Finite element method based analysis and modeling in rotordynamics

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2 Finite element method based analysis and modeling in rotordynamics A thesis submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Master of Science In the Department of Mechanical Engineering of the College of Engineering and Applied Sciences by Bradley Weiler B.S. Purdue University April 212 Committee Chair: J. Kim, Ph.D.

3 Abstract Rotordynamics refers to the analysis and study of the vibratory motion of rotating systems. The application of finite element analysis (FEA) to rotordynamics allows for modeling rotors that have complex geometry, which however requires sound understanding of basic concept and theory of rotordynamics and the constraints of existing FEA software. This study is to implement FEA based rotordynamics analysis to test, validate and compare capabilities of the rotordynamics part of commercial software, ANSYS and NASTRAN. The formation of FEA matrices containing concepts specific to rotordynamics such as gyroscopic effects, Coriolis force, spin softening effects, internal and external damping, and circulatory matrices are discussed focusing on their effects on the stability. A comparison between using the lumped parameter modeling approach and solid elements to model a rotor disk, and the benefits of the respective approaches are also discussed. The capabilities and limitations of current versions of ANSYS Workbench and NASTRAN are discussed with suggested improvements. While transient, harmonic, and modal analysis are all covered, the main focus is on modal analysis to specifically discuss some unique issues that rotordynamic modal analysis presents. Analyses show that the directivity information has to be used in the analysis for correct tracking of critical speeds and instability, which is a problem not encountered in non-rotating systems. The method to handle rotating structural damping in current commercial FEA software is discussed, and its implication is discussed using an analytical model of a simple system. For example, it is shown that modal solvers of current commercial software cannot properly handle frequency dependent structural damping. ii

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5 Acknowledgements I would like to thank my advisor Dr. Jay Kim for his guidance throughout my thesis work. His support and out my thesis work. His support and valuable insights have been instrumental during this process. His arrangement of financial support during my study at the University of Cincinnati is greatly appreciated and has enabled me to be in the position that I am today. I am grateful to both Dr. Kim and Dr. Randall Allemang for the wide range of courses they offer in theoretical and experimental vibration analysis. Their teachings have allowed me to gain a wealth of knowledge and experience in the field of study I am most passionate about. I would also like to thank the faculty members for whom I have served as a Teaching Assistant and the people at the University of Cincinnati Simulation Center who have given me great experiences. Mention must also be made of the incredible support and encouragement that my parents and family have made without which I may never have made the decision to attend graduate school. The advice and support of my fellow graduate students here is something I am extremely grateful for and which has helped me greatly. Finally, I must thank my first mentor in industry, Jim Gutknecht, who introduced me to the challenges of analyzing vibrations of rotating structures and helped develop my fascination with the subject. iv

6 Table of Contents Abstract... ii Acknowledgements... iv List of Figures... vii List of Symbols... ix 1. Introduction Motivation and Significance Fundamental Concepts of Rotordynamics Stationary Coordinate System Rotating Coordinate System Gyroscopic Effect Coriolis Effect Circulatory Matrix Spin Softening and Stress Stiffening Rotor Whirl and Critical Speeds Finite Element Modeling ANSYS Workbench NASTRAN Finite Element Modeling Finite Element Matrices Lumped Parameter Modeling Solid Element Modeling Element Selection ANSYS Workbench and NASTRAN Rotor Model Stationary Reference Frame Inertia and Stiffness Matrices Gyroscopic Matrix Bearings and the Non-Rotating Damping Matrix Modeling Bearings as Springs and Dampers Bearing Elements with User-Defined Stiffness and Damping Modeling Journal Bearings v

7 3.4. Rotating Damping Matrix Circulatory Matrix ANSYS Implementation of the Stationary Reference Frame NASTRAN Implementation of the Stationary Reference Frame Rotating Reference Frame Similarities to the Stationary Frame Coriolis Matrix Spin Softening and Stress Stiffening Matrices Circulatory Matrix due to Non-Rotating Damping ANSYS Workbench Implementation of the Rotating Reference Frame NASTRAN Implementation of the Rotating Reference Frame Rotordynamic Analysis Modal Analysis Complex Eigenvalues Campbell Diagrams and Critical Speeds Mode Tracking Directivity and Complex Coordinates Descriptions Harmonic Analysis Transient Analysis Damping and Stability Rotating Damping Mechanisms Viscous Damping in Rotors Structural Damping in Rotors Viscous Damping and Effect on Stability Structural Damping and Effect on Stability Conclusion Summary and Recommendations Future Work References vi

8 List of Figures Figure Stationary Reference Frame... 3 Figure Rotating Reference Frame... 4 Figure Centrifugal Force... 7 Figure 1-4 Whirling... 8 Figure NASTRAN.bdf File Format Figure Beam Element Figure Point Mass Element Figure Solid Elements Figure Rotor Model Figure Bearing as Springs and Dampers Figure CBUSH and PBUSH Data Entry Cards Figure Rigid Body Element RBE2 Card Figure Squeeze Film Damper NLRSFD Data Entry Card Figure Circulatory Matrix Effect on Non-Dimensionalized Natural Frequencies Figure ROTORG Data Entry Card Figure ROTOR Data Entry Card Figure RGYRO Data Entry Card Figure RSPINR and RSPINT Data Entry Cards Figure 5-1 ANSYS Workbench Mode Shapes Figure NASTRAN Mode Shapes Figure EIGC Data Entry Figure Root Locus Plot Figure 5-5 ANSYS Workbench Campbell Diagram Figure Campbell Diagram with Positive and Negative Frequencies... 5 Figure CAMPBLL and DDVAL Data Entry Card Figure Campbell Diagram without Directivity Considered Figure NASTRAN Complex Eigenvalue Results Figure ANSYS Workbench Frequency Response Function Figure APDL Commands for Unbalance Force Figure FREQ Data Entry Card Figure UNBALNC Data Entry Card Figure TSTEPNL Data Entry Card... 6 Figure Sign of Circular Structural Frequency Figure Stability Effect of Viscous Rotating Damping in ANSYS Workbench Stationary Reference Frame Figure 6-3 Stability Effect of Viscous Rotating Damping in NASTRAN Stationary Reference Frame... 7 Figure 6-4 Stability Effect of Viscous Rotating Damping in ANSYS Workbench Rotating Reference Frame Figure Stability Effect of Structural Rotating Damping in NASTRAN Stationary Reference Frame vii

9 Figure Stability Effect of Structural Rotating Damping in ANSYS Stationary Reference Frame Figure Stability Effect of Structural Rotating Damping in ANSYS Rotating Reference Frame viii

10 List of Symbols [ ] - matrix containing all degrees of freedom at every node for all elements [ ] el - matrix containing all degrees of freedom at every node of an element [ ] g generalized matrix containing all matrices with the same dimensionality [ ] n - matrix containing all degrees of freedom for a node [ ] nt - matrix containing translational degrees of freedom for a node [A], [B] linearization matrices for modal analysis [B n ] circulatory matrix due to non-rotating damping [B r ] circulatory matrix due to rotating damping [C] damping matrix [C n ] non-rotating damping matrix [C r ] Coriolis matrix [C r ] rotating damping matrix c - damping c qr damping in translational degree of freedom q due to velocity in translational degree of freedom r D - diameter t ( ) derivative with respect to time {F} - nodal force vector [F C/S ] force due to damping/stiffness in the fixed/rotating frame F/R [G] - gyroscopic matrix j = 1 J nq rotational inertia for rotational degree of freedom about direction q at node n k - damping [K] stiffness matrix [K c ] spin softening matrix [K n ] non-rotating stiffness matrix ix

11 [K r ] rotating stiffness matrix [I] identity matrix [M] inertia/mass matrix m - mass m n mass at node n [N] - shape function matrix, applied to a single element {q}, {q }, {q } nodal vectors (displacements, velocities, and accelerations respectively) in the stationary reference frame R radius {r} complex coordinates description about displacement in X and Y directions [S] stress stiffening matrix including centrifugal force terms t time [T] transformation matrix to change from stationary coordinates to rotating coordinates [T] T transpose of the transformation matrix {u}, {u }, {u } - nodal vectors (displacements, velocities, and accelerations respectively) in the rotating reference frame XYZ global, stationary coordinate system xyz rotating coordinate system α mass matrix coefficient for Rayleigh damping β stiffness matrix coefficient for Rayleigh damping η Structural or hysteretic damping ratio φ q rotation about axis q {φ} complex coordinates description of rotation about axes X and Y λ complex eigenvalue σ stability, real part of complex eigenvalue {ψ} eigenvector Ω rotor spin speed {Ω} rotor spin speed vector: {Ω x, Ω y, Ω z } x

12 [Ω] rotor spin speed matrix Ω rotational acceleration {Ω } rotational acceleration vector: {Ω x, Ω y, Ω z } ω circular or whirl frequency, imaginary part of complex eigenvalue xi

13 1. Introduction 1.1. Motivation and Significance Analyzing and predicting vibration behavior of structures is important in the design and development of mechanical systems. A rotor refers to a mechanical system in which at least one part rotates with a very high angular momentum. Vibration analysis of rotor systems requires special knowledge because of several unique behaviors of the structure stemming from rotation effect, which is not observed in non-rotating structures. These concepts specific to rotors are the gyroscopic effect, Coriolis effect, spin softening, rotating damping, and mode directivity. The concepts of the gyroscopic effect, Coriolis effect, and spin softening are critical in the rotordynamics field and have been discussed at length in literature [1-3]. Another unique aspect of rotordynamics is the effect of rotating damping. In non-rotating structures damping is always a stabilizing factor. Damping reduces the magnitude of vibration at resonance and makes the system more stable. In rotating systems, damping present in the rotating part of the system can act as a destabilizing influence in certain situations [4,5]. In non-rotating vibrating systems, the motion of the system oscillates harmonically along a linear path. In rotating systems, vibration is actually a whirling, or circular, motion. Whether the whirling motion is in the same direction of the rotation (forward motion) or opposite direction of the rotation (backward motion) is important in the analysis of the system. Application of FEA modeling enables analysis of more complex rotor systems, however care must be taken to ensure correct implementation of the analysis. Early rotor finite element models were limited to beam and point mass elements because it is very simple to define rotational degrees of freedom and gyroscopic effects. With the development of Geradin and Kill [6] of gyroscopic matrices 1

14 for solid elements, finite element models of rotors have expanded to more complex models [7-1]. Finite element models of rotating structures utilize similar modal, frequency response, and transient solvers as non-rotating models and have been developed to determine overall rotor motion and predict natural frequencies and Campbell diagrams. Commercially available finite element programs such as ANSYS Workbench and NASTRAN have been shown to match rotor natural frequencies determined via rotor testing [11]. Discussion of rotor stability, the effect of rotor internal damping, and mode directivity are found in literature [1-2], but discussions are mostly based on analytical models. The implementation of these concepts into the finite element method requires additional knowledge and understanding. To understand how these concepts should be implemented in the finite element method, a broader discussion of rotordynamics finite element modeling is needed. The construction of ANSYS Workbench and NASTRAN rotor models and the implementation of code internal matrices are discussed in the opening sections of this thesis. Using this as a foundation for further analysis; rotor internal damping, stability, and mode directivity in finite element modeling is then addressed. Limitations and inaccuracies in the ANSYS Workbench and NASTRAN implementation of these concepts are shown along with suggested changes that would allow for improved simulation capabilities Fundamental Concepts of Rotordynamics Stationary Coordinate System The stationary coordinate system is a global, fixed reference frame. It is defined by using the three principle axes X, Y, and Z. In rotordynamics, it is common for the axis of rotation for the rotor to align along the Z axis (Figure 1-1). 2

15 Figure Stationary Reference Frame The equation of motion for the stationary reference frame can be derived using either the Newton- Euler method or the Lagrange method and takes the form of Eq. 1.1 [1, 2, 12]. [M]{q } + (Ω[G] + [C r + C n ]){q } + ([K] + Ω[B r ]){q} = {F} 1.1 The above equation relates the motion of the rotor {q} to its inertia [M], stiffness [K], and external forces {F}. The damping of the rotor is comprised of rotating damping [C r ] and non-rotating damping [C n ]. The circulatory term Ω[B r ] occurs due to the transformation of the rotating damping term into the stationary reference frame. The gyroscopic moment in the system due to the rotation of the system is accounted for by the term Ω[G] Rotating Coordinate System The rotating coordinate system is a moving coordinate system that rotates with rotor spin speed (Figure 1-2). 3

16 Figure Rotating Reference Frame The equation of motion in the rotating reference frame can be derived by transforming the stationary reference frame equation of motion to the moving coordinate frame [1, 2, 12]. This transformation relates the acceleration in the rotating reference frame to the acceleration in the stationary reference frame. {u } = {q } 2{Ω} {u } {Ω} ({Ω} {u}) {Ω } {u} 1.2 When combined with the mass matrix, the three additional terms needed to describe the acceleration in the rotating reference frame create frame dependent forces. These forces are not actually acting on the body, but are present due to the constraint of viewing the rotor in the noninertial frame. The three forces are the Coriolis force, centrifugal force, and Euler force and are created by the terms 2{Ω} {u }, {Ω} ({Ω} {u}), and {Ω } {u} respectively. The Euler force is zero for rotors with a constant spin speed. The rotating reference frame equation of motion (assuming a constant spin speed) is [M]{u } + ([C or ] + [C r ] + [C n ]){u } + ([K] [K c ] + [S] + Ω[B n ]){u} = {F} 1.3 4

17 In the above equation of motion, the Coriolis force is referenced by the matrix [C or ]. The centrifugal force is represented by the spin softening matrix [K c ]. The stress stiffening matrix [S] represents changes in the stiffness of the structure due to geometric changes or changes in the stress field of the structure. The circulatory matrix Ω[B n ] is due to the fact that non-rotating damping acts in the stationary reference frame. When transformed into rotating reference frame a skew-symmetric circulatory matrix term proportional to displacement vector appears Gyroscopic Effect The gyroscopic effect refers to the gyroscopic moment that occurs due to the angular velocity of a body. A body with rotational inertia that undergoes an angular velocity experiences this gyroscopic moment. For a disk with polar inertia J p, rotation with angular velocity Ω about the Z-axis, the kinetic energy from the gyroscopic couple can be approximated [12] as E Gk = J p Ω Y X 1.4 This approximation assumes that the rotation about the X and Y axis is small and also allows for the gyroscopic couples from those rotations to be neglected. Translating the kinetic energy expression for the gyroscopic couple to its moment equivalent results in the gyroscopic moment term Ω[G]{q } = Ω [ X Y Z J p X J p Y ] { Z } 1.5 The gyroscopic effect is a skew-symmetric matrix and is directly proportional to the rotor spin speed which results in rotor natural frequencies changing with spin speed. The gyroscopic effect is only present in the stationary reference frame. When it is translated to the rotating reference frame, the effect is part of the reference frame motion and thus does not show up in the equation of motion. 5

18 For elements without rotational degrees of freedom, the gyroscopic effect is derived in a similar manner. The kinetic energy from rotation is determined for the element, and this kinetic energy term is then transformed into translational degrees of freedom Coriolis Effect The Coriolis effect captures some of the kinetic energy associated with a mass as it rotates about an axis of rotation in the rotating reference frame. When viewed in the rotating reference frame, the entire system is rotating with rotor spin speed Ω and the kinetic energy associated with this rotation is added to the system. To account for this additional energy, the Coriolis force, spin softening force, and Euler force are added to the rotating reference frame equation of motion [12]. The spin softening and Euler forces are dependent on the displacement vector of the rotor and change the apparent stiffness of the rotor. The Coriolis force 2{Ω} {u } is dependent on the velocity vector of the system, but is skew-symmetric and therefore affects the rotor natural frequencies more than the rotor stability. The Coriolis effect is significantly easier to implement than the gyroscopic effect for finite element nodes that do not have rotational degrees of freedom Circulatory Matrix Damping in rotordynamics is defined as two separate contributions: rotating and non-rotating damping. Rotating damping results in two phenomena [1, 4, 5]. The first is the dissipation of energy which is accounted for by the rotating damping matrix [C r ]. The second phenomenon is the transformation of energy from the rotation of the system to vibration. This transformation of energy to vibration is accounted for by the circulatory matrix [B r ] in the stationary reference frame and by the rotating damping matrix itself in the rotating reference frame. Non-rotating damping always results in the dissipation of energy from the system. In the stationary reference frame it is 6

19 accounted for via the non-rotating damping matrix [C n ], but in the rotating reference frame a circulatory matrix due to non-rotating damping [B n ] is required to counteract the implied rotation that the non-rotating damping matrix is multiplied by. Both circulatory matrices are the result of the transformation of their respective damping matrices from one reference frame to the other. For both reference frames, the circulatory matrix is proportional to rotor spin speed and skewsymmetric in nature. In the stationary reference frame, the circulatory matrix is proportional to the rotating damping matrix and can be a destabilizing contribution leading to excessive system vibration. In the rotating reference frame, the circulatory matrix is proportional to the non-rotating damping matrix and is a stabilizing contribution used to correctly account for the effect of nonrotating damping Spin Softening and Stress Stiffening Spin softening is the apparent softening of the stiffness of a structure due to the centrifugal force when viewed in the rotating reference frame. Spin softening is a negative contribution to the generalized stiffness matrix and as a result lower the natural frequency of the system. The centrifugal force is a frame dependent force that appears only when a system is viewed in the rotating reference frame (Figure 1-3). Figure Centrifugal Force 7

20 Stress stiffening refers to the changing stiffness of a structure due to changes in the geometric properties or stress field of the structure. In rotating structures, as a mass moves outward the stiffness relating that mass to other points in the structure is deflected resulting in a changing stiffness field. The change in the stress field due to a radial displacement away from the axis of rotation is proportional to Ω squared [13]. When both spin softening and stress stiffening effects are active in a model, the stress stiffening effects will be greater than the spin softening effects [14]. Lumped parameter rotor models will not show the effects of spin softening or stress stiffening Rotor Whirl and Critical Speeds Rotor whirling refers to the motion of the center of the geometry of the shaft [1]. Rotor whirling can occur in the same or opposite direction as rotor spin. Rotor whirling in the same direction as rotor spin is forward whirling; rotor whirling in the opposite direction is backward whirling (Figure 1-4). Figure 1-4 Whirling A critical speed is defined as a rotor speed at which the frequency of a forcing function corresponds to a natural frequency of the system [1, 15]. Not all intersections of forcing frequencies and natural 8

21 frequencies are excited. It is possible for a forcing function and mode to be completely uncoupled resulting in no resonance or for a mode to be sufficiently damped resulting in very little excitation [1, 15]. Unbalance mass in a rotor provides an excitation at a frequency equal to rotor spin speed Ω. Lateral modes, which are typically lower in frequency than torsional modes for realistic rotors, tend to be well excited by unbalance mass [16]. Because of the particular danger of unbalance mass in exciting modes, the spin speed at which an unbalance excitation frequency is equal to a natural frequency of a rotor is referred to as a flexural critical speed [1]. Unbalance mass tends to only excite forward whirl modes, but it is possible to excite a backward whirl mode with an unbalance excitation when there is a mixed mode (two shafts spinning in opposite directions) or a ball bearing with large differences in vertical and horizontal mount stiffness is present in the structure [15]. Subcritical speeds are rotor spin speeds below the flexural critical speed of a mode. Supercritical speeds are rotor spin speeds above the flexural critical speed of a mode Finite Element Modeling Finite element modeling is the process of dividing a structure into many subdivided elements. These elements share points called nodes. The mass and rotating inertia of an element is distributed to its nodes and the stiffness and damping relationship between nodes of the same element provide internal forces that act upon the nodes. From the inertia of at the nodes, the internal forces acting upon the nodes, and external forces acting upon the nodes an equation of motion for each degree of freedom can be written for each node. mq = f external f internal = f external cq kq 1.6 mq + cq + kq = f external 1.7 The equations of motion for all of the degrees of freedom at every node can be solved simultaneously when combined into a matrix. 9

22 Two commercially available finite element software programs are ANSYS and NASTRAN. Both programs have developed rotordynamic specific code to handle the unique challenges of modeling rotating systems. ANSYS Workbench 18. and NASTRAN version 216 are examined in the following chapters in order to discuss the specifics of finite element rotordynamics and the issues related to it. While the process of building a rotating system model in these programs will be discussed, the focus will be on the implementations of rotordynamic analysis equations ANSYS Workbench ANSYS Workbench is a program that can be used to build a finite element model (pre-processor), process a finite element analysis, and post-process the results. The pre-processor and postprocessor functions are handled by the program ANSYS Workbench and ANSYS Mechanical which is a subprogram of ANSYS Workbench. When a model is built in ANSYS Workbench it is writing to an input file using ANSYS Parametric Design Language (APDL) code. Several rotordynamic related functions are not available in ANSYS Workbench and must be added as separate APDL command lines. The APDL command lines specific to rotordynamic analysis will be discussed using the format command name, variable 1, variable 2, with the variables defined following the introduction of the command NASTRAN NASTRAN is a program utilized to perform analysis on finite element models. It does not have a preprocessor or a post-processor although several programs are available to perform those functions and create a NASTRAN input file or manipulate a NASTRAN output file. A NASTRAN input file, or.bdf file, is a text file that defines the entire analysis. The input file is passed to NASTRAN where it is read 1

23 and the analysis is solved based on the instructions it contains. The.bdf file is separated into three required sections in which the different parts of the analysis are defined. These sections are the executive control section, case control section, and bulk data section. The executive control section is used to define the type of analysis. The case control section is used to define analysis subcases, define sets, and define output requests. The bulk data section defines the geometry, elements, materials, loads, and boundary conditions. Figure NASTRAN.bdf File Format The executive control section is used to define controls for the entire analysis. The executive control section is where the type of analysis is chosen and allows for optional input including diagnostics and time limits on the analysis. The type of analysis is specified by the line SOL # where # is an integer number indicating the solver being utilized. The line CEND indicates the end of the executive control section and the beginning of the case control section. Inputs to the case control section related to rotordynamic analysis are short inputs that take the form of an equation. The case control section is used to define analysis subcases, define outputs, and define sets that detailed in the bulk data section. The rotordynamic related case control entries 11

24 will be discussed in the following chapters with the format case control entry = CCN where CCN is an integer value case control number that matches a bulk data entry. The bulk data inputs are referred to as bulk data cards and have multiple variables to be defined which must be placed in defined columns of the.bdf file. The order of the bulk data entries is not important to the analysis. Bulk data cards that are utilized in rotordynamic analysis will be shown as figures with the bulk data card name in the first column and the names of the other variables in the other numbered columns. The relevant variables of the bulk data cards will also be discussed. The NASTRAN solver produces several output files. The main output file is the.f6 file and contains printed output such as displacements, stresses, eigenvalues, and eigenvectors. The.f6 file is a text file and can be read using any text editor. The output data can also be read by a post-processing software to analyze the results of the analysis. 12

25 2. Finite Element Modeling 2.1. Finite Element Matrices Finite element analysis is the process of dividing a structure into discrete sections called elements that share points called nodes. The nodes of a single element are related to each other by element stiffness and damping. This stiffness and damping relationship between nodes of an element is determined by material properties, the type of element, and the shape of the element. The mass of the element is distributed to the node points and the rotational inertia of the nodes is determined by the element shape and type. Elements can be beams, pipes, point masses, shells, or solid element and each element type makes different assumptions in their stiffness relationship definition between nodes. The motions (accelerations, velocities, and displacements) of the nodes due to internal and external forces are written as a set of equations. These nodal equations are combined into matrix form for easier manipulation. [M]{q} + [C]{q} + [K]{q} = {F} 2.1 When utilizing finite element analysis for rotor systems, the process is the same except that rotordynamic analysis requires additional internal element relationships which form the gyroscopic, Coriolis, circulatory, and spin softening terms. These additional internal element relationships are all proportional to rotor spin speed Ω or rotor spin speed squared Ω 2. Only the gyroscopic matrix is calculated directly using the element shape. The other rotordynamic specific matrices are manipulations of existing mass, damping, and stiffness matrices. Two modeling approaches are common for finite element analysis of rotors. The lumped parameter approach models a rotor using beam, pipe, and point mass elements and is a simplifying approximation of rotor behavior. The solid element approach uses solid and shell elements and is more generally accurate but more calculation time intensive. Both of these modeling approaches 13

26 can be utilized in ANSYS Workbench and NASTRAN. The matrix equations of each approach contain the same terms but the matrices are much different Lumped Parameter Modeling Lumped parameter modeling simplifies the spatial description of a model by distributing the physical behavior into elements that approximate the behavior of the whole system. In rotordynamics, a rotor shaft is approximated as a series of beam elements and rotor disks are approximated as point mass elements located at a node of the shaft beam elements. Beam elements consist of two nodes that each have six total degrees of freedom: three linear and three rotational (Figure 2-1). The mass, rotational inertia, and gyroscopic effect of the beam element are distributed to the two nodes. The relationship between the relative velocities and displacements of the two nodes are described by the element s damping and stiffness respectively. Beam element stiffness definitions typically take the form of an Euler-Bernoulli beam or Timoshenko beam definition. Pipe elements are identical to beam elements in their formulation but do not have a solid cross-sectional area. Figure Beam Element 14

27 Point mass elements only add inertia terms of mass and rotational inertia to the system. A point mass element does not have stiffness since it consists of a single node with six degrees of freedom (Figure 2-2). Point mass elements have non-zero gyroscopic matrices in the stationary reference frame if a polar inertia term is defined for the axis of rotation. Non-zero damping matrices are present for point mass element if the Rayleigh damping definition is used and the α value in the Rayleigh damping is non-zero. Figure Point Mass Element 2.3. Solid Element Modeling Solid element modeling or three dimensional modeling of rotor systems separates the rotor model into discrete three dimensional elements. Solid elements consist of multiple nodes and form either tetrahedrons or quadratic solids (Figure 2-3). Unlike beam and point mass elements, solid element nodes only have translational degrees of freedom and do not have any rotational degrees of freedom. The stiffness and damping definitions of solid elements define the relationships between the nodes of the element. 15

28 Figure Solid Elements Shell elements are sometimes used in coordination with solid elements in finite element rotor models. Shell elements can be used with the deformation of the element in one direction can be neglected. The use of shell elements requires special care for attaching them to the surfaces of solid elements to ensure that the degrees of freedom at shared nodes are the same Element Selection There are several considerations to take into account when deciding which element types are best for the simulation of rotor behavior. Element types have an effect on calculation time, the accuracy of the model, and determine whether an existing geometric model can be used or a new model must be created. Matrix equations with fewer total degrees of freedom take less time to calculate solutions. This can be important if computational power is limited or an approximate solution is needed quickly. Beam and point mass models have six degrees of freedom at each node but the number of nodes on a beam and point mass model is limited. Solid elements have three degrees of freedom at each node but can have at least four nodes for each element. Models consisting of solid elements will have many more elements than an equivalent lumped parameter model consisting of beam and point 16

29 mass elements. This results in significantly higher computational requirements for solid element models. Solid element models are generally more accurate than an equivalent lumped parameter model [8]. Lumped parameter models can have difficulty finding complex modes since they have fewer total degrees of freedom than solid element models. Modeling large radius disks as point masses ignores the effect that the stiffness of the disk and the deformation of the disk have. The final advantage of solid element modeling is that is allows for the use of importing and meshing existing geometry files that may be used to otherwise analyze the rotor parts. Lumped parameter models mesh one dimensional line bodies into beams and add point masses along the line at points where the beam elements are connected to each other. Rotor parts are likely already modeled as three dimensional parts and three dimensional models when a rotordynamic analysis takes place. These models can be easily meshed into solid elements while a new model would need to be created to utilize beam and point mass elements. ANSYS Workbench allows the importing of existing geometry into a model. Meshing follows the geometry. Line bodies are meshed into beams, two dimensional bodies are meshed into shells, and three dimensional bodies are meshed into solid elements. Point mass elements are added to a model in ANSYS Workbench at the ends of line bodies or corners of two dimensional and three dimensional bodies by inserting them into the geometry in ANSYS Mechanical. Rigid body sections can alternatively be used to model disks as point masses considered as one element. Rigid bodies are created in ANSYS Workbench by changing the stiffness behavior of a body from flexible to rigid and prevent any deformation of the body. Rigid bodies in rotordynamic analysis can be used to 17

30 represent a disk or other non-shaft part that deform very little during the current analysis. However, the connection of the rigid body to meshed parts can result in excessive penetration of meshed elements and lead to errors in the simulation if this connection is not properly controlled. Rigid body elements also cannot be used for rotating parts in the stationary reference frame within ANSYS Workbench as they do not have a gyroscopic matrix definition. Due to these issues the use of rigid body disks or other parts should be modeled as point masses in ANSYS Workbench instead. In NASTRAN, elements are created by defining the position of the grid points of the element in the coordinate system. Beam and pipe elements are created using at least two grid points, shell elements are defined using several coplanar grid points, and solid elements are created using four or more non-coplanar grid points. Point masses are added to a single grid point. Unlike ANSYS Workbench, the types of beam, pipe, shell, or solid elements are specified by the user when meshing geometry in NASTRAN. NASTRAN currently limits the rotating reference frame to solid elements. Rigid bodies cannot be part of the rotating system in NASTRAN ANSYS Workbench and NASTRAN Rotor Model The rotor model used to compare the rotordynamics capabilities of ANSYS Workbench and NASTRAN is a modified Jeffcott rotor model shown in Figure 2-4. The original Jeffcott rotor was the first rotor model developed to study some fundamental behaviors of rotors such as critical speed, natural frequency and whirling of the rotor. 18

31 Figure Rotor Model This modified Jeffcott rotor model consists of a single shaft with a diameter of.1 m and length of.2 m is divided into twenty equal beam elements. At the midpoint of the rotor shaft there is a rotor disk with a diameter of.1 m and a thickness of.2 m modeled as a point mass. Two isotropic bearings are located at the ends of the shaft that are modeling as a combination of a spring with stiffness 1.x1 8 N/m and a damper with damping coefficient 5 N/m-s. The shaft and disk have a density of 785 kg/m 3, an elasticity of 2.x1 11 Pa, and a Poisson s ratio of.3. Rotating damping was given three alternative definitions for comparison purposes: no rotating damping, Rayleigh damping with α of and β of 5.x1-6, and structural damping with the loss factor η of 5.x

32 3. Stationary Reference Frame The stationary reference frame is used when simulating axisymmetric rotating bodies, performing multi-shaft analysis, or the rotating shaft is attached to a non-rotating support structure. In the stationary reference frame each element s matrices are formed with degrees of freedom that align with a global fixed XYZ frame. The equations of motion relating the nodal accelerations {q } in the global XYZ coordinates to the internal forces due to element deformations {q } and {q} and external forces {F} are put in matrix form. [M]{q } + (Ω[G] + [C r + C n ]){q } + ([K] + Ω[B r ]){q} = {F} 3.1 Any structure being analyzed via the finite element method requires an inertia matrix [M] and stiffness matrix [K]. The damping matrix is divided into rotating damping [C r ] and non-rotating damping [C n ] which contain the damping contributions of the rotating structure and non-rotating support structure respectively. The gyroscopic matrix [G] and circulatory matrix [B r ] only contain contributions for elements that are rotating Inertia and Stiffness Matrices The inertia matrix [M] for any element is a diagonal matrix consisting of the masses and rotational inertias at the nodes of the element. The inertia matrix of each element is formed by distributing the mass of the element to each node and directly applying the rotational inertias to the nodes where required. The mass of beam elements, shell elements, and solid elements are calculated by multiplying the density of the element s material by the volume of the element. The mass is distributed to the nodes of each element based on the shape of the element. The mass at each node is applied equally to all three translational degrees of freedom at that node by default. Rotating inertias are calculated and added to nodes of beam elements based on the geometry of the element. 2

33 m 1 J 1X m 1 m 1 J 1Y [M] el,beam = J 1Z m m 2 m 2 J 2X [ J 2Y J 2Z ] Solid elements do not have rotating inertias since they do not have rotating degrees of freedom. [M] el,solid = m 1 m 1 m 1 m n 3.3 [ m n m n ] Point mass elements are defined with a mass that is applied to all three translational degrees of freedom and with rotating inertias for all three rotational degrees of freedom. [M] el,pt.mass = m 1 m 1 m 1 J 1X 3.4 [ J 1Y J 1Z ] The stiffness matrix [K] is defined as a function of each element s material elasticity and shape function. This matrix defines the stiffness between nodes of an element and is highly dependent on the element shape for beam, shell, and solid elements. Point mass elements do not have a rotating stiffness matrix [K r ] since they only have one node. The stiffness matrix can be subdivided in rotating [K r ] and non-rotating [K n ] matrices. The internal forces from the rotating stiffness matrix [K r ] are due to the displacement of nodes in the rotating structure. [F K ] R = [K r ]{u}

34 Since this internal force occurs in the rotating reference frame it must be transformed into stationary coordinates using the transform matrix [T] that defines the relationship between stationary and rotating coordinate systems. {u} = [T]{q} 3.6 [F K ] R = [T][F K ] F 3.7 By applying Eqs. 3.6 and 3.7 the internal force in the stationary reference frame is determined. [F K ] F = [T] T [F K ] R = [T] T [K r ][T]{q} = [K r ]{q} 3.8 The transformation from the rotating reference frame to the stationary reference frame results in no changes to the rotating stiffness term. Similarly the non-rotating stiffness contribution transforms from the stationary reference frame to the rotating reference frame. While the transformations of the stiffness terms from one reference frame to another does not change the composition of the terms, this will not hold true for internal forces due to damping. When using ANSYS Workbench, material properties such as elasticity and density can be defined for a material under engineering data in the Workbench project schematic and then assigned to the geometric bodies in ANSYS Mechanical. When no material properties are defined or they are not assigned to the bodies, ANSYS Workbench will use default material properties for structural steel to create the inertia and rotating stiffness matrices or each element. In NASTRAN materials must defined and every element must be assigned a material. From these material definitions and the element shapes, the inertia and stiffness matrices are created. 22

35 3.2. Gyroscopic Matrix The gyroscopic term Ω[G] is formed on an element by element basis in ANSYS Workbench and NASTRAN. In ANSYS Workbench the gyroscopic effect is defined only for the point mass element MASS21, beam and pipe elements BEAM188, BEAM189, PIPE288, PIPE289, shell elements SHELL181 and SHELL281, solid elements SOLID185, SOLID186, SOLID187, SOLID272, SOLID273, and a superelement containing grouped together elements MATRIX5. Applying a rotational velocity to any other element types in the stationary frame will result in an error and prevent the simulation from successfully occurring. NASTRAN defines the gyroscopic effect for point mass elements CONM1 and CONM2, beam element CBEAM, axisymmetric elements CQUADX and CTRIAX, shell elements CQUAD4, CQUAD8, CTRIA3, and CTRIA6, and solid elements CTETRA, CPENTA, and CHEXA. NASTRAN also allows the use of beam elements CBAR in rotordynamic analyses but CBAR elements do not include any gyroscopic effects. All other element types cannot be part of a rotor in NASTRAN. The gyroscopic matrix for point mass and beam elements can be determined directly from the polar inertia of the elements, but solid elements do not have a polar inertia defined in their inertia matrix and their gyroscopic effect must be determined using a different approach. Both ANSYS Workbench and NASTRAN use the kinetic energy approach developed by Geradin & Kill [6] to determine the gyroscopic matrix for their elements. The kinetic energy of each element due to gyroscopic effects is defined and from the kinetic energy expression the gyroscopic matrix of the element is then determined by taking the derivation of the kinetic energy with respect to nodal displacement vector. 23

36 E Gk {q} = Ω[G]{q } Bearings and the Non-Rotating Damping Matrix Bearings connect rotor shafts to a non-rotating structure that support the structure and remove energy from the rotating system by applying a source of non-rotating damping helping the stability of the rotor system. There are several methods that can be used to model bearings for rotordynamic analysis. Bearings can be modeled as a set of springs and dampers, a combined spring and damper bearing element, or as a fluid bearing with changing stiffness and damping characteristics based on the position of the rotor. For each rotor node that they are connected to, bearings add stiffness and damping to the two translational degrees of freedom in the bearing s plane which is orthogonal to the rotor s axis. For all of modeling approaches, the damping of a bearing is part of the non-rotating damping matrix [C n ] and the stiffness of the bearing is added to the stiffness matrix [K] via its external stiffness matrix [K n ] contribution. Point masses can be added to the bearing nodes reflect the inertia effect of the bearing housing Modeling Bearings as Springs and Dampers The simplest modeling technique for bearings in rotordynamic analysis is to approximate the overall stiffness and damping of a bearing as a set of springs and dampers that are orthogonal to the rotor axis (Figure 3-1). If the shaft is modeled using beam elements a spring and a damper are attached to a node of the beam element and a fixed point. Two sets of springs and dampers are required to define the stiffness and damping in both translational directions. If the shaft is modeled using solid elements, multiple sets of springs and dampers can be used to constrain in-plane nodes at the location of the bearing. For simple models this approach is preferable in ANSYS Workbench. 24

37 Figure Bearing as Springs and Dampers Bearing Elements with User-Defined Stiffness and Damping ANSYS Workbench contains a bearing element COMBI214 with two degrees of freedom at each node and no bending or torsion consideration. The user defined stiffness and damping definition of this bearing element is the default option when creating a bearing in ANSYS Workbench. The bearing plane, stiffness, and damping definitions are all inputs for the element. Use of the COMBI214 element also allows for cross-coupling the stiffness or the damping between the two directions. The bearing element in ANSYS Workbench is scoped to two nodes. The first node is on the shaft itself and the second node is a stationary non-rotating point which defaults to a coincident ground. While the bearing element is not rotating in theory, because the nodes of the element are coincident and one node is connected to the rotating shaft the motion of the bearing element is determined to be rotating by ANSYS Workbench. As a result the damping of the bearing element is added to the rotating damping matrix instead of the non-rotating damping matrix. To avoid this incorrect application of bearing damping in FEA a rigid body element can be used. The rigid body element is used to connect the translational motion but not rotational motion of a node on the rotating structure to a coincident node that is not part of the rotating structure. This effectively 25

38 decouples the rotation of the rotor from the rotation of the support structure. Within ANSYS Workbench this should be possible though the use of a remote point connection, but use of the remote point connection still adds bearing damping to the rotating damping matrix instead of the non-rotating damping matrix. Due to this issue, bearings in ANSYS Workbench are best modeled as a combination of springs and dampers instead of using the built-in bearing elements. In NASTRAN a CBUSH element is used to connect two coincident nodes together or a rotor node to ground. The stiffness and damping of the CBUSH element are defined by its associated material property definition PBUSH (Figure 3-2). This bearing element definition does not allow for crosscoupling stiffness or damping terms. Figure CBUSH and PBUSH Data Entry Cards In the NASTRAN bearing element card CBUSH, EID is the element identification number, PID is the property identification number, GA and GB are the primary method of defining the grid points of the bearing, CID is coordinate system of the bearing, and the remainder of the card is used as an alternative method to scope the bearing location. The NASTRAN bearing property card PBUSH defines the stiffnesses Ki along each of the six primary directions along the coordinate system of the element, as well as damping Bi and structural damping GEi along these directions. A mass M can be added and recovery stress and strain components can be defined via the SA, ST, EA, and ET 26

39 variables. NASTRAN also requires that the bearing not be explicitly connected to a point on the rotor. A coincident node must be created with linked displacements but separate rotations using a rigid body connection card RBE2 (Figure 3-3). Figure Rigid Body Element RBE2 Card The element identification number EID must be a new, unused element value, the rotor grid point is GN, the linked degree of freedom are defined at CM as 123 for all translational degrees of freedom, and GM1 is the coincident grid point that is not defined as part of the rotor. Separation of the rotor from the bearing element is required to avoid adding the damping from the bearing to the rotating damping matrix instead of the non-rotating damping matrix Modeling Journal Bearings Journal bearings are non-linear bearings that change stiffness and damping based on the current position of the center of the shaft. ANSYS Workbench uses a Reynold s equation approach to model journal bearings when a bearing element is created using the following two ANSYS APDL command lines: et, Node ID, element type and keyopt, KEYOPT1, KEYOPT2, KEYOPT3. In these command lines, the NodeID is an integer number specifying the rotor node the bearing connects to, a journal bearing is created with element type combi214, KEYOPT1 is set to 1 for limited output results of a journal bearing or 2 for full output results, KEYOPT2 is set to for XY plane, 1 for YZ plane, or 2 for XZ plane, and KEYOPT3 is set to for a symmetric bearing or 1 for an asymmetric bearing. 27

40 For either journal bearing output requirements, the geometry is defined by the radial clearance C, bearing length L, and rotor radius R. These variables are defined as real constants using the ANSYS APDL command r, NodeID, Clearance, Length, Radius, Veloc1, Veloc2, PertInc, ThetaInc, OmgPrec. When the full output requirements are required, the perturbation increment for stiffness and damping calculation PertInc and theta increment for integration ThetaInc must be defined. Veloc1 and Veloc2 are not used in rotordynamic analysis. OmgPrec is only used in the special case of a squeeze film damper having synchronous precession and defines the rotational velocity for that case. In ANSYS Workbench this journal bearing definition cannot be used in Modal analysis. NASTRAN allows for squeeze film damper behavior to be approximated for transient analysis. The squeeze film damper creates a force that is added to the force vector {F}. Input data required to calculate this force are the journal diameter BDIA, clearance BCLR, length BLEN, oil viscosity VISCO, location of ports THETA1 and THETA2, and the boundary pressures PRES1 and PRES2. These variables are input using the NLRSFD card (Figure 3-4). Figure Squeeze Film Damper NLRSFD Data Entry Card 3.4. Rotating Damping Matrix Rotating damping represents all of the damping within the rotating structure as opposed to damping due to bearing supports and the stator. Rotor damping can be defined as Rayleigh damping applied to the rotating structure, structural damping applied to the rotating structure, or a viscous damper 28

41 element connecting two nodes that are part of the rotating structure. As will be shown, rotating damping results in the rotating damping matrix [C r ] and the circulatory matrix due to rotating damping [B r ]. These two matrices have a large effect on rotor stability and the different damping models affect stability differently. Due to its implication to the system stability, rotating damping is crucial in the analysis of rotordynamics. The internal force at a node from rotating damping occurs in the rotating reference frame. Like the internal force at a node from rotating stiffness, this force must be transformed to the stationary reference frame for use in the stationary reference frame equation of motion. [F C ] R = [C r ]{u } 3.1 Since the internal force from damping acts upon the velocity, the relationship between the velocity vectors in the two coordinate frames is derived using the first time derivative of Eq {u } = [T]{q } + ([T]){q} 3.11 dt This results in an internal force due to rotating damping with two terms. [F C ] F = [T] T [F C ] R = [T] T [C r ][T]{q } + [T] T [C r ] ([T]){q} 3.12 dt The first term in Eq results in the rotating damping matrix [C r ] multiplied by the nodal velocity vector {q }. The second term in the equation is the contribution from the circulatory matrix due to rotating damping in the stationary reference frame Circulatory Matrix As shown in the preceding section, the circulatory matrix in the stationary reference frame is a result of the transformation of rotating damping from the rotating coordinate system to the stationary coordinate system. For a rotor aligned along the z-direction the transformation matrix [T] relating the rotating xyz frame to the global XYZ frame can be written as 29

42 cosωt sinωt [T] = [ sinωt cosωt ] Inserting the transformation matrix into the circulatory matrix term in Eq gives cosωt sinωt cosωt sinωt Ω[B r ] nt = [ sinωt cosωt ] [C r ] nt ([ sinωt cosωt ]) 3.14 dt 1 1 cosωt sinωt sinωt cosωt = Ω [ sinωt cosωt ] [C r ] nt [ cosωt sinωt ] 1 1 Note that the transformation of the rotational degrees of freedom has the same relationship as the transformation of the translational degrees of freedom and that equation 3.14 can be expanded to include all six degrees of freedom at each node. Most implementations of the circulatory matrix will assume that the rotating damping matrix is symmetric about the rotation axis without crosscoupling terms. c r [C r ] nt = [ c r ] 3.15 With this assumption, the circulatory matrix due to rotating damping can be further reduced producing the skew-symmetric circulatory matrix. 1 c r c r [B r ] nt = [J][C r ] nt = [ 1 ] [ c r ] = [ c r ] 3.16 The rotating damping defined for rotational degrees of freedom φ x and φ y are likewise multiplied by the skew-symmetric matrix [J]. In ANSYS Workbench the circulatory matrix is calculated on an element basis instead of a nodal basis. Instead of defining damping at each node after the stiffness and inertia contributions of an element have been distributed to the nodes, ANSYS Workbench defines damping for the element based on the damping model and the element s stiffness or inertia contributions. The damping of an 3

43 element is distributed to its nodes in the same manner as the stiffness matrix. Because of this, ANSYS Workbench defines the circulatory matrix on an elemental basis and then distributes the circulatory matrix to the nodes. In its implementation, ANSYS Workbench assumes that the damping is symmetric, but allows for the rotor to rotate about an axis that does not align with one of the X, Y, or Z axes. The circulatory matrix contribution is thus Ω z Ω y Ω[B r ] el = [C r ] el [ Ω z Ω x ] 3.17 Ω y Ω x This element based circulatory matrix definition is then distributed to the element s nodes in the same manner as the stiffness matrix. NASTRAN calculates the circulatory matrix on a node by node basis. NASTRAN aligns rotors along their local z axis and calculates the circulatory matrix at each node with the rotating damping matrix [C r ] n and a transformation matrix [T I ] where Ω[B r ] n = Ω([T I ][C r ] n + [C r ] n [T I ]) 3.18 = Ω.5 ([ [C r ] n + [C r ] n ] [ ]) This definition removes all cross-coupled damping terms from the circulatory matrix and preserves the definition given in Eq For solid element nodes the transformation matrix [T I ] is the three by three upper left submatrix of the full six by six matrix. The nodal circulatory matrices are then combined to form the global circulatory matrix in NASTRAN. Since the circulatory matrix is multiplied by the rotational velocity, its effect on natural frequencies and stability increases with spin speed Ω (Figure 3-5). 31

44 Figure Circulatory Matrix Effect on Non-Dimensionalized Natural Frequencies 3.6. ANSYS Implementation of the Stationary Reference Frame ANSYS Workbench implementation of the stationary reference frame can only be accomplished by utilizing APDL commands. The command used is CORIOLIS, 1, --, --, 1, RotDamp where RotDamp is an option to activate the circulatory matrix due to rotating damping. To activate the circulatory matrix RotDamp is set to 1, to deactivate it RotDamp is set to. By default the circulatory matrix is not activated in ANSYS Workbench. The axis of rotation and the bodies included in the rotating structure are selected during the process of defining a rotational velocity NASTRAN Implementation of the Stationary Reference Frame Within NASTRAN the stationary reference is utilized whenever the ROTORG card is present in the input file or the ROTOR card is present and the fixed reference frame is specified as part of the ROTOR card. The ROTORG card (Figure 3-6) is used for lumped parameter modeling. 32

45 Figure ROTORG Data Entry Card In the ROTORG card, ROTORID is a positive integer number that serves as a rotor identification number, and GRID1, GRID2, and GRIDn are grid point (nodes) that are present in the rotor. The first format of the ROTORG card for identified the nodes requires individually listed all of the grid points. The second format defines minimum (GRID1) and maximum (GRID2) grid point identification numbers and defines the increment (INC) by which successive grid points are defined. All grid points in a ROTORG rotor must be collinear. The ROTOR card (Figure 3-7) is used for solid element modeling and is similar to the ROTORG card in that it requires a rotor identification number ROTORID and grid points to be defined. Figure ROTOR Data Entry Card The FRAME is set to FIX for a stationary reference frame or ROT for a rotating reference frame. LYTPE is the list type that is entered as either ELEM for elements or PROP for properties and is followed by element or property numbers (ID1, ID2, etc.) that define the rotor. The rotor axis is defined by two nodes identified as grid points GID1 and GID2. 33

46 Rotors defined by ROTOR and ROTORG cards require addition NASTRAN cards to have rotordynamic effects calculated. RGYRO is both a case control card, RGYRO = n, and a bulk data entry card (Figure 3-8) and is used to define the rotation speed of a reference rotor. Figure RGYRO Data Entry Card In the RGYRO bulk data entry card, RID is an integer number equal to that of the RGYRO case control card. REFROTR is used to refer to the rotor identification number defined in a ROTOR or ROTORG card. SYNCFLG specifies whether the analysis is asynchronous or synchronous and is set to ASYNC or SYNC respectively. The minimum and maximum rotation speeds of a synchronous analysis are set via SPDLOW and SPDHIGH, while the rotation speed of the rotor is defined by SPEED for an asynchronous analysis. SPDUNIT determines the units of the rotation speed as either revolutions per minute or hertz, and is input as RPM or FREQ. ROTRSEID is only used if the rotor is part of a superelement and matches the identification number of that superelement. Each rotor in an analysis must be given a RSPINR card for modal analysis and frequency response analysis or a RSPINT card for transient analysis. The RSPINR and RSPINT cards (Figure 3-9) must be defined for rotordynamic effects to be calculated even if the rotor is the same as the reference rotor defined by the RGYRO card. 34

47 Figure RSPINR and RSPINT Data Entry Cards The RSPINR and RSPINT data cards define a positive spin direction from grid point 1 (GRIDA) to grid point 2 (GRIDB) for a rotor identified by its rotor identification number ROTORID. SPTID refers to a table of spin speeds if it is an integer value or it is a multiplier of the reference rotor spin speed defined in the RGYRO card if it is a real value. If the rotor is the same as the reference rotor SPTID should be set to 1.. For transient analysis, SPTID must be a table. SPDOUT in the transient analysis card is a variable used to output rotor speed versus time. SPDUNT defines the units for the spin speed the same as the RGYRO card. ROTRSEID is only required if the rotor is part of a superelement and matches that superelement identification number. GR is used to define the structural damping ratio for the rotor, ALPHAR1 and ALPHAR2 define the α and β Rayleigh damping coefficients for the rotor. The circulatory matrix contribution can be turned off within a NASTRAN analysis using the bulk data entry MDLPRM RDBOTH 2. If this line is not added to the NASTRAN input file, the circulatory matrix will be considered. 35

48 4. Rotating Reference Frame The rotating reference frame is used when simulating non-axisymmetric rotating bodies, performing single shaft analysis, or there is no non-rotating support structure included in the model. In the rotating reference frame each element s matrices are formed with degrees of freedom that align with a rotating xyz frame that moves with the rotor s angular velocity Ω. The equations of motion at each node are put into matrix form for the rotating reference frame resulting in Eq [M]{u } + ([C or ] + [C r ] + [C n ]){u } + ([K] [K c ] + [S] + Ω[B n ]){u} = {F} 4.1 The internal forces in this equation of motion are aligned with the rotating coordinate system instead of the fixed reference frame. The Coriolis effect [C or ], spin softening matrix [K c ], and circulatory matrix due to non-rotating damping Ω[B n ] are unique to the rotating reference frame while the gyroscopic matrix and circulatory matrix due to rotating damping seen in the stationary reference frame vanish. The stress stiffening matrix [S] can also be applied in the stationary reference frame, but is discussed for the rotating reference frame where it is more often implemented by finite element models. NASTRAN limits the rotating reference frame to solid elements only while ANSYS Workbench allows any element type to be used. ANSYS Workbench does not take into account the circulatory matrix due to non-rotating damping. Neglecting this term contributes to differences seen when comparing results between a stationary reference frame analysis and rotating reference frame analysis Similarities to the Stationary Frame The inertia matrix [M] and stiffness matrix [K] are formed similarly to their stationary reference frame counterparts. The internal force due to rotating damping occurs in the rotating reference 36

49 frame and doesn t need to be transformed like in the stationary reference frame. This removes the circulatory matrix due to rotating damping and allows the internal force from rotating damping to be defined by only the rotating damping matrix [C r ]. The internal force from non-rotating damping occurs in the stationary reference frame. It is transformed to the rotating reference frame using the same transformation relationship used for transforming the rotating damping force to the stationary frame. This transformation results in a non-rotating damping matrix [C n ] formed the same as in the stationary reference frame and a circulatory matrix term with an internal force equivalent to Ω[B n ]{u} Coriolis Matrix The Coriolis matrix of each element is a skew-symmetric matrix representing the kinetic energy of an element as it rotates about the axis of rotation. The Coriolis matrix is calculated on an elemental basis and this calculation takes into account the mass of the element. For both ANSYS Workbench and NASTRAN the Coriolis matrix of an element is calculated by the equation Ω z Ω y m [C or ] el = 2[Ω][M m ] el = 2 [ Ω z Ω x ] [ m ] 4.1 Ω y Ω x m where Ω x, Ω y, and Ω z represent the x, y, and z contributions of the rotational speed. For simulations where the rotational velocity Ω is aligned solely along the z-direction this simplifies to the skewsymmetric form m [C or ] el = 2Ω [ m ]

50 The element s shape function is then used to distribute the x, y, and z components of the elemental Coriolis matrix to each of the element s nodes similarly to how the inertia and stiffness matrices are created. ANSYS Workbench calculates the Coriolis matrix for any element type while NASTRAN limits the calculation of the Coriolis matrix only to elements that are allowed in rotordynamic simulations Spin Softening and Stress Stiffening Matrices The spin softening matrix, sometimes referred to as the centrifugal softening matrix, accounts for the effect of centrifugal load in the rotating reference frame. The spin softening matrix is derived from the term {Ω} ({Ω} {u}). In ANSYS Workbench the spin softening matrix is calculated on an elemental basis using the elemental mass matrix [M m ] el and the rotational velocity matrix [Ω]. Ω z Ω y Ω z Ω y m [K c ] el = [Ω][Ω][M m ] el = [ Ω z Ω x ] [ Ω z Ω x ] [ m ] 4.4 Ω y Ω x Ω y Ω x m For simulations where the rotational velocity Ω is aligned solely along the z-direction this simplifies to m [K c ] el = Ω 2 [ m ] 4.5 The element s shape function is then used to distribute these terms to the nodal version of the element s spin softening matrix. NASTRAN calculates the spin softening matrix on a nodal basis for the translational degrees of freedom using the same procedure. In ANSYS Workbench and NASTRAN the spin softening effect is included for all analyses that take place in the rotating reference frame. 38

51 The stress stiffening matrix is used to consider the effects that the stress state or geometric changes of an element has on its stiffness. When an element sees large amounts of deformation the increase of the element s stiffness due to the stress state must be considered for accurate results. This increase in stiffness is accounted for in the stress stiffening matrix [S]. In the rotating reference frame, this is most likely to occur due to centrifugal load and which results in a differential increase in stiffness proportional to Ω 2. In ANSYS Workbench an analysis calculating the stress field of the elements must be simulated separately and added as a pre-stress for stress stiffening effects to be included. NASTRAN automatically accounts for the differential increase in stiffness due to centrifugal load but will not account for any other changes to the stiffness matrix due to the stress field of the elements. The effects of spin softening and stress stiffening due to centrifugal load are only seen in solid element models. Stress stiffening increases natural frequencies while spin softening decreases natural frequencies. For models that incorporate both stress stiffening and spin softening the stress stiffening effect is larger than that of spin-softening Circulatory Matrix due to Non-Rotating Damping The circulatory matrix in the rotating frame is due to non-rotating damping. This circulatory matrix has a stabilizing effect on rotor motion. It is required to offset the non-rotating damping matrix which is implied to rotate with a spin speed of Ω in the rotating reference frame since it creates a force with the rotating coordinates velocity vector {u }. Using the same transformation relationship between the rotating reference frame and stationary reference frame defined by Eq. 3.6 and the 39

52 same procedure as used for determining the circulatory matrix due to rotating damping in the stationary reference frame results in a similar circulatory matrix [B n ]. Ω[B n ] nt = Ω[J][C n ] nt 4.6 In NASTRAN, the circulatory matrix in the rotating reference frame is calculated using the same transformation matrix [T I ] used for the circulatory matrix in the stationary reference frame. Ω[B n ] n = Ω([T I ][C n ] n + [C n ] n [T I ]) 4.7 ANSYS Workbench does not account for the circulatory matrix due to non-rotating damping. The circulatory matrix due to non-rotating damping contributes stability to a system. Without this stability in ANSYS Workbench models, the stability results of simulations in the rotating reference frame are inaccurate ANSYS Workbench Implementation of the Rotating Reference Frame In ANSYS Workbench, the rotating reference frame is the default reference frame when rotordynamic effects are activated. Spin softening effects are included whenever rotordynamic effects are active in ANSYS Workbench. Stress stiffening effects can only be included as a prestressed field when the results from a previously simulated stress analysis are attached to the current simulation. The circulatory matrix due to non-rotating damping is never considered by ANSYS Workbench in its rotating reference frame equation of motion. 4

53 4.6. NASTRAN Implementation of the Rotating Reference Frame In NASTRAN, the rotating reference frame can only be used for solid element models. It is implemented when the FRAME variable of the ROTOR card is set to FIX. RGYRO and RSPINR or RSPINT bulk data cards must also be included in the simulation for rotating effects to be included. Spin softening effects and stress stiffening due to centrifugal load are automatically included in a NASTRAN rotating reference frame analysis. The circulatory matrix due to non-rotating damping is considered unless specifically turned off. 41

54 5. Rotordynamic Analysis With finite element models modal analysis, harmonic analysis, or transient analysis can be used. Modal analysis is used to determine rotor modes, critical speeds, and mode stability. Harmonic analysis determines rotor response to a harmonic forcing function that is either synchronous with rotor speed or asynchronous. Transient analysis models rotor behavior with time and can be particularly useful in modeling rotor response during start up or following a sudden change to the rotor system such as a blade loss. Between these three analysis types most rotordynamic related issues can be studied. In ANSYS Workbench the analysis type is chosen when a new model is created. Each analysis type has several solvers that can then be utilized. NASTRAN determines the analysis type based on the solver which is explicitly stated in the executive control section of a NASTRAN input file. 5.1 Modal Analysis Modal analysis is the process of determining the natural frequencies of a system and the corresponding mode shapes of the system. The natural frequencies and mode shapes are determined by finding the solutions to the equation of motion when the external forces are zero. The nodal displacement vector is written as consisting of a vector of coefficients {ψ} multiplied by a frequency dependent term e jλt. {q} = {ψ}e jλt 5.1 By taking the time derivative of Eq. 5.1 and substituting it into the equation of motion for the stationary reference frame (Eq. 3.1), the equation of motion becomes ( λ 2 [M] + jλ(ω[g] + [C r + C n ]) + ([K] + Ω[B r ])){ψ}e jλt = {}

55 In order to solve for the frequencies λ and vectors {ψ} that are the non-trivial solutions to Eq. 5.2, terms must be grouped into generalized mass, damping, and stiffness matrices containing all of the terms with the same dimensions. For the stationary reference frame these generalized matrices are defined as [M] g = [M] 5.3 [C] g = Ω[G] + [C r + C n ] 5.4 [K] g = [K] + Ω[B r ] 5.5 The rotating reference frame equation of motion is similarly grouped into generalized mass, damping, and stiffness matrices. The modal equation is now a classic eigenvalue equation. ( λ 2 [M] g + jλ[c] g + [K] g ){ψ} = {} 5.6 The eigenvalue λ solutions can now be extracted utilizing an eigensolver method. Several of these methods exist including the complex Lanczos method, Upper Hessenberg method, and QR reduction method. Once the eigenvalues have been determined, the corresponding eigenvectors {ψ} can be solved for when the eigenvalues are put back into the eigenvalue equation. The eigenvectors are the non-zero solutions to Eq. 5.6 and represent the mode shapes of the system. In some eigenvalue solvers, the eigenvectors are solved for in the complex modal subspace and then transformed back. Eigenvectors can be multiplied by a constant and still be valid solutions to the eigenvalue equation. As a result eigenvectors are often normalized using a specific method so that they can be more readily compared. The eigenvector can be normalized to the mass matrix, so that the largest 43

56 component of the eigenvector is a unit value, or so that the sum of the squares of each component of the eigenvector is equal to a unit value. ANSYS Workbench outputs mode shapes as animations (Figure 5-1) but does not include the numerical values of the eigenvectors in the solver output. The mode shapes in ANSYS Workbench are normalized to the mass matrix by default. Figure 5-1 ANSYS Workbench Mode Shapes NASTRAN outputs mode shape data as the complex eigenvector consisting of translational and rotational motions in the six degrees of freedom at each node point (Figure 5-2). NASTRAN normalizes the eigenvectors so that one of the components of the eigenvector has a unit value. 44

57 Figure NASTRAN Mode Shapes To perform a modal analysis of a rotordynamic system in ANSYS Workbench, the option for a damped system must be selected as Yes under the analysis settings and solver controls. If this option is set to No instead, the gyroscopic matrix, damping matrices, circulatory matrices, and the Coriolis matrix are not included in the analysis and the analysis is that of an undamped, non-rotating system. The number of frequencies to be extracted and the number of rotational speeds at which these frequencies are to be extracted at are also set in the analysis settings. ANSYS Workbench utilizes two algorithms for finding eigenvalues of damped systems. The first algorithm is referred to by ANSYS Documentation as the damped method and utilizes the full generalized damping matrix. The quadratic equation 5.6 is transformed to a linear version. [A] = [ [C] g [M] g [I] [] ]

58 [B] = [ [K] g [] [] [I] ] 5.8 (λ [A] + [B]){φ} =, where λ = j λ 5.9 This linearized version of the eigenvalue equation is then solved using the Block Lanczos eigenvalue and eigenvector extraction method. The second method used by ANSYS Workbench is the QR damped method. The QR damped method finds the eigenvalue solutions to the undamped non-rotating system, transforms the equation of motion to the modal coordinates of the undamped modes, and then extracts the modes of the transformed damped equation of motion. The QR algorithm is utilized to reduce the degrees of freedom using in solving the damped equation of motion by eliminating degrees of freedom that are not well activated by the undamped modes. The QR damped method is best for models with a large number of degrees of freedom and models that are lightly damped with symmetric stiffness matrices. The eigenvalue extraction method is selected in the solver controls portion of the analysis settings in ANSYS Workbench. The Full Damped option will utilize the damped method, the Reduced Damped option will utilize the QR damped method, and the Programmed Controlled option allows ANSYS Workbench to internally determine which of the two methods to use based on the size of the matrices and the number of frequencies being extracted. In NASTRAN two methods are used for rotordynamic modal analysis. These two methods are the complex Lanczos and Upper Hessenberg algorithms. The complex Lanczos method is similar to damped method utilized by ANSYS Workbench while the Upper Hessenberg method reduces the full 46

59 generalized matrices to Upper Hessenberg matrices before further reducing them to triangular matrices using QR transformation. Both methods are options under solver SOL 17 and are selected along with the number of frequencies to extract using the complex eigenvalue extraction method selection case control entry CMETHOD = CCN and the complex eigenvalue extraction bulk data entry EIGC (Figure 5-3). Figure EIGC Data Entry In the complex eigenvalue extraction bulk data entry, the set identification number (SID) must match the integer CCN of the complex extraction method selection case control entry. The method of extraction (METHOD) is set to CLAN for the complex Lanczos method and HESS for the Upper Hessenberg method. The eigenvectors are normalized (NORM) either by scaling the eigenvector such that the maximum component has a unit value for the real part and a zero value for the imaginary part or by scaling the eigenvalue so that a component at a point identified by point G and degree of freedom C has a unit value for the real part and a zero value for the imaginary part. The first normalization option is selected by setting NORM to MAX and is the default setting if no input is entered for NORM. The second option is selected by setting NORM to POINT. Inputs for point G and degree of freedom C are only required if the normalization method is the point method. The convergence criterion is defined as E. ND is the number of eigenfrequencies to be extracted. The second line of the complex eigenvalue extraction bulk data entry is used to specify the shift points for the eigenvalue extraction search of the complex Lanczos method by defining the real and imaginary parts of the shift point j (ALPHAAJ and OMEGAAJ), the maximum block size (MBLKSZ), initial blocksize (IBLKSZ), and frequency of solve (KSTEPS). This continuation line may be repeated for 47

60 multiple shift points and NJi is used to define the number of frequencies to be extracted for the shift and ND is not required. If the continuation line is not used, shifts will be automatically calculated by the solver Complex Eigenvalues The complex eigenvalues extracted by an eigensolver contain information of both the modal frequencies of the system and the stability of each mode. The imaginary value of the complex eigenvalue is the modal frequency ω and real value of the complex eigenvalue is the stability value σ. Each modal frequency represents a damped natural frequency of the system at a given rotor spin speed. The stability value determines whether that particular mode is stable or unstable at that rotor spin speed. A negative stability value indicates that the mode is stable at that spin speed and a positive stability value indicates that the mode is unstable. A root locus plot shows the change in complex frequency of a mode with increasing rotor spin speeds (Figure 5-4). Figure Root Locus Plot 48

61 5.1.2 Campbell Diagrams and Critical Speeds A Campbell diagram plots the natural frequencies of the rotor as a function of rotor spin speed. All the points of the same mode are connected to each other to form a line showing how the natural frequencies change with rotor spin speed. Critical speeds are the rotor spin speeds at which the resonance of a natural frequency occurs due to an excitation force. Campbell diagrams typically plot the excitation frequency equal to the rotor spin speed (first order excitation) to determine the flexural critical speeds of the rotor. The flexural critical speeds are also important in analyzing mode stability. When the rotor spin speed is above the flexural critical speed of a forward whirl mode, rotating damping is destabilizing for that mode. If this mode is then excited by an asynchronous forcing function, the rotor itself can become unstable. On a Campbell diagram it is possible to identify flexural critical speeds as a crossing between a natural frequency line and the first order excitation order line (Figure 5-5). Figure 5-5 ANSYS Workbench Campbell Diagram 49

62 Most Campbell diagrams do not consider the sign of the natural frequency and plot only the magnitude of the frequency. Backward whirl modes are typically not excited by rotor spin speed and therefore the crossing between a backward whirl frequency line and the first order excitation line is usually not a flexural critical speed [15]. Therefore it is necessary to plot the natural frequencies with the sign of the natural frequency to better understand the modal frequencies of the system (Figure 5-6). Figure Campbell Diagram with Positive and Negative Frequencies The data for a Campbell diagram is created in ANSYS Workbench when the Campbell diagram option in the analysis settings is activated and the number of rotor speeds and the frequencies to be extracted are defined. ANSYS Workbench creates a Campbell diagram using the absolute value of natural frequencies and identifies flexural critical speeds for both forward whirl and backward whirl modes. 5

63 A Campbell diagram is not created directly by NASTRAN. Instead the data required for producing a Campbell diagram is determined and this data can then be post-processed to create a Campbell diagram. To create the data for a Campbell diagram in NASTRAN three additional entries are required. The case control entry CAMPBELL = CCN must be added as well as the bulk data entries CAMPBLL and DDVAL (Figure 5-7). Figure CAMPBLL and DDVAL Data Entry Card For the CAMPBLL bulk data card CID is the case identification number and must match the integer value in the case control entry CAMPBELL, VPARM is the variable parameter and must be set to SPEED, DDVALID is the identification number for the DDVAL table entry (a table of discrete digital values), and TYPE should be set to either FREQ or RPM. The DDVAL bulk entry card is a list of discrete design variable values which in this case is the list of rotor spin speeds used to find the eigenvalues used to create the Campbell diagram. The ID must match the DDVALID of the CAMPBLL card and the DVAL entries are a list of the rotor spin speeds Mode Tracking One issue with Campbell diagrams is accurately applying mode tracking to ensure that the natural frequencies of the same mode at different rotor spin speeds are grouped together. This is necessary 51

64 to accurately produce lines connecting the multiple points of the same mode. If the mode shapes are highly complex or mode tracking is not used, the natural frequencies of different modes may be grouped together resulting in inaccurate critical speed estimations. ANSYS Workbench does not use mode tracking. When creating a Campbell diagram, ANSYS Workbench sorts modes based on the mode shapes to group together different points of the same mode. If the same modes are not calculated at each rotor spin speed, ANSYS Workbench will not be able to accurately group modes and will have an inaccurate Campbell diagram. NASTAN utilizes a looping system to determine the natural frequencies of the same mode at each rotor spin speed. Only the natural frequencies found at the first rotor spin speed are returned by the eigensolver for the later rotor spin speed values. The similarity of a mode shape from one spin speed to the next is used to accurately track the modes. Issues in accurate mode tracking can still arise if the mode shape changes significantly from one rotor spin speed to the next which can happen if two consecutive rotor spin speeds are far apart or the rotating reference frame is used and a forward whirl mode changes to a backward whirl mode Directivity and Complex Coordinates Descriptions Eigenvalue solutions to the modal equation are paired and take the form e jω 1t where ω 1 > and e jω 2t where ω 2 <. These solutions imply that ω 1 is a forward whirl mode since it is a positive frequency and that ω 2 is a backward whirl mode since it is a negative frequency. An alternative mathematical solution to the modal equation is the pair of e jω 3t where ω 3 > and e jω 4t where ω 4 <. These solutions lack the implied directional information of ω 1 and ω 2 due to the added 52

65 negative sign in the solution. Further examination of these two sets of paired solutions reveals that the solutions are mathematically equivalent [17]. ω 1 = ω ω 2 = ω When solving a rotordynamic analysis, eigenvalue solvers will find both sets of results. When plotted this leads to misleading and overcrowded Campbell diagram (Figure 5-8). Figure Campbell Diagram without Directivity Considered ANSYS Workbench and NASTRAN both solve for both sets of solutions. Despite solving for both sets of solutions, ANSYS Workbench outputs only positive frequency results and determines the direction of each mode from their eigenvector. NASTRAN outputs both sets of solutions (Figure 5-9) and the data must then be post-processed to identify the rotor modes with the correct directivity. 53

66 Recognizing that eigenvalues occur in pairs and that the absolute value of a backward whirl mode in the stationary reference frame will be lower than its companion forward whirl mode allows this determination to be easily made. In Figure 5-9, it can be determined that eigenvalues 1, 4, 5, and 8 are the modes with the correct directivity. Figure NASTRAN Complex Eigenvalue Results Directivity issues can be avoided by utilizing complex variables to model the rotor [17-2]. For a complex variable definition the translational deformations are coupled together as one complex variable and the rotational deformations about the translational directions are coupled together by another complex variable. {r} = {x} + j{y} 5.12 {φ} = {φ y } j{φ x } 5.13 The use of complex coordinates also reduces the number of degrees of freedom at each node from six to four which by halving the number of rotor modes that are related to lateral deformation and non-torsional rotation. The reduction in overall degrees of freedom leads to reduced solver time. By utilizing complex coordinates the direction of the positive whirl frequency is explicitly defined and the duplicated solutions are not calculated when extracted eigenvalues which reduces postprocessing time. ANSYS Workbench and NASTRAN do not utilize complex coordinates when modeling rotors but instead apply all six degrees of freedom to each node (or three degrees of freedom for solid elements). Complex coordinates can only be used for rotors with symmetric mass, 54

67 damping, and stiffness matrices. Since most rotors found in turbomachinery are symmetric, this requirement does not reduce the value of implementing complex coordinates for modal analysis very much. 5.2 Harmonic Analysis Harmonic analysis is used to determine the response of the system due to a harmonic excitation force. All forces in a harmonic analysis are a function of frequency. Synchronous response analysis means that the force excitation frequency ω is the same as the rotor spin speed Ω. Synchronous response analysis is often used to determine the response of a rotor due to mass unbalance. Asynchronous response means that the force excitation frequency ω is not related to the rotor spin speed Ω. Asynchronous response analysis is typically used for when the rotor is excited by an external force. Both types of harmonic analysis determine the frequency response function between the displacement of a node and a forcing function at a node. The stationary reference frame and rotating reference frame apply harmonic analysis in a similar manner. The displacement at each degree of freedom is considered as a magnitude and phase component. q(t) = q(ω)e jωt 5.14 Applying this definition to the acceleration and velocity terms results in Eq for the stationary reference frame and Eq for the rotating reference frame. ( ω 2 [M] + jω([g] + [C n ] + [C r ]) + ([K] + Ω[B r ])){q(ω)} = {F(ω)} 5.15 ( ω 2 [M] + jω([c or ] + [C n ] + [C r ]) + ([K] [K c ] + [S] + Ω[B n ])){u(ω)} = {F(ω)}

68 The magnitude and phase of the displacement vector due to the force vector is then calculated to determine the harmonic response. The frequency response function is the ratio between the force at one node and the displacement at one node. The frequency response function has a magnitude and a phase. In ANSYS Workbench, the response solution can be used to create the frequency response function (Figure 5-1). Figure ANSYS Workbench Frequency Response Function For a realistic rotor, the center of mass of the rotor is not aligned with the axis of rotation and the distance between the two of them is defined as the eccentricity ε. As the rotor spins, the center of mass rotates about the rotor s axis of rotation which creates an unbalance force that is synchronous with the rotor spin speed. This mass unbalance leads to rotor vibration. The amount of vibration due to mass unbalance can be simulated using synchronous response analysis. Harmonic response analysis is one of the options available at model creation in ANSYS Workbench. ANSYS Workbench allows for two types of harmonic response analyses, full and modesuperposition. The full analysis utilizes the entire finite element model to determine the response. 56

69 The mode-superposition determines the modal responses of a specified number of modes and adds the modal responses together. The mode-superposition method requires that the results of an existing ANSYS Workbench modal analysis precede the harmonic analysis. A harmonic response analysis in ANSYS Workbench will apply any forces in the model as harmonic forces. The response is determined for a range of frequencies defined in the analysis settings by a minimum frequency, maximum frequency, and the number of increments. By default ANSYS Workbench assumes that the harmonic response is asynchronous. For synchronous response the APDL command SYNCHRO must be added. The SYNCHRO command takes the form of SYNCHRO, Ratio, Cname where Ratio is the ratio between the harmonic frequency ω and the rotor spin speed Ω and Cname is the component name of rotor. An unbalance force can be added to a node using APDL commands where f is the magnitude of the unbalance force (Figure 5-11). Figure APDL Commands for Unbalance Force For the full matrix harmonic response analysis ANSYS Workbench utilizes three solvers, one direct matrix solver and two iterative, convergence based solvers. The direct solver is the SPARSE solver and uses the most amount of memory but can take the least amount of time. The iterative solvers are the Jacobi Conjugate Gradient (JCG) solver and the Incomplete Cholesky Conjugate Gradient (ICCG) solver which is more robust than the JCG solver but more time consuming. 57

70 In NASTRAN the solver SOL 18 is used for harmonic response analysis in rotordynamics. SOL 18 is a direct matrix solver and will use the entire finite element matrix to calculate the response vector. For asynchronous response analysis, the case control FREQUENCY = SID and bulk data entry FREQ (Figure 5-12) must be used where SID is the set identification number and must match for the case control and bulk data entries. Any static forces in the NASTRAN model will be applied at those frequencies. Figure FREQ Data Entry Card For synchronous response analysis, the synchronous option must be selected in the RGYRO bulk data entry. All static forces will be converted to harmonic forces with a frequency equal to the rotor spin speeds. The minimum and maximum frequency values are identified in the RGYRO data entry and the spin speeds are created in a DDVAL table that is referenced by the RSPINR data entry. For unbalance mass loads, NASTRAN has developed a specific bulk data card. The case control DLOAD = SID and bulk data entry UNBALNC are utilized for unbalance mass (Figure 5-13). Figure UNBALNC Data Entry Card 58