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The new airfoil model NACA0015, modal analysis and flutter properties
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Applied Mathematical Modelling 0 0 0 (2017) 1–9
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The new airfoil model NACA0015, modal analysis and flutter propertiesR Jan Kozanek a,∗, Vaclav Vlcek b, Igor Zolotarev a a b
Institute of Thermomechanics AS CR, Department of Dynamics and Vibrations, Dolejskova 5, 18200 Prague 8, Czechia Institute of Thermomechanics AS CR, Department of Fluid Dynamics, Dolejskova 5, 18200 Prague 8, Czechia
a r t i c l e
i n f o
Article history: Received 31 March 2016 Revised 26 January 2017 Accepted 16 February 2017 Available online xxx Keywords: Aeroelasticity Subsonic flow Self-excited vibration Modal analysis Kinematics
a b s t r a c t The experimental airfoil model NACA0015 was used to study aeroelastic phenomena during self-excited profile vibration. It provides data for control of aeroelastic calculation programs at subsonic speeds of the stream. The model movability is two-dimensional with two-degree of freedom dynamic system, one in pitch and the second in plunge, and is proposed to be a dynamic system having two near corresponding eigenfrequencies. To quantitatively evaluate flow field using interferometry, a special test section design and profile was constructed. It utilized a large visual field for the optical system together with the option of changing support stiffness for both degrees of freedom. In this paper experimental results from the range of Reynolds numbers Re = (2.63–2.83) 105 are published. The identified eigenvalues and eigenmodes for zero flow velocity are compared with measured flutter properties (frequency, modes and time evolutions) of the airfoil. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Vibration of the profile in fluid flow is an important phenomenon in many technical applications of aerodynamics, airplanes, wind turbines, helicopters, compressors and turbines. This research belongs to the broader field of investigation of complex questions regarding the stability of aerodynamic systems. Vibrations arise from interactions between the fluid flow and vibrating solid body, and differ significantly depending on the extent of the wing surface where separation occurs. If separation is only in the vicinity of the trailing edge, self-excited vibration of the profile is called flutter. If the region of the separated flow is great, the regime is called stall flutter. One of the most often studied flows, vibrating body interactions, occurs in the two-dimensional plunge and pitch airfoil model (Fig. 1). Another frequently used simplified approach to solving the problem of flutter, is to use external forced oscillation acting on the profile. This eliminates complications which may occur when tuning model parameters influencing self-excitation vibration for the required range of Reynolds numbers or Mach numbers. It also reduces randomness corresponding to variability of the flow field. However, during fluttering and stall-fluttering, a number of phenomena are significantly complicated by the oscillation mechanism. In stall-flutter regime the flow field around the profile has a very complex structure. The coupling between the self excited profile and the flow field is also very complex, and the resulting frequency may significantly differ from the initial eigenfrequencies (corresponding to zero air flow velocity) of the two degrees of freedom dynamic system. For R ∗
13th International Conference on Dynamical Systems: Theory and Applications. Corresponding author. E-mail addresses: [email protected] (J. Kozanek), [email protected] (V. Vlcek), [email protected] (I. Zolotarev).
http://dx.doi.org/10.1016/j.apm.2017.02.039 0307-904X/© 2017 Elsevier Inc. All rights reserved.
Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Fig. 1. Two-degree of freedom dynamic system — the airfoil model.
Fig. 2. The support scheme a) and measured air profile b).
example, in our earlier experiment the increase of the resulting frequency was almost the sum of both eigenfrequencies. On the other hand, in the majority of our experiments the resulting oscillation tends to have a frequency between the eigenfrequencies of the initial two degrees of freedom dynamic system. We note that the phase shift between the pitch and plunge profile vibration is significantly influenced by the stiffness of their support. A further complicating phenomenon is the spontaneous jump from a less stable oscillation to a more stable one, which is accompanied by a change of only one of the two eigenfrequencies. The change is realized during one period, the oscillation amplitude strongly increased and phase shifted by omission of a half wavelength. The frequency of the flutter does not change. Mentioned parameters of the dynamic system also affect the range of occurrence of self-excitation in the area of Reynolds and Mach numbers. A more detailed overview of individual variants of the aeroelastic phenomena associated with the described effects and the approximate status of their knowledge is listed in [1]. In both non-steady and steady cases the chord of the profile was used for evaluation of the Reynolds number as the geometry characteristic. The aeroelasticity and stability problems of profiles have been studied for a long time – [2,3]. Nevertheless the selfexcited vibrations in our experimental configuration were not yet investigated and published. Here we publish in detail the experiments, where self-excited vibrations were successfully achieved without the use of an external forced oscillation. 2. The new airfoil model NACA0015 and the experimental equipment The new airfoil model NACA0015 was constructed to generate self-excited vibrations in subsonic air flow. For these aerodynamic experiments the suction type aerodynamic tunnel of the Institute of Thermomechanics AS (with Mach numbers air flow M = 0.2−0.4 - see the DSTA 2015 conference contribution [4]) was used. Our fluttering profile is supported as a two-degree of freedom dynamic system, one for profile rotation (pitch) and the second for translation (shift) motion with mutually near and adjustable eigenfrequencies. Brief description of the layout of the new experimental facility, the results of the modal analysis and the first measurements are described in [5]. The measured air profile located in the wind tunnel is shown in Fig. 2. Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Fig. 3. The new airfoil model NACA0015 with two modifications of rotational springs.
Fig. 4. Airfoil rotational and transversal support and the experimental equipment.
Compared with older constructions [6,7], the translation springs were not changed, but the rotational support was realized using a new coil spring. The corresponding pitch eigenfrequency can be tuned by the coil spring parameters, i.e. by the spring wire length and by its diameter. The eigenfrequency of the transversal mode is influenced by using the grub screws [7]. The new airfoil model with a chord length of 59 mm, thickness of 8.85 mm and width of 76.6 mm, was divided into five parts (Fig. 3). Two ball bearings enable the profile to rotate; the rotation axis of profile is located at 1/3 of the chord. The weight of the profile is approximately 38 g. The support of the profile with measuring device are shown in Fig. 4. The magnetic rotary encoder RM08 was used for pitch angle recording. The frame translation was measured by the magnetic non-contact linear encoder LM13TCD. The strain gauge with full bridge placed on the flat spring measured the translation motion. Local pressure on the surface of the profile was indicated with four semiconductor pressure sensors MPXH6115. Transducers were coupled in pairs on both profile surfaces. Signals were registered with DEWETRON desktop PC software. The optical measurements could be measured by using the high-speed MotionPro X3 camera at 10 0 0 frames per second. Velocity of the flow field was determined by a wind tunnel Machmeter. The interferometer visualization is possible in an area with a diameter of 160 mm. The flutter was initialized by a 1.7 mm initial deviation of the frame. 3. Modal analysis of the two-degree of freedom airfoil support The special support of our airfoil is proposed as a dynamic system with two-degree of freedom, one degree of freedom in rotation (pitch of the airfoil about the axis situated at 1/3 of the profile chord) and the second in its translation mode (shift of the whole airfoil axis). For zero flow velocity, the eigenfrequencies of the above modes are relatively close to one another. Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Fig. 5. The excited and measured points 1,2 on our measured profile during modal analysis. Table 1 Initial tensioning corresponding to the different grub screw positions p1, p2, p3. p d [mm]
p1 0.23
p2 0.58
p3 0.93
For the successful implementation of aerodynamic experiments in the flow of air it is very useful to have the opportunity to influence the closeness of these eigenfrequencies. This is because when increasing air flow velocity, usually the lower eigenfrequency increases and the higher eigenfrequency decreases. On the other hand, in order to generate the self-excited airfoil flutter in some predestined Mach number interval, there is an optimal gap between the eigenfrequencies. We wanted to implement subsonic air flow with Mach numbers lower than M = 0.4. In our experimental support configuration we can modify the rotational eigenfrequency using different rotational springs, and the translation frequency by the initial pretensioning of the transversal flat strip springs. In the described experiment we made a second modification of the rotational springs (torsion bar from Fig. 3) and investigated the effect of the pre-tensioning of the transversal flat strip sprigs (Fig. 4). To do this, we made a modal analysis of this simple dynamic system for zero air flow velocity for three grub screw positions which gradually increased the stiffness of the transversal flat strip springs. The results were complex eigenvalues and corresponding eigenmodes. Imaginary parts of eigenvalues (eigenfrequencies corresponding to the transversal mode) increased, yielding quantitative results of how the system changes. The real parts of identified eigenvalues (the damping values) are the additional information about the possibilities of achieving self-excited airfoil vibration in subsonic air flow regime. The identification of complex eigenmodes corresponding to rotational and translation vibrations indicate “more complex” normalized eigenmodes for rotational motion which is perhaps related to the dry friction of this support. The effect of errors of dynamic measurement on these identification results was greater than that of the eigenvalue identification. The modal analysis in the idle state (zero air flow velocity) was carried out in the Dynamics and Vibration laboratory of the Institute of Thermomechanics AS CR – see [5]. The pulse-hammer excitation was realized by the 4519 B&K force sensor and measurements were performed with the acceleration pick-up DeltaTron 4519. The time signals were discretized, registered and evaluated by the PULSE B&K measurement system with card 7537A. There are two (marked 1, 2) excited and measured points in the centre of rotation and on the trailing edge − see Fig. 5. The measured transfer functions vj,k (f), j,k=1,2 as the function of excitation frequency f are complex and can be evaluated numerically by the parametric regression identification method [8]
v j,k ( f ) =
2 av, j,k + h( f ), i f − sv
(1)
v=1
where vj,k (f) is the complex transfer function (excitation in k-point and pick-up placed in j-point), sv is the complex eigenvalue and av,j,k is the corresponding complex modal contribution and h (f) is the influence of the other modes and experimental perturbations. The example of the measured frequency responses v( f ) is in Fig. 6 3.1. Pre-tensioning of the flat springs Two flat strip springs define transversal elastic support of the profile and their rigidities can be tuned by using grub screws. The three adjustments of the grub screws marked as p1, p2, p3 we define three pre-stress support deformations of the flat strip springs (see Table 1). The identified complex eigenvalues and eigenmodes for three dynamic systems corresponding to the different initial pre-stress support deformations are summarized in Tables 2−8. The rotational eigenmode with very small amplitude was neglected. Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Table 2 Identified modal parameters av,j,k and eigenvalues sv for the initial tensioning p1. (pick-up,excitation)
(1,1)
(1,2)
(2,1)
(2,2)
av,j,k
−0.0 0 02−0.0 03i 0.0 0 03−0.0 05i
−0.003−0.017i 0.003 + 0.005i 0.001 + 0.009i
0.0 03−0.0 05i 0.002 + 0.003i
0.019−0.24i 0.030 + 0.015i
sv [Hz]
−1.1 + 14.26i −0.6 + 17.13i
−0.5 + 13.37i −0.2 + 14.78i −0.3 + 17.25i
−0.5 + 13.53i −0.9 + 17.99i
−0.6 + 13.29i −0.2 + 14.45i
Table 3 Normalized eigenvectors vT1 and vT2 corresponding to rotational and transversal eigenmodes for the initial tensioning p1.
vT1 vT2
[0.0197 − 0.0141i, 1] [1, − 1.7816 + 0.3069i]
[a1,2,1 , a1,2,2 ]/a1,2,2 [a2,1,1 , a2,1,2 ]/a2,1,1
Table 4 Identified modal parameters av,j,k and eigenvalues sv for the initial tensioning p2. (pick-up,excitation)
(1,1)
(1,2)
(2,1)
(2,2)
av,j,k
−0.0 0 0 0−0.0 01i 0.0 0 02−0.0 02i
−0.0 0 02−0.0 09i 0.001 + 0.004i −0.0 0 02 + 0.004i
−0.003−0.0045i 0.001 + 0.003i
0.030−0.23i 0.010 + 0.032i
sv [Hz]
−0.9 + 14.40i −0.7 + 17.56i
−0.5 + 13.71i −0.2 + 14.91i −0.3 + 17.66i
−0.6 + 13.64i −0.7 + 18.19i
−0.8 + 13.36i −0.3 + 14.44i
Table 5 Normalized eigenvectors vT1 and vT2 corresponding to rotational and transversal eigenmodes for the initial tensioning p2.
vT1 vT2
[0.0176 − 0.0153i, 1] [1, − 1.9901 + 0.0990i]
[a1,2,1 , a1,2,2 ]/a1,2,2 [a2,1,1 , a2,1,2 ]/a2,1,1
Table 6 Identified modal parameters av,j,k and eigenvalues sv for the initial tensioning p3. (pick-up,excitation)
(1,1)
(1,2)
(2,1)
(2,2)
av,j,k
−0.0 0 03−0.0 01i 0.0 0 04−0.0 02i
−0.0 0 04−0.0 08i 0.0016 + 0.004i −0.0 0 02 + 0.004i
−0.004−0.003i −0.0 0 02 + 0.002i
0.085−0.23i −0.053 + 0.024i
sv [Hz]
−1.0 + 14.56i −0.9 + 17.630i
−0.6 + 13.69i −0.3 + 14.96i −0.6 + 17.76i
−0.6 + 13.83i −0.8 + 18.28i
−0.9 + 13.33i −0.4 + 14.29i
Table 7 Normalized eigenvectors vT1 and vT2 corresponding to rotational and transversal eigenmodes for the initial tensioning p3.
vT1 vT2
[a1,2,1 , a1,2,2 ]/a1,2,2 [a2,1,1 , a2,1,2 ]/a2,1,1
[0.0058 − 0.0195i, 1] [1, − 1.9423 + 0.2885i]
Table 8 Resulting eigenvalues sv for the largest amplitude of vibration, initial pre-stress support deformations p1, p2, p3. sv [Hz]
Rotational eigenmode
Transversal eigenmode
p1 p2 p3
−0.6 + 13.29i −0.8 + 13.36l −0.9 + 13.33i
−0.6 + 17.13i −0.7 + 17.56i −0.9 + 17.63i
Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Fig. 6. The example of the frequency responses ν (f) in two measured configurations (pick-up, excitation) (a)−(1,1), (b)−(2,2) for initial pre-stress support deformation p2.
Fig. 7. The starting phase of the flutter vibration, initial deviation was 1.7 mm.
For the following aerodynamic experiments with nonzero air flow velocity the geometrical deformation of the flat strip springs corresponds to the fixed position p2.
4. Aerodynamic experiments with the airfoil profile NACA0015 Aerodynamic experiments were realized in the Laboratory of the Institute of Thermomechanics AS near Novy Knin for subsonic air flow with Mach numbers M = 0.2–0.4 and Reynolds numbers (2.63–2.83)·105 . The results, involving flutter regime, with M = 0.21 and Re = 2.76 105 , will be presented as an example in this chapter. The measured parameters were shift [mm] of the rotation centre and pitch angle [deg] as a function of time. The frequency spectrum was evaluated for rotational vibration in different time moments. Profile kinematics for two other flow velocities M = 0.195 and M = 0.213 added supplementary information for the profile motion. Pressure of the fluid flow on the profile surface was measured in 4 surface points but the results are not presented here.
4.1. Shift of the centre of rotation in flutter regime The shift of the centre of rotation in the starting phase of the flutter vibration for Mach number M = 0.21 is shown in Fig. 7 (data No. 2911–21). The initial deviation of the frame for flutter starting was 1.7 mm, the flutter frequency was 15.2 Hz. Approximately 1.5 s after the flutter initialization, the steady state vibration was registered. The response on the interruption of the fluid flow we can see in Fig. 8. Periodic beat vibration, visible on the record, is probably connected with the eigenfrequency of the profile for zero fluid flow (see Table 8) and the above-mentioned flutter frequency. Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Fig. 8. The shift response on the interruption of the fluid flow.
Fig. 9. Time record of the pitch angle corresponding to the start of flutter in Fig. 7.
Fig. 10. Spectral analysis of the pitch angle record in the time period (7–9) s.
4.2. Pitch angle analysis The behavior of the pitch angle time records was more complicated compared to the shift of centre of rotation. This also concerns the starting period of flutter and the steady state period of shift motion. The time record of the pitch angle corresponding to the time period from Fig. 7 is depicted in Fig. 9. The initial deviation of the frame for the start of flutter caused a pitch angle of about −9 [deg]. Time record (the amplitude increase and steady state part) in Fig. 9 is similar to Fig. 7, but in Fig. 9 we can see significant polyharmonic vibration (7.8 Hz and 15.1 Hz), verified by spectral analysis in the time period (7−9) s – see Fig. 10. The 15.1 Hz vibration during the Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Fig. 11. Spectral analysis of the pitch angle record in the time period (26.5−28.6) s – after the interruption of the fluid flow.
Fig. 12. Airfoil motion in (pitch angle, shift) axis, (a) M = 0.195, (b) M = 0.210, c) M = 0.213.
start of flutter corresponds to the mutual approach of different rotational (13.3 Hz) and transversal (17.6 Hz) eigenfrequencies in the case of zero air flow velocity. This is caused by the interaction of the fluid flow with the vibrating airfoil. On the other hand, a new generated frequency 7.8 Hz with smaller amplitude corresponds to some subharmonic vibration with half frequency. When the flow in tunnel was suddenly closed, the fluid flow influence decreased and the airfoil polyharmonic vibration with frequencies 12.9 Hz and 16.1 Hz approached the case of zero air flow velocity – see Fig. 11. Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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4.3. Profile kinematics The previous paragraphs studied the time records – shift and pitch angle of the vibrating airfoil for one case with flutter for M = 0.210. Another possibility is to describe their kinematic motion simultaneously in the same graphs with new axes (pitch angle, shift), where time becomes the parameter of the curve. The Mach number interval of the flutter occurrence in our configuration was M = (0.195–0.215). Profile kinematics will be presented for three flow velocities M = 0.195, M = 0.210, M = 0.213 from this interval. For these velocities the 1 s time intervals of the vibrations (Data No. 2911) are depicted in Fig. 12. The two loop shapes are evident. Both loops are created in counter-clockwise direction and simultaneously the left loop follows the right one. The kinematics of these flutter experiments where airfoil rotational eigenfrequency was lower than transversal shows two loops motion. 5. Conclusions The experimental device for the aerodynamic research of the airfoil model NACA0015 behavior in the air flow was reconstructed and its operation was verified. The principal aim was to generate self-excited motion of the airfoil in subsonic air flow (Mach numbers M = 0.2−0.4) in an aerodynamic tunnel. For measurement of flow parameters optical methods, mainly interferometric, will be used. The measuring and data registration system were extended and improved. The spectral and modal properties of the new airfoil support were identified for different configurations of the flat strip springs. As an example, the experimental results describing the appearance of the flutter vibration for the air flow velocity with Mach number M = 0.21, the steady state vibration, and response to the interruption of the air flow were presented and analyzed. The profile kinematics for three flow velocities M = 0.195, M = 0.210, and M = 0.213 showed double loop vibration in comparison with the results obtained in earlier measurements [9,10]. Acknowledgments The authors have been supported by the Project of the Institute of Thermomechanics AS CR, v. v. i. No 903099 and by RVO 61388998. References [1] P. Sidlof, V. Vlcek, M. Stepan, Experimental investigation of flow-induced vibration of a pitch-plunge NACA 0015 airfoil under deep dynamic stall, J. Fluids Struct. 67 (2016) 48–59. [2] R.D. Blevins, Flow-Induced Vibration, Van Nostrand Reinhold, Amsterdam, 1990. [3] Y.C. Fung, An Introduction to the Theory of Aeroeleasticity, Dover Publications, Inc, Mineola, New York, 1993. [4] J. Kozanek, V. Vlcek, I. Zolotarev, M. Stepan, The dynamic and flutter properties of the new airfoil model NACA0015, in: J. Awrejcewicz, P. Kazmierczak, J. Mrozowski, P. Olejnik (Eds.), Proceedings of the Dynamical systems−Control and Stability DSTA 2015, Technical University of Lodz, 2015, pp. 341–350. [5] V. Vlcek, M. Stepan, I. Zolotarev, J. Kozanek, Innovation of the experimental facility for the study of flutter in the Institute of Thermomechanics AS CR and some results obtained from initial experiments, Applied Mechanics and Materials, vol. 821, Trans Tech Publications, 2016, pp. 144–151. [6] V. Vlcek, J. Kozanek, Preliminary interferometry measurements of flow field around fluttering NACA0015 profile, Acta Technica 56 (2011) 379–387. [7] J. Kozanek, V. Vlcek, I. Zolotarev, Vibrating Profile in the Aerodynamic Tunnel — Identification of the Start of Flutter, J. Appl. Nonlinear Dyn. 3 (4) (2014) 317–323. [8] J. He, Z.-F. Fu, Modal Analysis, Butterworth Heinemann, Oxford, 2004. [9] I. Zolotarev, V. Vlcek, J. Kozanek, Experimental results of a fluttering profile in the wind tunnel, in: C. Meskell, G. Bennett (Eds.), Flow-induced Vibration, School of Engineering Trinity College, Dublin, 2012, pp. 677–680. [10] V. Vlcek, M. Stepan, I. Zolotarev, J. Kozanek, Experimental Investigation of the Flutter Incidence Range for Subsonic Flow Mach Numbers, in: Proceedings of the 21-st International Conference Engineering Mechanics 2015, Svratka, Czech Republic, 2015, pp. 350–351.
Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Applied Mathematical Modelling 0 0 0 (2017) 1–9
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
The new airfoil model NACA0015, modal analysis and flutter propertiesR Jan Kozanek a,∗, Vaclav Vlcek b, Igor Zolotarev a a b
Institute of Thermomechanics AS CR, Department of Dynamics and Vibrations, Dolejskova 5, 18200 Prague 8, Czechia Institute of Thermomechanics AS CR, Department of Fluid Dynamics, Dolejskova 5, 18200 Prague 8, Czechia
a r t i c l e
i n f o
Article history: Received 31 March 2016 Revised 26 January 2017 Accepted 16 February 2017 Available online xxx Keywords: Aeroelasticity Subsonic flow Self-excited vibration Modal analysis Kinematics
a b s t r a c t The experimental airfoil model NACA0015 was used to study aeroelastic phenomena during self-excited profile vibration. It provides data for control of aeroelastic calculation programs at subsonic speeds of the stream. The model movability is two-dimensional with two-degree of freedom dynamic system, one in pitch and the second in plunge, and is proposed to be a dynamic system having two near corresponding eigenfrequencies. To quantitatively evaluate flow field using interferometry, a special test section design and profile was constructed. It utilized a large visual field for the optical system together with the option of changing support stiffness for both degrees of freedom. In this paper experimental results from the range of Reynolds numbers Re = (2.63–2.83) 105 are published. The identified eigenvalues and eigenmodes for zero flow velocity are compared with measured flutter properties (frequency, modes and time evolutions) of the airfoil. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Vibration of the profile in fluid flow is an important phenomenon in many technical applications of aerodynamics, airplanes, wind turbines, helicopters, compressors and turbines. This research belongs to the broader field of investigation of complex questions regarding the stability of aerodynamic systems. Vibrations arise from interactions between the fluid flow and vibrating solid body, and differ significantly depending on the extent of the wing surface where separation occurs. If separation is only in the vicinity of the trailing edge, self-excited vibration of the profile is called flutter. If the region of the separated flow is great, the regime is called stall flutter. One of the most often studied flows, vibrating body interactions, occurs in the two-dimensional plunge and pitch airfoil model (Fig. 1). Another frequently used simplified approach to solving the problem of flutter, is to use external forced oscillation acting on the profile. This eliminates complications which may occur when tuning model parameters influencing self-excitation vibration for the required range of Reynolds numbers or Mach numbers. It also reduces randomness corresponding to variability of the flow field. However, during fluttering and stall-fluttering, a number of phenomena are significantly complicated by the oscillation mechanism. In stall-flutter regime the flow field around the profile has a very complex structure. The coupling between the self excited profile and the flow field is also very complex, and the resulting frequency may significantly differ from the initial eigenfrequencies (corresponding to zero air flow velocity) of the two degrees of freedom dynamic system. For R ∗
13th International Conference on Dynamical Systems: Theory and Applications. Corresponding author. E-mail addresses: [email protected] (J. Kozanek), [email protected] (V. Vlcek), [email protected] (I. Zolotarev).
http://dx.doi.org/10.1016/j.apm.2017.02.039 0307-904X/© 2017 Elsevier Inc. All rights reserved.
Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Fig. 1. Two-degree of freedom dynamic system — the airfoil model.
Fig. 2. The support scheme a) and measured air profile b).
example, in our earlier experiment the increase of the resulting frequency was almost the sum of both eigenfrequencies. On the other hand, in the majority of our experiments the resulting oscillation tends to have a frequency between the eigenfrequencies of the initial two degrees of freedom dynamic system. We note that the phase shift between the pitch and plunge profile vibration is significantly influenced by the stiffness of their support. A further complicating phenomenon is the spontaneous jump from a less stable oscillation to a more stable one, which is accompanied by a change of only one of the two eigenfrequencies. The change is realized during one period, the oscillation amplitude strongly increased and phase shifted by omission of a half wavelength. The frequency of the flutter does not change. Mentioned parameters of the dynamic system also affect the range of occurrence of self-excitation in the area of Reynolds and Mach numbers. A more detailed overview of individual variants of the aeroelastic phenomena associated with the described effects and the approximate status of their knowledge is listed in [1]. In both non-steady and steady cases the chord of the profile was used for evaluation of the Reynolds number as the geometry characteristic. The aeroelasticity and stability problems of profiles have been studied for a long time – [2,3]. Nevertheless the selfexcited vibrations in our experimental configuration were not yet investigated and published. Here we publish in detail the experiments, where self-excited vibrations were successfully achieved without the use of an external forced oscillation. 2. The new airfoil model NACA0015 and the experimental equipment The new airfoil model NACA0015 was constructed to generate self-excited vibrations in subsonic air flow. For these aerodynamic experiments the suction type aerodynamic tunnel of the Institute of Thermomechanics AS (with Mach numbers air flow M = 0.2−0.4 - see the DSTA 2015 conference contribution [4]) was used. Our fluttering profile is supported as a two-degree of freedom dynamic system, one for profile rotation (pitch) and the second for translation (shift) motion with mutually near and adjustable eigenfrequencies. Brief description of the layout of the new experimental facility, the results of the modal analysis and the first measurements are described in [5]. The measured air profile located in the wind tunnel is shown in Fig. 2. Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Fig. 3. The new airfoil model NACA0015 with two modifications of rotational springs.
Fig. 4. Airfoil rotational and transversal support and the experimental equipment.
Compared with older constructions [6,7], the translation springs were not changed, but the rotational support was realized using a new coil spring. The corresponding pitch eigenfrequency can be tuned by the coil spring parameters, i.e. by the spring wire length and by its diameter. The eigenfrequency of the transversal mode is influenced by using the grub screws [7]. The new airfoil model with a chord length of 59 mm, thickness of 8.85 mm and width of 76.6 mm, was divided into five parts (Fig. 3). Two ball bearings enable the profile to rotate; the rotation axis of profile is located at 1/3 of the chord. The weight of the profile is approximately 38 g. The support of the profile with measuring device are shown in Fig. 4. The magnetic rotary encoder RM08 was used for pitch angle recording. The frame translation was measured by the magnetic non-contact linear encoder LM13TCD. The strain gauge with full bridge placed on the flat spring measured the translation motion. Local pressure on the surface of the profile was indicated with four semiconductor pressure sensors MPXH6115. Transducers were coupled in pairs on both profile surfaces. Signals were registered with DEWETRON desktop PC software. The optical measurements could be measured by using the high-speed MotionPro X3 camera at 10 0 0 frames per second. Velocity of the flow field was determined by a wind tunnel Machmeter. The interferometer visualization is possible in an area with a diameter of 160 mm. The flutter was initialized by a 1.7 mm initial deviation of the frame. 3. Modal analysis of the two-degree of freedom airfoil support The special support of our airfoil is proposed as a dynamic system with two-degree of freedom, one degree of freedom in rotation (pitch of the airfoil about the axis situated at 1/3 of the profile chord) and the second in its translation mode (shift of the whole airfoil axis). For zero flow velocity, the eigenfrequencies of the above modes are relatively close to one another. Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Fig. 5. The excited and measured points 1,2 on our measured profile during modal analysis. Table 1 Initial tensioning corresponding to the different grub screw positions p1, p2, p3. p d [mm]
p1 0.23
p2 0.58
p3 0.93
For the successful implementation of aerodynamic experiments in the flow of air it is very useful to have the opportunity to influence the closeness of these eigenfrequencies. This is because when increasing air flow velocity, usually the lower eigenfrequency increases and the higher eigenfrequency decreases. On the other hand, in order to generate the self-excited airfoil flutter in some predestined Mach number interval, there is an optimal gap between the eigenfrequencies. We wanted to implement subsonic air flow with Mach numbers lower than M = 0.4. In our experimental support configuration we can modify the rotational eigenfrequency using different rotational springs, and the translation frequency by the initial pretensioning of the transversal flat strip springs. In the described experiment we made a second modification of the rotational springs (torsion bar from Fig. 3) and investigated the effect of the pre-tensioning of the transversal flat strip sprigs (Fig. 4). To do this, we made a modal analysis of this simple dynamic system for zero air flow velocity for three grub screw positions which gradually increased the stiffness of the transversal flat strip springs. The results were complex eigenvalues and corresponding eigenmodes. Imaginary parts of eigenvalues (eigenfrequencies corresponding to the transversal mode) increased, yielding quantitative results of how the system changes. The real parts of identified eigenvalues (the damping values) are the additional information about the possibilities of achieving self-excited airfoil vibration in subsonic air flow regime. The identification of complex eigenmodes corresponding to rotational and translation vibrations indicate “more complex” normalized eigenmodes for rotational motion which is perhaps related to the dry friction of this support. The effect of errors of dynamic measurement on these identification results was greater than that of the eigenvalue identification. The modal analysis in the idle state (zero air flow velocity) was carried out in the Dynamics and Vibration laboratory of the Institute of Thermomechanics AS CR – see [5]. The pulse-hammer excitation was realized by the 4519 B&K force sensor and measurements were performed with the acceleration pick-up DeltaTron 4519. The time signals were discretized, registered and evaluated by the PULSE B&K measurement system with card 7537A. There are two (marked 1, 2) excited and measured points in the centre of rotation and on the trailing edge − see Fig. 5. The measured transfer functions vj,k (f), j,k=1,2 as the function of excitation frequency f are complex and can be evaluated numerically by the parametric regression identification method [8]
v j,k ( f ) =
2 av, j,k + h( f ), i f − sv
(1)
v=1
where vj,k (f) is the complex transfer function (excitation in k-point and pick-up placed in j-point), sv is the complex eigenvalue and av,j,k is the corresponding complex modal contribution and h (f) is the influence of the other modes and experimental perturbations. The example of the measured frequency responses v( f ) is in Fig. 6 3.1. Pre-tensioning of the flat springs Two flat strip springs define transversal elastic support of the profile and their rigidities can be tuned by using grub screws. The three adjustments of the grub screws marked as p1, p2, p3 we define three pre-stress support deformations of the flat strip springs (see Table 1). The identified complex eigenvalues and eigenmodes for three dynamic systems corresponding to the different initial pre-stress support deformations are summarized in Tables 2−8. The rotational eigenmode with very small amplitude was neglected. Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Table 2 Identified modal parameters av,j,k and eigenvalues sv for the initial tensioning p1. (pick-up,excitation)
(1,1)
(1,2)
(2,1)
(2,2)
av,j,k
−0.0 0 02−0.0 03i 0.0 0 03−0.0 05i
−0.003−0.017i 0.003 + 0.005i 0.001 + 0.009i
0.0 03−0.0 05i 0.002 + 0.003i
0.019−0.24i 0.030 + 0.015i
sv [Hz]
−1.1 + 14.26i −0.6 + 17.13i
−0.5 + 13.37i −0.2 + 14.78i −0.3 + 17.25i
−0.5 + 13.53i −0.9 + 17.99i
−0.6 + 13.29i −0.2 + 14.45i
Table 3 Normalized eigenvectors vT1 and vT2 corresponding to rotational and transversal eigenmodes for the initial tensioning p1.
vT1 vT2
[0.0197 − 0.0141i, 1] [1, − 1.7816 + 0.3069i]
[a1,2,1 , a1,2,2 ]/a1,2,2 [a2,1,1 , a2,1,2 ]/a2,1,1
Table 4 Identified modal parameters av,j,k and eigenvalues sv for the initial tensioning p2. (pick-up,excitation)
(1,1)
(1,2)
(2,1)
(2,2)
av,j,k
−0.0 0 0 0−0.0 01i 0.0 0 02−0.0 02i
−0.0 0 02−0.0 09i 0.001 + 0.004i −0.0 0 02 + 0.004i
−0.003−0.0045i 0.001 + 0.003i
0.030−0.23i 0.010 + 0.032i
sv [Hz]
−0.9 + 14.40i −0.7 + 17.56i
−0.5 + 13.71i −0.2 + 14.91i −0.3 + 17.66i
−0.6 + 13.64i −0.7 + 18.19i
−0.8 + 13.36i −0.3 + 14.44i
Table 5 Normalized eigenvectors vT1 and vT2 corresponding to rotational and transversal eigenmodes for the initial tensioning p2.
vT1 vT2
[0.0176 − 0.0153i, 1] [1, − 1.9901 + 0.0990i]
[a1,2,1 , a1,2,2 ]/a1,2,2 [a2,1,1 , a2,1,2 ]/a2,1,1
Table 6 Identified modal parameters av,j,k and eigenvalues sv for the initial tensioning p3. (pick-up,excitation)
(1,1)
(1,2)
(2,1)
(2,2)
av,j,k
−0.0 0 03−0.0 01i 0.0 0 04−0.0 02i
−0.0 0 04−0.0 08i 0.0016 + 0.004i −0.0 0 02 + 0.004i
−0.004−0.003i −0.0 0 02 + 0.002i
0.085−0.23i −0.053 + 0.024i
sv [Hz]
−1.0 + 14.56i −0.9 + 17.630i
−0.6 + 13.69i −0.3 + 14.96i −0.6 + 17.76i
−0.6 + 13.83i −0.8 + 18.28i
−0.9 + 13.33i −0.4 + 14.29i
Table 7 Normalized eigenvectors vT1 and vT2 corresponding to rotational and transversal eigenmodes for the initial tensioning p3.
vT1 vT2
[a1,2,1 , a1,2,2 ]/a1,2,2 [a2,1,1 , a2,1,2 ]/a2,1,1
[0.0058 − 0.0195i, 1] [1, − 1.9423 + 0.2885i]
Table 8 Resulting eigenvalues sv for the largest amplitude of vibration, initial pre-stress support deformations p1, p2, p3. sv [Hz]
Rotational eigenmode
Transversal eigenmode
p1 p2 p3
−0.6 + 13.29i −0.8 + 13.36l −0.9 + 13.33i
−0.6 + 17.13i −0.7 + 17.56i −0.9 + 17.63i
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Fig. 6. The example of the frequency responses ν (f) in two measured configurations (pick-up, excitation) (a)−(1,1), (b)−(2,2) for initial pre-stress support deformation p2.
Fig. 7. The starting phase of the flutter vibration, initial deviation was 1.7 mm.
For the following aerodynamic experiments with nonzero air flow velocity the geometrical deformation of the flat strip springs corresponds to the fixed position p2.
4. Aerodynamic experiments with the airfoil profile NACA0015 Aerodynamic experiments were realized in the Laboratory of the Institute of Thermomechanics AS near Novy Knin for subsonic air flow with Mach numbers M = 0.2–0.4 and Reynolds numbers (2.63–2.83)·105 . The results, involving flutter regime, with M = 0.21 and Re = 2.76 105 , will be presented as an example in this chapter. The measured parameters were shift [mm] of the rotation centre and pitch angle [deg] as a function of time. The frequency spectrum was evaluated for rotational vibration in different time moments. Profile kinematics for two other flow velocities M = 0.195 and M = 0.213 added supplementary information for the profile motion. Pressure of the fluid flow on the profile surface was measured in 4 surface points but the results are not presented here.
4.1. Shift of the centre of rotation in flutter regime The shift of the centre of rotation in the starting phase of the flutter vibration for Mach number M = 0.21 is shown in Fig. 7 (data No. 2911–21). The initial deviation of the frame for flutter starting was 1.7 mm, the flutter frequency was 15.2 Hz. Approximately 1.5 s after the flutter initialization, the steady state vibration was registered. The response on the interruption of the fluid flow we can see in Fig. 8. Periodic beat vibration, visible on the record, is probably connected with the eigenfrequency of the profile for zero fluid flow (see Table 8) and the above-mentioned flutter frequency. Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Fig. 8. The shift response on the interruption of the fluid flow.
Fig. 9. Time record of the pitch angle corresponding to the start of flutter in Fig. 7.
Fig. 10. Spectral analysis of the pitch angle record in the time period (7–9) s.
4.2. Pitch angle analysis The behavior of the pitch angle time records was more complicated compared to the shift of centre of rotation. This also concerns the starting period of flutter and the steady state period of shift motion. The time record of the pitch angle corresponding to the time period from Fig. 7 is depicted in Fig. 9. The initial deviation of the frame for the start of flutter caused a pitch angle of about −9 [deg]. Time record (the amplitude increase and steady state part) in Fig. 9 is similar to Fig. 7, but in Fig. 9 we can see significant polyharmonic vibration (7.8 Hz and 15.1 Hz), verified by spectral analysis in the time period (7−9) s – see Fig. 10. The 15.1 Hz vibration during the Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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Fig. 11. Spectral analysis of the pitch angle record in the time period (26.5−28.6) s – after the interruption of the fluid flow.
Fig. 12. Airfoil motion in (pitch angle, shift) axis, (a) M = 0.195, (b) M = 0.210, c) M = 0.213.
start of flutter corresponds to the mutual approach of different rotational (13.3 Hz) and transversal (17.6 Hz) eigenfrequencies in the case of zero air flow velocity. This is caused by the interaction of the fluid flow with the vibrating airfoil. On the other hand, a new generated frequency 7.8 Hz with smaller amplitude corresponds to some subharmonic vibration with half frequency. When the flow in tunnel was suddenly closed, the fluid flow influence decreased and the airfoil polyharmonic vibration with frequencies 12.9 Hz and 16.1 Hz approached the case of zero air flow velocity – see Fig. 11. Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039
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4.3. Profile kinematics The previous paragraphs studied the time records – shift and pitch angle of the vibrating airfoil for one case with flutter for M = 0.210. Another possibility is to describe their kinematic motion simultaneously in the same graphs with new axes (pitch angle, shift), where time becomes the parameter of the curve. The Mach number interval of the flutter occurrence in our configuration was M = (0.195–0.215). Profile kinematics will be presented for three flow velocities M = 0.195, M = 0.210, M = 0.213 from this interval. For these velocities the 1 s time intervals of the vibrations (Data No. 2911) are depicted in Fig. 12. The two loop shapes are evident. Both loops are created in counter-clockwise direction and simultaneously the left loop follows the right one. The kinematics of these flutter experiments where airfoil rotational eigenfrequency was lower than transversal shows two loops motion. 5. Conclusions The experimental device for the aerodynamic research of the airfoil model NACA0015 behavior in the air flow was reconstructed and its operation was verified. The principal aim was to generate self-excited motion of the airfoil in subsonic air flow (Mach numbers M = 0.2−0.4) in an aerodynamic tunnel. For measurement of flow parameters optical methods, mainly interferometric, will be used. The measuring and data registration system were extended and improved. The spectral and modal properties of the new airfoil support were identified for different configurations of the flat strip springs. As an example, the experimental results describing the appearance of the flutter vibration for the air flow velocity with Mach number M = 0.21, the steady state vibration, and response to the interruption of the air flow were presented and analyzed. The profile kinematics for three flow velocities M = 0.195, M = 0.210, and M = 0.213 showed double loop vibration in comparison with the results obtained in earlier measurements [9,10]. Acknowledgments The authors have been supported by the Project of the Institute of Thermomechanics AS CR, v. v. i. No 903099 and by RVO 61388998. References [1] P. Sidlof, V. Vlcek, M. Stepan, Experimental investigation of flow-induced vibration of a pitch-plunge NACA 0015 airfoil under deep dynamic stall, J. Fluids Struct. 67 (2016) 48–59. [2] R.D. Blevins, Flow-Induced Vibration, Van Nostrand Reinhold, Amsterdam, 1990. [3] Y.C. Fung, An Introduction to the Theory of Aeroeleasticity, Dover Publications, Inc, Mineola, New York, 1993. [4] J. Kozanek, V. Vlcek, I. Zolotarev, M. Stepan, The dynamic and flutter properties of the new airfoil model NACA0015, in: J. Awrejcewicz, P. Kazmierczak, J. Mrozowski, P. Olejnik (Eds.), Proceedings of the Dynamical systems−Control and Stability DSTA 2015, Technical University of Lodz, 2015, pp. 341–350. [5] V. Vlcek, M. Stepan, I. Zolotarev, J. Kozanek, Innovation of the experimental facility for the study of flutter in the Institute of Thermomechanics AS CR and some results obtained from initial experiments, Applied Mechanics and Materials, vol. 821, Trans Tech Publications, 2016, pp. 144–151. [6] V. Vlcek, J. Kozanek, Preliminary interferometry measurements of flow field around fluttering NACA0015 profile, Acta Technica 56 (2011) 379–387. [7] J. Kozanek, V. Vlcek, I. Zolotarev, Vibrating Profile in the Aerodynamic Tunnel — Identification of the Start of Flutter, J. Appl. Nonlinear Dyn. 3 (4) (2014) 317–323. [8] J. He, Z.-F. Fu, Modal Analysis, Butterworth Heinemann, Oxford, 2004. [9] I. Zolotarev, V. Vlcek, J. Kozanek, Experimental results of a fluttering profile in the wind tunnel, in: C. Meskell, G. Bennett (Eds.), Flow-induced Vibration, School of Engineering Trinity College, Dublin, 2012, pp. 677–680. [10] V. Vlcek, M. Stepan, I. Zolotarev, J. Kozanek, Experimental Investigation of the Flutter Incidence Range for Subsonic Flow Mach Numbers, in: Proceedings of the 21-st International Conference Engineering Mechanics 2015, Svratka, Czech Republic, 2015, pp. 350–351.
Please cite this article as: J. Kozanek et al., The new airfoil model NACA0015, modal analysis and flutter properties, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.039