OS-V: 1300 Flutter Analysis of an AGARD 445.6 Wing

Flutter analysis of an AGARD 445.6 wing model is performed using the PK method.

The results are validated against experimental data from a NASA technical memorandum¹.

Model Files

Before you begin, copy the file(s) used in this problem to your working directory.

Benchmark Model

Dimension details of the model:

Dimension: Value (m)
Span: 0.762
Root Chord Length: 0.5587
Tip Chord Length: 0.3682

The structural domain consists of 200 CQUAD4 elements with linear orthotropic material properties.

E1: 3.151E+09
E2: 4.162E+08
NU12: 0.31
G12: 4.392E+08
RHO: 381.980

Flutter analysis is performed for a set of Mach numbers (M) = {0.338, 0.499, 0.678, 0.901, 0.96, 1.072, 1.141} for a velocity range of [100, 500] m/s. As can be seen from a figure in ^¹, the variation of mass ratios across Mach Number suggests difference in flow conditions across the experiments. Hence, the density ratios are varied for each Mach number in separate simulations. More specifically, the flow density follows in the referenced FOI report ^², which is also listed as:


Mach	ρ_f(kg/m³)
0.338	1.09854
0.499	0.42770
0.678	0.20818
0.901	0.09945
0.96	0.06338
1.072	0.05514
1.141	0.07833

Comparison of Normal Modes

Mode shape and mode frequency comparisons are as follows. The results from OptiStruct are in agreement with the reference results^¹.


Reference results¹

Mode 1	F = 9.5992 Hx	Mode 2	F = 38.1650 Hz	Mode 3	F = 48.3482 Hz	Mode 4	F = 91.5448 Hz


Results from OptiStruct

Mode 1	F = 9.4589 Hz	Mode 2	F = 39.5289 Hz	Mode 3	F = 49.2213 Hz	Mode 4	F = 94.8019 Hz

Flutter Analysis

From the .flt file of the first Mach number (M = 0.338) simulation, the flutter point (where damping changes its sign) corresponding to the lowest mode is identified as the 2^nd mode with a velocity between 120 m/s to 124 m/s.

Note: By definition, instability (flutter or divergence) occurs when the damping values are zero. At this point, if the frequency is zero, then the instability is due to divergence. Otherwise, the instability is due to flutter.

Figure 2. Flutter Analysis Summary from .flt File

Plotting the v-g curve, the velocity at this flutter point is 123.19 m/s. This is the most critical flutter point that needs to be avoided for M = 0.338.

Figure 3. Identify Flutter Points. The flutter point corresponding to the lowest velocity is also visually identified.

Figure 4. Identify Frequency Value at Critical Flutter Point from v-f Curve

Plotting the v-f plot for the 2^nd mode (corresponding to the critical flutter point), the frequency value for 2^nd mode at a velocity of 123.19 m/s is determined as 24.19 Hz. In the same way, the flutter speed and flutter frequency determination was repeated for M = {0.499, 0.678, 0.901, 0.96, 1.072, 1.141}.

Comparison of Flutter Speed Coefficient

The flutter speed coefficient (or flutter speed index, FSI) $(\frac{v}{b_{s} ω_{α} \sqrt{\bar{μ}}})$ is calculated from OptiStruct and plotted against M and compared against the reference plot from Figure 16(a) on page 66¹.

Where,

$v$: Flutter velocity
$b_{s}$: Streamwise semi chord length at wing root = $\frac{0.5587}{2}$ m
$ω_{α}$: Natural circular frequency of the first uncoupled torsional mode $= 2 π f = 2 π (39.5289)$ rad/s (This is the 2^nd normal mode for this wing)
$\bar{μ}$: Mass ratio; This value was determined for each Mach number from Figure 15 on page 65¹

\frac{v}{b_{s} ω_{α} \sqrt{\bar{μ}}}

Figure 5. Flutter Speed Coefficient versus M Comparison Between Experimental Reference and OptiStruct Flutter Analysis

Comparison of Flutter Frequency Ratio

The flutter frequency ratio $(\frac{ω}{ω_{α}})$ is calculated from OptiStruct and plotted against M. This is compared against the reference plot from Figure 16(b) on page 67¹.

\frac{ω}{ω_{α}}

Figure 6. Flutter Frequency Ratio versus M Comparison Between Experimental Reference and OptiStruct Flutter Analysis

Observations

The flutter speed coefficient and flutter frequency ratio from OptiStruct are comparable to experimental reference data.
The discrepancy between OptiStruct and experimental data is expected and consistent with what was reported in Figure 5 of ^³ regarding DLM method.
In realistic conditions, for M ~ 0.75 and above, local pockets of supersonic flow could occur around the structure. This intermediate regime is denoted as transonic.
In the flutter speed coefficient versus M plot (Figure 5), the experimental reference data shows a reduction in flutter speed coefficient around M = 1.0 and this is called the transonic dip.
OptiStruct flutter analysis is capable of capturing the descent of this dip.

Reference

¹ E. Carson Yates Jr, “AGARD Standard Aeroelastic Configurations for Dynamic Response. Candidate Configuration I.-WING 445.6,” NASA Technical Memorandum I00492. 1987

² Pahlavanloo, P., “Dynamic aeroelastic simulation of the AGARD 445.6 wing using Edge,” FOI Technical Report FOI-R-2259-SE. 2007

³ Beaubien, R. J., Nitzsche, F., and Feszty, D., “Time and frequency domain flutter solutions for the AGARD 445.6 wing,” Paper IF-102, IFASD. 2005