Frequency-specific phase synchronization analysis of a stall-induced aeroelastic system undergoing 2:1 internal resonance in a low-speed wind tunnel

Abstract

Wind tunnel experiments are performed on a pitch-plunge aeroelastic system possessing structural nonlinearity in the stiffness and subjected to dynamic stall conditions. With an increase in the flow speed (U), we observe a well-expected transition from decaying responses to limit cycle oscillations wherein the flutter frequencies of pitch and plunge coalesce perfectly close to the pitch natural frequency, indicating the onset of stall flutter. Interestingly, a further increase in the U yields a secondary frequency peak, emerging approximately at half of the primary frequency peak, accompanied by an intermittent period-1–period-2 (P1–P2) behaviour. Another transition is observed as the flow speed is increased further—the pitch-plunge motion starts exhibiting a beat-like response. We demonstrate that this particular behaviour is ‘2:1 internal resonance (IR)’ and is attributed to a specific combination of structural parameters: the ratio (${\bar{\omega }}$) of natural frequencies of plunge and pitch ($f_y$ and $f_\alpha$, respectively) being 0.44, and the structure possessing quadratic nonlinearity, resulting as the aerodynamic loads tune the flutter frequencies to become commensurate for a 2:1 IR. We show that a typical phase synchronization analysis is inadequate to explain the atypical transitions. To that end, we resort to a frequency-specific synchronization analysis wherein the pitch and plunge time signals are decomposed into two signals of specific frequency range. This reveals that the higher frequency signal (HFS) is always perfectly synchronized. On contrary, the lower frequency signal (LFS) is asynchronous during the intermittent regime and has strong synchrony at the onset of beats. The physical insights presented in this study can be useful for the design of increasingly slender aeroelastic structures that are often subjected to dynamic stall.

A effect of \({\bar{\omega }}\) on response dynamics

The experiments are performed to observe the change in stall flutter characteristics with frequency ratio. Four combinations of ${\bar{\omega }}$ (0.38, 0.44, 0.55, and 0.60) are considered by varying the plunge stiffness, keeping pitch stiffness the same. The natural frequencies of pitch and plunge modes are obtained from the free vibration test and are shown in Fig. 20a–d.

Fig. 20

Determination of natural frequencies of plunge and pitch modes for different frequency ratios

Subsequently, the flutter experiments are carried out for each case by systematically increasing the flow speed. The frequency response for the same is plotted in the range 6–17 m/s and is shown in Fig. 21. For each case, we see that the oscillations are pitch dominant as the flutter frequency is close to $f_{\alpha }$ for both pitch and plunge (see Fig. 21a–d). Upon increasing the flow speed, the dominant peaks remain almost stationary (albeit a slight shift not exceeding 0.2 Hz can be observed for higher speeds) for all but one case where ${\bar{\omega }} = 0.44$ (Fig. 21b). For this case, we observe a secondary peak emerging at higher speeds, approximately at half the frequency of the primary peak (and close to $f_{y}$). This secondary peak gradually gains strength and becomes dominant. For all other cases we observe period-1 stall flutter LCOs although, for the case ${\bar{\omega }} = 0.38$, there is a very small secondary peak (Fig. 21a) which may give raise to similar phenomena at much higher air-speeds which was not possible to achieve with the present experimental conditions. Interestingly, for both the cases (${\bar{\omega }} =$ 0.38 and 0.44), the frequency ratio is below and close to 0.5, making a possibility of IR.

Fig. 21

Variation of modal frequencies with flow speed for different frequency ratios