Aeromodal Element

Note: prepared by Alessandro Scotti.

This element is used to model an aerodynamic modal element, i.e. an unsteady aerodynamic model that inherits the structural motion from a modal element Its definition is very similar to that of the pure modal element, but it also includes some data representing unsteady aerodynamics in the time domain trough the residualization matrices. This element is defined as follows:

    <element_type> ::= aeromodal
    <element_arglist> ::= <label> , 
        <modal node> ,  
        <reference modal joint> ,
        (Mat3x3)<orientation> ,
        <reference chord> ,
        <number of aerodynamic states> ,
        [ rigid , ]
        <state space modal matrices file>

With this formulation, anytime an aeromodal element is defined, the user needs to declare the number of modal aerodynamic elements in use in the control data section. An air properties card definition is also required.

The keyword rigid indicates that the generalized aerodynamic forces provided by the model include global forces and moments associated to the rigid body motion of the underlying modal element (FIXME: untested).

There is also an optional gust model which is totally undocumented; for further information, please contact the Author(s).

The .fea file includes the state space model in form of matrices A, B, C, D₀, D₁ and D₂, according to the representation

$$\dot{{\boldsymbol{x}}}$$ = Ax + Bq

$$\left(\vphantom{ \boldsymbol{C}\boldsymbol{x} + \boldsymbol{D}_0 ... ...frac{2V_{\infty}}{c} \right) ^2 \boldsymbol{D}_2 \ddot{\boldsymbol{q}} }\right. {\frac{{2V_{\infty}}}{{c}}} \dot{{\boldsymbol{q}}} \left(\vphantom{ \frac{2V_{\infty}}{c} }\right. {\frac{{2V_{\infty}}}{{c}}} \left.\vphantom{ \frac{2V_{\infty}}{c} }\right)^{2}_{} \ddot{{\boldsymbol{q}}} \left.\vphantom{ \boldsymbol{C}\boldsymbol{x} + \boldsymbol{D}_0 ... ...frac{2V_{\infty}}{c} \right) ^2 \boldsymbol{D}_2 \ddot{\boldsymbol{q}} }\right)$$ f

where q are the modal variables that describe the structural motion, and f are the unsteady aerodynamic forces that apply to the structural dynamics equations.

The file is formatted as follows:

    *** MATRIX A
    (<na> x <na> coefficients)
    *** MATRIX B
    (<na> x <ns> coefficients)
    *** MATRIX C
    (<ns> x <na> coefficients)
    *** MATRIX D0
    (<ns> x <ns> coefficients)
    *** MATRIX D1
    (<ns> x <ns> coefficients)
    *** MATRIX D2
    (<ns> x <ns> coefficients)

Example.

    aeromodal: Wing, Wing, Wing,
        eye,
        131.25, 10, "ha145b.fea";

The aeromodal element is declared with the label Wing. This element is attached to a modal node, also labeled with the name Wing, and the reference modal joint it is referred to is named Wing as well. The orientation of the aerodynamic reference with respect to the nodal reference is here expressed by the matrix eye. The aerodynamic element chord is 131.25 (inches!). This quantity must be consistent with the system chosen to define the whole model (S.I., for example; in this case, British Units). The next field, 10, indicates the number of states needed to use the aerodynamic model. The string ha145b.fea is the name of the file that contains the state space model matrices, obtained with an approximation chosen by the user. In this particular case, a 10 states Padé approximation has been chosen. This example is taken from the Bisplinghoff Ashley Halfman (BAH) Jet Transport Wing cantilevered wing with modal aerodynamic frequency responce, computed by a double-lattice method at Mach 0.0. Data were extracted from the MSC-NASTRAN aeroelastic example file, named ha145b, while the aerodynamic state-space fitting has been computed using a Padé polynomial approximation (by Pasinetti & Mantegazza). All quantities are expressed in inches and pounds.