9489 E. Ironwood Square DriveScottsdale, AZ 85258Tel (480) 945-9988 Fax (480) 945-6588info@zonatech.comwww.zonatech.com

ZONA TECHNOLOGY INC

The ZONA Technology, Inc. corporate logo and ZAERO are trademarks of the ZONA Technology, Inc. in the United States and other countries. All other trademarks belong to their respected owners. This documentation, as well as the ZAEROsoftware, are furnished under license and may be used only in accordance with terms of such license.

© 2020 ZONA Technology, Inc. All Rights Reserved Worldwide.

Edition 9.3.4

v. 9.3

ZAER

O

ZONAA

pplications Manual Vol. 1

v. 9.3

ZAERO

ZONA TECHNOLOGY INC

Applications Manual Vol. 1Engineers’ Toolkit for Aeroelastic Solutions

Lorem ipsum

ZONA TECHNOLOGY INC

ZAEROVersion 9.3

APPLICATION’S MANUAL (Vol. I)

ZONA 01 – 17.0 January 2017

© 2020 ZONA Technology, Inc. All rights reserved. Fourth Edition 10-18

DISCLAIMER

THE MATERIAL PRESENTED IN THIS TEXT IS FOR ILLUSTRATIVE AND EDUCATIONAL PURPOSES ONLY, AND IS NOT INTENDED TO BE EXHAUSTIVE OR TO APPLY TO ANY PARTICULAR ENGINEERING PROBLEM OR DESIGN. ZONA TECHNOLOGY, INC. ASSUMES NO LIABILITY OR RESPONSIBILITY TO ANY PERSON OR COMPANY FOR DIRECT OR INDIRECT DAMAGES RESULTING FROM THE USE OF ANY INFORMATION

MSC.PATRAN is a registered trademark of the MSC Software Corporation. MSC.NASTRAN is a registered tradem ark o f the MSC Software Corporation. MSC.NASTRAN is an enhanced, proprietary version developed an d m aintained by t h e MSC Corporation. MSC.ARIES is a trademark of MSC. I-DEAS and FEMAP are trademark s o f St ruct ural Dy namics Research Corporations. TECPLOT is a trademark of TECPLOT Inc. Other product names and trademarks are the pro perty of their respective owners.

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ZONA Technology, Inc. • 9489 E. Ironwood Square Drive • Scottsdale, AZ 85258-4578 Tel: (480) 945-9988 • Fax: (480) 945-6588 • E-mail: info@zonatech.com

TABLE OF CONTENTS i

TABLE OF CONTENTS

Vol. 1 PAGE 1.0 INTRODUCTION ............................................................................................ 1-1

1.1 WHAT IS ZAERO .......................................................................................................................1-1 1.2 ZAERO/UAIC MODULE...........................................................................................................1-3 1.3 HIGH-FIDELITY GEOMETRY (HFG) MODULE ........................................................................1-6 1.4 3D SPLINE MODULE...................................................................................................................1-7 1.5 ZONA DYNAMIC MEMORY AND DATABASE MANAGEMENT SYSTEM ..............................1-7 1.6 BULK DATA INPUT .....................................................................................................................1-8 1.7 GRAPHIC DISPLAY CAPABILITY ...............................................................................................1-8 1.8 FLUTTER MODULE .....................................................................................................................1-8 1.9 ZAERO/ASE: AEROSERVOELASTICITY (ASE) MODULE ...................................................1-8

1.10 TRIM MODULE ..........................................................................................................................1-10 1.11 MLOADS MODULE.................................................................................................................1-10 1.12 ELOADS MODULE ..................................................................................................................1-11 1.13 GLOADS MODULE .................................................................................................................1-11 1.14 NLFLTR MODULE...................................................................................................................1-12 2.0 FLUTTER SAMPLE CASES ......................................................................... 2-1

2.1 CASE 1: SUBSONIC (M=0.45) FLUTTER ANALYSIS OF A 15-DEGREE SWEPTBACK WING(HA145E) 2-1

2.2 CASE 2: FLUTTER ANALYSIS OF A CROPPED DELTA WING – BODY (CROP) ............. 2-32 2.3 CASE 3: LOW SUPERSONIC (M=1.3) FLUTTER ANALYSIS OF A 15-DEGREE

SWEPTBACKWING (HA145FB) WITH AND WITHOUT THICKNESS EFFECTS . 2-50 2.4 CASE 4: HIGH SUPERSONIC (M=3.0) FLUTTER ANALYSIS OF A 15-DEGREE

SWEPTBACK WING (HA145G) WITH AND WITHOUT THICKNESS EFFECTS ................ 2-66 2.5 CASE 5: F-16 AIRCRAFT WITH EXTERNAL STORES (F16MA41) .....................................2-85 2.6 CASE 6: TRANSONIC FLUTTER ANALYSIS OF THE AGARD STANDARD 445.6 USING

THE ZTRAN METHOD............................................................................................2-105 2.7 CASE 7: FLUTTER ANALYSIS OF THE AGARD 445.6 WEAKENED WING USING THE

ZTAW MODULE ......................................................................................................2-151

ii TABLE OF CONTENTS

TABLE OF CONTENTS (cont.) PAGE 3.0 ASE SAMPLE CASES ...................................................................................3-1

3.1 CASE 1: OPEN-LOOP FLUTTER AND CONTINUOUS GUST ANALYSIS

OF THE CROPPED DELTA WING CASE (CROPASE)........................................... 3-1 3.2 CASE 2: OPEN-LOOP AND CLOSED-LOOP ASE STABILITY ANALYSIS OF A GENERIC ADVANCED FIGHTER AIRCRAFT (AFA)................ 3-23

4.0 STATIC AEROELASTIC / TRIM SAMPLE CASES .....................................4-1

4.1 CASE 1: ASYMMETRIC ROLLING PULLOUT AT M=0.9, (Q=1200 PSF)........................... 4-30 4.2 CASE 2: SYMMETRIC 1-G LEVEL FLIGHT AT M=0.9, (Q=40 PSF).................................. 4-35 4.3 CASE 3: OVER-DETERMINED TRIM SYSTEM WITH INDUCED DRAG

AND STRESS MINIMIZATION................................................................................. 4-37 5.0 HOW TO IMPORT THE USER SUPPLIED GENERALIZED

AERODYNAMIC MATRICES AND STRUCTURAL MATRICES ........... 5-1

5.1 CREATION OF MATRIX ENTITIES ........................................................................................... 5-1 5.2 IMPORTING A MATRIX BY DIRECT MATRIX INPUT ............................................................ 5-3 5.3 NOTES FOR IMPORTING THE STRUCTURAL MATRICES....................................................... 5-3 5.4 NOTES FOR IMPORTING THE GENERALIZED AERODYNAMIC MATRICES ......................... 5-3

6.0 TRANSIENT MANEUVER LOADS SAMPLE CASES ................................6-1

6.1 SYMMETRIC TRANSIENT MANEUVER LOADS OF THE OPEN-LOOP FSW CONFIGURATION (M144OPEN.INP) ...................................................................................... 6-4 6.2 SYMMETRIC TRANSIENT MANEUVER LOADS OF THE CLOSED-LOOP FSW CONFIGURATION (M144CLOSE.INP) .................................................................................. 6-36

7.0 TRANSIENT EJECTION LOADS SAMPLE CASES ....................................7-1

7.1 SUBCASE 1: THE ELOADS ANALYSIS OF THE OPEN-LOOP FSW CONFIGURATION........ 7-5 7.2 SUBCASE 2: THE ELOADS ANALYSIS OF THE CLOSED-LOOP FSW CONFIGURATION . 7-10

TABLE OF CONTENTS iii

TABLE OF CONTENTS (cont.) Vol. 2 PAGE 8.0 DISCRETE GUST LOADS SAMPLE CASES ............................................. 8-1

8.1 CASE 1: 2-D AIRFOIL SUBJECTED TO A SHARP-EDGED GUST AT M = 0.0 .......................8-3 8.2 CASE 2: OPEN-LOOP AND CLOSED-LOOP DISCRETE GUST RESPONSE ANALYSIS

OF A GENERIC BUSINESS JET (GBJ) ...................................................................8-37 8.3 CASE 3: OPEN-LOOP AND CLOSED-LOOP CONTINUOUS GUST RESPONSE ANALYSIS OF A GENERIC BUSINESS JET (GBJ)....................................................................8-166

9.0 TRANSIENT RESPONSE OF NONLINEAR OPEN/CLOSED- LOOP

AEROELASTIC SYSTEMS: SAMPLE CASES OF THE NLFLTR MODULE ...... 9-1 9.1 3 D.O.F. AIRFOIL WITH FREEPLAY............................................................................................9-1 9.2 STRUT-BRACED WING SUBJECTED TO DISCRETE GUST......................................................9-43 9.3 FOLDING WING WITH BILINEAR STIFFNESS IN HINGES .......................................................9-95 9.4 3 D.O.F. AIRFOIL WITH FREEPLAY MODELED BY CLOSED-LOOP SYSTEM .....................9-180 9.5 FOLDWING WING WITH FREEPLAY IN OUTBOARD HINGE ................................................ 9-233 9.6 FOLDWING WING WITH WINGTIP STORE AND FREEPLAY IN OUTBOARD HINGE .......... 9-233 10.0 PARAMETRIC FLUTTER ANALYSIS FOR MASSIVE NUMBER OF

AIRCRAFT WITH STORES CONFIGURATIONS .................................... 10-1 10.1 FORMULATIONS OF THE FLTPRAM AND AFLTPRM MODULES .....................................10-1 10.2 DESCRIPTION OF THE FSW+STORE MODEL..........................................................................10-4 10.3 SUBCASE 1: FLUTTER ANALYSIS FOR THREE SYMMETRIC STORE CONFIGURATIONS ...........10-6 10.4 SUBCASE 2: FLUTTER ANALYSIS FOR THREE ASYMMETRIC STORE CONFIGURATIONS USING HALF-SPAN MODEL ........................................................................................................10-9 11.0 WHIRL FLUTTER SAMPLE CASES.................................................................... 11-1 11.1 CASE 1: D-1807 PROPELLER ...................................................................................................11-1 11.2 CASE 2: BAH WING WITH NACELLE AND PROPELLER........................................................11-18

INTRODUCTION 1-1

Chapter 1

INTRODUCTION

1.1 WHAT IS ZAERO

ZAERO is a software system that integrates the essential disciplines required for aeroelastic design/analysis. The bulk of ZAERO is comprised of eleven (11) essential modules under a ZONA Dynamic Memory Database Management (ZDM) system. The eleven modules include: High-Fidelity Geometry (HFG); 3-D Spline; Unified AIC (UAIC); Modal Data Importer; Flutter with Sensitivity; AeroServoElasticity (ASE) for closed-loop stability analysis; Static Aeroelastic/Trim Analysis, Transient Maneuver LOADS (MLOADS); Ejection LOADS (ELOADS); Gust LOADS (GLOADS) Analysis for continuous gust/discrete gust and Nonlinear Flutter(NLFLTR). The main features of the ZAERO system include (see Figure 1.1):

• High Fidelity Geometry (HFG) module to model full aircraft with stores/nacelles 1

• Flight regimes that cover all Mach numbers including transonic/hypersonic ranges 2

• Unified Mach AIC (UAIC) matrices as archival data entities for repetitive structural design/analysis 3

• Matched/non-matched point flutter solutions using K / g methods with true damping 4

• Built-in Flutter Mode Tracking procedure with structural parametric sensitivity analysis 5

• Aeroservoelastic (ASE) analysis with margin analysis for SISO/MIMO control systems 6

• Trim analysis for static aeroelasticity/flight loads 7

• Dynamic Loads Analysis including transient maneuver loads (MLOADS), ejection loads (ELOADS), and discrete/continuous gust loads (GLOADS) 8 , 9 , 10

• 3D Spline module provides accurate FEM/Aero displacements and forces transfer 11

• Modal Data Importer to process NASTRAN/I-DEAS/ANSYS/ABAQUS etc. modal output 12

1-2 INTRODUCTION

• Dynamic Memory & Database Management (ZDM) Systems establish subprogram modularity 13

• Open architecture allows user direct access to data entities 14

• Bulk Data Input minimizes user learning curve while relieving user input burden 15

• Provides graphic display capability of aerodynamic models, CP’s, flutter modes and flutter curves for use with PATRAN/FEMAP/TECPLOT/ANSYS/EXCEL/etc. 16

• Executive control allows massive flutter/ASE/Trim/Dynamic Loads inputs and solution outputs 17

• Nonlinear Flutter Analysis for open/closed loop system using discrete time-domain state space approach (NLFLTR) 18

• NASKLINK module to export ZAERO aerodynamic data to MSC.NASTRAN 19

Figure 1.1 Main Features of the ZAERO Software System

Aerodynamic Model Definition • CAERO7 • BODY7

• SPLINE1 • SPLINE2 • SPLINE3 • ATTACH

FEM/Aero Spline Input

Flight Condition Definition

Bulk Data Entry: MKAEROZ - Mach numbers - List of reduced frequencies - Method flag for ZONA6,

ZONA7,ZTAIC, ZONA7U - Mean flow conditions”

in terms of α , β ,p,q,r, and δ .

HFG Module

3D Spline Module

UAIC Module

ZDM Database

• UAIC matrices of M,k pairs • Gust force vectors • Control surface aerodynamic

force vectors • 3 - D spline matrix

User Direct Access to Data Entities

Executive Control • FLUTTER • ASE • TRIM • NLFLTR • MLOADS • ELOADS • GLOADS

Aeroelastic Analysis & Sensitivity

Flutter (g - method)

Maneuver Loads

(MLOADS)

Ejection Loads

(ELOADS) Gust Loads (GLOADS)

Aeroservo - elasticity

(ASE)

Flight Loads

(TRIM)

Modal Data Importer

Structural Finite Element (FEM) Modal Output File (MSC, ASTROS, IDEAS, ELFINI, ANSYS, NE)

Graphic/Analysis Output (PATRAN, FEMAP, TECPLOT,

ANSYS, EXCEL, PEGASUS)

Sensitivity

1

11

3

2 15

15

15

14 17

13

12

16

4

7

9

5

6

8

10

18

Nonlinear Flutter

(NLFLTR) NASLINK

19

MSC.Nastran

Aerodynamic Model Definition • CAERO7 • BODY7

• SPLINE1 • SPLINE2 • SPLINE3 • ATTACH • SPLINE1 • SPLINE2 • SPLINE3 • ATTACH

FEM/Aero Spline Input

Flight Condition Definition

MKAEROZ - Mach numbers - List of reduced frequencies - Method flag for ZONA6,

ZONA7,ZTAIC, ZONA7U - Mean flow conditions

in terms of α , β ,p,q,r, and δ .

HFG Module HFG Module

3D Spline Module 3D Spline Module

UAIC Module UAIC Module

ZDM Database

• UAIC matrices of M,k pairs • Gust force vectors • Control surface aerodynamic

force vectors • 3 - D spline matrix

User Direct Access to Data Entities

Executive Control • FLUTTER • ASE • TRIM • NLFLTR • MLOADS • ELOADS • GLOADS

Aeroelastic Analysis & Sensitivity

Flutter (g - method)

Maneuver Loads

(MLOADS)

Ejection Loads

(ELOADS) Gust Loads (GLOADS)

Aeroservo - elasticity

(ASE)

Flight Loads

(TRIM)

Modal Data Importer Modal Data Importer

Structural Finite Element (FEM) Modal Output File (MSC, ASTROS, IDEAS, ELFINI, ANSYS, NE)

Graphic/Analysis Output (PATRAN, FEMAP, TECPLOT,

ANSYS, EXCEL, PEGASUS)

Sensitivity

1

11

3

2 15

15

15

14 17

13

12

16

4

7

9

5 5

6

8

10

18 18

Nonlinear Flutter

(NLFLTR) NASLINK

19

MSC.Nastran

INTRODUCTION 1-3

The ZAERO system does not provide the structural finite element solutions. It imports externally computed structural free vibration solutions (or, the normal modes solutions) generated by other structural finite element codes. The Modal Data Importer module of ZAERO is developed to directly process the output files of five commercial finite element programs: MSC.NASTRAN, NX.NASTRAN, NE.NASTRAN, ANSYS, ABAQUS, ASTROS, and I-DEAS. For other finite element codes, a “free format” for modal data input is available in the Modal Data Importer module. The Modal Data Importer processes the finite element output file to obtain the structural grid point locations for spline, the coordinate transformations for relating the local/global to the basic coordinate system, the modes, the natural frequencies, the generalized mass matrix and the generalized stiffness matrix of the structural finite element model. As an option, the user can also directly input the generalized mass and stiffness matrices as well as the mode shapes into ZAERO. Thus, there is virtually no burden to the user for importing the structural finite element solutions to ZAERO. In the following subsections, the main features of ZAERO listed above are discussed and some sample cases associated with each feature are presented.

1.2 ZAERO/UAIC MODULE One of the major strengths of the ZAERO software system is its ability to generate Unified Aerodynamic Influence Coefficient (UAIC) matrices for a complete aircraft configuration at any Mach number. Five aerodynamic codes (also referred to as methods) are incorporated in the ZAERO software system that covers the entire Mach number range. These are ZONA6, ZTAIC, ZTRAN, ZSAP, ZONA7, and ZONA7U. Figure 1.2 compares the capability and applicability of the ZAERO/UAIC module over that of other commercially available aeroelastic analysis software packages such as MSC.NASTRAN. In addition, there are two AIC correction methods available in the UAIC module; the force/moment correction method and the downwash weighting method. These two AIC correction methods generate the so-called “AIC Weighting Matrix” which corrects the AIC matrix so that the “corrected” AIC matrix can generate forces/moments or unsteady pressures to match the given set of forces/moments or unsteady pressures, respectively.

Figure 1.2 Capability/Applicability of the ZAERO/UAIC Module

1-4 INTRODUCTION

The functionality and main features of each of these methods is presented as follows.

• ZONA6: Subsonic Unsteady Aerodynamics

- Functionality: Generates steady/unsteady subsonic aerodynamics for wing-body/aircraft configurations with external stores/nacelles including the body-wake effect.

- Main Features:

Any combinations of planar/nonplanar lifting surfaces with arbitrary bodies including fuselage+stores+tip missiles.

Higher-order panel formulation for lifting surfaces than the Doublet Lattice Method (DLM).

High-order paneling allows high-fidelity modeling of complex aircraft with arbitrary stores/tip missile arrangement.

• ZSAP: Sonic Acceleration Potential Method

- Functionality: Generates steady/unsteady aerodynamics at sonic speed (M = 1.0) for wing-body/aircraft configurations with external stores/nacelles.

- Main Features:

Any combinations of planar/nonplanar lifting surfaces with arbitrary bodies including fuselage+stores+tip missiles.

Compute the steady/unsteady aerodynamics at exactly Mach one.

Paneling scheme is identical to that of ZONA6/ZONA7, i.e. ZSAP shares the same aerodynamic model as ZONA6/ZONA7.

• ZTAIC: Transonic Unsteady Aerodynamics using a Transonic Equivalent Strip Method

- Functionality: Generates unsteady transonic modal AIC using a Transonic Equivalent Strip (TES) method with externally-provided steady mean pressure.

- Main Features:

While using steady pressure input (provided by measurement or Computational Fluid Dynamics codes), grid generation is not required, and the correct unsteady shock strength and position are ensured.

The modal AIC of ZTAIC as an archival data entity allows: repetitive aeroelastic analysis and structures design.

Easily applicable to the K / P-K / g methods for flutter analysis.

Readily integrated with ZONA6/ZONA7 as a unified subsonic/transonic/supersonic AIC method for complex aircraft configurations.

INTRODUCTION 1-5

Additional input required to be provided externally to ZONA6 or ZONA7 is the steady pressure data. Only.

• ZTRAN: Transonic Unsteady Aerodynamics using an Overset Field-Panel Method

- Functionality: Generates unsteady transonic AIC matrix that has the identical form to those by ZONA6/ZONA7.

- Main Features:

ZTRAN solves the 3D time-linearized transonic small disturbance equation using an overset field-panel method.

The surface box modeling is identical to that of ZONA6/ZONA7. Only a few additional input parameters are required to generate the field panels (volume cells).

The variant coefficients in the time-linearized transonic small disturbance are provided by the Computational Fluid Dynamics (CFD) steady solutions.

The overset field-panel scheme allows the modeling of complex configurations such as whole aircraft with external stores without extensive field panel generation efforts.

ZTRAN generated AIC matrix has the same form as that of ZONA6/ZONA7; a transonic counterpart of ZONA6/ZONA7.

• ZONA7: Supersonic Unsteady Aerodynamics

- Functionality: Generates steady/unsteady supersonic aerodynamics for wing-body/aircraft configurations with external stores/nacelles.

- Main Features:

Any combinations of planar/nonplanar lifting surfaces with arbitrary bodies including fuselage+stores+tip missiles.

Panel formulation for lifting surface is identical to that of ZONA51 – which is now available as an industrial standard method for supersonic flutter analysis in MSC.NASTRAN.

High-order paneling allows high-fidelity modeling of complex aircraft with arbitrary stores/tip missile arrangement.

• ZONA7U: Hypersonic Unsteady Aerodynamics

- Functionality: Generates unified hypersonic and supersonic steady/unsteady aerodynamics for wing-body/aircraft configurations with external stores/nacelles. ZONA7U can be also used to perform aeroheating analysis in hypersonic flow.

1-6 INTRODUCTION

- Main Features:

Nonlinear thickness effects of ZONA7U yields good agreement with Euler solution and test data.

Steady solutions approach linear and Newtonian limits.

It can handle point nose as well as blunt nose bodies.

Results/formulation are superior to Unsteady Linear Theory and Piston Theory.

It includes a streamline module and a hypersonic boundary layer module for aeroheating analysis.

Unified with ZONA7 and is therefore applicable to all Mach numbers > 1.0.

Additional input required to ZONA7 is only the wing root and tip sectional profile thickness.

• AIC Correction Module: (The ZTAW Method)

- Functionality: Generates a corrected AIC matrix to match the given set of forces/moments or unsteady pressures

- Main Features:

The AIC correction module computes the AIC weighting matrix using a ZONA Transonic AIC weighting (ZTAW) method that adopts a successive kernel expansion procedure.

The ZTAW method is an improved AIC correction method over the previous correction methods such as the force/moment correction method by Giesing et al and the downwash correction method by Pitt and Goodman. With in-phase pressures obtained by wind-tunnel measurement or CFD, ZTAW yields accurate out-of-phase and higher-frequency pressures resulting in well-correlated aeroelastic solutions whereas the previous method yield erroneous out-of-phase pressure in terms of shock jump behavior.

The ZTAW method also has an option to receive user’s weighted AIC input for either pressure matching or force/moment matching.

1.3 HIGH-FIDELITY GEOMETRY (HFG) MODULE The HFG module is capable of modeling full aircraft configuration with stores and/or nacelles. A complex aircraft configuration can be represented by the HFG module by means of wing-like and body-like definitions. Any modifications to the HFG module, such as input geometry enhancements, will have minimal impact on other general modules.

INTRODUCTION 1-7

1.4 3D SPLINE MODULE The 3D Spline module establishes the displacement/force transferal between the structural Finite Element Method (FEM) model and the ZAERO aerodynamic model. It consists of four spline methods that jointly assemble a spline matrix. These four spline methods include: (a) Thin Plate Spline; (b) Infinite Plate Spline; (c) Beam Spline, and (d) Rigid Body Attachment methods. The spline matrix provides the x, y and z displacements and slopes at all aerodynamic grids.

1.5 ZONA DYNAMIC MEMORY AND DATABASE MANAGEMENT SYSTEM

The ZONA Dynamic Memory and Database Management System (ZDM) consists of the following five parts:

• Matrix Entity Manager

The matrix entity manager is designed to store and retrieve very large, often sparse, matrices. It minimizes disk storage requirements while allowing algorithms developed by ZONA Tech. to perform matrix operations of virtually unlimited size.

• Relational Entity Manager

Relational entities are essentially tables. Each relation has data stored in rows (called entries) and columns (called attributes). Each attribute is given a descriptive name, a data type, and constraints on the values that the attributes may assume (i.e. integer, real or character data). These definitions are referred to as the schema of the relation.

• Unstructured Entity Manager

There are many times that a software module requires temporary, or scratch, disk space while performing tasks. The data generated within these tasks are generally "highly-local" and, due to the modular nature of the software, are not to be passed through arguments to other modules within the system. To effectively accommodate the transfer of this type of data, ZDM supports an unstructured database entity type composed of "records" that may contain any arbitrary collection of data.

• Dynamic Memory Manager

The dynamic memory manager consists of a suite of utility routines to allocate and release blocks of dynamic memory. The Dynamic Memory Manager provides the capability of developing an engineering software system that allows operations to be performed on data that would normally exceed the size of available memory.

• Engineering Utility Modules

Engineering utility modules contain a pool of routines that perform operations on matrix database entities. These operations include matrix decomposition, eigenvalue solver, matrix multiplication, matrix partitioning/merging, etc. These routines first check the property of a given matrix and then select the appropriate numerical technique to perform a particular matrix operation.

1-8 INTRODUCTION

1.6 BULK DATA INPUT ZAERO utilizes the bulk data input format, similar to that of NASTRAN and ASTROS. This type of input format has the advantage of: (a) minimizing the user learning curve; (b) relieving user input burden and (c) automated input error detection.

1.7 GRAPHIC DISPLAY CAPABILITY ZAERO provides for graphic interface with several commercialized graphic packages. Graphical data in output files containing the aerodynamic model, unsteady pressures (CP), interpolated structural modes and flutter modes can be displayed via TECPLOT, I-DEAS, FEMAP or PATRAN. Flutter curves (V-g and V-f diagrams) can be displayed via typical spreadsheet programs (such as Microsoft Excel) and X-Y plotting packages.

1.8 FLUTTER MODULE The ZAERO flutter module contains two flutter solution techniques: the K-method and the g-method. The g-method is ZONA’s newly developed flutter solution method that generalizes the K-method and the P-K method for true damping prediction. It is shown that the P-K method is only valid at the conditions of zero damping, zero frequency, or linearly varying generalized aerodynamic forces (Qhh) with respect to reduced frequency. In fact, if Qhh is highly nonlinear, it is shown that the P-K method may produce unrealistic roots due to its inconsistent formulation.

The flutter module has a built-in atmospheric table as an option to perform matched-point flutter analysis. Sensitivity analysis with respect to the structural parameters is also included in the g-method.

1.9 AEROSERVOELASTICITY (ASE) MODULE Functionality: Performs the aeroservoelastic analysis with margin analysis for SISO/MIMO

control systems. The ASE module was developed based on the publications by Prof. Moti Karpel of Technion – Israel Institute of Technology and revised and reformulated by Dr. Zhicun Wang of ZONA. Please refer to the following technical reports and papers for the detailed formulation of the ASE module: (1) Karpel, M., Moulin B., “Development of the Aerodynamics/Aeroservoelastic Modules in

ASTROS Volume IV – Aeroservoelastic Disciplines in ASTROS Theoretical Manual (F33615-96-C-3217)”, Air Force Research Laboratory, Wright-Patterson AFB, OH, February 4, 1999.

(2) Karpel, M. and Strul, E., “Minimum-State Unsteady Aerodynamic Approximations with Flexible Constraints,” Journal of Aircraft, Vol. 33, No. 6, 1996, pp. 1190-1196.

(3) Karpel, M. and Wieseman, C. D., “Modal Coordinates for Aeroelastic Analysis with Large Local Structural Variations,” Journal of Aircraft, Vol. 31, No. 2, 1994, pp. 396-403.

(4) Karpel, M., “Size-Reduction Techniques for the Determination of Efficient Aeroservoelastic Models,” Academic Press, Advances in Control and Dynamic Systems, Vol. 54, 1992, pp. 263-295.

(5) Karpel, M., “Extension to the Minimum-State Aeroelastic Modeling Method,” AIAA Journal, Vol. 29, No. 11, 1991, pp. 2007-2009.

INTRODUCTION 1-9

(6) Hoadley, S.T. and Karpel, M., “Applications of Aeroservoelastic Modeling Using Minimum-State Unsteady Aerodynamic Approximations'', Journal of Guidance, Control and Dynamics, Vol. 14, No. 6, 1991, pp. 1267-1276.

(7) Karpel, M., “Sensitivity Derivatives of Flutter Characteristics and Stability Margins for Aeroservoelastic Design,” Journal of Aircraft, Vol. 27, No. 4, 1990, pp. 368-375.

(8) Karpel, M., “Time-Domain Aeroservoelastic Modeling Using Weighted Unsteady Aerodynamic Forces,” Journal of Guidance, Control, and Dynamics, Vol.13, No. 1, 1990, pp. 30-37.

(9) Karpel, M., “Design for Active Flutter Suppression and Gust Alleviation Using State-Space Aeroelastic Modeling,” Journal of Aircraft, Vol. 19, No. 3, 1982, pp. 221-227.

(10) Karpel, M., “Design for Active and Passive Flutter Suppression and Gust Alleviation,” NASA CR-3482, 1981.

(11) Zhicun Wang and Ping-Chih Chen. "Adapted K-Method for Frequency-Domain ASE Control Stability Margin Analysis," 55th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA SciTech Forum, (AIAA 2014-1198)

(12) Zhicun Wang and Ping-Chih Chen. "Accurate Rational Function Approximation for Time-Domain Gust Analysis," 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA SciTech Forum, (AIAA 2017-0414)

- Main Features:

The ASE module performs stability analysis of either the open-loop system or the closed-loop system. It is formulated by the state space equation that includes the modeling of the sensors, actuators and control system. Figure 1.3 illustrates the computational procedures of the ASE module to construct the closed-loop state-space equation.

Figure 1.3 Computational Procedures of the ASE Module

The ASE module first generates the state space equation of the aeroelastic system by combining the generalized structural matrices with the generalized aerodynamic force matrix. The output equation of the state space equation is defined by the sensors. The generalized aerodynamic force matrix is transformed into the rational function using the rational function approximation (RFA) technique so that it can be easily incorporated into the aeroelastic state space equation. Two methods are available in the ASE module to perform the RFA; the Roger's method and the Minimum State Method.

Next, the aeroelastic state space equation is combined with the state space equation of the actuator model to construct the state space equation of the plant model. This plant model is then combined with

1-10 INTRODUCTION

the state space equation of the control system to form the state space equation of the vehicle model. The control system can be represented by a serious of single-input-single-output (SISO) and/or multi-input-multi-output (MIMO) control elements. An upstream control element and a downstream control element can be connected by either a fixed gain connection or a variable gain connection. Finally, the closed-loop state space equation can be obtained by applying the gain matrix to the vehicle system. Damping and frequency of the closed-loop system at each density and velocity pair can be computed by solving the eigenvalues of the closed-loop system matrix. Meanwhile, gain margin and phase margin of the closed-loop system also can be evaluated. Bode diagrams also can be generated.

The state space equations of the aeroelastic model, the plant model, the vehicle model and the closed loop model can be exported from the ASE module and imported into other control system design codes.

A frequency-domain approach is also incorporated in the ASE module that does not require the rational function approximation of the generalized aerodynamic force matrix. Because RFA could introduce error, by avoiding the RFA the frequency-domain solution is considered as the exact solution that can be used to validate the solution computed by the state space approach.

1.10 TRIM MODULE - Functionality: Performs the static aeroelastic/trim analysis for solving the trim system and

computing the flight loads.

- Main Features:

It employs the modal approach for solving the trim system of the flexible aircraft. The modal approach formulates a reduced-order trim system that can be solved with much less computational cost than the so-called “direct method”.

It is capable of dealing with the determined trim system as well as the over-determined trim system (more unknowns than the trim equations). The solution of the over-determined trim system is obtained by using an optimization technique which minimizes a user-defined objective function while satisfying a set of constraint functions.

For a symmetric configuration (symmetric about the x-z plane), it requires only the modeling of one half of the configuration even for the asymmetric flight conditions.

It generates the flight loads on both sides of the configuration in terms of forces and moments at the structural finite element grid points for subsequent detailed stress analysis.

1.11 MLOADS MODULE - Functionality: Performs the transient maneuver loads analysis due to the pilot input command.

- Main Features:

It is formulated in the state space form for either the open loop system or closed loop system. The rigid body degrees of freedom are transformed into the airframe states so that the sub-matrices associated with the airframe states in the state space matrices are in the same definition with those of the flight dynamics.

INTRODUCTION 1-11

A frequency-domain approach is also incorporated in the MLOADS module that does not require the rational function approximation of the generalized aerodynamic force matrix.

It computes the time histories of the maneuver loads of flexible airframe in the presence of control system. These maneuver loads include the time histories of component loads, grid point loads, etc. Based on these time histories of loads, the user can identify the critical maneuver load conditions.

It outputs the transient maneuver loads at each time step in terms of NASTRAN FORCE and MOMENT bulk data cards either by the mode displacement method or the mode acceleration method for subsequent detailed stress analysis.

1.12 ELOADS MODULE - Functionality: Performs the transient ejection loads analysis due to store ejections.

- Main Features:

It allows multiple store ejections (in sequential scheduling) while the aircraft is maneuvering due to pilot input commands.

It accounts for the effects of the sudden reduction in aircraft weight due to the separation of the stores from the aircraft.

It is formulated in the state-space form for either an open-loop or closed-loop system.

It outputs the transient loads at each time step in terms of NASTRAN FORCE and MOMENT bulk data cards either by the mode displacement method or the mode acceleration method for subsequent detailed stress analysis.

1.13 GLOADS MODULE - Functionality: Performs a transient response analysis for discrete gust analysis or solves the

power spectral density (PSD) of the component loads for continuous gust analysis.

Please refer to the following technical paper for the detailed formulation of the GLOADS module. (1) Karpel, M., Moulin, B., and Chen, P.C., “Dynamics Response of Aeroservoelastic Systems to

Gust Excitation,” Journal of Aircraft, Vol. 42, No. 5, September – October 2005. pp. 1264-1272.

(2) Zhicun Wang and Ping-Chih Chen. "Adapted K-Method for Frequency-Domain ASE Control Stability Margin Analysis," 55th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA SciTech Forum, (AIAA 2014-1198)

(3) Zhicun Wang and Ping-Chih Chen. "Accurate Rational Function Approximation for Time-Domain Gust Analysis," 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA SciTech Forum, (AIAA 2017-0414)

1-12 INTRODUCTION

- Main Features:

It includes various options for defining the discrete gust profile such as one-minus-cosine, sine, sharp-edged gust, and arbitrary gust profiles for discrete gust and Dryden’s or Von Karman’s gust spectrum for continuous gust.

Three approaches are available for the discrete gust analysis; the frequency-domain approach, the hybrid approach and the state space approach. Only the frequency-domain approach is available for the continuous gust analysis.

For computing the component loads due to discrete gust, the GLOADS module has three options; the mode displacement (MD) method; the summation of forces (SOF) method and the rational function approximation method. The PSD of component loads due to continuous gust also can be computed either by the MD method or the SOF method.

It outputs the forces at structural grid points in terms of NASTRAN FORCE and MOMENT bulk data cards at each time step for the discrete gust analysis and at each frequency step for the continuous gust analysis.

1.14 NLFLTR MODULE - Functionality: Performs nonlinear flutter analysis for open/closed loop system using discrete

time-domain state space approach.

- Main Features:

Both nonlinear structures and nonlinear control system can be specified.

Nonlinearities can be specified as a function of multiple user defined nonlinear parameters such as displacements, velocities, accelerations, element forces, modal values and control system outputs.

Results are the transient response of the structures and the control system.

SUBSONIC FLUTTER ANALYSIS (HA145E) 2-1

Chapter 2

FLUTTER SAMPLE CASES

2.1 CASE 1: SUBSONIC (M=0.45) FLUTTER ANALYSIS OF A 15-DEGREE SWEPTBACK WING (HA145E)

• Purpose: Demonstrate a wing only, subsonic (i.e., ZONA6 method) flutter case using the g- and

K- flutter solution methods.

• Description of Input:

A 15 degree sweptback wing (modified HA145E case from the MSC.NASTRAN Aeroelastic Analysis User’s Guide, Version 68) is considered for this case. The structural and aerodynamic models are shown in Figure 2.1. The input data for this example are shown in Listing 2.1.

(a) Structural Model (b) Aerodynamic Model

Figure 2.1 15 Degree Sweptback Wing

- Executive Control

The ‘ASSIGN FEM=’ Executive Control Command specifies the structural modal data to be imported. Listing 2.2 shows the ‘HA145E.F06’ file containing free vibration solutions, which is generated using MSC/ NASTRAN. Refer to the MSC.NASTRAN Aeroelastic Analysis User’s Guide, Version 68, for a description of the structural model.

2-2 SUBSONIC FLUTTER ANALYSIS (HA145E)

- Case Control

The case control includes a single subcase for flutter computation (FLUTTER = 100). Sorted bulk data echo is requested through ECHO = SORT.

- Aerodynamic Parameters / Flight Conditions

The AEROZ bulk data card specifies a symmetric model about the x-z plane. It specifies mass and length units as ‘slinch’ and ‘inch’, respectively. A reference chord of 2.07055 inches is used.

The MKAEROZ bulk data card specifies a freestream Mach number of 0.45 and 11 reduced frequencies from 0.05 to 0.50. The program inserts the reduced frequency of 0.00 (required for g-method calculations), if not included by the user. The identification number of the MKAEROZ bulk data card (IDMK = 80) is referred to by FIXMDEN to obtain the Mach number and its associated aerodynamics for non-matched point flutter calculations. ZONA6 subsonic method is used for generating Aerodynamic Influence Coefficients (AIC) (METHOD = 0), and it is saved (SAVE = SAVE) in the file named ‘HA145_AIC.45’.

- Aerodynamic Model

One CAERO7 wing macroelement is defined with 5 chordwise and 7 spanwise evenly cut which generate 4 by 6 aerodynamic boxes. Root and tip chord lengths are both 2.07055 inches with a 5.5251 inch semispan length. The wing tip x- and y- coordinates are located at 1.48044 and 5.5251 inches, respectively, establishing a 15 degree leading edge sweep angle.

- Spline

A SPLINE1 bulk data card is used to spline the aerodynamic wing model to the structure. A PANLST2 bulk data card is referenced by SETK = 101 and a SET1 bulk data card by SETG = 100. The PANLST2 defines the wing macroelement to be splined (CAERO7 with WID of 101), and splines all of the wing aerodynamic boxes (101 through 124) to the structural grid points listed in the SET1 bulk data card (see Input Data Listing 2.1 for SET1 GRID point id’s and Figure 2.1(a)).

- Flutter

A FLUTTER bulk data card with SETID = 100 requests a symmetric flutter case. The FIX = 100 entry specifies the identification number of a FIXMDEN bulk data card. All structural modes are used for the flutter analysis (NMODE = 0). The modal damping is obtained from a TABDMP1 bulk data card (with the identification number of 10), which specifies one percent structural damping. The FIXMDEN bulk data card specifies density as 1.073*10-7 slinch/in3 and the velocity list. The velocity is normalized by 20.20 to change the units to knots.

- Plot Files

For post-processing the output, a number of plot files are requested. PLTAERO bulk data card generates the aerodynamic model in Tecplot format (Figure 2.1). The flutter mode plot of the wing is requested using the PLTFLUT bulk data card. The unsteady pressure coefficients for Mode 1 at the reduced frequency of 0.12 (IK = 6) are generated using the PLTCP bulk data card. The PLTVG bulk data card generates the data file (for X-Y plots) containing frequency and damping as a function of velocity.

SUBSONIC FLUTTER ANALYSIS (HA145E) 2-3

• Description of Output:

The output data for this example are shown in Listing 2.3. The main features of the output are discussed below.

The structural natural frequencies and generalized mass (Table 2.1) for the first four modes generated by MSC.NASTRAN are read in from file ‘HA145E.F06’. The g-method predicts a flutter speed of 479 ft/sec and flutter frequency of 112 Hz. These values are nearly identical to those predicted by the K-method. The ZAERO (g- and K-methods) flutter speed/frequency compare well with the experimental (Tuovila, W.J., NACA RM L55E11, 1955) and MSC.NASTRAN results (Table 2.2).

Table 2.1 Natural Frequencies and Generalized Mass of Case HA145E

Mode No.

MSC.NASTRAN Natural Frequency

(Hz) Generalized

Mass 1 34.3439 2.4855E-05 2 210.000 9.0881E-06 3 260.429 8.5232E-06 4 634.761 7.9439E-06

Table 2.2 Flutter Results of Case HA145E (M = 0.45)

Method Vf (ft/s) ff (Hz) Test (NACA RM L55E11) 495 120 MSC.NASTRAN (KE-method) 483 113 ZAERO (ZONA6, G-method) 479 112 ZAERO (ZONA6, K-method) 480 110

The flutter results using the g-method are shown in the frequency versus velocity and damping versus velocity diagrams of the output. It is observed that for this example, Mode 2 damping becomes unstable, leading to flutter. A number of plot data files are requested for post-processing. The damping and frequency variations for the Mode 2 (PLTVG bulk data card), obtained using the g- and K-methods, are shown in Figure 2.2. These plots are generated by using the VG1.PLT file. The remaining plots are made using the Tecplot software. The flutter mode plot of the wing (PLTFLUT bulk data card) is shown in Figure 2.3, using only three deformed aerodynamic models for clarity. The real and imaginary parts of the unsteady pressure coefficients for Mode 1 (PLTCP bulk data card) at the reduced frequency of 0.12 (IK = 6) are presented in Figure 2.4.

This case was run on a PC with 2.66 GHz CPU speed. The log file (at the end of the output data) shows the total CPU time of 2.5 seconds, out of which the generation of AIC takes about 2.0 seconds.

2-4 SUBSONIC FLUTTER ANALYSIS (HA145E)

400 420 440 460 480 500-0.2

-0.1

0

0.1

Velocity, ft/s

Dam

ping

, gZAERO (K-method)

ZAERO (g-method)

400 420 440 460 480 50050

100

150

200

Velocity, ft/s

Freq

uenc

y, H

z

ZAERO (K-method)

ZAERO (g-method)

Figure 2.2 Flutter Results of Case HA145E (M=0.45)

XY

Z

Figure 2.3 Flutter Mode of Case HA145E (M=0.45)

X

Y

Z

RE(CP)

0.040.020

-0.02-0.04-0.06-0.08-0.1-0.12-0.14-0.16-0.18-0.2-0.22-0.24-0.26-0.28-0.3-0.32-0.34

IM(CP)

-0.1-0.125-0.15-0.175-0.2-0.225-0.25-0.275-0.3-0.325-0.35-0.375-0.4-0.425-0.45-0.475-0.5-0.525-0.55-0.575-0.6

(a) real part (b) imaginary part

Figure 2.4 Unsteady Cp of Case HA145E : (M=0.45, Mode 1, k=0.12)

SUBSONIC FLUTTER ANALYSIS (HA145E) 2-5

• Input Data Listing:

Listing 2.1 Input Data for the 15 Degree Sweptback Wing (HA145E) $ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ $ $ $ Z A E R O I N P U T (HA145E.INP) $ $ $ $ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ $ $ THIS CASE DEMONSTRATES A SINGLE WING, SUBSONIC FLUTTER CASE USING $ K AND G FLUTTER SOLUTION METHODS. $ $Begin Executive Control Section ASSIGN FEM=ha145e.f06, PRINT=0,FORM=MSC,BOUND=SYM DIAG 1 CEND $Begin Case Control Section TITLE= SUBSONIC FLUTTER ANALYSIS (15 DEGREE SWEPTBACK WING) ECHO = SORT SUBCASE = 1 SUBTITLE=ZONA6 METHOD LABEL=MACH NUMBER = 0.45, NON-MATCH POINT FLUTTER ANALYSIS FLUTTER=100 BEGIN BULK $ $ * AERO PARAMETERS / FLIGHT CONDITIONS * $ $...1..|...2...|...3...|...4...|...5...|...6...|...7...|...8...|...9...|...10..| $ $ ACSID XZSYM FLIP FMMUNIT FMLUNIT REFC REFB REFS +ABC AEROZ 0 YES NO SLIN IN 2.07055 11.0502 22.8800 +A $+ABC REFX REFY REFZ +A 0.5176 0. 0. $ $ IDMK MACH METHOD IDFLT SAVE PRINT $ MKAEROZ 80 .45 0 0 SAVE HA145E_AIC.45 -3 +MK1 $ FREQ1 FREQ2 ETC $ +MK1 0.05 0.08 0.10 0.11 0.12 0.14 0.16 0.18 +MK2 +MK2 0.20 0.25 0.50 $ $ $ * WING MACROELEMENT * $ $ $ $ WID LABEL ACOORD NSPAN NCHORD LSPAN ZTAIC PAFOIL7 $ CAERO7 101 WING 0 7 5 +CA101 $ XRL YRL ZRL RCH LRCHD ATTCHR $ +CA101 .0 .0 .0 2.07055 0 0 +CA102 $ XTL YTL ZTL TCH LTCHD ATTCHT $ +CA102 1.48044 5.52510 0.0 2.07055 0 0 $ $ $ * SURFACE SPLINE FIT ON THE WING * $ $ $ $ EID MODEL CP SETK SETG DZ EPS $ SPLINE1 100 WING 101 100 0.0 $ $ $ SETID MACROID BOX1 BOX2 ETC $ PANLST2 101 101 101 THRU 124 $ $ $ SID G1 G2 ETC $ SET1 100 2 4 6 8 9 11 13 +S1 +S1 15 18 20 22 24 25 27 29 +S2 +S2 31 34 36 38 40 $ $ $ * NON-MATCHED POINT FLUTTER ANALYSIS * $ $ $ SETID SYM FIX NMODE TABDAMP MLIST CONMLST FLUTTER 100 SYM 100 0 10 $ $ TABDMP1 10 G +TAB1 +TAB1 0.0 0.01 1000. 0.01 $...1..|...2...|...3...|...4...|...5...|...6...|...7...|...8...|...9...|...10$ $ SETID IDMK DEN FTMUNIT FTLUNIT VREF FLUTTF PRINT $ FIXMDEN 100 80 1.0726-7SLIN IN 12.00 0 +FL1 $ V1 V2 V3 ETC $ +FL1 4000. 4200. 4400. 4600. 4800. 5000. 5200. 5400. +FL2 +FL2 5600. 5700. 5800. 5900. 6000. 6200. 6400.

2-6 SUBSONIC FLUTTER ANALYSIS (HA145E)

$ $ $ * PLOT AERO MODEL BY PLTAERO * $ $ $ PLTAERO 11 YES 0 TECPLOT AERO1.PLT $ $ $ $ $ * PLOT CP BY PLTCP * $ $ $ PLTCP 3000 SYM 80 6 1 TECPLOT CP1.PLT $ $ $ * PLOT FLUTTER MODE BY PLTFLUT * $ $ $ PLTFLUT 10 100 1 8 .3 TECPLOT FLUT1.PLT PLTMODE 10 SYM 1 .3 TECPLOT MODE1.PLT PLTMODE 20 SYM 2 .3 PATRAN MODE2.PLT $ $ * V-G PLOT * $ $ PLTVG 11 100 V VG1.PLT $ $ ENDDATA

Listing 2.2 Modal Data for the 15 Degree Sweptback Wing (HA145E.F06; MSC.NASTRAN) SOL 103 CEND 1 MARCH 25, 1999 MSC.NASTRAN 4/28/98 PAGE 3 0 0 C A S E C O N T R O L D E C K E C H O CARD COUNT 1 ECHO = SORT 2 SPC = 1 $ WING ROOT DEFLECTIONS AND PLATE IN-PLANE ROTATIONS FIX ED 3 METHOD = 10 $ MODIFIED GIVENS METHOD OF REAL EIGENVALUE EXTRACTION 4 DISP = ALL 5 BEGIN BULK 0 INPUT BULK DATA CARD COUNT = 194 1 MARCH 25, 1999 MSC.NASTRAN 4/28/98 PAGE 4 0 0 S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 1- ASET1 3 1 THRU 8 2- ASET1 3 10 THRU 16 3- ASET1 3 18 THRU 24 4- ASET1 3 26 THRU 40 5- CQUAD4 1 1 1 2 10 9 +M00000 6- +M00000 0.0 0.0 .041 .041 7- CQUAD4 2 1 2 3 11 10 +M00001 8- +M00001 0.0 0.0 .041 .041 9- CQUAD4 3 1 3 4 12 11 +M00002 10- +M00002 0.0 0.0 .041 .041 11- CQUAD4 4 1 4 5 13 12 +M00003 12- +M00003 0.0 0.0 .041 .041 13- CQUAD4 5 1 5 6 14 13 +M00004 14- +M00004 0.0 0.0 .041 .041 15- CQUAD4 6 1 6 7 15 14 +M00005 16- +M00005 0.0 0.0 .041 .041 17- CQUAD4 7 1 7 8 16 15 +M00006 18- +M00006 0.0 0.0 .041 .041 19- CQUAD4 8 1 9 10 18 17 20- CQUAD4 9 1 10 11 19 18 21- CQUAD4 10 1 11 12 20 19 22- CQUAD4 11 1 12 13 21 20 23- CQUAD4 12 1 13 14 22 21 24- CQUAD4 13 1 14 15 23 22 25- CQUAD4 14 1 15 16 24 23 26- CQUAD4 15 1 17 18 26 25 27- CQUAD4 16 1 18 19 27 26 28- CQUAD4 17 1 19 20 28 27 29- CQUAD4 18 1 20 21 29 28 30- CQUAD4 19 1 21 22 30 29 31- CQUAD4 20 1 22 23 31 30 32- CQUAD4 21 1 23 24 32 31 33- CQUAD4 22 1 25 26 34 33 +M00007 34- +M00007 .041 .041 0.0 0.0 35- CQUAD4 23 1 26 27 35 34 +M00008 36- +M00008 .041 .041 0.0 0.0

SUBSONIC FLUTTER ANALYSIS (HA145E) 2-7

37- CQUAD4 24 1 27 28 36 35 +M00009 38- +M00009 .041 .041 0.0 0.0 39- CQUAD4 25 1 28 29 37 36 +M00010 40- +M00010 .041 .041 0.0 0.0 41- CQUAD4 26 1 29 30 38 37 +M00011 42- +M00011 .041 .041 0.0 0.0 43- CQUAD4 27 1 30 31 39 38 +M00012 44- +M00012 .041 .041 0.0 0.0 45- CQUAD4 28 1 31 32 40 39 +M00013 46- +M00013 .041 .041 0.0 0.0 47- EIGR 10 MGIV 4 +ER 48- +ER MAX 49- GRID 1 0.0 0.0 0.0 50- GRID 2 .211491 .7893 0.0 1 MARCH 25, 1999 MSC.NASTRAN 4/28/98 PAGE 5 0 S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 51- GRID 3 .422983 1.5786 0.0 52- GRID 4 .634474 2.3679 0.0 53- GRID 5 .845966 3.1572 0.0 54- GRID 6 1.05746 3.9465 0.0 55- GRID 7 1.26895 4.7358 0.0 56- GRID 8 1.48044 5.5251 0.0 57- GRID 9 .258819 0.0 0.0 58- GRID 10 .47031 .7893 0.0 59- GRID 11 .681802 1.5786 0.0 60- GRID 12 .893293 2.3679 0.0 61- GRID 13 1.10478 3.1572 0.0 62- GRID 14 1.31628 3.9465 0.0 63- GRID 15 1.52777 4.7358 0.0 64- GRID 16 1.73926 5.5251 0.0 65- GRID 17 1.03528 0.0 0.0 66- GRID 18 1.24677 .7893 0.0 67- GRID 19 1.45826 1.5786 0.0 68- GRID 20 1.66975 2.3679 0.0 69- GRID 21 1.88124 3.1572 0.0 70- GRID 22 2.09273 3.9465 0.0 71- GRID 23 2.30422 4.7358 0.0 72- GRID 24 2.51572 5.5251 0.0 73- GRID 25 1.81173 0.0 0.0 74- GRID 26 2.02322 .7893 0.0 75- GRID 27 2.23471 1.5786 0.0 76- GRID 28 2.44621 2.3679 0.0 77- GRID 29 2.6577 3.1572 0.0 78- GRID 30 2.86919 3.9465 0.0 79- GRID 31 3.08068 4.7358 0.0 80- GRID 32 3.29217 5.5251 0.0 81- GRID 33 2.07055 0.0 0.0 82- GRID 34 2.28204 .7893 0.0 83- GRID 35 2.49353 1.5786 0.0 84- GRID 36 2.70502 2.3679 0.0 85- GRID 37 2.91652 3.1572 0.0 86- GRID 38 3.12801 3.9465 0.0 87- GRID 39 3.3395 4.7358 0.0 88- GRID 40 3.55099 5.5251 0.0 89- MAT1 1 9.2418+63.4993+6 0.097464 90- PARAM COUPMASS1 91- PARAM GRDPNT 17 92- PARAM WTMASS .0025901 93- PSHELL 1 1 .041 1 1 94- SPC1 1 6 1 THRU 40 95- SPC1 1 12345 9 96- SPC1 1 12345 25 ENDDATA 0 TOTAL COUNT= 97 1 MARCH 25, 1999 MSC.NASTRAN 4/28/98 PAGE 6 0 O U T P U T F R O M G R I D P O I N T W E I G H T G E N E R A T O R 0 REFERENCE POINT = 17 M O * 4.000018E-02 6.352747E-20 0.000000E+00 0.000000E+00 0.000000E+00 -1.105025E-01 * * 6.352747E-20 4.000018E-02 0.000000E+00 0.000000E+00 0.000000E+00 2.960875E-02 * * 0.000000E+00 0.000000E+00 4.000018E-02 1.105025E-01 -2.960875E-02 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.105025E-01 4.070249E-01 -1.090611E-01 0.000000E+00 * * 0.000000E+00 0.000000E+00 -2.960875E-02 -1.090611E-01 4.038716E-02 0.000000E+00 * * -1.105025E-01 2.960875E-02 0.000000E+00 0.000000E+00 0.000000E+00 4.474121E-01 * S * 1.000000E+00 0.000000E+00 0.000000E+00 *

2-8 SUBSONIC FLUTTER ANALYSIS (HA145E)

* 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 4.000018E-02 0.000000E+00 2.762550E+00 0.000000E+00 Y 4.000018E-02 7.402154E-01 0.000000E+00 0.000000E+00 Z 4.000018E-02 7.402154E-01 2.762550E+00 0.000000E+00 I(S) * 1.017562E-01 2.726544E-02 0.000000E+00 * * 2.726544E-02 1.847031E-02 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.202265E-01 * I(Q) * 1.098881E-01 * * 1.033837E-02 * * 1.202265E-01 * Q * 9.582864E-01 2.858097E-01 0.000000E+00 * * -2.858097E-01 9.582864E-01 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * 1 MARCH 25, 1999 MSC.NASTRAN 4/28/98 PAGE 7 0 *** SYSTEM INFORMATION MESSAGE 6916 (DFMSYN) DECOMP ORDERING METHOD CHOSEN: BEND, ORDERING METHOD USED: BEND *** USER INFORMATION MESSAGE 5458 (REIG) MODIFIED GIVENS METHOD IS FORCED BY USER . 1 MARCH 25, 1999 MSC.NASTRAN 4/28/98 PAGE 8 0 R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED NO. ORDER MASS STIFFNESS 1 1 4.656517E+04 2.157896E+02 3.434399E+01 2.485460E-05 1.157359E+00 2 2 1.741004E+06 1.319471E+03 2.100004E+02 9.088139E-06 1.582249E+01 3 3 2.677550E+06 1.636322E+03 2.604287E+02 8.523230E-06 2.282137E+01 4 4 1.590670E+07 3.988320E+03 6.347608E+02 7.943937E-06 1.263618E+02 5 5 2.676045E+07 5.173051E+03 8.233166E+02 0.0 0.0 6 6 6.705253E+07 8.188561E+03 1.303250E+03 0.0 0.0 7 7 1.020573E+08 1.010234E+04 1.607838E+03 0.0 0.0 8 8 2.060991E+08 1.435615E+04 2.284853E+03 0.0 0.0 9 9 2.839796E+08 1.685169E+04 2.682030E+03 0.0 0.0 10 10 3.717570E+08 1.928100E+04 3.068667E+03 0.0 0.0 11 11 4.330577E+08 2.081004E+04 3.312021E+03 0.0 0.0 12 12 5.752656E+08 2.398470E+04 3.817283E+03 0.0 0.0 13 13 6.478300E+08 2.545251E+04 4.050892E+03 0.0 0.0 14 14 7.371570E+08 2.715063E+04 4.321158E+03 0.0 0.0 15 15 1.062141E+09 3.259050E+04 5.186939E+03 0.0 0.0 16 16 1.340560E+09 3.661366E+04 5.827244E+03 0.0 0.0 17 17 1.504211E+09 3.878416E+04 6.172690E+03 0.0 0.0 18 18 2.016248E+09 4.490264E+04 7.146478E+03 0.0 0.0 19 19 2.896543E+09 5.381955E+04 8.565646E+03 0.0 0.0 20 20 3.734667E+09 6.111192E+04 9.726265E+03 0.0 0.0 21 21 3.846096E+09 6.201690E+04 9.870296E+03 0.0 0.0 22 24 4.477242E+09 6.691220E+04 1.064941E+04 0.0 0.0 23 25 4.639454E+09 6.811354E+04 1.084061E+04 0.0 0.0 24 26 4.942809E+09 7.030511E+04 1.118941E+04 0.0 0.0 25 31 5.256086E+09 7.249887E+04 1.153855E+04 0.0 0.0 26 35 5.759773E+09 7.589317E+04 1.207877E+04 0.0 0.0 27 37 6.100987E+09 7.810882E+04 1.243140E+04 0.0 0.0 28 36 6.657556E+09 8.159385E+04 1.298606E+04 0.0 0.0 29 34 7.332475E+09 8.562988E+04 1.362842E+04 0.0 0.0 30 33 8.441387E+09 9.187702E+04 1.462268E+04 0.0 0.0 31 32 9.166372E+09 9.574117E+04 1.523768E+04 0.0 0.0 32 30 1.070425E+10 1.034613E+05 1.646638E+04 0.0 0.0 33 29 1.173995E+10 1.083511E+05 1.724461E+04 0.0 0.0 34 28 1.259991E+10 1.122493E+05 1.786504E+04 0.0 0.0 35 27 1.455259E+10 1.206341E+05 1.919952E+04 0.0 0.0 36 23 1.897902E+10 1.377644E+05 2.192588E+04 0.0 0.0 37 22 2.018797E+10 1.420844E+05 2.261343E+04 0.0 0.0 1 MARCH 25, 1999 MSC.NASTRAN 4/28/98 PAGE 9 0 1 MARCH 25, 1999 MSC.NASTRAN 4/28/98 PAGE 10

SUBSONIC FLUTTER ANALYSIS (HA145E) 2-9

0 EIGENVALUE = 4.656517E+04 CYCLES = 3.434399E+01 R E A L E I G E N V E C T O R N O . 1 POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G 0.0 0.0 -3.320767E-03 5.275748E-02 -2.311046E-02 0.0 2 G 0.0 0.0 3.354167E-02 2.932303E-02 -1.964025E-02 0.0 3 G 0.0 0.0 1.197589E-01 1.662287E-01 -4.147353E-02 0.0 4 G 0.0 0.0 2.415489E-01 1.236907E-01 -2.596768E-02 0.0 5 G 0.0 0.0 3.895444E-01 2.266178E-01 -4.942432E-02 0.0 6 G 0.0 0.0 5.599717E-01 1.821056E-01 -3.271172E-02 0.0 7 G 0.0 0.0 7.364756E-01 2.416311E-01 -4.689491E-02 0.0 8 G 0.0 0.0 9.197437E-01 2.006610E-01 -3.359312E-02 0.0 9 G 0.0 0.0 0.0 0.0 0.0 0.0 10 G 0.0 0.0 3.985384E-02 8.988190E-02 -3.247913E-02 0.0 11 G 0.0 0.0 1.283573E-01 1.182844E-01 -2.095009E-02 0.0 12 G 0.0 0.0 2.494006E-01 1.707260E-01 -3.638510E-02 0.0 13 G 0.0 0.0 4.009324E-01 1.918168E-01 -3.647194E-02 0.0 14 G 0.0 0.0 5.693350E-01 2.126955E-01 -4.017603E-02 0.0 15 G 0.0 0.0 7.477078E-01 2.169071E-01 -3.920391E-02 0.0 16 G 0.0 0.0 9.293225E-01 2.215306E-01 -4.025509E-02 0.0 17 G 0.0 0.0 -2.937914E-04 6.190476E-02 -8.185685E-04 0.0 18 G 0.0 0.0 5.363604E-02 7.274584E-02 -2.740528E-03 0.0 19 G 0.0 0.0 1.427034E-01 1.454802E-01 -1.611947E-02 0.0 20 G 0.0 0.0 2.734128E-01 1.727809E-01 -2.510017E-02 0.0 21 G 0.0 0.0 4.279284E-01 2.013101E-01 -3.278478E-02 0.0 22 G 0.0 0.0 5.990526E-01 2.123529E-01 -3.604479E-02 0.0 23 G 0.0 0.0 7.780459E-01 2.201141E-01 -3.867503E-02 0.0 24 G 0.0 0.0 9.601673E-01 2.201419E-01 -3.893496E-02 0.0 25 G 0.0 0.0 0.0 0.0 0.0 0.0 26 G 0.0 0.0 4.663620E-02 1.196191E-01 1.940306E-02 0.0 27 G 0.0 0.0 1.527655E-01 1.500145E-01 -9.815941E-03 0.0 28 G 0.0 0.0 2.906154E-01 1.892013E-01 -1.920878E-02 0.0 29 G 0.0 0.0 4.520696E-01 2.050554E-01 -2.915488E-02 0.0 30 G 0.0 0.0 6.267195E-01 2.186406E-01 -3.503649E-02 0.0 31 G 0.0 0.0 8.077031E-01 2.194385E-01 -3.752929E-02 0.0 32 G 0.0 0.0 9.904701E-01 2.227564E-01 -3.894404E-02 0.0 33 G 0.0 0.0 -3.744977E-03 2.764643E-02 2.112055E-02 0.0 34 G 0.0 0.0 4.227901E-02 1.043890E-01 2.184120E-02 0.0 35 G 0.0 0.0 1.548405E-01 1.764728E-01 -9.414680E-03 0.0 36 G 0.0 0.0 2.953975E-01 1.730397E-01 -1.472252E-02 0.0 37 G 0.0 0.0 4.598230E-01 2.263945E-01 -3.197226E-02 0.0 38 G 0.0 0.0 6.353986E-01 2.007433E-01 -3.072660E-02 0.0 39 G 0.0 0.0 8.179795E-01 2.402105E-01 -4.210586E-02 0.0 40 G 0.0 0.0 1.000000E+00 2.000586E-01 -3.453320E-02 0.0 1 MARCH 25, 1999 MSC.NASTRAN 4/28/98 PAGE 11 0 EIGENVALUE = 1.741004E+06 CYCLES = 2.100004E+02 R E A L E I G E N V E C T O R N O . 2 POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G 0.0 0.0 1.107332E-02 -2.260668E-01 8.574074E-02 0.0 2 G 0.0 0.0 -1.414919E-01 -1.058193E-01 -5.301957E-03 0.0 3 G 0.0 0.0 -4.007863E-01 -5.091912E-01 -5.855032E-02 0.0 4 G 0.0 0.0 -6.050647E-01 -7.694189E-02 -2.867363E-01 0.0 5 G 0.0 0.0 -6.624407E-01 -2.089252E-01 -3.248615E-01 0.0 6 G 0.0 0.0 -5.821251E-01 1.932655E-01 -4.811796E-01 0.0 7 G 0.0 0.0 -3.570981E-01 1.159731E-01 -4.723542E-01 0.0 8 G 0.0 0.0 -7.326158E-02 3.253082E-01 -5.398714E-01 0.0 9 G 0.0 0.0 0.0 0.0 0.0 0.0 10 G 0.0 0.0 -1.446287E-01 -3.232221E-01 4.263464E-02 0.0 11 G 0.0 0.0 -3.784626E-01 -2.700317E-01 -1.332725E-01 0.0 12 G 0.0 0.0 -5.368559E-01 -2.138972E-01 -2.361298E-01 0.0 13 G 0.0 0.0 -5.745575E-01 -3.738778E-02 -3.616104E-01 0.0 14 G 0.0 0.0 -4.620206E-01 1.019581E-01 -4.421384E-01 0.0 15 G 0.0 0.0 -2.318032E-01 2.202560E-01 -4.951098E-01 0.0 16 G 0.0 0.0 6.338770E-02 2.507957E-01 -5.138512E-01 0.0 17 G 0.0 0.0 -8.396643E-04 -1.929060E-01 -5.599579E-03 0.0 18 G 0.0 0.0 -1.403477E-01 -1.577093E-01 -6.282534E-02 0.0 19 G 0.0 0.0 -2.714325E-01 -2.098485E-01 -1.464372E-01 0.0 20 G 0.0 0.0 -3.485831E-01 -8.095299E-02 -2.510886E-01 0.0 21 G 0.0 0.0 -2.981144E-01 4.920985E-02 -3.500597E-01 0.0 22 G 0.0 0.0 -1.210223E-01 1.818425E-01 -4.342829E-01 0.0 23 G 0.0 0.0 1.500929E-01 2.477379E-01 -4.856598E-01 0.0 24 G 0.0 0.0 4.641854E-01 2.736363E-01 -5.148948E-01 0.0 25 G 0.0 0.0 0.0 0.0 0.0 0.0 26 G 0.0 0.0 -6.794432E-02 -1.815082E-01 -1.232568E-01 0.0 27 G 0.0 0.0 -1.554098E-01 -9.816097E-02 -1.533911E-01 0.0 28 G 0.0 0.0 -1.528569E-01 4.911089E-03 -2.520888E-01 0.0

2-10 SUBSONIC FLUTTER ANALYSIS (HA145E)

29 G 0.0 0.0 -2.636060E-02 1.529317E-01 -3.479257E-01 0.0 30 G 0.0 0.0 2.173962E-01 2.450575E-01 -4.343207E-01 0.0 31 G 0.0 0.0 5.304149E-01 2.874658E-01 -4.905873E-01 0.0 32 G 0.0 0.0 8.653485E-01 2.852452E-01 -5.145663E-01 0.0 33 G 0.0 0.0 1.004413E-02 -3.865173E-02 -5.667925E-02 0.0 34 G 0.0 0.0 -3.825525E-02 -1.303979E-01 -1.227094E-01 0.0 35 G 0.0 0.0 -1.152494E-01 -1.173702E-01 -1.502222E-01 0.0 36 G 0.0 0.0 -8.781209E-02 7.332725E-02 -2.545322E-01 0.0 37 G 0.0 0.0 6.338259E-02 1.431831E-01 -3.390636E-01 0.0 38 G 0.0 0.0 3.312269E-01 3.028386E-01 -4.435413E-01 0.0 39 G 0.0 0.0 6.570766E-01 2.549826E-01 -4.840532E-01 0.0 40 G 0.0 0.0 1.000000E+00 3.335835E-01 -5.236316E-01 0.0 1 MARCH 25, 1999 MSC.NASTRAN 4/28/98 PAGE 12 0 EIGENVALUE = 2.677550E+06 CYCLES = 2.604287E+02 R E A L E I G E N V E C T O R N O . 3 POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G 0.0 0.0 2.114405E-03 1.097659E-02 1.560848E-03 0.0 2 G 0.0 0.0 1.986177E-02 4.924958E-02 5.667208E-02 0.0 3 G 0.0 0.0 6.028285E-02 1.016693E-01 1.542817E-01 0.0 4 G 0.0 0.0 1.323192E-01 1.805242E-01 2.828980E-01 0.0 5 G 0.0 0.0 2.601769E-01 2.973404E-01 3.864959E-01 0.0 6 G 0.0 0.0 4.728811E-01 4.399830E-01 4.779978E-01 0.0 7 G 0.0 0.0 7.355467E-01 4.746437E-01 5.630959E-01 0.0 8 G 0.0 0.0 1.000000E+00 4.938532E-01 6.007993E-01 0.0 9 G 0.0 0.0 0.0 0.0 0.0 0.0 10 G 0.0 0.0 2.551313E-03 3.454826E-02 6.988558E-02 0.0 11 G 0.0 0.0 1.791450E-02 7.838506E-02 1.701921E-01 0.0 12 G 0.0 0.0 5.546071E-02 1.507140E-01 3.131657E-01 0.0 13 G 0.0 0.0 1.576512E-01 2.960207E-01 4.080524E-01 0.0 14 G 0.0 0.0 3.465791E-01 4.133718E-01 5.025063E-01 0.0 15 G 0.0 0.0 5.893694E-01 4.738452E-01 5.697677E-01 0.0 16 G 0.0 0.0 8.437764E-01 4.800567E-01 6.100507E-01 0.0 17 G 0.0 0.0 -3.491340E-04 -5.233242E-02 7.806516E-03 0.0 18 G 0.0 0.0 -5.833368E-02 -5.130282E-02 8.907135E-02 0.0 19 G 0.0 0.0 -1.493535E-01 -6.783146E-02 2.609330E-01 0.0 20 G 0.0 0.0 -2.188800E-01 7.632377E-02 3.917430E-01 0.0 21 G 0.0 0.0 -1.930001E-01 2.267826E-01 4.935344E-01 0.0 22 G 0.0 0.0 -6.678205E-02 3.666732E-01 5.606208E-01 0.0 23 G 0.0 0.0 1.327621E-01 4.394740E-01 6.049351E-01 0.0 24 G 0.0 0.0 3.624147E-01 4.669136E-01 6.279791E-01 0.0 25 G 0.0 0.0 0.0 0.0 0.0 0.0 26 G 0.0 0.0 -1.448625E-01 -2.932250E-01 1.413140E-01 0.0 27 G 0.0 0.0 -3.904501E-01 -1.701667E-01 3.604765E-01 0.0 28 G 0.0 0.0 -5.610465E-01 -1.699963E-02 4.885510E-01 0.0 29 G 0.0 0.0 -6.075832E-01 1.886168E-01 5.722555E-01 0.0 30 G 0.0 0.0 -5.242109E-01 3.358437E-01 6.155127E-01 0.0 31 G 0.0 0.0 -3.486536E-01 4.334817E-01 6.325694E-01 0.0 32 G 0.0 0.0 -1.321459E-01 4.514872E-01 6.430338E-01 0.0 33 G 0.0 0.0 -2.928282E-03 -1.261122E-01 -1.285228E-03 0.0 34 G 0.0 0.0 -1.885662E-01 -2.