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2001-36

Rotor-dynamics Analysis Process

Mohammad A. Heidari, Ph.D. David L. Carlson Ted Yantis Technical Fellow Principal Engineer Principal Engineer The Boeing Company The Boeing Company The Boeing Company Seattle, Washington 98124-2207 Seattle, Washington 98124-2207 Seattle, Washington 98124-2207

ABSTRACT

This paper presents the Boeing rotor-dynamics analysis procedures using MSC.Nastran. These analytical procedures are typically used in analyses supporting airplane and propulsion system designs, but they are applicable to any structure in which the rotating parts can be idealized as a collection of rigid disks and line elements co-linear with an axis of rotation. Each rotating component is modeled separately in its own superelement, which allows structures to be modeled with an arbitrary number of components rotating at different speeds about different axes. The standard MSC.Nastran solutions for frequency response, complex eigenvalue, and transient analyses have been enhanced by Boeing to include both synchronous and non-synchronous gyroscopic effects. Gyroscopic matrices are used in the steady state solutions, and are included in the mass and damping matrices depending on whether the solution is synchronous or non-synchronous. Specific topics discussed in this paper are: gyroscopic matrices, equations of motion, and damping representation. An example problem is presented which illustrates the application of the rotor-dynamics analysis procedures. This example is a representative finite element model of an engine mounted on a wing. The accuracy of the nonlinear transient analysis technique is validated with test/analysis correlation results for an actual propulsion system during the fan blade loss event.

Notation Ip polar moment of inertia about the axis of spin

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Mx moment about the x-axis My moment about the y-axis &θ x rotational velocity about the x-axis

&θ y rotational velocity about the y-axis

Ω speed of rotation about the z-axis (engine axis)

oiΩ constant

Ri shaft i speed ratio relative to the reference speed

REFΩ reference speed, for example, N1

p complex eigenvalue solution g uniform structural damping coefficient ξi modal damping coefficient for mode i

ω i frequency of mode i

g j structural damping coefficient for superelement j

S ij percent of the total strain energy that superelement j contributes in mode i

[ ]K ( )[ ] [ ]1 4+ +ig K i K complex system stiffness matrix

[K4] assembled structural damping matrix based on element structural damping

[ ]B system damping matrix, including gyroscopic terms

[ ]Φ complex system eigenvectors

( ) U ω steady state response vector

( ) P ω input forcing function for frequency response analysis (e.g. rotor unbalance)

( ) P t input forcing function for transient analysis

( ) N t nonlinear force vector

1.0 Introduction In recent years, airplane certification requirements have been mandated requiring rotor-dynamics failure events to be rigorously analyzed. In order to satisfy these new requirements, the scope and complexity of the finite element models of the airplane and propulsion systems have increased significantly. Typical solutions performed at Boeing are to identify engine critical speeds, and to ultimately design airplane/engine/nacelle structures which tolerate the extreme conditions such as fan blade loss events.

The changes and additions to the standard MSC.Nastran solution DMAPs are referred to in this paper as “The Boeing Rotor Deliverables”. These DMAPs are updated by Boeing whenever a new version of MSC.Nastran is released. All procedures described in this paper have notations used by MSC.Software in their MSC.Nastran Documentation. More details regarding the standard MSC.Nastran procedures and equations described herein can be found in Chapter 9 of Reference (5). “The Boeing Rotor Deliverables” require that each rotor be placed in its own superelement. The polar moment of inertias from the mass matrix are then used to build the gyro matrices. 2.0 Theory The rotor-dynamic superelement analysis procedure can be summarized by the flowing steps: Step 1 – Superelement matrix generation and assembly Step 2 – Superelement dynamic reduction Step 3 – Develop and reduce the gyroscopic matrices Step 4 – Repeat steps 1-3 for each superelement Step 5 – Assemble and reduce the final analysis matrices

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Step 6 – Perform the rotor-dynamics analyses Step 7 – Extract and evaluate the results The steps 1,2,4, and 5 are adequately documented in Reference (5), and step 7 is post-processing of MSC.Nastran output. The theory behind Steps 3, and 6 are summarized in this section. There is no requirement about the reduction of the entire model to the analysis set except that each rotor must be placed in its own superelement. The analysis set must contain enough information about the rotors to adequately define their behavior. The analysis set must also contain all nonlinearities. It is prudent to minimize the size of the residual superelement for economical reasons, but enough generalized coordinates, and physical freedoms need to be included to define the rest of the system.

2.1 Development of the Gyroscopic Matrices: The gyroscopic forces experienced by an axisymmetric rigid body motion rotating about the axis of symmetry, here assumed to be the z-axis, can be represented by:

−

Ω=

y

x

p

p

y

x

II

MM

θθ&&

00

(1)

The matrix gyro capability available with the “The Boeing Rotor Deliverables”, the rotational speed of rotor i is:

REFioii R Ω+Ω=Ω (2)

The assumption in “The Boeing Rotor Deliverables” is that the spinning structure can be modeled as an assembly of rigid rotating bodies aligned on a common spin axis and connected by flexible beam type elements . The individual gyroscopic matrices can be assembled into superelement gyro matrices and to be reduced to the superelement boundary. This process is similar to that used for the mass and damping matrices, and described in Section (9) of Reference (5). The superelement gyroscopic forces can be described as:

[ ] [ ]( ) UGBP GYROREFGYROGYRO&Ω+= (3)

Where,

[ ]BGYRO are the coefficients from equations (1) and (2) related to oiΩ

[ ]GGYRO are the coefficients from equations (1) and (2) related to Ri, the rotor speed ratios

&U are rotational velocities normal to the spin axis

The gyroscopic matrices, [BGYRO] and [GGYRO], for the residual structure can then be added to the damping and mass matrices as appropriate for the type of analysis being done, as described below. For synchronous analyses such as synchronous frequency response or complex modes, the reference speed, REFΩ , is equal to the speed of excitation

(frequency response) or the frequency of the mode (complex eigenvalues), in both cases denoted by ω . In this

case, [BGYRO], the matrix of gyroscopic coefficients related to the constant speed term of Equation 2 ( oiΩ ), is added

to the system damping matrix and the proportional part, [GGYRO], is added to the system mass matrix, as described in Section 2.2. For non-synchronous analysis types, such as non-synchronous frequency response, complex modes, or transient response, the reference speed REFΩ , is constant. Hence, REFΩ must be defined by the analyst, and the

term [BGYRO] + REFΩ [GGYRO] is added to the system damping matrix, as described in Section 2.2.

2.2 Analyses on the Residual Structure The total system analyses are performed using the reduced residual structure matrices. Here, the unknowns are any retained physical freedoms and the generalized freedoms. The equations used for the various analysis types are illustrated in this section.

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2.2.1 Complex Eigenvalue Analysis Complex eigenvalue analyses can be performed for constant analysis speed (i.e., REFΩ=Ω ) or as synchronous

analyses, where the reference spin speed is assumed to be equal to the mode frequencies. Complex normal mode frequencies and vectors are extracted by means of the following equation:

[ ] [ ] [ ]( )[ ] ωiapMpBpK +==Φ++ ,02 (4)

Where,

[ ] [ ] [ ]B B BGYRO= + , Synchronous

[ ] [ ] [ ] [ ]GYROREFGYRO GBBB Ω++= , Asynchronous

[ ] [ ] [ ]GYROGiMM −= , Synchronous

[ ] [ ]M M= , Asynchronous

“The Boeing Rotor Deliverables” has the capability to step through a table of reference speeds, performing a complete asynchronous analysis for each speed. This capability enables the analyst to construct a Campbell diagram which is a plot of complex frequencies versus rotation speed. 2.2.2 Frequency Response Analysis Steady state frequency response analyses can be performed for constant analysis speed ( REFΩ=Ω ) or as

synchronous analyses, where the reference spin speed is assumed to be equal to the frequency of the forcing function. The equation of motion for steady state frequency response analysis can be written as:

[ ] [ ] [ ]( ) ( ) ( ) ωωωω PUMBiK =−+ 2 (5)

Damping and mass matrices in equations (4) and (5) are the same.

2.2.3 Transient Analysis Transient analyses are by definition asynchronous, and the individual rotor spin speeds may be functions of time. The gyroscopic forces may be modeled by using the matrix gyro approach described for the linear analysis types, or by defining nonlinear forces (NOLINs) which allow continuous variation of the rotor speeds with time, or by using a combination of the two approaches. The equation of motion for transient response analysis is:

[ ] ( ) [ ] ( ) [ ] ( ) ( ) ( ) K U t B U t M U t P t N t+ + = +& && (6)

In the typical engine fan blade loss analyses ( ) N t is used to model forces due to nonlinear interfaces and

gyroscopic forces. These forces are functions of displacements or velocities from the previous time steps. [ ]B

which is the system damping matrix may also include gyroscopic terms for non-synchronous linear analyses. All damping components such as structural damping, Rayleigh damping, or Direct Modal Damping as discussed in

Section 2.4 can be included in [ ]B and the equation is:

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]KMGBKW

KW

gBB GYROREFGYRO βα ++Ω++++= 4

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3 (7)

Properties that explicitly vary with time, such as failure or breakage analysis, are most efficiently modeled by using transfer functions or direct matrix input in order to directly add terms to the [K], [B], and [M] matrices. “The Boeing

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Rotor Deliverables” allows this type of input to be changed for each residual structure subcase in SOL 129 which is nonlinear transient analysis.

2.3 Breakage Simulation Breakage or fusing mechanism is a local non-linear event that occurs by design to relieve structural loading during a severe event such as engine fan blade loss. Because of the size of the typical engine model and the number of time steps required for the typical Fan Blade Loss analysis, non-linear materials and elements are not usually included in the model. Instead, non-linear effects are simulated by one or both of the following methods:

(a) Include the non-linearities on the force vector side of the equations of motion, as with the circular gap (i.e., NLRGAP) or NOLIN gyroscopic moment calculations. This method still requires iterations for convergence, but often it is the only efficient option for large models which is a standard MSC.Nastran capability.

(b) Incorporate the non-linear effects in the transient analysis by explicitly changing the structural matrices at a given time in the analysis. If the approximate timing of the event is known and only a gross representation of the event is required, this method is an efficient analysis technique, since it adds no requirement for additional iterations. The required changes to the structural matrices can be accomplished by including DMIG matrices or alternatively by using transfer functions, to the stiffness, mass, or damping matrices. For SOL 129, “The Boeing Rotor Deliverables” allows various transfer functions and DMIG matrices to be used in each residual structure subcase. This capability is typically used to model the breakage that sometimes occurs in a fan blade loss event. The undamaged structure stiffness can be added to the analysis in the first subcase, and replaced at the beginning of a subsequent subcase with a DMIG matrix or transfer function with null coefficients. Figure 1 shows this process graphically. 2.4 Direct Modal Damping Formulation Direct Modal Damping is a way to use a modal damping formulation in direct transient or direct frequency response analyses. The modal damping coefficients are calculated, redistributed to physical coordinates, and added to the viscous damping matrix. For structures that have significant response at frequencies other than the forcing frequency, Direct Modal Damping is more accurate as compared to structural damping because structural damping is mostly accurate correct at only one frequency (specified in MSC.Nastran by parameters W3 and W4). Direct Modal Damping can also be tailored so that the modal damping coefficient for each mode reflects the damping characteristics of the part of the structure being exercised by that mode. Starting with the basic equation of motion:

[ ] [ ] [ ] ( ) M X C X K X F t&& &+ + = (8)

We can express the displacements as a linear combination of the independent eigenvectors:

[ ] X q= φ (9)

Applying the transformation gives:

[ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] ( ) tFqKqCqM TTTT φφφφφφφ =++ &&& (10)

Since [ ]φ are mass normalized, equation (10) can be expressed as:

[ ] [ ][ ] [ ] ( ) tFqqCqI TT φωφφ =

++

O

O&&&

O

O2

(11)

We can define a diagonal modal damping matrix [D] such that: Dii i i= ξ ω (12)

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If a different structural damping coefficient is assumed for each superelement, one rational way to calculate the equivalent overall modal damping coefficient for each mode would be to sum the superelement structural damping coefficients, using the fraction of strain energy that each superelement contributes to the mode as weighting factors:

ξiij j

j

S g= ∑ 2

(13)

To calculate the matrix [C] that corresponds to matrix [D], start with the identity:

[ ] [ ] [ ][ ]D CT

= φ φ (14)

Since [ ]φ is mass normalized:

[ ] [ ] [ ]φ φ−

=1 T

M

We can solve for [C]:

[ ] [ ][ ][ ][ ] [ ]C M D MT

= φ φ (15)

This calculated matrix [C] can then be used directly in equation (8) or can be added to matrix [ ]B in equation (6).

The disadvantages of Direct Modal Damping are that it creates a very full viscous damping matrix, and all the modes of response must be calculated. These disadvantages can be circumvented somewhat by combining structural damping with Direct Modal Damping. In order to combine Direct Modal Damping with structural damping, the normal practice is to use modes only up to ω max , the frequency at which the structural damping gives the correct value.

The modal damping coefficients are then adjusted so that they have their full value at a modal frequency of 0.0 Hertz, and decrease in a linear fashion to g=0.0 at frequency ω max . This is to account for the fact that the structural

damping effect increases linearly from g=0.0 at 0.0 Hertz to the correct value at frequency ω max . The effectiveness

of the structural damping will continue to increase for frequencies above ω max . Figure 2 shows how structural and

Direct Modal Damping can be combined to give the desired damping level as a function of the frequency content of the structural responses. Assuming that equation (13) is used to calculate the overall damping for each mode, the appropriate modal damping coefficients to account for the addition of structural damping with a coefficient of gmax can be calculated by:

ξω

ωiij j

j

iS g

g=

−∑ 2 max

max

(16)

Where ω max is given by PARAM, W3 in MSC.Nastran.

3.0 Enhancements to SOL 129:

A number of enhancements have been added to SOL 129 (direct non-linear transient analysis), in “The Boeing Rotor Deliverables”. These have all been geared toward enhancing the accuracy of fan blade off analyses by allowing changes in various analysis parameters with each subcase. The parameters that can be changed include the following:

Direct input matrices: (K2PP, B2PP, M2PP). Direct input matrices can be used to model gyroscopic effects, structural properties such as stiffness, etc., or transfer functions and constraint equations via. LaGrange multipliers. Any of these quantities can change in a piecewise fashion with time.

Transfer Functions: Transfer functions are exactly equivalent to direct input matrices, and can be changed in the same way. Both transfer functions and direct input matrices are commonly used to model LaGrange multipliers and breakage simulation described in Section 2.3.

Damping related parameters: such parameters are G, W3, and W4. Structural damping is only correct at one frequency of response in transient analyses. That frequency is defined by parameters W3 and W4, and it is often desirable to change W3 and W4 as the engine rotors spool down and the unbalance load changes frequency. Another way to enhance the damping simulation is to use the Direct Modal Damping capability described earlier.

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Parameter N1SPD: Changing this parameter allows the rotor speeds used for any matrix gyroscopic effect calculations to change in a piecewise fashion with time, to account for engine spool-down. Typically, the small high-speed rotors are modeled with matrix gyro using this piecewise linear time varying technique, which is much more numerically efficient than the non-linear force technique. In order to capture the effect of continuously varying spin speed accurately, the gyroscopic effects of the low speed rotor which is normally the larger rotor are usually modeled with NOLINs or a combination of NOLINs and matrix gyro.

4.0 An Example Problem The Prototype Engine Rotor Dynamics Model is shown in Figure 3. It is a fictitious engine model developed by Boeing and Pratt & Whitney to test analytical techniques. This model contains high bypass ratio engine with three spools, Nacelle structure comprised of inlet, fan cowl, non-deployable thrust reverser, nozzle and plug, and strut with typical links to the engine and beam type wing. The cross section of the finite element model is shown in Figure 4.

The rotor schematic is shown in Figure 5. Each bearing consists of scalar springs attaching a case point to the rotor. The case point represents the average motion of the case at the bearing axial location, and is attached to the case by an RBE3 element. The rotors are modeled with beam type elements and concentrated mass elements at the engine centerline.

“The Boeing Rotor Deliverables” for MSC.Nastran Version 70.5 are applied for this example. Gyroscopic effects are defined using the matrix methods discussed in Section 2.1. For this model, the forcing function, or unbalance load, is at the fan rotor speed (N1). The low rotor (N2) and high rotor (N3) are assumed to be linear functions of N1. Complex eigenvalue, frequency response, and transient response analyses were performed.

Both synchronous and non-synchronous complex eigenvalue analyses were performed. In the synchronous run the reference (N1) speeds are equal to the mode frequencies. (One-per-Rev) line in Figure 8 connects the modal frequencies from synchronous run. The non-synchronous run loops through N1 speeds from 0 to 4000 RPM, performing analyses at 500 RPM intervals, calculating a set of eigenvalues for each spin rate. The results are plotted in the Campbell diagram shown in Figure 6. Gyroscopic effects cause modal frequencies corresponding to the rotating shafts increase (forward whirl) and decrease (backward whirl) with the shaft rotational speed. Critical speed as it is shown in Figure 6 is the crossing of the (one-per-Rev) line with the forward whirl line. In the frequency response analysis, a 1.0 ounce-in unbalance was applied to the FAN stage. Fan displacements are shown in Figure 7.

The structure is assumed to be in a steady state condition at the start of the transient analysis using SOL 129. To simulate this, we have a 1.0-second "run up" to bring the model from the unloaded state to the steady state. The "run up" in this model consists of thrust building up to the pre-event level. The run up can usually be handled as a static analysis in the first residual structure subcase with only the displacement dependant non-linearities and stiffnesses active (TSTATIC=1). When using structural damping, there needs to be a representative frequency (W4) selected at which the structural damping will be correct. In actuality, the frequency of the unbalance would be a reasonable choice except that it needs to be constant within a subcase. To get around this, it is normal to break up the event into additional subcases and to make a stepwise linear function of W4 see Figure 8. An alternative to the structural damping used in this example would be Direct Modal Damping, discussed in Section 2.4.

The applied loads for transient analysis in Figure 9 represent the types of input supplied by the engine manufacturer who derives them from test or analysis which are highly proprietary to simulate a fan blade loss event. The unbalance loads are applied at the FAN location, seizure torque caused by engine deceleration is applied as uniform tangential forces to the engine case, and thrust load is applied in fore-aft direction to the engine case.

Figure 10 shows solution 129 fan lateral displacement and strut to wing interface loads with structural and Direct Modal Damping methods.

5. Test/Analysis Correlation The fan blade loss test/analysis correlation studies are performed during the engine certification test to validate the finite element model and analysis process. The analytical loads are computed based on a nonlinear transient analysis of the integrated finite element model for the engine/test stand configuration. Figure 11 shows a schematic of the strut hardware which integrates the engine/nacelle structure with the test-stand structure. Figures 12 show the strut

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interface loads with the engine rear mount and the test stand structure. Results are non-dimensional mainly to protect the proprietary information.

6. Conclusion The Boeing rotor-dynamic analysis procedures with MSC.Nastran are presented in this paper. The standard MSC.Nastran solutions for frequency response, complex eigenvalue, and transient analyses have been enhanced with “The Boeing Rotor Deliverables” to include the gyroscopic effects. SOL 129 is enhanced to simulate engine behavior during fan blade loss events. Analysis results for a fictit ious engine model are presented. The accuracy of the nonlinear transient analysis technique is validated with test/analysis correlation results for an actual propulsion system during the fan blade loss event.

7. Acknowledgments Authors would like to thank Dr. Ray Frick from Pratt & Whitney for providing example engine model. 8. References

(1) H Bedrossian, N. Veikos, Rotor-Disk System Gyroscopic Effect in MSC.Nastran Dynamic Solutions, Presented at the 1982 MSC.Nastran Users’ Conference.

(2) D. Bella and M. Reymond, Eds., MSC.Nastran DMAP Module Dictionary, Version 68, The MacNeal Schwendler Corporation, Los Angeles, CA, 1994.

(3) D. Herting, MSC.Nastran Advanced Dynamic Analysis User’s Guide, The MacNeal Schwendler Corporation, Los Angeles, CA, 1997.

(4) K. Kilroy, Ed., MSC.Nastran Quick Reference Guide, Version 70.5 , The MacNeal Schwendler Corporation, Los Angeles, CA, 1998.

(5) R. Lahey, M. Miller, and M. Reymond, Eds., MSC.Nastran Version 68 Reference Manual, The MacNeal Schwendler Corporation, Los Angeles, CA, 1995.

(6) G. Sitton, MSC.Nastran Basic Dynamic Analysis User’s Guide, The MacNeal Schwendler Corporation, Los Angeles, CA, 1997.

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Interface Stiffness Subcase 1 (K2PP=DMIG1) Subcase 2 (K2PP=DMIG2) … Time

Figure 1 Stiffness Change with Time (Subcase) for Fuse Modeling

Effective Damping Desired Damping Level = Structural + Modal g Structural Damping Adjusted Modal Damping Level

Response Frequency ω max

Figure 2 Structural and Direct Modal Damping in Transient Analysis

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Figure 3 Example , Prototype Engine Rotor Dynamics Model

Figure 4 Prototype Engine Rotor Dynamics Model Cross Section

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Figure 5 Prototype Engine Rotor Dynamics Model Rotor Schematic

Figure 6 Campbell Diagram for example proble m

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Figure 7 Example, Frequency Response Fan Displacements

Figure 8 Fan Rotor Speed History

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Figure 9 Input Forcing Functions to Simulate Fan Blade Loss Event

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Figure10 SOL 129, Comparison of Damping Methods

Figure11 Schematic of Fan Blade -out Test Strut

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Figure 12 – Test/analysis correlation of strut interface loads with engine rear mount and test-stand

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