                            
                              (
                              B
                              )
                            
                          
                        
                        
                          ∂
                          
                            X
                            ^
                          
                        
                      
                    
                    ]
                  
                
                
                  
                    r
                    ^
                  
                  =
                  
                    
                      r
                      ^
                    
                    A
                  
                
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
          ≡
          
            
              ∑
              
                n
                =
                1
              
              
                n
                b
              
            
            ⁢
            
              
                α
                yn
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        47
        )
      
    
  
  
    
      
        
          α
          z
        
        =
        
          
            
              ∑
              
                n
                =
                1
              
              
                n
                b
              
            
            ⁢
            
              
                
                  
                    1
                    2
                  
                  ⁡
                  
                    [
                    
                      
                        
                          ∂
                          
                            W
                            yn
                            
                              (
                              B
                              )
                            
                          
                        
                        
                          ∂
                          
                            X
                            ^
                          
                        
                      
                      -
                      
                        
                          ∂
                          
                            W
                            xn
                            
                              (
                              B
                              )
                            
                          
                        
                        
                          ∂
                          
                            Y
                            ^
                          
                        
                      
                    
                    ]
                  
                
                
                  
                    r
                    ^
                  
                  =
                  
                    
                      r
                      ^
                    
                    A
                  
                
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
          ≡
          
            
              ∑
              
                n
                =
                1
              
              
                n
                b
              
            
            ⁢
            
              
                α
                zn
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        48
        )
      
    
  




Similarly, the rotating part will experience an infinitesimal rigid-body rotation




  
    
      
        
          β
          ≡
          
            
              
                β
                x
              
              ⁢
              i
            
            +
            
              
                β
                y
              
              ⁢
              j
            
            +
            
              
                β
                z
              
              ⁢
              k
            
          
        
        =
        
          
            1
            2
          
          ⁢
          
            ∇
            
              ×
              
                
                  ∑
                  
                    n
                    =
                    1
                  
                  
                    n
                    s
                  
                
                ⁢
                
                  
                    
                      W
                      n
                      
                        (
                        S
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          ~
                        
                        A
                      
                      )
                    
                  
                  ⁢
                  
                    
                      q
                      n
                      
                        (
                        S
                        )
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        49
        )
      
    
  




where





  
    
      
        
          β
          x
        
        =
        
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
                
                  
                    1
                    2
                  
                  ⁡
                  
                    [
                    
                      
                        
                          ∂
                          
                            W
                            zn
                            
                              (
                              S
                              )
                            
                          
                        
                        
                          ∂
                          y
                        
                      
                      -
                      
                        
                          ∂
                          
                            W
                            yn
                            
                              (
                              S
                              )
                            
                          
                        
                        
                          ∂
                          z
                        
                      
                    
                    ]
                  
                
                
                  
                    r
                    ~
                  
                  =
                  
                    
                      r
                      ~
                    
                    A
                  
                
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
          ≡
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
                β
                xn
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        50
        )
      
    
  
  
    
      
        
          β
          y
        
        =
        
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
                
                  
                    1
                    2
                  
                  ⁡
                  
                    [
                    
                      
                        
                          ∂
                          
                            W
                            xn
                            
                              (
                              S
                              )
                            
                          
                        
                        
                          ∂
                          z
                        
                      
                      -
                      
                        
                          ∂
                          
                            W
                            zn
                            
                              (
                              S
                              )
                            
                          
                        
                        
                          ∂
                          x
                        
                      
                    
                    ]
                  
                
                
                  
                    r
                    ~
                  
                  =
                  
                    
                      r
                      ~
                    
                    A
                  
                
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
          ≡
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
                β
                yn
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        51
        )
      
    
  
  
    
      
        
          β
          z
        
        =
        
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
                
                  
                    1
                    2
                  
                  ⁡
                  
                    [
                    
                      
                        
                          ∂
                          
                            W
                            yn
                            
                              (
                              S
                              )
                            
                          
                        
                        
                          ∂
                          x
                        
                      
                      -
                      
                        
                          ∂
                          
                            W
                            xn
                            
                              (
                              S
                              )
                            
                          
                        
                        
                          ∂
                          y
                        
                      
                    
                    ]
                  
                
                
                  
                    r
                    ~
                  
                  =
                  
                    
                      r
                      ~
                    
                    A
                  
                
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
          ≡
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
                β
                zn
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        52
        )
      
    
  




Note that β in (49) is written in the rotating coordinates xyz. Transformnation of β back to the inertia coordinates XYZ via (19) results in




  
    
      
        
          
            
              β
              =
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    1
                  
                  
                    n
                    s
                  
                
                ⁢
                
                  [
                  
                    
                      
                        (
                        
                          
                            
                              β
                              xn
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              ω
                              3
                            
                            ⁢
                            t
                          
                          -
                          
                            
                              β
                              yn
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              ω
                              3
                            
                            ⁢
                            t
                          
                        
                        )
                      
                      ⁢
                      I
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    
                      (
                      
                        
                          
                            β
                            xn
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            ω
                            3
                          
                          ⁢
                          t
                        
                        +
                        
                          
                            β
                            yn
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            ω
                            3
                          
                          ⁢
                          t
                        
                      
                      )
                    
                    ⁢
                    J
                  
                  +
                  
                    
                      β
                      zn
                    
                    ⁢
                    K
                  
                
                ]
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        53
        )
      
    
  




The angular displacements of A relative to A′ are

ξ≡ξxI+ξyJ+ξzK  (54)

where





  
    
      
        
          ξ
          x
        
        =
        
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
                [
                
                  
                    
                      β
                      xn
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      3
                    
                    ⁢
                    t
                  
                  -
                  
                    
                      β
                      yn
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      3
                    
                    ⁢
                    t
                  
                
                ]
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
          -
          
            
              ∑
              
                n
                =
                1
              
              
                n
                b
              
            
            ⁢
            
              
                α
                xn
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
          -
          
            γ
            x
          
        
      
    
    
      
        (
        55
        )
      
    
  
  
    
      
        
          
            ξ
            y
          
          =
          
            
              
                ∑
                
                  n
                  =
                  1
                
                
                  n
                  s
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        β
                        xn
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        ω
                        3
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        β
                        yn
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        ω
                        3
                      
                      ⁢
                      t
                    
                  
                  ]
                
                ⁢
                
                  
                    q
                    n
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
            -
            
              
                ∑
                
                  n
                  =
                  1
                
                
                  n
                  b
                
              
              ⁢
              
                
                  α
                  yn
                
                ⁢
                
                  
                    q
                    n
                    
                      (
                      B
                      )
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
            -
            
              γ
              y
            
          
        
        ⁢
        
          

        
        ⁢
        and
      
    
    
      
        (
        56
        )
      
    
  
  
    
      
        
          ξ
          z
        
        =
        
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
                β
                zn
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
          -
          
            
              ∑
              
                n
                =
                1
              
              
                n
                b
              
            
            ⁢
            
              
                α
                zn
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
          -
          
            γ
            z
          
        
      
    
    
      
        (
        57
        )
      
    
  




4. Vector Notation

To keep track of the lengthy derivation, it is beneficial to adopt the following vector notations. Define the vector of generalized coordinates as

q(t)=[qBT(t),qST(t)]T  (58)

where

qB(t)=[q1(B),q2(B), . . . ,qnb(B)]T  (59)

and

qS(t)=[q1(S),q2(S), . . . ,qns(S)]T  (60)

Also, define the vector of shaker excitations as

fs(t)=[sx,sy,sz,γx,γy,γz]T  (61)

With the vector notation, the linear and angular bearing deformations can be rewritten as




  
    
      
        
          
            Δ
            b
          
          ≡
          
            (
            
              
                
                  
                    Δ
                    x
                  
                
              
              
                
                  
                    Δ
                    y
                  
                
              
              
                
                  
                    Δ
                    z
                  
                
              
              
                
                  
                    ξ
                    x
                  
                
              
              
                
                  
                    ξ
                    y
                  
                
              
            
            )
          
        
        =
        
          
            
              h
              b
            
            ⁢
            
              
                f
                s
              
              ⁡
              
                (
                t
                )
              
            
          
          +
          
            
              
                B
                b
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              q
              ⁡
              
                (
                t
                )
              
            
          
        
      
    
    
      
        (
        62
        )
      
    
  




where





  
    
      
        
          h
          b
        
        =
        
          [
          
            
              
                
                  -
                  1
                
              
              
                0
              
              
                0
              
              
                0
              
              
                
                  -
                  
                    p
                    z
                  
                
              
              
                
                  p
                  y
                
              
            
            
              
                0
              
              
                
                  -
                  1
                
              
              
                0
              
              
                
                  p
                  z
                
              
              
                0
              
              
                
                  -
                  
                    p
                    x
                  
                
              
            
            
              
                0
              
              
                0
              
              
                
                  -
                  1
                
              
              
                
                  -
                  
                    p
                    y
                  
                
              
              
                
                  p
                  x
                
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                0
              
              
                
                  -
                  1
                
              
              
                0
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                0
              
              
                0
              
              
                
                  -
                  1
                
              
              
                0
              
            
          
          ]
        
      
    
    
      
        (
        63
        )
      
    
  




and

Bb(t)≡Bb(0)+Bb(1) cos ω3t+Bb(2) sin ω3t  (64)

with





  
    
      
        
          B
          b
          
            (
            0
            )
          
        
        =
        
          [
          
            
              
                
                  -
                  
                    
                      W
                      x
                      
                        (
                        B
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          ^
                        
                        A
                      
                      )
                    
                  
                
              
              
                0
              
            
            
              
                
                  -
                  
                    
                      W
                      y
                      
                        (
                        B
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          ^
                        
                        A
                      
                      )
                    
                  
                
              
              
                0
              
            
            
              
                
                  -
                  
                    
                      W
                      z
                      
                        (
                        B
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          ^
                        
                        A
                      
                      )
                    
                  
                
              
              
                
                  
                    W
                    z
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        ~
                      
                      A
                    
                    )
                  
                
              
            
            
              
                
                  -
                  
                    α
                    x
                  
                
              
              
                0
              
            
            
              
                
                  -
                  
                    α
                    y
                  
                
              
              
                0
              
            
          
          ]
        
      
    
    
      
        (
        65
        )
      
    
  
  
    
      
        
          B
          b
          
            (
            1
            )
          
        
        =
        
          [
          
            
              
                0
              
              
                
                  
                    W
                    x
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        ~
                      
                      A
                    
                    )
                  
                
              
            
            
              
                0
              
              
                
                  
                    W
                    y
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        ~
                      
                      A
                    
                    )
                  
                
              
            
            
              
                0
              
              
                0
              
            
            
              
                0
              
              
                
                  β
                  x
                
              
            
            
              
                0
              
              
                
                  β
                  y
                
              
            
          
          ]
        
      
    
    
      
        (
        66
        )
      
    
  
  
    
      and
    
    
      
        
      
    
  
  
    
      
        
          B
          b
          
            (
            2
            )
          
        
        =
        
          [
          
            
              
                0
              
              
                
                  -
                  
                    
                      W
                      y
                      
                        (
                        S
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          ~
                        
                        A
                      
                      )
                    
                  
                
              
            
            
              
                0
              
              
                
                  
                    W
                    x
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        ~
                      
                      A
                    
                    )
                  
                
              
            
            
              
                0
              
              
                0
              
            
            
              
                0
              
              
                
                  -
                  
                    β
                    y
                  
                
              
            
            
              
                0
              
              
                
                  β
                  x
                
              
            
          
          ]
        
      
    
    
      
        (
        67
        )
      
    
  




In (65), (66), and (67),

Wi(B)({tilde over (r)}A)≡[Wil(B)({tilde over (r)}A),Wi2(B)({tilde over (r)}A), . . . ,Winb(B)({tilde over (r)}A)]  (68)

and

Wi(S)({tilde over (r)}A)≡[Wi1(S)({tilde over (r)}A),Wi2(S)({tilde over (r)}A), . . . ,Wins(S)({tilde over (r)}A)]  (69)

where i=x, y, z. Also,

αi≡[αil,αi2, . . . ,αinb]  (70)

and

βi≡[βil,βi2, . . . ,βins]  (71)

where i=x, y.

5. Generalized Forces

Generalized forces in the unified approach consist of generalized bearing forces, generalized forces from the stationary part, generalized forces from the rotating part, and generalized damping forces. They are described individually as follows.
5.1 Generalized Bearing Forces
For each bearing, the bearing forces consist of three force components and two moment components. Moreover, the bearing forces are linear combination of spring forces and damping forces described by




  
    
      
        
          
            F
            b
          
          ≡
          
            (
            
              
                
                  
                    F
                    xb
                  
                
              
              
                
                  
                    F
                    yb
                  
                
              
              
                
                  
                    F
                    zb
                  
                
              
              
                
                  
                    M
                    xb
                  
                
              
              
                
                  
                    M
                    yb
                  
                
              
            
            )
          
        
        =
        
          
            
              -
              
                K
                
                  b
                  ⁢
                  
                    
                  
                
              
            
            ⁢
            
              Δ
              b
            
          
          -
          
            
              C
              b
            
            ⁢
            
              Δ
              b
            
          
        
      
    
    
      
        (
        72
        )
      
    
  




where Kb and Cb are 5×5 stiffness and damping matrices. The virtual work done by the bearing forces is

δWb=ΣFbTδΔb=−δqTQb  (73)

where the summation sums over all the bearings and Qb is the generalized bearing force vector. Substitution of (62) and (72) into (73) yields

Qb=KB(t)q+HB(t)fs+CB(t){dot over (q)}+GB(t){dot over (f)}s  (74)

where

KB(t)≡ΣBbT(t)[KbBb(t)+Cb{dot over (B)}b(t)]  (75)

HB(t)≡ΣBbT(t)Kbhb  (76)

CB(t)≡ΣBbT(t)CbBb(t)  (77)

GB(t)≡ΣBbT(t)Cbhb  (78)

Although linear bearings are described above, and used in the preferred embodiment of the present method, it will be appreciated that the method would also work using non-linear bearing models, such as those encountered in fluid-dynamic bearings in hard disk drives under sever shock environments.
5.2 Generalized Forces from Stationary Part
Let us consider a concentrated load fB in the form of

fB=ƒBxÎ+ƒByĴ+ƒBz{circumflex over (K)}  (79)

The force fB is fixed in the shaker frame and acts on a point B on the stationary part with position vector

{circumflex over (r)}B≡{right arrow over (OB)}=x1Î+y1Ĵ+z1{circumflex over (K)}  (80)

According to (12), the displacement UB of B can be arranged in a matrix form

uB=BBq+DBfs  (81)

where





  
    
      
        
          B
          B
        
        =
        
          [
          
            
              
                
                  
                    W
                    x
                    
                      (
                      B
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        ^
                      
                      B
                    
                    )
                  
                
              
              
                0
              
            
            
              
                
                  
                    W
                    y
                    
                      (
                      B
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        ^
                      
                      B
                    
                    )
                  
                
              
              
                0
              
            
            
              
                
                  
                    W
                    z
                    
                      (
                      B
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        ^
                      
                      B
                    
                    )
                  
                
              
              
                0
              
            
          
          ]
        
      
    
    
      
        (
        82
        )
      
    
  
  
    
      
        
          D
          B
        
        =
        
          [
          
            
              
                1
              
              
                0
              
              
                0
              
              
                0
              
              
                
                  z
                  1
                
              
              
                
                  -
                  
                    y
                    1
                  
                
              
            
            
              
                0
              
              
                1
              
              
                0
              
              
                
                  -
                  
                    z
                    1
                  
                
              
              
                0
              
              
                
                  x
                  1
                
              
            
            
              
                0
              
              
                0
              
              
                1
              
              
                
                  y
                  1
                
              
              
                
                  -
                  
                    x
                    1
                  
                
              
              
                0
              
            
          
          ]
        
      
    
    
      
        (
        83
        )
      
    
  




Therefore, the virtual work done is

δWB=δuBTfB=δqTBBTfB  (84)

5.3 Generalized Forces from Rotating Part
The contents of this section are primarily for concentrated loads. The virtual work done by distributed forces can always be represented in terms of concentrated loads through superposition. For the rotating part, two types of concentrated loads often appear in real applications. One is fixed in space and takes the form of

fR(1)=ƒRx(1)I+ƒRy(1)J+ƒRz(1)K  (85)

The other is rotating with the rotor and takes the form of

FR(2)=ƒRx(2)i+ƒRy(2)j+ƒRz(2)K  (86)

In addition, the concentrated force acts on point R with position vector {tilde over (r)}R. Moreover, {tilde over (r)}R can be represented in terms of both inertia XYZ frames or rotating xyz frames as follows.

{tilde over (r)}R≡xRI+yRJ+zRK≡{tilde over (x)}Ri+{tilde over (y)}Rj+{tilde over (z)}Rk  (87)

If the concentrated force is fixed in space as in (85), (xR, yR, zR) are constants and ({tilde over (x)}R, {tilde over (y)}R, {tilde over (z)}R) are functions of time. In contrast, if the concentrated force is rotating in space as in (86), (xR, yR, zR) are functions of time and ({tilde over (x)}R, {tilde over (y)}R,{tilde over (z)}R) are constants. According to (26), the virtual displacement of {tilde over (r)}R is




  
    
      
        
          δ
          ⁢
          
            
          
          ⁢
          
            
              R
              R
              
                (
                S
                )
              
            
            ⁡
            
              (
              
                
                  
                    r
                    ~
                  
                  R
                
                ,
                t
              
              )
            
          
        
        =
        
          
            ∑
            
              n
              =
              1
            
            
              n
              s
            
          
          ⁢
          
            
          
          ⁢
          
            
              
                W
                n
                
                  (
                  S
                  )
                
              
              ⁡
              
                (
                
                  
                    r
                    ~
                  
                  R
                
                )
              
            
            ⁢
            δ
            ⁢
            
              
            
            ⁢
            
              
                q
                n
                
                  (
                  S
                  )
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
      
    
    
      
        (
        88
        )
      
    
  




There are two issues worth noting in (88). First, the virtual displacement is written in terms of the rotating coordinates xyz because of the mode shapes Wn(S)({tilde over (r)}R). Second, the argument {tilde over (r)}R inside the mode shape is a function of time for space-fixed force fR(1)(t), and is a constant for rotating force fR(2)(t).
Based on (88), the virtual work done by space-fixed force fR(1)(t) is

δWR(1)=δqTBR(1)(t)fR(1)  (89)

where





  
    
      
        
          
            B
            R
            
              (
              1
              )
            
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              [
              
                
                  
                    0
                  
                  
                    
                      
                        W
                        x
                        
                          (
                          S
                          )
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              ~
                            
                            R
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        )
                      
                    
                  
                
                
                  
                    0
                  
                  
                    
                      
                        W
                        y
                        
                          (
                          S
                          )
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              ~
                            
                            R
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        )
                      
                    
                  
                
                
                  
                    0
                  
                  
                    
                      
                        W
                        z
                        
                          (
                          S
                          )
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              ~
                            
                            R
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        )
                      
                    
                  
                
              
              ]
            
            T
          
          ⁡
          
            [
            
              
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      3
                    
                    ⁢
                    t
                  
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      3
                    
                    ⁢
                    t
                  
                
                
                  0
                
              
              
                
                  
                    
                      -
                      sin
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      3
                    
                    ⁢
                    t
                  
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      3
                    
                    ⁢
                    t
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
              
            
            ]
          
        
      
    
    
      
        (
        90
        )
      
    
  




Similarly, the virtual work done by rotating force fR(2)(t) is

δWR(2)=δqTBR(2)fR(2)  (91)

where





  
    
      
        
          B
          R
          
            (
            2
            )
          
        
        =
        
          
            [
            
              
                
                  0
                
                
                  
                    W
                    x
                    
                      (
                      S
                      )
                    
                  
                
                
                  
                    (
                    
                      
                        r
                        ~
                      
                      R
                    
                    )
                  
                
              
              
                
                  0
                
                
                  
                    W
                    y
                    
                      (
                      S
                      )
                    
                  
                
                
                  
                    (
                    
                      
                        r
                        ~
                      
                      R
                    
                    )
                  
                
              
              
                
                  0
                
                
                  
                    W
                    z
                    
                      (
                      S
                      )
                    
                  
                
                
                  
                    0
                    ⁢
                    
                      (
                      
                        
                          r
                          ~
                        
                        R
                      
                      )
                    
                  
                
              
            
            ]
          
          T
        
      
    
    
      
        (
        92
        )
      
    
  




Note that BR(2) is a constant matrix.

5.4 Generalized Damping Forces
For rotating machines, damping is very difficult to model accurately and effectively. If the damping model is too simple, predictions on forced response amplitude will be inaccurate. If the damping model is too complicated, it could become impractical to apply. As a compromise, the damping model used in this section assumes proportional damping and modal viscous damping factors. For the stationary part, assume that the damping force density (damping force per unit volume) is




  
    
      
        
          
            F
            B
          
          ⁡
          
            (
            
              
                r
                ^
              
              ,
              t
            
            )
          
        
        =
        
          
            -
            2
          
          ⁢
          
            ρ
            B
          
          ⁢
          
            
              ∑
              
                n
                =
                1
              
              
                n
                b
              
            
            ⁢
            
              
            
            ⁢
            
              
                ζ
                n
                
                  (
                  B
                  )
                
              
              ⁢
              
                ω
                n
                
                  (
                  B
                  )
                
              
              ⁢
              
                
                  W
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  
                    r
                    ^
                  
                  )
                
              
              ⁢
              
                
                  
                    q
                    .
                  
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        93
        )
      
    
  




where ρB is the density of the stationary part and ξn(B) is the viscous damping factor of each mode. In this case, the model allows some flexibility through ξn(B) while keeping the complexity of the model under control. The virtual work done by the damping force is





  
    
      
        
          
            ∫
            
              V
              b
            
            
              
            
          
          ⁢
          
            
              (
              
                
                  
                    F
                    B
                  
                  ·
                  δ
                
                ⁢
                
                  
                
                ⁢
                
                  R
                  P
                
              
              )
            
            ⁢
            
              
            
            ⁢
            
              d
              
                V
                b
              
            
          
        
        =
        
          
            -
            2
          
          ⁢
          
            
              ∑
              
                n
                =
                1
              
              
                n
                b
              
            
            ⁢
            
              
            
            ⁢
            
              
                ζ
                n
                
                  (
                  B
                  )
                
              
              ⁢
              
                ω
                n
                
                  (
                  B
                  )
                
              
              ⁢
              
                
                  
                    q
                    .
                  
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              δ
              ⁢
              
                
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    B
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        94
        )
      
    
  




where (10) and (12) have been used.

For the rotating part, there are two types of damping—internal damping and external damping. For asymmetric rotors, external damping is very difficult to model accurately and remains largely open. Therefore, we will focus on internal damping only. For internal damping, assume that




  
    
      
        
          
            F
            S
            int
          
          ⁡
          
            (
            
              
                r
                ~
              
              ,
              t
            
            )
          
        
        =
        
          
            -
            2
          
          ⁢
          
            
          
          ⁢
          
            ρ
            S
          
          ⁢
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
            
            ⁢
            
              
                ζ
                n
                int
              
              ⁢
              
                ω
                n
                
                  (
                  S
                  )
                
              
              ⁢
              
                
                  W
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  
                    r
                    ~
                  
                  )
                
              
              ⁢
              
                
                  
                    q
                    .
                  
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        95
        )
      
    
  




where ρS is the density of the rotating part, and ξnint is the internal viscous damping factor of each mode. The virtual work done by the internal damping force is





  
    
      
        
          
            ∫
            
              V
              S
            
            
              
            
          
          ⁢
          
            
              (
              
                
                  
                    F
                    S
                    int
                  
                  ·
                  δ
                
                ⁢
                
                  
                
                ⁢
                
                  R
                  S
                
              
              )
            
            ⁢
            
              d
              
                V
                S
              
            
          
        
        =
        
          
            -
            2
          
          ⁢
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
            
            ⁢
            
              
                ζ
                n
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
              ⁢
              
                ω
                n
                
                  (
                  S
                  )
                
              
              ⁢
              
                
                  
                    q
                    .
                  
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              δ
              ⁢
              
                
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
        
      
    
    
      
        (
        96
        )
      
    
  




where (23) and (26) have been used. Finally, the virtual work done by all the damping forces is arranged in a matrix form.

δWd=−δqTD{dot over (q)}  (97)

where

D=2×diag[ξ1(B)ω1(B), . . . ,ξnb(B)ωnb(B),ξ1intω1(S), . . . ,ξnsintωns(S)]  (98)

6. Gyroscopic Equations of Motion

With the kinetic energy, potential energy, and generalized forces of the system, one can derive the gyroscopic equations of motion through use of Lagrangian equations. The resulting equation of motion is

{umlaut over (q)}+[G+CB(t)+D]{dot over (q)}+[K+KB(t)]q=BR(1)(t)fR(1)+BR(2)fR(2)+BBTfB+fc−H{umlaut over (f)}s−GB(t){dot over (f)}s−HB(t)fs  (99)

where,





  
    
      
        G
        =
        
          [
          
            
              
                0
              
              
                0
              
            
            
              
                0
              
              
                
                  
                    G
                    22
                  
                  ⁡
                  
                    (
                    
                      n
                      ,
                      m
                    
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        100
        )
      
    
  




with G22(n,m)=2ω3gnm given in (32).

Also,




  
    
      
        K
        =
        
          [
          
            
              
                
                  ω
                  B
                
              
              
                0
              
            
            
              
                0
              
              
                
                  
                    ω
                    S
                  
                  -
                  
                    
                      ω
                      3
                      2
                    
                    ⁢
                    
                      
                        K
                        λ
                      
                      ⁡
                      
                        (
                        
                          n
                          ,
                          m
                        
                        )
                      
                    
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        101
        )
      
    
  




where

ωB≡diag[{ω1(B)}2,{ω2(B)}2, . . . ,{ωnb(B)}2]  (102)

ωS=diag[{ω1(S)}2,{ω2(S)}2, . . . ,{ωns(S)}2]  (103)

and Kλ(n,m)=λnm defined in (29).

Moreover,




  
    
      
        H
        =
        
          [
          
            
              
                
                  J
                  a
                  
                    (
                    B
                    )
                  
                
              
              
                
                  J
                  b
                  
                    (
                    B
                    )
                  
                
              
            
            
              
                0
              
              
                0
              
            
          
          ]
        
      
    
    
      
        (
        104
        )
      
    
  




where

Ja(B)=[Ja1(B),Ja2(B), . . . ,Janb(B)]T  (105)

and

Jb(B)=[Jb1(B),Jb2(B), . . . ,Jbnb(B)]T  (106)

Finally, fc is defined as

fc=ω32[0,Jc1(S),Jc2(S), . . . ,Jcn5(S)]T  (107)

Note that the constant loading fc results from centrifugal forces due to rotor vibration. Nevertheless, fc does not affect vibration response of the rotor. Instead, it redefines the equilibrium position.
7. Special Case of Asymmetric, Flexible, Rotationally Periodic Rotors
The mathematical development shown above may be simplified for the special case of an asymmetric, flexible and rotationally periodic rotor. This simplified model may be appropriate, for example, for modeling jet engines wherein the rotating blades describe a rotationally periodic rotor. For an initial analysis, the effects of a flexible housing and bearings are not included. With these simplifying assumptions, a finite element analysis on the corresponding stationary rotor may be done to extract natural frequencies and mode shapes. The vibration of the spinning rotor may then be represented in terms of the mode shapes and their modal response in a coordinate system that is rotating with the spinning rotor. Through the use of Lagrange's equation, a modal equation of motion may be derived, using the finite element analysis to calculate the gyroscopic matrix, centrifugal stiffening matrix, and generalized modal excitation vector. Finally, the equation of motion may be solved numerically to obtain the modal response, which, in turn will lead to the physical response of the rotor for a rotor-based observer or for a ground-based observer through a coordinate transformation.
The derivation described above may be applied directly to this simplified special case. The resulting equation of motion is

{umlaut over (q)}+[G+D]{dot over (q)}+[ωS−ω32Kλ]q=BR(1)(t)fR(1)+BR(2)fR(2)+fc  (108)

In (108), the generalized coordinate vector q onlly includes qb, because the stationary part is not considered. Moreover, G(n,m)=2ω3gmn given in (32). Also,

ωS=diag[{ω1(S)}2,{ω2(S)}2, . . . ,{ωn5(S)}2]  (109)

and K80(n,m)=λnm defined in (29). Finally, fc is defined as

fc=ω32[Jc1(S),Jc2(S), . . . ,Jcn5(S)]T  (110)

Note that the constant loading fc results from centrifugal forces due to rotor vibration. Nevertheless, fc does not affect vibration response of the rotor. Instead, it redefines the equilibrium position.
The equation of motion in (108) is valid for asymmetric rotors with arbitrary geometry and complexity. Theoretically, using (108) to find the response of a rotating asymmetric rotors consists of the following three steps. The first step is to conduct a finite element simulation on a stationary rotor with the same boundary conditions. In the finite element analysis, natural frequencies, mode shapes, and nodal mass are extracted to calculate the coefficient matrices. The second step is to calculate the modal response q(t) from (108). The third step is to reconstruct the temporal response through (26). Although the procedure may look straightforward, there are several critical issues that need to be explained further. They include gyroscopic coupling, modal analysis, axisymmetric rotors, and balanced and unbalanced modes.
7.1 Gyroscopic Coupling
Vibration modes qm(t) and qn(t) are gyroscopically coupled, if gnm defined in (32) is not zero. According to (32), the gyroscopic coupling effect will disappear if the mode shapes Wm(S)({tilde over (r)}), Wn(S)({tilde over (r)}), and the z-axis are co-planar. For example, and referring now to FIG. 6, flexural vibration of a rotating disk 140 with evenly spaced radial slots 142 will not have gyroscopic coupling. This is because the mode shapes of the slotted disk 140 in flexural vibration are all in the z direction, causing gnm to vanish. In contrast, a spinning fork 150, as shown in FIG. 7, will have significant gyroscopic coupling between its radial bending mode, FIG. 7(a), and torsional mode, FIG. 7(b). This is because the bending mode shape, torsional mode shape, and the z-axis are orthogonal to one another.
7.2 Modal Analysis
The theory of modal analysis assumes that the system is time-invariant, i.e., the equation of motion has constant coefficients. Under this assumption, natural frequencies and mode shapes exist from which one can calculate frequency response functions (“FRF”). Conversely, if FRF can be measured experimentally, natural frequencies and mode shapes can be extracted from the measured FRF.
For spinning asymmetric rotors, the equation of motion only has constant coefficients in the rotating frame xyz. This implies that the theory of modal testing is only valid when both the input force and the output displacement are co-rotating with the rotor. For example, a strain gage on the spinning fork 150 in FIG. 7 can give output strain co-rotating with the fork 150. In practical applications, it is often more desirable to know the relationship between a ground-based input force and a ground-based vibration output. For example, vibration monitoring of rotating machines often avoids the use of a slip ring and adopts ground-based measurements.
To explain modal analysis of spinning asymmetric rotors with ground-based input and output, assume that a ground-based input force fR(1)(t) fixed in the space is applied at {tilde over (r)}R given in (87). Moreover, the ground-based observer has an output location C with position vector

{tilde over (r)}C=xCI+yCJ+zCK={tilde over (x)}Ci+{tilde over (y)}Cj+{tilde over (z)}Ck  (111)

Note that (xC,yC,zC) are constants and ({tilde over (x)}C, {tilde over (y)}C,{tilde over (z)}C) are functions of time because the output location is fixed in space. According to (26), the displacement of the output point C is




  
    
      
        
          
            u
            C
            
              (
              S
              )
            
          
          ⁡
          
            (
            
              
                
                  
                    r
                    ~
                  
                  C
                
                ⁡
                
                  (
                  t
                  )
                
              
              ,
              t
            
            )
          
        
        =
        
          
            
              ∑
              
                n
                =
                1
              
              
                n
                s
              
            
            ⁢
            
              
            
            ⁢
            
              
                
                  W
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        ~
                      
                      C
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  )
                
              
              ⁢
              
                
                  q
                  n
                  
                    (
                    S
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
          
          =
          
            
              
                B
                C
                
                  (
                  1
                  )
                
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              q
              ⁡
              
                (
                t
                )
              
            
          
        
      
    
    
      
        (
        112
        )
      
    
  




where uC(S)({tilde over (r)}C(t),t) is the displacement vector in the inertia XYZ frame and





  
    
      
        
          
            B
            C
            
              (
              1
              )
            
          
          ⁡
          
            (
            t
            )
          
        
        ≡
        
          
            [
            
              
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      3
                    
                    ⁢
                    t
                  
                
                
                  
                    
                      -
                      sin
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      3
                    
                    ⁢
                    t
                  
                
                
                  0
                
              
              
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      3
                    
                    ⁢
                    t
                  
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      3
                    
                    ⁢
                    t
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
              
            
            ]
          
          ⁢
          
            

          
          ⁢
          
            
          
          [
          
            
              
                
                  
                    W
                    x1
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          ~
                        
                        C
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    )
                  
                
              
              
                
                  
                    W
                    x2
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          ~
                        
                        C
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    )
                  
                
              
              
                ⋯
              
              
                
                  
                    W
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      
                        n
                        s
                      
                    
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          ~
                        
                        C
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    W
                    y1
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          ~
                        
                        C
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    )
                  
                
              
              
                
                  
                    W
                    y2
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          ~
                        
                        C
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    )
                  
                
              
              
                ⋯
              
              
                
                  
                    W
                    
                      y
                      ⁢
                      
                        
                      
                      ⁢
                      
                        n
                        s
                      
                    
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          ~
                        
                        C
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    W
                    z1
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          ~
                        
                        C
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    )
                  
                
              
              
                
                  
                    W
                    z2
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          ~
                        
                        C
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    )
                  
                
              
              
                ⋯
              
              
                
                  
                    W
                    
                      z
                      ⁢
                      
                        
                      
                      ⁢
                      
                        n
                        s
                      
                    
                    
                      (
                      S
                      )
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          ~
                        
                        C
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        113
        )
      
    
  




The modal analysis of spinning asymmetric rotors with ground-base input and output consists of three steps. The first step is time modulation through BR(1)(t) in (90). Note that the mode shape Wn(S)({tilde over (r)}) is periodic in its angular coordinate θ. Moreover, the fixed input location requires that θ=−ω3t. As a result, BR(1)(t) in (90) is a periodic function with fundamental frequency ω3, i.e.,




  
    
      
        
          
            B
            R
            
              (
              1
              )
            
          
          ⁡
          
            (
            t
            )
          
        
        ≡
        
          
            ∑
            
              n
              =
              
                -
                ∞
              
            
            ∞
          
          ⁢
          
            
              b
              n
            
            ⁢
            
              e
              
                j
                ⁢
                
                  
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                
                  ω
                  3
                
                ⁢
                t
              
            
          
        
      
    
    
      
        (
        114
        )
      
    
  




where j≡√{square root over (−1)} and bn are the Fourier coefficients depending only on the radial input location. Now consider the equation of motion (99). The Fourier transform of the modal input force is





  
    
      
        
          F
          ⁢
          
            {
            
              
                
                  B
                  R
                  
                    (
                    1
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              
                
                  f
                  R
                  
                    (
                    1
                    )
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
            
            }
          
        
        =
        
          
            ∑
            
              n
              =
              
                -
                ∞
              
            
            ∞
          
          ⁢
          
            
              b
              n
            
            ⁢
            
              
                
                  f
                  _
                
                R
                
                  (
                  1
                  )
                
              
              ⁡
              
                (
                
                  ω
                  -
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      3
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        115
        )
      
    
  




where fR(1)ω≡F{fR(1)(t)} is the Fourier transform of fR(1)(t). Note that the modulation in time domain through BR(1)(t) turns into a series frequency shift of fR(1)(ω). Because the modulation is not purely sinusoidal, the frequency shift contains not only the fundamental frequency ω3 but also its higher harmonics.

The second step is the traditional modal analysis of time-invariant systems. Assume that only the ground-based input force fR(1)(t) is present. A Fourier transform of (108) and use of (115) result in




  
    
      
        
          
            q
            _
          
          ⁡
          
            (
            ω
            )
          
        
        =
        
          
            
              [
              
                
                  
                    -
                    
                      ω
                      2
                    
                  
                  ⁢
                  I
                
                +
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ω
                    ⁡
                    
                      (
                      
                        G
                        +
                        D
                      
                      )
                    
                  
                
                +
                
                  (
                  
                    
                      ω
                      S
                    
                    -
                    
                      
                        ω
                        3
                        2
                      
                      ⁢
                      
                        K
                        λ
                      
                    
                  
                  )
                
              
              ]
            
            
              -
              1
            
          
          ⁢
          
            
              ∑
              
                n
                =
                
                  -
                  ∞
                
              
              ∞
            
            ⁢
            
              
                b
                n
              
              ⁢
              
                
                  
                    f
                    _
                  
                  R
                  
                    (
                    1
                    )
                  
                
                ⁡
                
                  (
                  
                    ω
                    -
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      
                        ω
                        3
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        116
        )
      
    
  




The third step is to transform q(ω) back to the ground base observer according to (112). In the time domain, the resulting displacement uC(S) is very complicated because of the modulation through BC(1)(t) in (112). The physics, however, can be explained better in the frequency domain as follows. Again, note that the mode shape Wn(S)({tilde over (r)}) is periodic in its angular coordinate θ. Moreover, the fixed output location requires that θ=−ω3t. As a result, BC(1)(t) in (113) is a periodic function with fundamental frequency ω3, i.e.,




  
    
      
        
          
            B
            C
            
              (
              1
              )
            
          
          ⁡
          
            (
            t
            )
          
        
        ≡
        
          
            ∑
            
              n
              =
              
                -
                ∞
              
            
            ∞
          
          ⁢
          
            
              c
              n
            
            ⁢
            
              e
              
                j
                ⁢
                
                  
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                
                  ω
                  3
                
                ⁢
                t
              
            
          
        
      
    
    
      
        (
        117
        )
      
    
  




where cn are the Fourier coefficients, depending on the radial output location. Therefore, a Fourier transform of (112) results in





  
    
      
        
          
            
              u
              _
            
            C
            
              (
              S
              )
            
          
          ⁡
          
            (
            ω
            )
          
        
        =
        
          
            ∑
            
              n
              =
              
                -
                ∞
              
            
            ∞
          
          ⁢
          
            
              c
              n
            
            ⁢
            
              
                q
                _
              
              ⁡
              
                (
                
                  ω
                  -
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      3
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        118
        )
      
    
  




Likewise, the modulation in time domain through BC(1)(t) turns into a series frequency shift of q(ω).
8. Application of Method to Design of Apparatus with Rotating Components
The vibrational analysis method described in detail above provides a method for analyzing the vibration of rotating machines with arbitrary geometry and complexity. The method provides several advantages over prior art methods, including (i) the method incorporates the gyroscopic effects that only appear in rotating machines; (ii) the method computationally couples the vibration of the rotor and the housing; (iii) the method can accommodate rotor and housing with arbitrary geometry; and, in particular, (iv) the method can be applied to asymmetric rotor geometry. It is contemplated that the method may be incorporated into existing finite element analysis software tools.
The method may be used to improve the design of apparatus having rapidly spinning components and to reduce the cost of such designs by providing an analytical framework for analyzing the vibrational response of the apparatus. The results of the vibrational analysis may be utilized by the designer to model many different physical designs and/or to modify existing designs, to improve the performance of the device, and without requiring, or at least reducing the amount of, prototyping and testing that must be done.
FIG. 8 shows a flow chart of the steps that may be undertaken by a design engineer during the development and design of an apparatus having a rapidly rotating component, such as a hard disk drive, a jet engine, or the like. A discretized model of the rotating component of the apparatus is generated 200, for example a mesh model of the component that is suitable for finite element analysis. A similar discretized model of a stationary component of the apparatus may also be generated 210. Using the discretized models, the natural frequency characteristics of the rotating component are calculated 202, which for convenience may be based on the natural frequency for the stationary rotor, undamped, and with free boundary conditions. The natural frequency characteristics of the stationary component are also calculated 212. The vibration mode shapes of the rotating component 204, and the vibration mode shapes of the stationary component 214 are then calculated. The centrifugal and gyroscopic effects are then included into the model 206. For example the finite element analysis may be used to calculate the gyroscopic matrix, centrifugal stiffening matrix. The force excitations are transformed into modal space 208 for the rotating part. Similarly, the force excitations for the stationary part are transformed into modal space 215.
The designer enters estimated/calculated bearing stiffness and damping matrices 216. The designer will also typically enter force and excitation parameters. The gyroscopic equations of motion are then solved 218 in modal space, based on the calculated vibration modes. The results of the modal space solution may then be transformed into real space 220 (i.e. physical three-dimensional) using a coordinate transformation, to provide the vibrational response of the initial design of the apparatus. The designer can then utilize this data to modify the design of the apparatus 222, to improve its vibrational characteristics. It will be apparent that the design engineer may modify the design specifically to achieve a desired outcome, for example by selectively changing the thickness, shape or stiffness of the components according to standard engineering principles, and iteratively modify the design. The designer may thereby analyze a large number of designs and select the design providing the desired characteristics.
Although the method has been illustrated for analyzing a single rotor arrangement, it will be apparent that the method can be extended in a straightforward manner to apparatus having multiple rotors, arranged in series, parallel or a network of series and parallel. Rotating machines with multiple rotors are common, including for example, turbine engines and mass data storage systems.
While the preferred embodiment of the invention has been illustrated and described, it will be appreciated that various changes can be made therein without departing from the spirit and scope of the invention.

## Claims

1. A method for improving the vibrational characteristics of a device having a rotating element, the method comprising:
(a) in a computer, generating a finite element model of a rotating element of a first version of a device while the rotating element is stationary, the rotating element having arbitrary geometry;
(b) selecting and calculating from said finite element model a set of independent vibration shape functions, each of the independent vibration shape functions being defined over the entire geometry for the rotating element while the rotating element is stationary;
(c) calculating the stationary natural frequencies of the rotating element corresponding to the independent vibration shape functions using the finite element model;
(d) calculating gyroscopic and stiffness coefficients of the rotating element from the independent vibration shape functions while the rotating element is rotating at a specified rotational speed;
(e) solving gyroscopic equations of motion that include the gyroscopic and stiffness coefficients in terms of generalized coordinates to obtain vibrational characteristics of the rotating element corresponding to the independent vibration shape functions;
(f) transforming the obtained vibrational characteristics from generalized coordinates into three-dimensional physical space; and
(g) modifying the first version of the device using the transformed vibrational characteristics.

2. The method of claim 1, wherein the independent vibration shape functions are stationary vibration mode shapes of the rotating element.

3. The method of claim 1, wherein at least some of the natural frequencies and said independent vibration shape functions are determined using a finite element analysis applied to the finite element model wherein the rotating element is stationary, undamped, and subject to free boundary conditions.

4. The method of claim 1, wherein the set of equations of motion includes externally applied generalized forces, damping forces, and centrifugal stiffening effects from the rotating element.

5. The method of claim 1, wherein the coefficient of the set of equation of motion are derived from the natural frequencies and the gyroscopic coefficients.

6. The method of claims 4 wherein the damping forces are modeled in the form of modal viscous damping factors.

7. The method of claim 1, wherein the equations of motion comprise constant coefficients.

8. A method for improving the vibrational characteristics of a device having a rotating element, a non-rotating element, and at least one bearing disposed between the rotating element and the non-rotating element, the method comprising the steps of:
(a) in a computer, generating a first finite element model of a rotating element of a first version of a device while the rotating element is stationary, the rotating element having an arbitrary geometry;
(b) selecting from the first finite element model a first set of independent vibration shape functions, each of the independent vibration shape functions being defined over the entire geometry of the rotating element while the rotating element is stationary;
(c) calculating the stationary natural frequencies of the rotating element corresponding to the first set of independent vibration shape functions using the first finite element model;
(d) calculating a first set of gyroscopic and stiffness coefficients of the rotating element from the first set of independent vibration shape functions while the rotating element is rotating at a specified rotational speed;
(e) in a computer, generating a second finite element model of a non-rotating element of the first version of the device, the non-rotating element having an arbitrary geometry;
(f) selecting from the second finite element model a second set of independent vibration shape functions defined over the entire geometry of the non-rotating element;
(g) calculating natural frequencies of the non-rotating element corresponding to the second set of independent vibration shape functions using the second finite element model;
(h) calculating a second set of stiffness coefficients of the non-rotating element from the second set of independent vibration shape functions;
(i) solving a set of equations of motion comprising the first set of gyroscopic and stiffness coefficients and the second set of stiffness coefficients in terms of generalized coordinates to obtain some of the vibrational characteristics of the rotating element corresponding to the first and second sets of independent vibration shape functions;
(j) transforming the obtained vibrational characteristics from generalized coordinates into three-dimensional physical space; and
(k) modifying the first version of the device using the transformed vibrational characteristics to generate a second version of the device.

9. The method of claim 8, wherein the first set of independent vibration shape functions are stationary vibration mode shapes of the rotating element.

10. The method of claim 8, wherein the step of selecting the first set of independent vibration shape functions for the rotating element comprises selecting a first set of independent vibration shape functions for the rotor when the rotor is stationary, undamped, and subject to free boundary conditions.

11. The method of claim 8, wherein the second set of independent vibration shape functions is vibration mode shapes of the non-rotating element.

12. The method of claim 8, wherein the second set of independent vibration shape functions is a portion of vibration mode shapes, defined over the non-rotating element of the device while the rotating element of the device is stationary.

13. The method of claim 8, wherein the non-rotating element is rigid.

14. The method of claim 8, wherein at least some of the natural frequencies and said independent vibration shape functions of the rotating element are determined using a finite element analysis applied to the model of the rotating element, wherein the rotating element is stationary, undamped, and subject to free boundary conditions.

15. The method of claim 8, wherein the set of equations of motion includes externally applied generalized forces, damping forces, and centrifugal stiffening effects from the rotating element.

16. The method of claim 8, wherein the set of equations of motion includes externally applied generalized forces, damping forces, and externally prescribed motion of the non-rotating element.

17. The method of claim 8, wherein the set of equation of motion includes generalized forces from the at least one bearing.

18. The method of claim 8, wherein the coefficient of the set of equation of motion are calculated from the natural frequencies of the rotating element and the non-rotating element and from the gyroscopic coefficients of the rotating element.

19. The method of claims 15, wherein the damping forces of the rotating element and the non-rotating element are modeled as modal viscous damping factors.

20. The method of claim 8, wherein the equations of motion comprise constant coefficients.

21. The method of claim 8, wherein the bearing model is a linear combinations of spring forces and damping forces.

22. The method of claim 21, wherein the spring forces and damping forces are nonlinear.

23. The method of claim 8, wherein the rotating element and the non-rotating element are modeled as linearly elastic elements.