High-Fidelity Geometry (HFG) Module

The HFG module is capable of modeling any full aircraft configuration with stores and/or nacelles. A complex aircraft configuration can be represented by the HFG module by means of wing-like (CAERO7) and body-like (BODY7) definitions. User-friendly CAERO7 and BODY7 macroelements make it easy to set up any arbitrary wing-body input.

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3D Spline Module

The 3D Spline module establishes the displacement/force transferal between the structural Finite Element Analysis (FEA) model and the ZAERO aerodynamic model. It contains four powerful 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 in three dimensions at all aerodynamic grids.

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ZONA's 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 to be developed that can 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
Often times 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 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 which 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.

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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. An example of this type of input format is shown below.

Graphic Display

ZAERO allows for the graphic interface with numerous commercialized graphic packages. Graphical data in output files containing the aerodynamic model, unsteady pressures, interpolated structural modes, flutter modes, and transient responses can be displayed via Patran, Femap, Ansys, Ideas, Elfini, Pegasus or Tecplot. V-g and V-f diagrams can be displayed via typical X-Y plotting packages such as Microsoft Excel.

Aerodynamic Model

View Animated Flutter Mode Shapes

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Flutter Module

ZAERO FLUTTER SOLUTION TECHNIQUES
▪ K-method
▪ g-method (true damping solutions)

FEATURES
▪ Built-in standard atmospheric table to perform matched-point flutter analyses
▪ Sensitivity analysis with respect to any user-defined structural parameters

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

3 Degree-Of-Freedom Airfoil at M=0.0

- A non-zero frequency “dynamic divergence speed” is well predicted by the g-method, the P-K method and the transient method.
- Both the g-method and the transient method capture two aerodynamic lag roots which are absent in the P-K method solution.
- The frequency vs. velocity (V-f) diagrams of the g-method and the transient method are in good agreement. The frequency of the free-free plunge mode computed by the P-K method remains zero (poor correlation in the V-f diagram with the g-method and transient method)

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Parametric Flutter Analysis

FUNCTIONALITY
Performs a parametric flutter analysis by executing a massive number of flutter/ASE analyses for various mass and stiffness distributions.

FEATURES
▪ Massive flutter analyses of open/closed loop systems with various structural mass and stiffness using the
   mass increment method. For n aircraft/store configurations, the flutter equation in physical coordinates {x} reads:

   where MB and KB are the mass and stiffness matrices of a baseline structure and ΔM and ΔK are the incremental    changes of mass and stiffness from the baseline structure to the i-th structure of interest

▪ Data mining the massive flutter results by automatically searching for the velocity-damping curve crossing at
   user-specified damping levels.
▪ Ease for post-processing using off-the-shelf graphic tool such as Tecplot. Shown in the figure below is the flutter
   speed vs. various pitch inertia and weight diagram of the store.
▪ Flags to indicate the severity of the flutter instability.

Flutter Speed vs. Various Store Pitch Inertia and Weight

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Sensitivity Analysis with Respect to Structural Parameters

ZAERO provides analytical sensitivities of flutter parameters (frequencies and damping) with respect to structural design variables. Using the same set of generalized coordinates, such as the normal modes of a baseline structure, sensitivities can be computed without returning to the finite-element model .

Symbolically, the analytical sensitivity can be expressed as  

where A and B can be derived using the orthogonality of the right and left eigenvectors of the flutter equation. Computing A and B requires additional structural information from the finite element code; the derivatives of the stiffness and mass matrices with respect to the design variables. These matrices are imported from the finite element code through a DMIG bulk data input card.

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TRIM: Static Aeroelastic Trim Module

FUNCTIONALITY
Performs the static aeroelastic / trim analysis for solving the trim system and computing the flight loads.

FEATURES
▪ Employs the modal approach for solving the trim system of the flexible aircraft. The modal approach establishes
    a reduced-order trim system that can be solved with much less computer time than the so-called “direct method”.
▪ Capable of dealing with a determined trim system as well as the over-determined trim system (i.e., more
   unknowns than the number of trim equations). The solutions of the over-determined trim system are
   obtained by using an optimization technique that minimizes a user-defined objective function while satisfying
   a set of user-defined constraint functions.
▪ For a symmetric configuration (about the x-z plane), the Trim Module requires only that one-half of the configuration
   be modeled, even for an asymmetric flight condition.
▪ Generates the flight loads on both sides of the configuration in terms of forces and moments at the structural
   finite element grid points. Output are Nastran FORCE and MOMENT bulk data cards that can be used in
   subsequent detailed stress analyses.

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ASE: Aeroservoelasticity Module

The ASE software, developed by Professor Moti Karpel of Technion – Israel Institute of Technology, has been integrated in the ZAERO software system as shown below.

FEATURES
▪ Rational-function approximation of the unsteady aerodynamic coefficient matrices
▪ State-space MIMO formulation
▪ Modular linear control modeling of most-general architecture
▪ Open- and closed-loop flutter analysis
▪ Input and Output singular values
▪ Augmentation of continuous-gust dynamics
▪ Structural gust response in statistical terms
▪ Fixed-modes parametric studies
▪ Sensitivity of flutter and control margins with respect to structural and control variables
▪ Frequency-domain stability analysis without rational function approximation

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Rational Function Approximation of the Unsteady Aerodynamics

The unsteady aerodynamic force coefficient matrices are approximated by a rational matrix function in the Laplace domain. The approximation formula is either the classic Roger’s formula:

where p is the non-dimensional Laplace variable, p = sb/V or the more general minimum-state formula

which results in significantly less subsequent aerodynamic states per desired accuracy. The approximation roots are selected by the user or determined by the code based on the frequency range of the input matrices. A direct least-square solution is used for Roger’s approximation, and a non-linear least-square is used for the minimum-state approximation. A physical-weighting algorithm may be used to weight the data terms according to aeroelastic measures of importance.

Sample Minimum State Fit of Generalized Aerodynamic Forces of a Generic Advanced Fighter Aircraft

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Aeroelastic State-Space Model

The generalized structural matrices and the aerodynamic approximation coefficient matrices are used to construct the time-domain state-space equation of motion of the open-loop aeroelastic system excited by control-surface motion,

where

where is the vector of generalized displacements, is the vector of aerodynamic lag states, is the vector of control-surface deflection commands. The system output vector may include structural displacements, velocities and accelerations.

Augmentation of control actuators of at least third order yields the plant equations

where includes aeroelastic and actuator states and is the vector of actuator inputs.

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Control System Model

The control system is modeled as an interconnection of four types of basic control elements:

▪ Single-Input-Single-Output (SISO) elements defined by s-domain transfer functions.
▪ Multi-Input-Multi-Output (MIMO) elements defined by individual state-space matrices
   that can be imported from external control synthesis codes.
▪ Junction elements (JNC) that are actually zero-order elements connecting some inputs with some outputs by
  .
▪ Variable control gains which form the control gain matrix when the system is closed.  Control margins, singular
   values and sensitivity analyses are performed with respect to these gains.

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The ASE Model

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GLOADS: Transient Discrete and Continuous Gust Loads

FUNCTIONALITY
Performs a transient discrete gust loads or continuous gust response analysis of an aircraft structure subjected to air gust.

FEATURES
▪ 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.

▪ For the discrete gust analysis, it includes three options to model the gust profile; the frequency-domain approach,
   the state-space approach, and the hybrid approach where the discrete gust loads are obtained by inverse Fouier
   transform and the system matrix by state-space formulation:

▪ The GLOADS modules state space equations provide accurate displacement time history; thereby circumventing
    the unreasonably large displacement response problem of the Fourier transform method in NASTRAN ▪ 
▪ 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.

2-D Thin Airfoil Subjected to a Sharp-Edged Gust

Validation of the Discrete Gust Module with 2-D Classical Theory
(Excellent agreement is seen while NASTRAN fails to provide satisfactory results)

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MLOADS: Transient Maneuver Loads

FUNCTIONALITY
Performs a transient maneuver loads analysis due to pilot input commands.

FEATURES
▪ Formulated in state space form for either an open-loop or closed-loop system. The rigid body degrees of freedom
   are transformed into the airframe states so that sub-matrices associated with the airframe states within the
   state space matrices are defined the same as the flight dynamic states.
▪ Allows you to replace the program-computed sub-matrices associated with the airframe states by those
   supplied by flight dynamic engineers. This ensures that the time response of the airframe states is
   in close agreement with those of the flight dynamic analysis.
▪ Computes the time histories of the maneuver loads of a flexible airframe in the presence of a control system.
   These maneuver loads include the time histories of component loads, grid point loads, etc. Based on
   these time histories of the loads, you can identify the critical maneuver load conditions.
▪ 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.

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ELOADS: Transient Ejection Loads

FUNCTIONALITY
Performs a transient ejection loads analysis due to store ejection.

FEATURES
▪ Allows for multiple store ejections (in sequential scheduling) while the aircraft is maneuvering under pilot input
   commands.
▪ Accounts for the effects of the sudden reduction in aircraft weight due to the separation of stores from the aircraft.
▪ Formulated in state-space form for either an open-loop or closed-loop system.
▪ 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.

     Aircraft Response due to Ejection Force

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NLFLTR: Nonlinear Flutter Module

FUNCTIONALITY
Performs a simulation of the transient response of open and closed-loop aeroelastic systems that include:
(1) nonlinear structures, (2) a nonlinear control system, and (3) large-amplitude unsteady aerodynamics
(externally imported from other CFD software).

FEATURES
▪ 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.
▪ Discrete time-domain state space equations at each distinct value of the nonlinear parameters are pre-computed.
   During the time-integration computation, updated state-space equations are obtained through interpolation.
▪ Outputs NASTRAN FORCE and MOMENT bulk data cards at a given time step for subsequent detailed stress analysis.

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ZAERO Validation: F-16 MA41 Limit Cycle Oscillation (LCO)

▪ The F-16 configuration MA41 experiences LCO from Mach 0.6 to 1.0 while configuration MA43 is free from LCO.
▪  ZAERO (the ZONA6 subsonic linear method) predicts a hump mode damping curve that falls within 1-2%
    damping (M=0.6 to 1.0) for configuration MA41 while configuration MA43 remains below 1%.
▪ The onset of LCO as predicted by ZAERO (i.e., with the hump mode falling between 1-2% damping) correlates
    very well with flight test data.

                   ZAERO Flutter Solutions of F-16 Configuration MA41 at M = 0.6, 0.8, 0.9 and 1.2

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ZAERO Validation: F-18 Limit Cycle Oscillation (LCO)

▪ ZAERO's transonic unsteady aerodynamic method (ZTAIC) is used to perform the flutter analysis.
▪ The steady pressure input to ZTAIC is obtained from the CFL3D Navier-Stokes solver.
▪ Unstable damping of Case 1 below approximately ranges from M = 0.85 to 1.0 with a frequency of 5.6 Hz which
    correlates very well with the flight test data.
▪ Similarly, good correlation is found for Case 2 below (LCO region at M>0.9 with a frequency 8.8 Hz) .

ZAERO Flutter Solutions of F/A-18 LCO Case 1 (at Altitude = 0 kft)

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