The PK-Method of Flutter Solution
1.7 Dynamic Aeroelastic Analysis
Dynamic aeroelasticity differs from the flutter analysis of the previous section in that the right-hand side ofEq. 1-118 is no longer zero. Instead, loading, which can be in either the frequency or the time domain, is applied. For both types of loading, NX Nastran performs the primary analyses in the frequency domain. If the user has supplied loadings in the time domain, Fourier Transform techniques are used to convert the loadings into the frequency domain, a frequency response analysis is performed and the computed quantities are transformed back to the time domain using Inverse Fourier Transform techniques. This section first describes the frequency response analysis that is the basis of all NX Nastran dynamic aeroelastic analysis and then discusses the special topics of transient response analysis and random response analysis.
Aeroelastic Frequency Response Analysis
Aeroelastic frequency response analysis in NX Nastran is performed in modal coordinates and has a basic equation of the form
Equation 1-139.
where all terms on the left-hand side are identical to those ofEq. 1-118and are defined with that equation. The right-hand side provides the loading in modal coordinates, which can be aerodynamic or nonaerodynamic in nature and is a function of the analysis frequency. Nonaerodynamic generalized loads, designated PHF(w), are obtained in the standard fashion from the loadings applied to physical coordinates and do not require further comment here. The aeroelastic (gust) portion of the loading does require further comment that is similar in nature to the discussion of the generalized aerodynamic matrices of the previous subsection.
A prerequisite to performing aerodynamic gust analysis is the availability of an aerodynamic matrix that provides the forces on the aerodynamic elements due to an applied downwash at any other element:
which can be transformed to modal coordinates using:
Since extra points cannot affect the gust loading, there are no generalized loadings associated with them so that matrix Qhj(which provides the generalized loadings in the modal set) is obtained by adding a null matrix onto the bottom of Qij.
The Qhjmatrix supplies the generalized aerodynamic forces due to the downwash vector at the collocation points. For the matrix to be useful in the gust analysis, two other steps are required. First, the Qhjand Qhhmatrices must be interpolated to all the frequencies required in the analysis from the discrete reduced frequencies at which the aerodynamics have been calculated. This is done using the Specialized Linear Interpolation technique of the previous subsection applied to the two matrices. The other step is the generation of the gust downwash matrix. This is a function of frequency and the geometry of the aerodynamic model:
Equation 1-140.
where:
wi = excitation frequency
gj = dihedral angle of the j-th aerodynamic element
xj = x-location of the j-th aerodynamic element in the aerodynamic coordinate system
xo = user-supplied offset distance for the gust
It is seen that this represents a one-dimensional gust field, i.e., the gust varies only in its x-coordinate.
Equation 1-141.
where:
= dynamic pressure
wg = gust scale factor
PP = user-supplied frequency variation of the gust. (This may also be obtained from a Fourier transform of the user-supplied discrete gust.)
and the total frequency dependent loading applied inEq. 1-139is
Equation 1-142.
The solution of Eq. 1-139 entails solving for the generalized displacements by
decomposition/forward-backward substitution techniques applied to the coupled set of complex equations. Because modal reduction techniques have been applied, the solution costs are typically modest. Once the generalized displacements have been computed, standard data recovery techniques can be used to determine physical displacements, velocities, stress, etc.
Aeroelastic Transient Response Analysis
As discussed in the introduction to this section, Aeroelastic Transient Analysis relies on Fourier transform techniques. Transient analysis by a Fourier transformation is separated into three phases. First, the loads (defined as a function of time) are transformed into the frequency domain. Second, the responses are computed in the frequency domain using the algorithm of the preceding subsection. Third, these responses (in the frequency domain) are transformed back to the time domain.
Two forms of the transform are considered, the Fourier series and the Fourier integral, which are defined using the following terminology.
The Fourier Series
The basic time interval is 0 < t < T, with the function periodic. The circular frequencies are given by
Equation 1-143.
Equation 1-144.
Equation 1-145.
The response at point j is given by
Equation 1-146.
The response in the time domain is
Equation 1-147.
The Fourier Integral
This is the limit as T ® ∞, Δf ® 0, with 2πnΔf® ∞, of the Fourier series. Here, w is a continuous variable.
Equation 1-148.
Equation 1-149.
Equation 1-150.
Both methods must be implemented numerically, which requires approximations that the user should appreciate. First, the inverse transform includes an infinite sum, for which only a finite number of terms are evaluated numerically; this approximation leads to truncation errors. Next, the inverse Fourier integral must be integrated numerically, which leads to integration errors. Finally, the number of frequencies at which the integrand is evaluated is limited by the cost of calculations. Each of these sources of error should be addressed separately by the user to ensure adequate accuracy in the final results.
Transformation of Loads
You specify loads in the same manner as given in the Transient Excitation Definition in the NX Nastran Basic Dynamic Analysis User’s Guide . The two general forms are the tabular, piecewise linear function [Eq. 1-151 below] and the general purpose function [Eq. 1-156below]. The transformation is given byEq. 1-148, in which it is assumed that the user defines a function which vanishes for t > T.
For piecewise linear tabular functions, a table of pairs xi, yi, i = 1, Nis prescribed, which defines
N− 1 time intervals. In addition, an X1shift and an X2scale factor are allowed. Thus, the time-dependent load at point j is given by
Equation 1-151.
where YTis the tabular function supplied by the user-defined by the N pairs (xi, yi), and Ajand tj are an amplitude factor and a delay, respectively, which may depend upon the point which is loaded. The transformed load is
Equation 1-152.
with
Equation 1-153.
Equation 1-154.
Equation 1-155.
The general purpose function is defined by
Equation 1-156.
where:
Equation 1-157.
The value of n is restricted to be an integer for transient analysis by the Fourier method. The transform is given by Equation 1-158. where: Equation 1-159. Equation 1-160. Equation 1-161.
Equation 1-162.
The second form is used for |z| < 0.1.
These loads, which are in the form required for frequency response, are transformed to the modal coordinates exactly as in the modal frequency response method.
Inverse Transformation of the Response
The response is found from a numerical approximation toEq. 1-146or fromEq. 1-143, the Fourier series result, which can be regarded as a special form of approximation to the integral. The quantity ˜u(w) is first calculated at a set of frequencies, wi, by the frequency response analysis. The wido not need to be equally spaced and the integral is evaluated only over the frequency range for which the frequency response has been performed. The frequency response is taken to be constant over the frequency interval resulting in a discrete inverse transform of the form:
Equation 1-163.
where Re denotes the real part and
C1 =
Cn =
CNFREQ = (wNFREQ− wNFREQ − 1)/2
A special case occurs if all the frequency intervals are equal to Δf and the first frequency is an integer multiple of Δf. In this case, the time step Δt is adjusted to make 1/ΔfΔt = an integer, thereby reducing the number of distinct values of sinwntmand coswntmused inEq. 1-163, and cn does not require recalculation at each frequency.
Practical Considerations for Use with the Fourier Methods
Some important practical considerations must be observed to use these methods successfully. To illustrate one problem, consider the response of a simple damped oscillator to a pulse (Figure 1-5). The upper three curves show the pulse and the response of the system, first, if it is very stable, and then if it is only slightly stable. Using the Fourier method, the pulse is replaced by a
series of pulses with period 1/(Δf). As can be seen, this gives an accurate representation if the system is very stable, but an incorrect impression if the system is only slightly stable. Guidelines that lead to valid results using the Fourier method include:
1. The system should be stable.
2. The forcing functions should be zero for some time interval to allow decay. 3. The frequency interval Δf ≤ 1/(Tpulse+ Tdecay)
Figure 1-5. Response of a Damped Oscillator to a Triangular Pulse
If the system has unstable modes, these will appear as a precursor before the pulse, just like a “stable mode in reverse time.”
The use of both equal frequency intervals and unequal intervals has been studied briefly and results are shown by Rodden, Harder, and Bellinger (1979) for a lightly damped, single degree-of-freedom oscillator. It has been found there that the combination of a few well-chosen values near the resonant frequencies and a uniformly spaced set of frequencies elsewhere produces reasonable results for the lightly damped example considered. However, further convergence studies on more general examples are needed.