In these cases, the rigid body shapes are defined geometrically by the locations of the grid points and orientation of their displacement coordinate systems. The structure may or may not be a free body. Also, mechanisms that produce free body differences in displacements or rotations are not included.
The parameter GRDPNT is used normally to calculate the total weight, inertia, and CG
properties of a structure. It is also used to generate geometric rigid body shapes. The matrix [Dl] as defined inEq. 5-50may be calculated directly from the geometry. The VECPLOT Module generates a 6 × 6 matrix, [Dl], for every grid point, g, from basic kinematics. The definition is
Equation 5-51.
where {ugi} is the vector of three displacements and three rotations in the local coordinate system. {uo} represents three displacements and three rotations at GRDPNT (or the origin) in basic coordinates. These 6 × 6 partitions are merged into a full-size matrix, [Dgo] with six rows for each grid point and six columns total. Note that the rows of [Dgo] are null for the scalar points and is a unit matrix for the six rows corresponding to the GRDPNT parameter.
If SUPORT data is present, the matrix is transformed to define motion relative to the supported degrees-of-freedom, {ur}. The internal calculations are given below
Partition:
Equation 5-52.
Invert the support partition, and multiply:
Advanced Dynamic Analysis User’s Guide 5-49
Equation 5-53.
and merge:
Equation 5-54.
These matrices may be calculated at the superelement level as opposed to assembled residual SE level required by the SUPORT partition method.
The applications include:
1. Static inertia relief solutions for SOL 101. The requirement to combine and reduce mass matrices is eliminated with this method, saving nearly 50% of the execution cost. For inertia relief effects, these solutions also require the input PARAM,INREL,–1 and a set of six SUPORT degrees-of-freedom.
2. Component Mode Synthesis for superelements provides an option for inertia relief vectors.
The advantage is improved accuracy in the final results with an equal or lesser number of component modes. These are deformation shapes that correspond to the six uniform acceleration loads on the constrained structure. The deformation shape vectors are appended to the mode shapes and are included as generalized degrees-of-freedom. The INRLM
parameter controls this option.
Note that in addition to the limitation of six free body motions, the geometric method will not be correct for scalar points and fluid analysis methods.
5.8 Aeroelastic Solutions
The NXNastranAeroelasticAnalysisUser’sGuide describes the theoretical aspects and the numerical techniques used to perform aeroelastic analyses. As described in“Overview of Aeroelastic Analysis”, the system is used for flutter, frequency response, gust response, and static analysis of aerodynamically loaded structures. An outline of the capability is given here.
The aeroelastic analyses use the following features:
Structural Model
Any of the existing NX Nastran structural finite elements (except axisymmetric and p-elements) can be used to build the structural model. The structural stiffness, mass, and damping matrices required by the aeroelastic analyses are generated by NX Nastran from geometric, structural, inertial, and damping data that you input for subsequent use in the various aeroelastic analyses.
Fluid/Structure Connections
Matrices of aerodynamic influence coefficients are computed only from the data describing the geometry of the aerodynamic finite elements. The choice of aerodynamic grid points for the aerodynamic model is independent of the location of the structural grid points. An automated interpolation procedure is provided to relate the aerodynamic to the structural
degrees-of-freedom. Splining techniques for both lines and surfaces are used to generate the transformation matrix from structural grid point deflections to aerodynamic grid point deflections where local streamwise slopes are also computed. The transpose of this matrix transfers the aerodynamic forces and moments at aerodynamic boxes to structural grid points.
Aerodynamic Theories
One subsonic and three supersonic lifting surface aerodynamic theories are available in NX Nastran, as well as Strip Theory. The subsonic theory is the Doublet-Lattice method, which can account for interference among multiple lifting surfaces and bodies. The supersonic theories are the Mach Box method, Piston Theory, and the ZONA51 method for multiple interfering lifting surfaces.
Static Aeroelastic Analysis
The structural load distribution on an elastic vehicle in trimmed flight is determined by solving the equations for static equilibrium. The SOL 144 and SOL 200 processes will calculate aerodynamic stability derivatives (e.g., lift and moment curve slopes and lift and moment coefficients due to control surface rotation) and trim variables (e.g., angle of attack and control surface setting) as well as aerodynamic and structural loads, structural deflections, and element stresses.
Modal Formulation
Dynamic aero solutions provide for modal reduction of the system matrices. The number of degrees-of-freedom required for accurate solutions to dynamic aeroelastic problems is generally far less than the number of physical degrees-of-freedom used in the finite element structural model. The number of independent degrees-of-freedom can be greatly reduced by using the (complex) amplitudes of a series of vibration modes as generalized coordinates, e.g., by Galerkin’s method. NX Nastran can compute the vibration modes and frequencies and make the transformation to modal coordinates. The matrices of aerodynamic influence coefficients are also transformed to generalized aerodynamic forces by use of the vibration eigenvectors.
Flutter Analysis
The dynamic aeroelastic stability problem, flutter, is solved in SOL 145, by any of three methods.
The traditional American flutter method developed by the Air Materiel Command (AMC) in 1942 is available in the first two methods. The first method is called the K-method and is a variation of the AMC method. The second method, called the KE-method, is more efficient from the point of view of tracking roots, but is limited in input (no viscous damping) and output (no eigenvectors).
The third method, called the PK-method, is similar to the British flutter method, which was developed by the Royal Aircraft Establishment.
Frequency Response
The coupling with aerodynamic loads has also been added to the existing NX Nastran structural modal frequency response capability, SOL 146. Analyses of frequency response to arbitrarily specified forcing functions can be carried out using the oscillatory aerodynamic loads from any of the available aerodynamic theories. Frequency response to a harmonic gust field can be calculated at subsonic speeds using the Doublet-Lattice method for wing/body interference, and by the ZONA51 method for interfering lifting surfaces at supersonic speeds.
Transient Response
Because unsteady aerodynamic loads are obtained only for steady-state harmonic motion, they are known only in the frequency- and not the time-domain. In SOL 146, Inverse Fourier Transform techniques provide the appropriate methods by which transient response is obtained from the frequency response. Both forward and inverse Fourier transforms are provided so that the time-varying forcing function or the gust profile can be transformed into the frequency
Advanced Dynamic Analysis User’s Guide 5-51
domain. Then, after convolution with the system frequency response, the inverse transform leads to the transient response of the system to the specified forcing function or gust profile.
Random Response
Stationary random response of the system, is available in SOL 146 from specified loadings and the power spectral densities of loads. Loads may be either specified force distributions or harmonic gust fields. The statistical quantities of interest in the response are ¯A, the ratio of standard deviations (rms values) of the response to that of the input loading, and No, the mean frequency of zero crossings (with a positive slope) of the response. The capability to compute these quantities was added to NX Nastran by modifying the existing random response module to include options to generate various atmospheric turbulence power spectra and to perform the calculation of No.
Design Sensitivities
The sensitivities of response parameters to changes in design variables are calculated by the perturbation techniques developed for structural optimization in NX Nastran and extended to include static aeroelasticity and flutter in SOL 200. The basic aeroelastic sensitivities that can be obtained include stability derivatives, trim variables, and flutter system dampings.
The synthetic response technique of NX Nastran optimization also permits the calculation of sensitivities of user-specified functions of those standard response quantities.
Aeroelastic Optimization
Optimization of aeroelastic characteristics can be combined with the other optimization features of NX Nastran in SOL 200, and vehicles can now be designed optimally for aeroelastic loads, flying qualities, and flutter, as well as for strength, vibration frequencies, and buckling characteristics.