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Flutter derivatives of a flat plate section and analysis of flutter instability at various wind angles of attack
Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104046
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Flutter derivatives of a flat plate section and analysis of flutter instability at various wind angles of attack Bo Wu a, b, Qi Wang a, b, *, Haili Liao a, b, Yulin Li c, Minghsui Li a, b a b c
Department of Bridge Engineering, Southwest Jiaotong University, Chengdu, 610031, China Key Laboratory for Wind Engineering of Sichuan Province, Chengdu, 610031, China Hunan Provincial Communications Planning, Survey & Design Institute Co., LTD, Changsha, 410000, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Flat plate Flutter derivatives Wind angles of attack Forced vibration test Flutter index
Flutter derivatives of a flat plate section with side ratio of 40 were obtained under various wind angles of attack via forced vibration test, based on which flutter instability was further analyzed in detail. Accuracy of the experimental setup was firstly verified by comparison of flutter derivatives between experimental and theoretical results at 0 wind angle of attack. Detail discussions on distributions of flutter derivatives under various wind angles of attack were conducted and potential influence of changing flutter derivatives on flutter performance was analyzed. It reveals the dramatic changing of A*2 with changing of wind angle of attack. The most significant change in the flutter derivatives is the sign change of A*2 from negative to positive at a 5.5 wind angle of attack. The accuracy of tested flutter derivatives was verified by comparing calculated critical wind speed with those from wind tunnel test. Flutter instability characteristics at various wind angles of attack were discussed, which can aid understanding on flutter behavior of flat plate section by considering effect of large wind angles of attack. Finally, characteristic of flutter index was discussed to demonstrate the flutter performance varying with reduced speed at various wind angles of attack. It remains stable when reduced wind speed less than 6 at wind angle of attack less than 3 , which can be directly used for predicting critical flutter speed of flat plate.
1. Introduction To achieve better understanding of the coupling interactions between fluids and structures, thin plates have been widely investigated in bridge engineering (Noda et al., 2003; Chen et al., 2005; Xu et al., 2014) and aerospace engineering (Theodorsen, 1935; Liaw, 1993; Vedeneev, 2012; Strangfeld et al., 2018; Yu et al., 2018), not only because of their simple aerodynamic geometry, but also as a result of the significant representative positions of thin flat plates on many bridge decks. The self-excited force model of a thin plate, which was theoretically proposed by Theodorsen in the landmark NACA report (Theodorsen, 1935), also highlights the important role of thin plates in wind engineering. Furthermore, the collapse of the old Tacoma Bridge in 1940 has attracted lots of attention of researchers on bridge flutter. Although a wide variety of representation conventions have been introduced to improve the modeling of self-excited forces, which appear to show specific advantages with respect to prediction and interpretation of flutter instability (Zasso, 1996; Virgoleux, 1992; Falco et al., 1992), the typical convention in terms of
flutter derivatives that proposed by Scanlan and Tomoko in 1971 (Scanlan and Tomko, 1971), is still the most accepted tool for flutter analysis. Following their proposals, an explicit nexus between flutter derivatives and the well-known Theodorsen function for thin plates has been built. Based on flutter derivatives which are functions of reduced velocity, the mechanism of certain complex aerodynamic phenomena (e.g., flutter and vortex-induced vibration) can be well explained from this perspective (Ge and Xiang, 2008; Matsumoto et al., 1995; Matsumoto, 2013). It is well known that flutter derivatives of a section are uniquely determined by the aerodynamic configuration and are independent of other parameters. Theoretical results for flutter derivatives can be obtained based on an ideal thin plate model assumption. For bridge decks, wind tunnel test is the most reliable means for qualifying flutter derivatives, despite the great advances in computational fluid dynamics (CFD). In recent years, flutter derivative identification methods, especially those based on wind tunnel testing, have been gradually developed. Because of the pure-mode oscillation assumption of Scanlan’s
* Corresponding author. Department of Bridge Engineering, Southwest Jiaotong University, Chengdu, 610031, China. E-mail addresses: [email protected] (B. Wu), [email protected] (Q. Wang), [email protected] (H. Liao), [email protected] (Y. Li), [email protected] (M. Li). https://doi.org/10.1016/j.jweia.2019.104046 Received 11 June 2019; Received in revised form 20 September 2019; Accepted 16 November 2019 Available online xxxx 0167-6105/© 2019 Elsevier Ltd. All rights reserved.
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104046
As mentioned before, a long-span bridge deck of box girder type with a relatively large aspect ratio (>10) is commonly considered to be an ideal thin plate model, and therefore aerodynamic behavior studies of these bridge decks by wind tunnel tests also adopt the thin plate section model (Gu et al., 2000, 2008; Zhu et al., 2009), considering the representation position and the important role of a thin plate in flutter analysis of bridge engineering for the investigation of flutter derivatives of a thin plate under various wind AOAs. In this study, flutter derivatives of a thin flat plate section model under various wind AOAs were investigated via a forced vibration test. With these flutter derivatives, the critical flutter speed can be derived based on the bimodal coupled flutter solution (e.g., Chen and Kareem, 2006; Chen, 2007). To verify the accuracy of flutter derivatives obtained from the forced vibration test, an additional free vibration section model test was introduced to determine the critical flutter speeds directly. Comparison of critical flutter speeds reveals the accuracy of the estimated flutter derivatives. Subsequently, the flutter instability characteristics of the flat plate section at various wind AOAs were discussed to interpret the effects of the changes in flutter derivatives on flutter behaviors at various wind AOAs, which can aid better understanding of the flutter behavior of a flat plate section by considering the effect of large wind AOAs. Finally, the curves of the flutter index of a thin plate were calculated to demonstrate the flutter performance varying with reduced speed at various wind AOAs and were found to be stable as a constant when wind AOA was less than 3 , which can be directly used for predicting the critical flutter speed of a flat plate.
method, the implementation of experimental techniques and identification accuracy for coupled related terms (e.g., crossed flutter derivatives) have been severely constrained. Thus, considerable efforts have been expended to simplify the experimental procedure and better identify flutter derivatives; many advanced identification methods have subsequently been proposed in recent decades (Shinozuka, 1982; Sarkar, 1992, 1994; Gu et al., 2000; Zhu et al., 2002). Shinozuka (1982) introduced the ARMA model which was widely used in practice (e.g., Jiang et al., 2019) in the system identification procedure, from which the aerodyanmic coefficient matrics of a two dimensional model of a suspension bridge can been extracterd coveniently. Sarkar (1992, 1994) proposed the modified Ibrahim time domain (MITD) method to identify all flutter derivatives from a vertical-torsional coupled free vibration test. Gu et al. (2000) and Zhu et al. (2002) treated the modal properties from the MITD method as initial values for an iterative procedure; in their method, flutter derivatives could further be estimated based on the unifying least squares (ULS) theory. In recent years, methods aiming at flutter derivative identification based on free vibration tests have seen significant improvement (Chowdhury and Sarkar, 2003; Andersen et al., 2018). Chowdhury and Sarkar (2003) proposed the modified iterative least squares (MILS) method that provides a more accurate prediction of the coupled stiffness and damping coefficient matrix than the iterative least squares (ILS) method. Andersen et al. (2018) suggested a hybrid system identification method to estimate the flutter derivatives of a 10:1 rectangular section based on a coupled free vibration test. These frameworks provide more accurate results of bridge flutter derivatives when conducting section-model free vibration wind tunnel tests. However, it is also noteworthy that certain approximations, concerning two distinct modes in free vibration that correspond to two sets of distinct frequency-dependent flutter derivatives, which make identification impossible to implement, are all implied in the methods mentioned above (Chen and Kareem, 2004; Xu et al., 2014). Therefore, forced vibration tests in which a section model can be rigidly supported in a mechanism and can be forced into precise motions with pre-set amplitudes and frequencies has become another optional experimental technique for the identification of flutter derivatives (Matsumoto, 1996; Haan et al., 2000; Chen et al., 2005). This is because the precise measurement of either motion-induced forces or instantaneous surface pressures and a rigorous algorithm with zero assumptions provide more reliable estimations of flutter derivatives. This experimental technique makes the identification of flutter derivatives easier than before. All previous identification studies perform well regarding the identification of flutter derivatives at a 0 wind angle of attack (AOA). Meanwhile, some reports focusing on the identification of flutter derivatives under irregular conditions have also been presented (e.g., Zhu et al., 2002; Gu and Xu, 2008). In addition, some reports have also aimed to investigate the lateral flutter derivatives of bridge decks based on wind tunnel tests instead of quasi-steady theory (e.g., Chen et al., 2002; Chowdhury and Sarkar, 2003). These studies indicate the important influence of external factors on flutter derivatives in bridge decks. However, flutter derivatives of a thin flat plate under various wind angles of attack (AOAs) have rarely been studied before. The wind AOA is an important factor affecting the aerodynamic performance of long-span bridges in mountainous areas. It is generally accepted that even a small fluctuation in wind or weak structural motion could lead to a variety of effective wind AOAs, which may further lead to significant changes in flutter performance; these conditions require accurate sets of flutter derivatives under various wind AOAs to satisfy the modulation of flutter derivatives by the instantaneous wind AOAs when conducting flutter calculations (Diana et al., 1995; Chen and Kareem, 2001). Tang et al. (2017) experimentally investigated the potential influence of the wind AOA on flutter performance of a truss girder and further numerically calculated the flutter derivatives under large wind AOAs. Therefore, the flutter derivatives of bridge decks under various wind AOAs should be estimated accurately before evaluating the influence of the wind AOAs and the turbulence on flutter performance.
2. Forced vibration test for flutter derivatives 2.1. Experimental setup The forced vibration test with a rigidly supported thin flat plate section model was conducted in the first testing section of the XNJD-1 Boundary Layer Wind Tunnel of Southwest Jiaotong University. The working section has a geometric size of 2.0 m in height and 2.4 m in width, which allows wind speeds varying from 0 m/s to 45 m/s continuously. The flow condition was smooth flow, with a turbulence intensity less than 1% (Fig. 1(a)). The thin plate section model is 1.1 m in length (L), 0.4 m in width (B), and 0.01 m in depth (D), which leads to an aspect ratio (B/D) of 40:1, as shown in Fig. 1(b). It is made of carbon so that it can be lightweight yet rigid. Its total weight is 2.2 kg. The test model is fixed to the floor with four linear actuators driven by motors, which are equipped with sensors to synchronously acquire the force and the displacement under a windaxis coordinate system. The section model is eccentrically supported in the mechanism such that four actuators will move synchronously when vertical excitation occurs, while for torsional excitation, only two actuators (No.1 and No.2) on the windward side move synchronously, and the other two (No.3 and No.4) on the leeward side remain still, as shown in Fig. 1(c). The oscillation frequency can be varied from 0 Hz to 20 Hz. The model is set up with a 10-mm vertical vibration amplitude and a 2 torsional vibration amplitude with a fixed vibration frequency of 2.5 Hz; the inflow speed varies from 4 m/s to 18 m/s with an interval of 1 m/s. The wind AOA varies from 0 to 7 with an interval of 1 , a schematic diagram of positive wind AOA with respect to wind direction and bridge transverse axis is shown in Fig. 1(c). Meanwhile, a 6-mm-thick rigid chamfered plate, 600 mm 200 mm in size and oriented parallel to the inflow, is fixed to each end of the model to ensure the plate model is under two-dimensional flow. The height from the section model to the floor is 1 m, so that the model will not be affected by the boundary conditions. 2.2. Theory for extraction of aerodynamic forces and flutter derivatives A 30-Hz cut-off frequency in low-pass filter mode is adopted for both force and displacement sensor signals. Assuming the detected forces on the four force sensors are F1 , F2 , F3 , and F4 , respectively, the total vertical 2
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Fig. 1. Experimental setup and measuring technique.
lift Ltotal and torsional moment Mtotal can be represented by the following equations: Ltotal ¼ F1 þ F2 þ F3 þ F4
(1a)
Mtotal ¼ ðF1 þ F2 Þ ΔS
(1b)
LðtÞ ¼ Ltotal L0 ∕L
(2a)
MðtÞ ¼ Mtotal M0 ∕L
(2b)
where Fi0 ði ¼ 1 4Þ are forces measured by the force sensors in still air conditions with a single degree of freedom (SDOF) for either vertical or torsion motion; L0 ¼ F10 þ F20 þ F30 þ F40 is the inertia lift force under still air; L is the length of the section model; and M0 ¼ ðF10 þF20 Þ ΔS is the inertia moment under still air. For a thin plate with single degree of freedom (SDOF) harmonic motion, the vertical and torsional motions can be expressed by:
where ΔS¼145 mm is the distance between the front actuator and the rear actuator. Both aerodynamic components and inertia components are included in the lift force Ltotal and torsional moment Mtotal in these equations. However, the inertial force is not associated with flutter derivatives and thus should be removed from the total one. The inertia force can be obtained under the same testing conditions, i.e., with the same driven frequency, phase lag, and amplitude, but tested in still air. Thereafter, the aerodynamic lift force L(t) and torsional moment M(t) per unit length can be expressed as:
hðtÞ ¼ h0 sinðωh tÞ
(3a)
αðtÞ ¼ α0 sinðωα tÞ
(3b)
where h0 and α0 are the vertical and torsional amplitudes, respectively; 3
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ωh ¼ 2π fh0 and ωα ¼ 2π fα0 are the circular frequencies of vertical and torsional motion, respectively. The aerodynamic lift force and torsional moment can be written as follows (Chen, 2007): 1 bα_ h h_ Lse ðtÞ ¼ ρU 2 ð2bÞ kH 1 þ kH 2 þ k2 H 3 α þ k 2 H 4 2 b U U
(4a)
h_ 1 bα_ h Mse ðtÞ ¼ ρU 2 2b2 kA1 þ kA2 þ k2 A3 α þ k2 A4 U 2 b U
(4b)
angle; ϕLH is the phase lag between the lift force and the heaving motion, and ϕMH is the phase lag between the moment and the heaving motion; ϕLP is the phase lag between the lift force and the torsional motion, and ϕMP is the phase lag between the moment and the torsional motion; hr0 ¼ h0 =b is the reduced amplitude of the vertical motion. 3. Test results for flutter derivatives 3.1. Aerostatic test Prior to forced vibration test, an aerostatic test was conducted to obtain the aerostatic coefficients of the thin plate model by wind tunnel test. The theoretical aerostatic coefficients for an ideal thin plate are expressed as follows: Lift coefficient
where ρ is the air density; U is the mean wind velocity; b ¼ B= 2 is the half width of the bridge deck; k ¼ ωb=U is the reduced frequency; ω is the circular frequency of motion; and H *i and A*i (i ¼ 1 4) are flutter derivatives, which are functions of reduced frequency. By substituting Eqs. (3a) and (3b) into Eqs. (4a) and (4b), the flutter derivatives can be derived as follows:
CL ¼ 2πα
(6)
Moment coefficient
H *1 ¼
CLH sinðφLH ϕLH Þ 2hr0 k 2
(5a)
H 2 ¼
CLP sinðφLP ϕLP Þ 2α0 k 2
(5b)
H 3 ¼
CLP cosðφLP ϕLP Þ 2α0 k 2
(5c)
H 4 ¼
CLH cosðφLH ϕLH Þ 2hr0 k 2
(5d)
A1 ¼
CMH sinðψ MH ϕMH Þ 2hr0 k 2
(5e)
A*2 ¼
CMP sinðψ MP ϕMP Þ 2α0 k2
(5f)
A*3 ¼
CMP cosðψ MP ϕMP Þ 2α0 k2
(5g)
A*4 ¼
CMH cosðψ MH ϕMH Þ 2hr0 k 2
(5h)
CM ¼ CL =4
(7)
Fig. 2 presents a comparison of the static coefficients of the thin plate model with the theoretical ones for an ideal plate, in which the wind AOAs vary from 10 ~ 10 (0.175–0.175 radian). Fig. 2 demonstrates that both the lift coefficient and moment coefficient show great agreement with theoretical values, in which the slopes of the static lift coefficient curves for experimental and theoretical values are, respectively, 5.83 and 2π, with a corresponding relative difference of only 7.2%. Regarding the static moment coefficients, the slopes for experimental and theoretical values are 1.47 and π=2, respectively, which induces a relative difference of only 6.4%. Therefore, the testing apparatus and the reliability of the model set up were validated. 3.2. Flutter derivatives at zero degree wind AOA Considering that the theoretical solution for an ideal thin plate at 0 wind AOA is known, the flutter derivatives of the thin plate model at 0 wind AOA were tested first to verify the feasibility and accuracy of the forced vibration experimental setup. The tested flutter derivatives at 0 wind AOA obtained from Eq. (5) are shown in Fig. 3. Chen (2007) proposed a simplified closed-from solution for flutter analysis of bridge decks that onset flutter speed can be approximately determined by flutter derivatives of A*1 , A*2 , A*3 , and H *3 , which emphasizes the most important roles of these four flutter derivative to a bridge deck’s (and a thin plate’s) flutter performance. Details about the simplified closed form solution can also be referred to Section 4.1. It is clear from Fig. 3 that these four critical derivatives are close to the theoretical values. The absolute values of A*1 and H *3 which are related to
where CLH is the unsteady lift coefficient generated by vertical motion, and φLH is the corresponding phase angle; CLP is the unsteady lift coefficient generated by torsional motion, and φLP is the corresponding phase angle; CMH is the unsteady moment coefficient generated by heaving motion, and ψ MH is the corresponding phase angle; CMP is the unsteady moment coefficient generated by torsional motion, and ψ MP is the corresponding phase
Fig. 2. Test results for CL and CM. 4
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Fig. 3. Flutter derivatives at zero wind AOA.
flutter morphological parameters (e.g., coupled aerodynamic damping, amplitude ratio and phase angle et al.) and A*2 which is related to uncoupled aerodynamic damping of the thin plate section model, are slightly higher than the theoretical values, while the uncoupled aerodynamic stiffness term of the torsional branch, i.e., the flutter derivative of A*3 , remains almost equal to the theoretical value. The other four derivatives, which are not significantly associated with the critical flutter speed, show a certain level of discrepancy with the theoretical results,
except for H *1 . These results illustrate that the aerodynamic characteristics of an ideal flat plate can be characterized by the thin flat plate section model; further, the accuracy of the forced vibration technique is verified. 3.3. Flutter derivatives under various wind AOAs By modifying the mean position of the two windward-side actuators, the wind AOAs can be modulated to satisfy harmonic motion under 5
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Fig. 4. Flutter derivatives at various wind AOAs (Dashed ellipse in plots of A2 and A4 : Partial enlargement drawing).
various test conditions. The tested flutter derivatives are shown in Fig. 4. By altering the wind AOA, the flutter derivatives of the thin flat plate model changed significantly. Fig. 4 also gives the enlarged drawings for A2 and A4 over a range of [6,8] of reduced wind speed to emphasizes the fundamental change of these two flutter derivatives under large AOAs, as shown in Fig. 4. Regarding the uncoupled related terms, i.e., H 1 , H 4 ; A2 , and A3 , it is inferred from the diagram of H 1 that heaving instability (i.e., galloping) will never occur because the trend of H 1 remains negative and increases
in absolute value with increasing wind AOA. The vertical motion-related stiffness term H 4 shows a significant increase in absolute value, which indicates a probable increase in the modal frequency of the vertical mode branch (Chen and Kareem, 2006). For the torsional mode branch, the potential influence of the wind AOA on the modal properties becomes more significant. While the stiffness-related term A3 is insensitive to changes in the wind AOA, the damping-related term A2 shows a great dependence on the wind AOA. For small wind AOAs (3 or less), A2 is negative and gradually decreases in absolute value as the wind AOA 6
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Fig. 5. Flutter derivatives at wind AOAs within [5 , 8 ].
increases. However, A2 tends to change sign at a 5 wind AOA, even though it is still negative, which indicates that the positive aerodynamic damping induced by torsional motion is close to zero. At a 7 wind AOA, A2 changes sign at a reduced speed of 7.5, which indicates negative aerodynamic damping caused by A2 and torsional flutter instability of the thin plate model. The A2 diagram in Fig. 4 reveals the essential change in the flutter modality of the thin plate for wind AOAs from 0 to 7 . Regarding the coupled motion-related terms, i.e., H 2 , H 3 ; A1 , and A4 , it is generally accepted that these four terms are associate with four coupled parameters between vertical motion and torsion motion, i.e., frequency, damping, phase difference, and amplitude ratio. Thus, it is difficult to quantitatively evaluate the influence of changing flutter derivatives on the coupled parameters resulting from changes in the wind AOA. However, some qualitative predictions can still be conducted using the changing flutter derivatives shown in Fig. 4. This figure demonstrates that the wind AOA had practically no or very little influence on A1 and A4 and H 3 , as compared with the other parameters when wind AOA 5 . At a 7 wind AOA, A1 and A4 may show a dramatic variation that further affects the modification of modal properties, especially the phase lag between motion-induced moment and vertical motion, according to the bimodal flutter theory (e.g., Chen and Kareem, 2006; Chen, 2007). Although H 2 shows a significant change at 7 wind AOA, compare to
lower wind AOAs, it can still be inferred from the H 3 diagram that this change in H 2 may not change the level of influence on coupled motion, as the change in H 3 at this wind AOA is relatively small. Overall, it is expected that the thin flat plate changes behavior with increasing wind AOAs may be related to different leading-edge separation patterns thus lead to a dramatic or even a fundamental change in the aerodynamic characteristics and flutter derivatives as compared with those at zero wind AOA. It is noted that further investigation on the flow characteristics analysis of flat plate under different wind angles of attack is of importance and will be discussed in the upcoming papers. 3.4. Refined tests for flutter derivatives As summarized in Section 3.3, since A1 and A2 vary significantly between wind AOAs of 5 and 7 , the model can be assumed to be blunt body between wind AOAs of 5 –7 . To further refine this study, especially for A1 and A2 , additional tests were carried out at wind AOAs of 5.5 , 6 , 6.5 , and 8 . Four dominant flutter derivatives determining the critical wind speed are shown in Fig. 5. As shown in Fig. 5, the uncoupled aerodynamic damping-related flutter derivative, i.e., A2 , shows significant wind AOA-dependent characteristics. At 5.5 , it takes values close to 0, which suggests the important role of uncoupled aerodynamic damping induced by torsional motion in flutter. When the reduced speed approaches 17, it changes sign from negative to positive. At each wind AOA larger than 5.5 , A2 increases rapidly at a given reduced wind speed, leading to a dramatic increasing in negative damping caused by torsional motion and thereby significantly increasing the chance of flutter occurrence. As the wind AOA increases, the critical reduced speed for the thin flat plate for flutter decreases. For A1 , when the reduced speed is below 10, there is a minor difference with a change in wind AOA. When this value is larger than 20, the difference becomes distinct: as the wind AOA increases, A1 decreases. The slope of the curve becomes stable, reaching almost zero, at a reduced speed of 12.5 when the wind AOA is above 7 , which reveals the reduced contribution of coupled aerodynamic damping effects induced by vertical
Fig. 6. Free vibration wind tunnel test for flutter. 7
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Table 1 Parameters under different testing cases. AOAs ( )
Group
m (kg)
I (kg⋅m2)
r=b
fh (Hz)
fα (Hz)
fα /fh
ξh
ξα
0/3/5/7
Group 1 Group 2 Group 3
6.43 6.43 6.43
0.133 0.176 0.216
0.720 0.828 0.916
2.35 2.35 2.34
2.68 3.26 3.77
1.14 1.39 1.61
0.27% 0.19% 0.17%
0.26% 0.23% 0.22%
Note: fh and fα are the natural frequencies in the vertical and torsional directions, respectively; ξh and ξα are structural damping ratios in the vertical and torsional directions, respectively.
motion above this reduced wind speed. A3 and H 3 still remain insensitive with increasing wind AOA, even at apparently large wind AOAs.
similar factor between the vertical mode and torsional mode. The section model wind tunnel test will reduce G to 1.
4. Verification of flutter derivatives by free vibration test
4.2. Critical flutter speed by free vibration wind tunnel test
To verify the accuracy of flutter derivatives for the thin plate model under various wind AOAs, a free vibration section model wind tunnel test is introduced concerning the critical flutter velocity. Conversely, the critical flutter speed can also be theoretically estimated via the flutter derivatives obtained from the forced vibration test by conducting flutter analysis. By comparing the critical flutter speeds from the two methods, the accuracy of the identified flutter derivatives under the forced vibration test can be determined.
Free vibration wind tunnel testing of the same thin flat plate in two degree of freedom (vertical-torsional) was carried out in the XNJD-2 boundary layer wind tunnel. A photograph of the sectional model is shown in Fig. 6. The upstream flow condition was smooth flow. Three groups of dynamic parameters were adopted in the test for full verification. Different dynamic parameter set-ups for the suspension system were designed by adjusting the distance between the springs and the dummy mass. The dynamic parameters for each case are summarized in Table 1. The initial wind AOAs for each case are set to be 0 , 3 , 5 , and 7 , respectively.
4.1. Theory regarding critical flutter speed and flutter derivatives The “bimodal coupled flutter analysis theory” which was proposed by Chen and Kareem (2006) and Chen (2007) and the “step-by-step method” which was proposed by Matsumoto (Matsumoto et al., 1995, 1996 and 1996) are efficient tools for the flutter prediction and dynamic mechanism explanation, by which the contribution of aerodynamic damping and its sub-terms to flutter can be well quantified, other flutter morphological parameters (e.g., frequency, phase difference et al.) as functions of reduced wind speed can be well predicted, thus these methods are widely applied in flutter analysis of long span bridges (e.g., Li et al., 2019; Khang et al., 2016). In this study, the bimodal coupled flutter analysis theory was selected as a tool for the validation of flutter derivatives and for the upcoming flutter analysis. Details concerning the methods are omitted here for the sake of brevity. Simplified bimodal coupled flutter solution (Chen, 2007), which can quantify the flutter performance with parameter γ and indicates how the onset speed changes with the reduced wind speed, is another means to estimate flutter performance. The simplified formulas for bimodal coupled bridge flutter based on closed-form solution are illustrated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 mr 1 2h ωα ρb3
Ucr ¼ γ ωs2 b
4.3. Test results and comparison Based on the parameters given in Table 1, the flutter critical wind speeds of the thin flat plate section model under various wind AOAs are obtained, as shown in Table 2. The critical flutter speeds, which can also be theoretically estimated based on the tested flutter derivatives using the bimodal coupled flutter theory as mentioned in section 4.1, are also presented. It is noted that Vtest and Vcal in Table 2 are critical reduced wind speeds which are obtained by means of free vibration wind tunnel test and theoretical calculation, respectively. It is undoubtedly clear that the critical wind speed decreases considerably with increasing wind AOA, which satisfies the regular flutter performance when it comes from a streamlined section to a bluff one. It is observed that theoretical critical wind speeds (i.e., Vcal ) appear to be in great agreement with those from the wind tunnel test (i.e., Vtest ), with a maximum difference of just 4.5%, corresponding to the case of Group 2 at a 7 wind AOA. The very agreeable results demonstrate the accuracy of the flutter derivatives. Thus, it is reasonable to conclude that the forced vibration technique used in this study for measuring self-excited forces and flutter derivatives under various wind AOAs is quite accurate. These flutter derivatives can be used to precisely predict the critical flutter speed. In addition, the flutter mechanism under various wind AOAs can also be interpreted using the flutter derivatives in this study.
(8)
where Ucr is the critical flutter speed; ωh and ωα are the circular frequency of vertical and torsional motion, respectively; m is the mass per unit pffiffiffiffiffiffiffiffi length; r ¼ I=m is the radius of gyration; I is the mass moment of inertia per unit length; ρ is the air density. γ is a flutter performance evaluation index for a given section that is referred to as the ‘flutter index’ and is expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ ¼ 1 C1 þ C2
C1 ¼ ðr = bÞG
2
k
Table 2 Critical flutter reduced wind speeds for Group1-3.
(9)
2
H *3
* kA1
,2 , 3 1 4 kA* þ 2kξα 1 þ υA* 2 υ5 2
3
AOA ( )
Group
0
Group Group Group Group Group Group Group Group Group Group Group Group
3
(10) 5
C2 ¼ ðb = rÞ k 2 A*3
(11) 7
where υ ¼ ρb4 =I is the non-dimensional effective mass moment of inertia; ξα is the structural damping ratio related to the torsional mode; G is a 8
1 2 3 1 2 3 1 2 3 1 2 3
r=b
fα (Hz)
fα /fh
Vtest
Vcal
Difference
0.720 0.828 0.916 0.720 0.828 0.916 0.720 0.828 0.916 0.720 0.828 0.916
2.68 3.26 3.77 2.68 3.26 3.77 2.68 3.26 3.77 2.68 3.26 3.77
1.14 1.39 1.61 1.14 1.39 1.61 1.14 1.39 1.61 1.14 1.39 1.61
6.13 10.36 12.64 6.07 10.06 12.52 5.78 7.92 10.55 4.68 6.02 6.57
6.29 10.78 12.98 5.82 9.67 12.23 5.97 8.08 10.87 4.88 6.29 6.78
2.61% 4.05% 2.69% 4.12% 3.88% 2.32% 3.29% 2.02% 3.03% 4.27% 4.48% 3.20%
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5. Discussion of the flutter mechanism under various wind Aoas
although the specific meanings of other parameters are omitted here for the sake of brevity (the reader can refer to the literatures (Chen and Kareem, 2006; Chen, 2007) if needed). As an example, the dynamic parameters for Group 1, as listed in Table 1, are used for illustration. Fig. 7 portrays the contributions of the three components listed in Eq. (12) to the total modal damping with increasing reduced wind velocity. The uncoupled terms undoubtedly decrease significantly because of the decrease in A2 at large wind AOAs, especially at a 7 wind AOA. The structural term has a slight influence on the flutter performance because of its slight contribution to total modal damping. However, it is unclear whether the detailed influence of flutter derivatives on the evolution of the coupled term can be determined, even though the coupled term appears to decrease with increasing wind AOAs, as shown in Fig. 7. More detailed investigation was done to analyze the coupled aerodynamic damping term associated with vertical motion in the torsional moment from the microscopic perspective. Fig. 8 portrays the values of sub-terms in coupled damping at critical wind speed Ucr. Changing the flutter derivatives leads to a slight decrease in X, and the coupled damping may decrease if other sub-terms remain constant. However, the modal frequency may influence the parameter Rd2, so that Ψ’ remains stable or even has a slight increase because of the increase in Rd2, which is attributed to the fact that the relative increase in Rd2 is larger than the relative decrease in X, as shown in Fig. 8. Another important parameter, sin ψ ’ , can be approximated to 1 under wind AOAs of 0 , 3 , and 5 , which indicates that coupled damping is uniquely determined by Ψ’ . However, when the wind AOA reaches 7 , the phase difference induced by A*1 =A*4 shows a dramatic change, leading to a sin ψ ’ value of 0.69, which is far less than 1. In this situation, coupled damping may be affected by Ψ’ and ψ ’ simultaneously. In summary, the weak flutter performance at large wind AOAs is mostly attributed to the decrease in uncoupled flutter derivative A2 . On the other hand, coupled damping is slightly changing with changing
It is clear from the literature (Chen and Kareem, 2006; Chen, 2007) that modal damping of the torsional modal branch plays the most important role in controlling flutter instability, as represented by the summation of three components as follows: Structural Term ⇔ ξα ðωα = ω2 Þ
(12)
Uncoupled Term ⇔ 0:5A*2 ρb4 I Coupled Term ⇔
0:5ρ2 b6 Ψ’ sinψ ’ mI
where ωα ¼ 2π fα is the structural circular frequency in the torsional direction; ω2 ¼ 2π f2 is the modal circular frequency in the torsional modal branch; and Ψ’ and ψ ’ , respectively, are represented as follows: Ψ’ ¼ Rd2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi * 2 * 2 * 2 * 2 H2 þ H3 A1 þ A4
ψ ’ ¼ tan1 A*1 A*4 þ ψ Rd2 ¼ ðω2 =ω1 Þ2
(13) (14)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ðω2 =ω1 Þ2 þ ½2ξ1 ðω2 =ω1 Þ2
(15)
where ω1 and ξ1 are the frequency and damping ratio influenced only by the uncoupled self-excited lift force caused by vertical motion; ψ is the phase difference between vertical and torsional motion which takes positive value when the vertical displacement leads the torsional displacement. The expressions of some necessary parameters are given above,
Fig. 7. Contribution of various force components to total torsional damping ratio. 9
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Fig. 8. Values of sub-terms at critical wind speeds Ucr under various wind AOAs.
flutter derivatives, the phase lag ψ ’ may undergo a dramatic change when the wind AOA is sufficiently large.
Table 3 Back calculated flutter indexes for Group1-3.
6. Flutter index under various wind AOAs With the critical flutter speed obtained from the wind tunnel test, the corresponding flutter index can also be back-calculated by: , γ ’ ¼ U ’cr
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ω2 mr ωα b 1 2h ωα ρb3
AOA ( )
Group
0
Group Group Group Group Group Group Group Group Group Group Group Group
3
(16)
5
where U ’cr is the critical flutter speed obtained from the wind tunnel test; and γ ’ is the flutter index related to testing cases. Table 3 shows the back calculated flutter indexes based on free vibration results. Similar to the trend of flutter wind speed, flutter index also shows a decreasing trend with increasing wind AOA, which shows that the flutter performance of a certain section can be directly characterized using the flutter index. Fig. 9 presents the curves of the flutter index as functions of reduced velocity in terms of flutter derivatives under wind AOAs of 0 , 3 , 5 , and 7 , which were calculated using the simplified coupled flutter theory (i.e., Eq. (8)-Eq. (11)). The back-calculated flutter indexes from the free vibration test are also presented, as shown by the hollow points in Fig. 9. The solid points as shown in Fig. 9(d) denote the flutter index that backcalculated from the critical wind speeds estimated using the bimodal coupled flutter theory. The curves of Groups D–F are plotted here but not defined yet, detail of Groups D-F can refer to Table 4. The back-calculated flutter indexes are also consistent with the
7
1 2 3 1 2 3 1 2 3 1 2 3
r=b
fα (Hz)
fα /fh
Vcr
Flutter index
0.720 0.828 0.916 0.720 0.828 0.916 0.720 0.828 0.916 0.720 0.828 0.916
2.68 3.26 3.77 2.68 3.26 3.77 2.68 3.26 3.77 2.68 3.26 3.77
1.14 1.39 1.61 1.14 1.39 1.61 1.14 1.39 1.61 1.14 1.39 1.61
6.13 10.36 12.64 6.07 10.06 12.52 5.78 7.92 10.55 4.68 6.02 6.57
0.4011 0.4210 0.4202 0.3948 0.4108 0.4153 0.3760 0.3283 0.2973 0.3008 0.2559 0.1966
theoretical curves (Groups 1–3) for wind AOAs of 0 , 3 , and 5 , as shown in Fig. 9(a)–(c). Fig. 9(d) reveals that the flutter index from bimodal coupled flutter calculation is almost identical to those from the test, while the curves from the simplified formulas are apparently different from the hollow points. It is mainly attributed to the fact that the approximation implied in the simplified theory, which is illustrated as H *3 A*1 þ H *2 A*4 H *3 A*1 , may not available due to the dramatic increase in H *2 at 7 wind AOA, as shown in Fig. 3, which may lead to a certain level of error in the simplified formulas, as presented in Fig. 9(d). Table 3 and Fig. 9 show that the flutter indexes are closely spaced when the wind AOA is relatively small, especially at 0 . However, the flutter indexes are spaced with significant discrepancies at large wind AOAs, as shown in Fig. 9. To recheck the stability of the flutter index of 10
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Fig. 9. Curves of flutter index at various wind AOAs.
For the section that tends to be a bluff body (i.e., with increasing wind AOAs), similar patterns of critical wind speed also exist for wind AOAs of 3 , 5 , and 7 , as shown in Fig. 10. However, the corresponding flutter indexes cannot remain stable and have space regions of 0.38–0.42, 0.30–0.38, and 0.2–0.37 for wind AOAs of 3 , 5 , and 7 , respectively. In light of this situation, the critical wind speed cannot directly be estimated using the flutter index anymore because of its uncertainty with respect to bluff bodies. The stability of the flutter index under various wind AOAs can be explained qualitatively using the curves of the flutter index shown in Fig. 9, in which the curves of Groups 1–3 and Groups D–F are presented as examples. As shown, at 0 wind AOA, the curves of the flutter index are similar to each other and reach stability when U=fB is greater than 6, which can easily lead to a stable flutter index even when the critical wind speeds are widely distributed. At 3 wind AOA, although the flutter index increases with increasing reduced speed, and the mean value is around 0.4 which can also lead to an acceptable result of critical flutter speed. At 5 wind AOA, the curves of flutter index show a “funnel-shaped” tendency, i.e., the flutter index reaches the bottom at a certain reduced speed, which implies that the dynamic parameters and flutter derivatives produce a mutual effect on the critical flutter speed. At 7 wind AOA, the flutter index decreases dramatically with increasing reduced wind speed. Thus, the flutter index cannot remain stable because a slight change in reduced velocity can induce a significant change in flutter index. Despite the uncertainty in flutter index under large wind AOAs, the curves in Fig. 9 can be good references for either better tailoring of bridge sections or better modifying of dynamic parameters and further achieve good aerodynamic characteristics, as well as enhanced flutter performance. Finally, the most important enlightenment in this study is that the flutter index of bridge girders may be stable because the dynamic parameters of long span bridges are not fluctuated dramatically, and further study will be conducted in future especially for the streamlined box girders.
Table 4 Parameters for extra flutter analysis. AOAs ( )
Group No.
m
I
r=b
fh (Hz)
fα (Hz)
fα /fh
0/3/5
A B C D E F
4.30 4.30 7.50 7.50 10.00 10.00
0.18 0.25 0.34 0.43 0.80 1.50
0.51 0.60 0.53 0.60 0.71 0.97
1.31 1.23 1.13 1.02 1.68 2.35
2.55 3.78 1.65 3.59 4.23 5.21
1.95 3.07 1.46 3.52 2.52 2.22
the thin flat plate section model under various wind AOAs, six distinct groups of dynamic parameters (as compared with Table 1) were used for an additional flutter analysis, as listed in Table 4. In addition, the width of the thin flat plate was also alternatively assumed to be 0.8 m. The modal damping ratio for all cases was assumed to be 0.0032. Fig. 10 summarizes the calculated results for critical wind speed as well as the corresponding flutter indexes. For comparison purposes, the flutter indexes of Groups 1–3, as shown in Table 3, are also presented. The critical wind speeds are widely spaced with respect to each other with a maximum wind speed of Ucr ¼ 32.7 m/s and a minimum wind speed of Ucr ¼ 5.9 m/s, for Groups A–F under 0 wind AOA, which reveals that different groups of dynamic parameters may dramatically influence the flutter performance of a system. However, the flutter indexes under 0 wind AOA are spaced closely together within a region of 0.401–0.421, which gives a flutter index approaching 0.416, as proposed by Selberg’s formula. These results illustrate that for a relatively streamlined section at small wind AOAs, its corresponding flutter index may remain relatively stable at values close to 0.41 and may be insensitive to the given dynamic parameters which induced the fluctuation of the critical flutter speed. Therefore, the critical wind speed may be directly estimated by setting the flutter index at 0.41 and avoiding complex flutter theoretical calculations.
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Fig. 10. Critical wind speed and flutter index for Groups A–F and Groups 1–3.
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7. Concluding remarks
Chen, X., Kareem, A., 2004. Efficacy of the implied approximation in the identification of flutter derivatives. J. Struct. Eng. ASCE 130 (12), 2070–7074. Chen, X., Kareem, A., 2006. Revisiting multimode coupled bridge flutter: some new insights. J. Eng. Mech. 132 (10), 1115–1123. Chen, X., 2007. Improved understanding of bimodal coupled bridge flutter based on closed-form solution. J. Eng. Mech. 133 (1), 22–31. Chen, Z.Q., Yu, X.D., Yang, G., et al., 2005. Wind-induced self-excited loads on bridges. J. Struct. Eng. ASCE 131 (12), 1783–1793. Chowdhury, A.G., Sarkar, P.P., 2003. A new technique for identification of eighteen flutter derivatives using a three-degree-of-freedom section model. Eng. Struct. 25, 1763–1772. Diana, G., Falco, M., Bruni, S., et al., 1995. Comparisons between wind tunnel tests on a full aeroelastic model of the proposed bridge over Stretto di Messina and numerical results. J. Wind Eng. Ind. Aerodyn. 54/55, 101–113. Falco, M., Curami, A., Zasso, A., 1992. Nonlinear effects in sectional model aeroelastic parameters identification. J. Wind Eng. Ind. Aerodyn. 41–44, 1321–1332. Ge, Y.J., Xiang, H.F., 2008. Computational models and methods for aerodynamic flutter of long-span bridges. J. Wind Eng. Ind. Aerodyn. 96, 1912–1924. Gu, M., Xu, S.Z., 2008. Effect of rain on flutter derivatives of bridge decks. Wind Struct. 11 (3), 209–220. Gu, M., Zhang, R.X., Xiang, H.F., 2000. Identification of flutter derivatives of bridge decks. J. Wind Eng. Ind. Aerodyn. 84, 151–162. Haan, F.L., Kareem, A., Szewczyk, A.A., 2000. Experimental Measurements of Spanwise Correlation of Self-Excited Forces on a Rectangular Cross Section. Volume of Abstract, Fourth Int. Colloq. On Bluff Body Aerodynamic and Application. (BBAA IV). Ruhr-University Bochum, Germany, pp. 439–442. Jiang, Y., Zhao, N., Peng, L., et al., 2019. A new hybrid framework for probabilistic wind speed prediction using deep feature selection and multi-error modification. Energy Convers. Manag., 111981 Khang, V., Seils, A., An, N., et al., 2016. An improvement of the step-by-step analysis method for study on passive flutter control of a bridge deck. Arch. Appl. Mech. 86 (3), 557–573. Li, K., Li, S., Ge, Y., et al., 2019. An investigation into the bimodal flutter details based on flutter derivatives’ contribution along the bridge deck’s surface. J. Wind Eng. Ind. Aerodyn. 192, 1–16. Liaw, D.G., 1993. Supersonic flutter of laminated thin plates with thermal effects. J. Aircr. 30 (1), 105–111. Matsumoto, M., Kobayashi, K., Niihara, Y., et al., 1995. Flutter mechanism and its stabilization of bluff bodies. In: Proc. 9th Inter. Conf. On Wind Eng., New Delhi, India. Matsumoto, M., 1996. Aerodynamic damping of prisms. J. Wind Eng. Ind. Aerodyn. 59, 159–175. Matsumoto, M., Kobayashi, Y., Shirato, H., 1996. The influence of aerodynamic derivatives on flutter. J. Wind Eng. Ind. Aerodyn. 60, 227–239. Matsumoto, M., 2013. Flutter and its application—flutter mode and shape navigation. J. Wind Eng. Ind. Aerodyn. 122, 10–20. Noda, M., Utsunomiya, H., Nagao, F., et al., 2003. Effects of oscillation amplitude on flutter derivatives. J. Wind Eng. Ind. Aerodyn. 91, 101–111. Sarkar, P.P., 1992. New-identification Methods Applied to the Response of Flexible Bridges to Wind. PhD thesis. The Johns Hopkins University, Baltimore, Md. Sarkar, P.P., 1994. Identification of aeroelastic parameters of flexible bridges. J. Eng. Mech. 120 (8), 1718–1741. Scanlan, R.H., Tomko, J., 1971. Airfoil and bridge deck flutter derivatives. J. Soil Mech. Found. Div. ASCE 97 (6), 1717–1737. Shinozuka, M., 1982. Identification of linear structure dynamic system. J. Soil Mech. Found. Div. ASCE 108 (6), 1371–1390. Strangfeld, C., Müller-Vahl, H., Nayeri, C.N., et al., 2018. Airfoil in a high amplitude oscillating stream. J. Fluid Mech. 793, 79–108. Tang, H.J., Li, Y.L., Wang, Y.F., et al., 2017. Aerodynamic optimization for flutter performance of steel truss stiffening girder at large angles of attack. J. Wind Eng. Ind. Aerodyn. 168, 260–270. Theodorsen, T., 1935. General Theory of Aerodynamic Instability and the Mechanism of Flutter, vol. 496. NACA TR No., Washington, D.C. Virgoleux, M., 1992. Wind design and analysis for the Normandy bridge. In: Larsen, A., Balkema (Eds.), Aerodynamics of Large Bridges. Rotterdam. Vedeneev, V.V., 2012. Panel flutter at low supersonic speeds. J. Fluids Struct. 29, 79–96. Xu, F.Y., Zhu, L.D., Ge, Y.J., et al., 2014. Some new insights to the identification of bridge deck flutter derivatives. Eng. Struct. 75, 418–428. Yu, Y.L., Amandolese, X., Fan, C.W., et al., 2018. Experimental study and modelling of unsteady aerodynamic forces and moment on flat plate in high amplitude pitch ramp motion. J. Fluid Mech. 846, 82–120. Zasso, A., 1996. Flutter derivatives: advantages of a new representation convention. J. Wind Eng. Ind. Aerodyn. 60, 35–47. Zhu, L.D., Xu, Y.L., Xiang, H.F., 2002. Tsing Ma bridge deck under skew winds—Part II: flutter derivatives. J. Wind Eng. Ind. Aerodyn. 90, 807–837. Zhu, Z.W., Chen, Z.Q., Gu, M., 2009. CFD based simulation of flutter characteristics of ideal thin plate with and without central slot. Wind Struct. 12 (1), 1–19.
The works described in this paper aim at the measurement of flutter derivatives and analysis of flutter instability of a 40:1 thin flat plate section model under various wind AOAs. Stability of flutter index under various wind AOAs were also discussed. Major concluding remarks can then be drawn as follows: (1) The coupled damping related term, i.e., A*2 , shows significant dependency on wind AOAs, which decreasing in absolute value with increasing wind AOAs and changes its sign from negative to positive at wind AOA of 5.5 . (2) A discussion on the mechanism of flutter under different wind AOAs is conducted. It is pointed out that major reason responses for the weakening of flutter performance at large wind AOA is the decreasing of flutter derivative A*2 , while other flutter derivatives have little influence on coupled damping. However, phase angle ψ ’ will has a dramatic change at 7 wind AOA since the dramatic change of A*1 and A*4 . (3) Flutter index is an effective index for evaluating flutter performance of thin flat plate when the wind AOA is less than 3 , and curves of flutter indexes in this study are still good references for either better tailoring bridge sections or changing groups of dynamic parameters, to achieve good aerodynamic characteristics and enhance flutter performance. Moreover, the flutter derivatives of the flat plate model under various wind AOAs may provide alternative choices for the estimations of flutter onset speed of streamlined box girders with large aspect ratios (>10) at the primary design stage when actual flutter derivatives are not experimentally or numerically identified. The future work of this study is to conduct visualized studies by means of computational fluid dynamics (CFD) technique or particle image velocimetry (PIV) technique to check the leading edge separation patterns of a flat plate under various wind AOAs and to figure out the differences and the similarities of flow characteristics between a flat plate and large aspect ratio (>10) streamlined box girders. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The supports for this work provided in part by the Major State Basic Research Development Program (NO. 2013CB036301) from China Ministry of Science and Technology and the National Natural Science Foundation of China (NO.51308478, NO. 51678508, No. 51778547 and NO. U1434205) are greatly acknowledged. References Andersen, M.S., Øiseth, O., Johansson, J., et al., 2018. Flutter derivatives from free decay tests of a rectangular B/D¼10 section estimated by optimized system identification methods. Eng. Struct. 156, 284–293. Chen, A.,R., He, X.F., Xiang, H.F., 2002. Identification of 18 flutter derivatives of bridge decks. J. Wind Eng. Ind. Aerodyn. 90, 2007–2022. Chen, X., Kareem, A., 2001. Nonlinear response analysis of long-span bridges under turbulent winds. J. Wind Eng. Ind. Aerodyn. 89, 1335–1350.
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Flutter derivatives of a flat plate section and analysis of flutter instability at various wind angles of attack Bo Wu a, b, Qi Wang a, b, *, Haili Liao a, b, Yulin Li c, Minghsui Li a, b a b c
Department of Bridge Engineering, Southwest Jiaotong University, Chengdu, 610031, China Key Laboratory for Wind Engineering of Sichuan Province, Chengdu, 610031, China Hunan Provincial Communications Planning, Survey & Design Institute Co., LTD, Changsha, 410000, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Flat plate Flutter derivatives Wind angles of attack Forced vibration test Flutter index
Flutter derivatives of a flat plate section with side ratio of 40 were obtained under various wind angles of attack via forced vibration test, based on which flutter instability was further analyzed in detail. Accuracy of the experimental setup was firstly verified by comparison of flutter derivatives between experimental and theoretical results at 0 wind angle of attack. Detail discussions on distributions of flutter derivatives under various wind angles of attack were conducted and potential influence of changing flutter derivatives on flutter performance was analyzed. It reveals the dramatic changing of A*2 with changing of wind angle of attack. The most significant change in the flutter derivatives is the sign change of A*2 from negative to positive at a 5.5 wind angle of attack. The accuracy of tested flutter derivatives was verified by comparing calculated critical wind speed with those from wind tunnel test. Flutter instability characteristics at various wind angles of attack were discussed, which can aid understanding on flutter behavior of flat plate section by considering effect of large wind angles of attack. Finally, characteristic of flutter index was discussed to demonstrate the flutter performance varying with reduced speed at various wind angles of attack. It remains stable when reduced wind speed less than 6 at wind angle of attack less than 3 , which can be directly used for predicting critical flutter speed of flat plate.
1. Introduction To achieve better understanding of the coupling interactions between fluids and structures, thin plates have been widely investigated in bridge engineering (Noda et al., 2003; Chen et al., 2005; Xu et al., 2014) and aerospace engineering (Theodorsen, 1935; Liaw, 1993; Vedeneev, 2012; Strangfeld et al., 2018; Yu et al., 2018), not only because of their simple aerodynamic geometry, but also as a result of the significant representative positions of thin flat plates on many bridge decks. The self-excited force model of a thin plate, which was theoretically proposed by Theodorsen in the landmark NACA report (Theodorsen, 1935), also highlights the important role of thin plates in wind engineering. Furthermore, the collapse of the old Tacoma Bridge in 1940 has attracted lots of attention of researchers on bridge flutter. Although a wide variety of representation conventions have been introduced to improve the modeling of self-excited forces, which appear to show specific advantages with respect to prediction and interpretation of flutter instability (Zasso, 1996; Virgoleux, 1992; Falco et al., 1992), the typical convention in terms of
flutter derivatives that proposed by Scanlan and Tomoko in 1971 (Scanlan and Tomko, 1971), is still the most accepted tool for flutter analysis. Following their proposals, an explicit nexus between flutter derivatives and the well-known Theodorsen function for thin plates has been built. Based on flutter derivatives which are functions of reduced velocity, the mechanism of certain complex aerodynamic phenomena (e.g., flutter and vortex-induced vibration) can be well explained from this perspective (Ge and Xiang, 2008; Matsumoto et al., 1995; Matsumoto, 2013). It is well known that flutter derivatives of a section are uniquely determined by the aerodynamic configuration and are independent of other parameters. Theoretical results for flutter derivatives can be obtained based on an ideal thin plate model assumption. For bridge decks, wind tunnel test is the most reliable means for qualifying flutter derivatives, despite the great advances in computational fluid dynamics (CFD). In recent years, flutter derivative identification methods, especially those based on wind tunnel testing, have been gradually developed. Because of the pure-mode oscillation assumption of Scanlan’s
* Corresponding author. Department of Bridge Engineering, Southwest Jiaotong University, Chengdu, 610031, China. E-mail addresses: [email protected] (B. Wu), [email protected] (Q. Wang), [email protected] (H. Liao), [email protected] (Y. Li), [email protected] (M. Li). https://doi.org/10.1016/j.jweia.2019.104046 Received 11 June 2019; Received in revised form 20 September 2019; Accepted 16 November 2019 Available online xxxx 0167-6105/© 2019 Elsevier Ltd. All rights reserved.
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As mentioned before, a long-span bridge deck of box girder type with a relatively large aspect ratio (>10) is commonly considered to be an ideal thin plate model, and therefore aerodynamic behavior studies of these bridge decks by wind tunnel tests also adopt the thin plate section model (Gu et al., 2000, 2008; Zhu et al., 2009), considering the representation position and the important role of a thin plate in flutter analysis of bridge engineering for the investigation of flutter derivatives of a thin plate under various wind AOAs. In this study, flutter derivatives of a thin flat plate section model under various wind AOAs were investigated via a forced vibration test. With these flutter derivatives, the critical flutter speed can be derived based on the bimodal coupled flutter solution (e.g., Chen and Kareem, 2006; Chen, 2007). To verify the accuracy of flutter derivatives obtained from the forced vibration test, an additional free vibration section model test was introduced to determine the critical flutter speeds directly. Comparison of critical flutter speeds reveals the accuracy of the estimated flutter derivatives. Subsequently, the flutter instability characteristics of the flat plate section at various wind AOAs were discussed to interpret the effects of the changes in flutter derivatives on flutter behaviors at various wind AOAs, which can aid better understanding of the flutter behavior of a flat plate section by considering the effect of large wind AOAs. Finally, the curves of the flutter index of a thin plate were calculated to demonstrate the flutter performance varying with reduced speed at various wind AOAs and were found to be stable as a constant when wind AOA was less than 3 , which can be directly used for predicting the critical flutter speed of a flat plate.
method, the implementation of experimental techniques and identification accuracy for coupled related terms (e.g., crossed flutter derivatives) have been severely constrained. Thus, considerable efforts have been expended to simplify the experimental procedure and better identify flutter derivatives; many advanced identification methods have subsequently been proposed in recent decades (Shinozuka, 1982; Sarkar, 1992, 1994; Gu et al., 2000; Zhu et al., 2002). Shinozuka (1982) introduced the ARMA model which was widely used in practice (e.g., Jiang et al., 2019) in the system identification procedure, from which the aerodyanmic coefficient matrics of a two dimensional model of a suspension bridge can been extracterd coveniently. Sarkar (1992, 1994) proposed the modified Ibrahim time domain (MITD) method to identify all flutter derivatives from a vertical-torsional coupled free vibration test. Gu et al. (2000) and Zhu et al. (2002) treated the modal properties from the MITD method as initial values for an iterative procedure; in their method, flutter derivatives could further be estimated based on the unifying least squares (ULS) theory. In recent years, methods aiming at flutter derivative identification based on free vibration tests have seen significant improvement (Chowdhury and Sarkar, 2003; Andersen et al., 2018). Chowdhury and Sarkar (2003) proposed the modified iterative least squares (MILS) method that provides a more accurate prediction of the coupled stiffness and damping coefficient matrix than the iterative least squares (ILS) method. Andersen et al. (2018) suggested a hybrid system identification method to estimate the flutter derivatives of a 10:1 rectangular section based on a coupled free vibration test. These frameworks provide more accurate results of bridge flutter derivatives when conducting section-model free vibration wind tunnel tests. However, it is also noteworthy that certain approximations, concerning two distinct modes in free vibration that correspond to two sets of distinct frequency-dependent flutter derivatives, which make identification impossible to implement, are all implied in the methods mentioned above (Chen and Kareem, 2004; Xu et al., 2014). Therefore, forced vibration tests in which a section model can be rigidly supported in a mechanism and can be forced into precise motions with pre-set amplitudes and frequencies has become another optional experimental technique for the identification of flutter derivatives (Matsumoto, 1996; Haan et al., 2000; Chen et al., 2005). This is because the precise measurement of either motion-induced forces or instantaneous surface pressures and a rigorous algorithm with zero assumptions provide more reliable estimations of flutter derivatives. This experimental technique makes the identification of flutter derivatives easier than before. All previous identification studies perform well regarding the identification of flutter derivatives at a 0 wind angle of attack (AOA). Meanwhile, some reports focusing on the identification of flutter derivatives under irregular conditions have also been presented (e.g., Zhu et al., 2002; Gu and Xu, 2008). In addition, some reports have also aimed to investigate the lateral flutter derivatives of bridge decks based on wind tunnel tests instead of quasi-steady theory (e.g., Chen et al., 2002; Chowdhury and Sarkar, 2003). These studies indicate the important influence of external factors on flutter derivatives in bridge decks. However, flutter derivatives of a thin flat plate under various wind angles of attack (AOAs) have rarely been studied before. The wind AOA is an important factor affecting the aerodynamic performance of long-span bridges in mountainous areas. It is generally accepted that even a small fluctuation in wind or weak structural motion could lead to a variety of effective wind AOAs, which may further lead to significant changes in flutter performance; these conditions require accurate sets of flutter derivatives under various wind AOAs to satisfy the modulation of flutter derivatives by the instantaneous wind AOAs when conducting flutter calculations (Diana et al., 1995; Chen and Kareem, 2001). Tang et al. (2017) experimentally investigated the potential influence of the wind AOA on flutter performance of a truss girder and further numerically calculated the flutter derivatives under large wind AOAs. Therefore, the flutter derivatives of bridge decks under various wind AOAs should be estimated accurately before evaluating the influence of the wind AOAs and the turbulence on flutter performance.
2. Forced vibration test for flutter derivatives 2.1. Experimental setup The forced vibration test with a rigidly supported thin flat plate section model was conducted in the first testing section of the XNJD-1 Boundary Layer Wind Tunnel of Southwest Jiaotong University. The working section has a geometric size of 2.0 m in height and 2.4 m in width, which allows wind speeds varying from 0 m/s to 45 m/s continuously. The flow condition was smooth flow, with a turbulence intensity less than 1% (Fig. 1(a)). The thin plate section model is 1.1 m in length (L), 0.4 m in width (B), and 0.01 m in depth (D), which leads to an aspect ratio (B/D) of 40:1, as shown in Fig. 1(b). It is made of carbon so that it can be lightweight yet rigid. Its total weight is 2.2 kg. The test model is fixed to the floor with four linear actuators driven by motors, which are equipped with sensors to synchronously acquire the force and the displacement under a windaxis coordinate system. The section model is eccentrically supported in the mechanism such that four actuators will move synchronously when vertical excitation occurs, while for torsional excitation, only two actuators (No.1 and No.2) on the windward side move synchronously, and the other two (No.3 and No.4) on the leeward side remain still, as shown in Fig. 1(c). The oscillation frequency can be varied from 0 Hz to 20 Hz. The model is set up with a 10-mm vertical vibration amplitude and a 2 torsional vibration amplitude with a fixed vibration frequency of 2.5 Hz; the inflow speed varies from 4 m/s to 18 m/s with an interval of 1 m/s. The wind AOA varies from 0 to 7 with an interval of 1 , a schematic diagram of positive wind AOA with respect to wind direction and bridge transverse axis is shown in Fig. 1(c). Meanwhile, a 6-mm-thick rigid chamfered plate, 600 mm 200 mm in size and oriented parallel to the inflow, is fixed to each end of the model to ensure the plate model is under two-dimensional flow. The height from the section model to the floor is 1 m, so that the model will not be affected by the boundary conditions. 2.2. Theory for extraction of aerodynamic forces and flutter derivatives A 30-Hz cut-off frequency in low-pass filter mode is adopted for both force and displacement sensor signals. Assuming the detected forces on the four force sensors are F1 , F2 , F3 , and F4 , respectively, the total vertical 2
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Fig. 1. Experimental setup and measuring technique.
lift Ltotal and torsional moment Mtotal can be represented by the following equations: Ltotal ¼ F1 þ F2 þ F3 þ F4
(1a)
Mtotal ¼ ðF1 þ F2 Þ ΔS
(1b)
LðtÞ ¼ Ltotal L0 ∕L
(2a)
MðtÞ ¼ Mtotal M0 ∕L
(2b)
where Fi0 ði ¼ 1 4Þ are forces measured by the force sensors in still air conditions with a single degree of freedom (SDOF) for either vertical or torsion motion; L0 ¼ F10 þ F20 þ F30 þ F40 is the inertia lift force under still air; L is the length of the section model; and M0 ¼ ðF10 þF20 Þ ΔS is the inertia moment under still air. For a thin plate with single degree of freedom (SDOF) harmonic motion, the vertical and torsional motions can be expressed by:
where ΔS¼145 mm is the distance between the front actuator and the rear actuator. Both aerodynamic components and inertia components are included in the lift force Ltotal and torsional moment Mtotal in these equations. However, the inertial force is not associated with flutter derivatives and thus should be removed from the total one. The inertia force can be obtained under the same testing conditions, i.e., with the same driven frequency, phase lag, and amplitude, but tested in still air. Thereafter, the aerodynamic lift force L(t) and torsional moment M(t) per unit length can be expressed as:
hðtÞ ¼ h0 sinðωh tÞ
(3a)
αðtÞ ¼ α0 sinðωα tÞ
(3b)
where h0 and α0 are the vertical and torsional amplitudes, respectively; 3
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ωh ¼ 2π fh0 and ωα ¼ 2π fα0 are the circular frequencies of vertical and torsional motion, respectively. The aerodynamic lift force and torsional moment can be written as follows (Chen, 2007): 1 bα_ h h_ Lse ðtÞ ¼ ρU 2 ð2bÞ kH 1 þ kH 2 þ k2 H 3 α þ k 2 H 4 2 b U U
(4a)
h_ 1 bα_ h Mse ðtÞ ¼ ρU 2 2b2 kA1 þ kA2 þ k2 A3 α þ k2 A4 U 2 b U
(4b)
angle; ϕLH is the phase lag between the lift force and the heaving motion, and ϕMH is the phase lag between the moment and the heaving motion; ϕLP is the phase lag between the lift force and the torsional motion, and ϕMP is the phase lag between the moment and the torsional motion; hr0 ¼ h0 =b is the reduced amplitude of the vertical motion. 3. Test results for flutter derivatives 3.1. Aerostatic test Prior to forced vibration test, an aerostatic test was conducted to obtain the aerostatic coefficients of the thin plate model by wind tunnel test. The theoretical aerostatic coefficients for an ideal thin plate are expressed as follows: Lift coefficient
where ρ is the air density; U is the mean wind velocity; b ¼ B= 2 is the half width of the bridge deck; k ¼ ωb=U is the reduced frequency; ω is the circular frequency of motion; and H *i and A*i (i ¼ 1 4) are flutter derivatives, which are functions of reduced frequency. By substituting Eqs. (3a) and (3b) into Eqs. (4a) and (4b), the flutter derivatives can be derived as follows:
CL ¼ 2πα
(6)
Moment coefficient
H *1 ¼
CLH sinðφLH ϕLH Þ 2hr0 k 2
(5a)
H 2 ¼
CLP sinðφLP ϕLP Þ 2α0 k 2
(5b)
H 3 ¼
CLP cosðφLP ϕLP Þ 2α0 k 2
(5c)
H 4 ¼
CLH cosðφLH ϕLH Þ 2hr0 k 2
(5d)
A1 ¼
CMH sinðψ MH ϕMH Þ 2hr0 k 2
(5e)
A*2 ¼
CMP sinðψ MP ϕMP Þ 2α0 k2
(5f)
A*3 ¼
CMP cosðψ MP ϕMP Þ 2α0 k2
(5g)
A*4 ¼
CMH cosðψ MH ϕMH Þ 2hr0 k 2
(5h)
CM ¼ CL =4
(7)
Fig. 2 presents a comparison of the static coefficients of the thin plate model with the theoretical ones for an ideal plate, in which the wind AOAs vary from 10 ~ 10 (0.175–0.175 radian). Fig. 2 demonstrates that both the lift coefficient and moment coefficient show great agreement with theoretical values, in which the slopes of the static lift coefficient curves for experimental and theoretical values are, respectively, 5.83 and 2π, with a corresponding relative difference of only 7.2%. Regarding the static moment coefficients, the slopes for experimental and theoretical values are 1.47 and π=2, respectively, which induces a relative difference of only 6.4%. Therefore, the testing apparatus and the reliability of the model set up were validated. 3.2. Flutter derivatives at zero degree wind AOA Considering that the theoretical solution for an ideal thin plate at 0 wind AOA is known, the flutter derivatives of the thin plate model at 0 wind AOA were tested first to verify the feasibility and accuracy of the forced vibration experimental setup. The tested flutter derivatives at 0 wind AOA obtained from Eq. (5) are shown in Fig. 3. Chen (2007) proposed a simplified closed-from solution for flutter analysis of bridge decks that onset flutter speed can be approximately determined by flutter derivatives of A*1 , A*2 , A*3 , and H *3 , which emphasizes the most important roles of these four flutter derivative to a bridge deck’s (and a thin plate’s) flutter performance. Details about the simplified closed form solution can also be referred to Section 4.1. It is clear from Fig. 3 that these four critical derivatives are close to the theoretical values. The absolute values of A*1 and H *3 which are related to
where CLH is the unsteady lift coefficient generated by vertical motion, and φLH is the corresponding phase angle; CLP is the unsteady lift coefficient generated by torsional motion, and φLP is the corresponding phase angle; CMH is the unsteady moment coefficient generated by heaving motion, and ψ MH is the corresponding phase angle; CMP is the unsteady moment coefficient generated by torsional motion, and ψ MP is the corresponding phase
Fig. 2. Test results for CL and CM. 4
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Fig. 3. Flutter derivatives at zero wind AOA.
flutter morphological parameters (e.g., coupled aerodynamic damping, amplitude ratio and phase angle et al.) and A*2 which is related to uncoupled aerodynamic damping of the thin plate section model, are slightly higher than the theoretical values, while the uncoupled aerodynamic stiffness term of the torsional branch, i.e., the flutter derivative of A*3 , remains almost equal to the theoretical value. The other four derivatives, which are not significantly associated with the critical flutter speed, show a certain level of discrepancy with the theoretical results,
except for H *1 . These results illustrate that the aerodynamic characteristics of an ideal flat plate can be characterized by the thin flat plate section model; further, the accuracy of the forced vibration technique is verified. 3.3. Flutter derivatives under various wind AOAs By modifying the mean position of the two windward-side actuators, the wind AOAs can be modulated to satisfy harmonic motion under 5
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Fig. 4. Flutter derivatives at various wind AOAs (Dashed ellipse in plots of A2 and A4 : Partial enlargement drawing).
various test conditions. The tested flutter derivatives are shown in Fig. 4. By altering the wind AOA, the flutter derivatives of the thin flat plate model changed significantly. Fig. 4 also gives the enlarged drawings for A2 and A4 over a range of [6,8] of reduced wind speed to emphasizes the fundamental change of these two flutter derivatives under large AOAs, as shown in Fig. 4. Regarding the uncoupled related terms, i.e., H 1 , H 4 ; A2 , and A3 , it is inferred from the diagram of H 1 that heaving instability (i.e., galloping) will never occur because the trend of H 1 remains negative and increases
in absolute value with increasing wind AOA. The vertical motion-related stiffness term H 4 shows a significant increase in absolute value, which indicates a probable increase in the modal frequency of the vertical mode branch (Chen and Kareem, 2006). For the torsional mode branch, the potential influence of the wind AOA on the modal properties becomes more significant. While the stiffness-related term A3 is insensitive to changes in the wind AOA, the damping-related term A2 shows a great dependence on the wind AOA. For small wind AOAs (3 or less), A2 is negative and gradually decreases in absolute value as the wind AOA 6
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Fig. 5. Flutter derivatives at wind AOAs within [5 , 8 ].
increases. However, A2 tends to change sign at a 5 wind AOA, even though it is still negative, which indicates that the positive aerodynamic damping induced by torsional motion is close to zero. At a 7 wind AOA, A2 changes sign at a reduced speed of 7.5, which indicates negative aerodynamic damping caused by A2 and torsional flutter instability of the thin plate model. The A2 diagram in Fig. 4 reveals the essential change in the flutter modality of the thin plate for wind AOAs from 0 to 7 . Regarding the coupled motion-related terms, i.e., H 2 , H 3 ; A1 , and A4 , it is generally accepted that these four terms are associate with four coupled parameters between vertical motion and torsion motion, i.e., frequency, damping, phase difference, and amplitude ratio. Thus, it is difficult to quantitatively evaluate the influence of changing flutter derivatives on the coupled parameters resulting from changes in the wind AOA. However, some qualitative predictions can still be conducted using the changing flutter derivatives shown in Fig. 4. This figure demonstrates that the wind AOA had practically no or very little influence on A1 and A4 and H 3 , as compared with the other parameters when wind AOA 5 . At a 7 wind AOA, A1 and A4 may show a dramatic variation that further affects the modification of modal properties, especially the phase lag between motion-induced moment and vertical motion, according to the bimodal flutter theory (e.g., Chen and Kareem, 2006; Chen, 2007). Although H 2 shows a significant change at 7 wind AOA, compare to
lower wind AOAs, it can still be inferred from the H 3 diagram that this change in H 2 may not change the level of influence on coupled motion, as the change in H 3 at this wind AOA is relatively small. Overall, it is expected that the thin flat plate changes behavior with increasing wind AOAs may be related to different leading-edge separation patterns thus lead to a dramatic or even a fundamental change in the aerodynamic characteristics and flutter derivatives as compared with those at zero wind AOA. It is noted that further investigation on the flow characteristics analysis of flat plate under different wind angles of attack is of importance and will be discussed in the upcoming papers. 3.4. Refined tests for flutter derivatives As summarized in Section 3.3, since A1 and A2 vary significantly between wind AOAs of 5 and 7 , the model can be assumed to be blunt body between wind AOAs of 5 –7 . To further refine this study, especially for A1 and A2 , additional tests were carried out at wind AOAs of 5.5 , 6 , 6.5 , and 8 . Four dominant flutter derivatives determining the critical wind speed are shown in Fig. 5. As shown in Fig. 5, the uncoupled aerodynamic damping-related flutter derivative, i.e., A2 , shows significant wind AOA-dependent characteristics. At 5.5 , it takes values close to 0, which suggests the important role of uncoupled aerodynamic damping induced by torsional motion in flutter. When the reduced speed approaches 17, it changes sign from negative to positive. At each wind AOA larger than 5.5 , A2 increases rapidly at a given reduced wind speed, leading to a dramatic increasing in negative damping caused by torsional motion and thereby significantly increasing the chance of flutter occurrence. As the wind AOA increases, the critical reduced speed for the thin flat plate for flutter decreases. For A1 , when the reduced speed is below 10, there is a minor difference with a change in wind AOA. When this value is larger than 20, the difference becomes distinct: as the wind AOA increases, A1 decreases. The slope of the curve becomes stable, reaching almost zero, at a reduced speed of 12.5 when the wind AOA is above 7 , which reveals the reduced contribution of coupled aerodynamic damping effects induced by vertical
Fig. 6. Free vibration wind tunnel test for flutter. 7
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Table 1 Parameters under different testing cases. AOAs ( )
Group
m (kg)
I (kg⋅m2)
r=b
fh (Hz)
fα (Hz)
fα /fh
ξh
ξα
0/3/5/7
Group 1 Group 2 Group 3
6.43 6.43 6.43
0.133 0.176 0.216
0.720 0.828 0.916
2.35 2.35 2.34
2.68 3.26 3.77
1.14 1.39 1.61
0.27% 0.19% 0.17%
0.26% 0.23% 0.22%
Note: fh and fα are the natural frequencies in the vertical and torsional directions, respectively; ξh and ξα are structural damping ratios in the vertical and torsional directions, respectively.
motion above this reduced wind speed. A3 and H 3 still remain insensitive with increasing wind AOA, even at apparently large wind AOAs.
similar factor between the vertical mode and torsional mode. The section model wind tunnel test will reduce G to 1.
4. Verification of flutter derivatives by free vibration test
4.2. Critical flutter speed by free vibration wind tunnel test
To verify the accuracy of flutter derivatives for the thin plate model under various wind AOAs, a free vibration section model wind tunnel test is introduced concerning the critical flutter velocity. Conversely, the critical flutter speed can also be theoretically estimated via the flutter derivatives obtained from the forced vibration test by conducting flutter analysis. By comparing the critical flutter speeds from the two methods, the accuracy of the identified flutter derivatives under the forced vibration test can be determined.
Free vibration wind tunnel testing of the same thin flat plate in two degree of freedom (vertical-torsional) was carried out in the XNJD-2 boundary layer wind tunnel. A photograph of the sectional model is shown in Fig. 6. The upstream flow condition was smooth flow. Three groups of dynamic parameters were adopted in the test for full verification. Different dynamic parameter set-ups for the suspension system were designed by adjusting the distance between the springs and the dummy mass. The dynamic parameters for each case are summarized in Table 1. The initial wind AOAs for each case are set to be 0 , 3 , 5 , and 7 , respectively.
4.1. Theory regarding critical flutter speed and flutter derivatives The “bimodal coupled flutter analysis theory” which was proposed by Chen and Kareem (2006) and Chen (2007) and the “step-by-step method” which was proposed by Matsumoto (Matsumoto et al., 1995, 1996 and 1996) are efficient tools for the flutter prediction and dynamic mechanism explanation, by which the contribution of aerodynamic damping and its sub-terms to flutter can be well quantified, other flutter morphological parameters (e.g., frequency, phase difference et al.) as functions of reduced wind speed can be well predicted, thus these methods are widely applied in flutter analysis of long span bridges (e.g., Li et al., 2019; Khang et al., 2016). In this study, the bimodal coupled flutter analysis theory was selected as a tool for the validation of flutter derivatives and for the upcoming flutter analysis. Details concerning the methods are omitted here for the sake of brevity. Simplified bimodal coupled flutter solution (Chen, 2007), which can quantify the flutter performance with parameter γ and indicates how the onset speed changes with the reduced wind speed, is another means to estimate flutter performance. The simplified formulas for bimodal coupled bridge flutter based on closed-form solution are illustrated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 mr 1 2h ωα ρb3
Ucr ¼ γ ωs2 b
4.3. Test results and comparison Based on the parameters given in Table 1, the flutter critical wind speeds of the thin flat plate section model under various wind AOAs are obtained, as shown in Table 2. The critical flutter speeds, which can also be theoretically estimated based on the tested flutter derivatives using the bimodal coupled flutter theory as mentioned in section 4.1, are also presented. It is noted that Vtest and Vcal in Table 2 are critical reduced wind speeds which are obtained by means of free vibration wind tunnel test and theoretical calculation, respectively. It is undoubtedly clear that the critical wind speed decreases considerably with increasing wind AOA, which satisfies the regular flutter performance when it comes from a streamlined section to a bluff one. It is observed that theoretical critical wind speeds (i.e., Vcal ) appear to be in great agreement with those from the wind tunnel test (i.e., Vtest ), with a maximum difference of just 4.5%, corresponding to the case of Group 2 at a 7 wind AOA. The very agreeable results demonstrate the accuracy of the flutter derivatives. Thus, it is reasonable to conclude that the forced vibration technique used in this study for measuring self-excited forces and flutter derivatives under various wind AOAs is quite accurate. These flutter derivatives can be used to precisely predict the critical flutter speed. In addition, the flutter mechanism under various wind AOAs can also be interpreted using the flutter derivatives in this study.
(8)
where Ucr is the critical flutter speed; ωh and ωα are the circular frequency of vertical and torsional motion, respectively; m is the mass per unit pffiffiffiffiffiffiffiffi length; r ¼ I=m is the radius of gyration; I is the mass moment of inertia per unit length; ρ is the air density. γ is a flutter performance evaluation index for a given section that is referred to as the ‘flutter index’ and is expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ ¼ 1 C1 þ C2
C1 ¼ ðr = bÞG
2
k
Table 2 Critical flutter reduced wind speeds for Group1-3.
(9)
2
H *3
* kA1
,2 , 3 1 4 kA* þ 2kξα 1 þ υA* 2 υ5 2
3
AOA ( )
Group
0
Group Group Group Group Group Group Group Group Group Group Group Group
3
(10) 5
C2 ¼ ðb = rÞ k 2 A*3
(11) 7
where υ ¼ ρb4 =I is the non-dimensional effective mass moment of inertia; ξα is the structural damping ratio related to the torsional mode; G is a 8
1 2 3 1 2 3 1 2 3 1 2 3
r=b
fα (Hz)
fα /fh
Vtest
Vcal
Difference
0.720 0.828 0.916 0.720 0.828 0.916 0.720 0.828 0.916 0.720 0.828 0.916
2.68 3.26 3.77 2.68 3.26 3.77 2.68 3.26 3.77 2.68 3.26 3.77
1.14 1.39 1.61 1.14 1.39 1.61 1.14 1.39 1.61 1.14 1.39 1.61
6.13 10.36 12.64 6.07 10.06 12.52 5.78 7.92 10.55 4.68 6.02 6.57
6.29 10.78 12.98 5.82 9.67 12.23 5.97 8.08 10.87 4.88 6.29 6.78
2.61% 4.05% 2.69% 4.12% 3.88% 2.32% 3.29% 2.02% 3.03% 4.27% 4.48% 3.20%
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5. Discussion of the flutter mechanism under various wind Aoas
although the specific meanings of other parameters are omitted here for the sake of brevity (the reader can refer to the literatures (Chen and Kareem, 2006; Chen, 2007) if needed). As an example, the dynamic parameters for Group 1, as listed in Table 1, are used for illustration. Fig. 7 portrays the contributions of the three components listed in Eq. (12) to the total modal damping with increasing reduced wind velocity. The uncoupled terms undoubtedly decrease significantly because of the decrease in A2 at large wind AOAs, especially at a 7 wind AOA. The structural term has a slight influence on the flutter performance because of its slight contribution to total modal damping. However, it is unclear whether the detailed influence of flutter derivatives on the evolution of the coupled term can be determined, even though the coupled term appears to decrease with increasing wind AOAs, as shown in Fig. 7. More detailed investigation was done to analyze the coupled aerodynamic damping term associated with vertical motion in the torsional moment from the microscopic perspective. Fig. 8 portrays the values of sub-terms in coupled damping at critical wind speed Ucr. Changing the flutter derivatives leads to a slight decrease in X, and the coupled damping may decrease if other sub-terms remain constant. However, the modal frequency may influence the parameter Rd2, so that Ψ’ remains stable or even has a slight increase because of the increase in Rd2, which is attributed to the fact that the relative increase in Rd2 is larger than the relative decrease in X, as shown in Fig. 8. Another important parameter, sin ψ ’ , can be approximated to 1 under wind AOAs of 0 , 3 , and 5 , which indicates that coupled damping is uniquely determined by Ψ’ . However, when the wind AOA reaches 7 , the phase difference induced by A*1 =A*4 shows a dramatic change, leading to a sin ψ ’ value of 0.69, which is far less than 1. In this situation, coupled damping may be affected by Ψ’ and ψ ’ simultaneously. In summary, the weak flutter performance at large wind AOAs is mostly attributed to the decrease in uncoupled flutter derivative A2 . On the other hand, coupled damping is slightly changing with changing
It is clear from the literature (Chen and Kareem, 2006; Chen, 2007) that modal damping of the torsional modal branch plays the most important role in controlling flutter instability, as represented by the summation of three components as follows: Structural Term ⇔ ξα ðωα = ω2 Þ
(12)
Uncoupled Term ⇔ 0:5A*2 ρb4 I Coupled Term ⇔
0:5ρ2 b6 Ψ’ sinψ ’ mI
where ωα ¼ 2π fα is the structural circular frequency in the torsional direction; ω2 ¼ 2π f2 is the modal circular frequency in the torsional modal branch; and Ψ’ and ψ ’ , respectively, are represented as follows: Ψ’ ¼ Rd2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi * 2 * 2 * 2 * 2 H2 þ H3 A1 þ A4
ψ ’ ¼ tan1 A*1 A*4 þ ψ Rd2 ¼ ðω2 =ω1 Þ2
(13) (14)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ðω2 =ω1 Þ2 þ ½2ξ1 ðω2 =ω1 Þ2
(15)
where ω1 and ξ1 are the frequency and damping ratio influenced only by the uncoupled self-excited lift force caused by vertical motion; ψ is the phase difference between vertical and torsional motion which takes positive value when the vertical displacement leads the torsional displacement. The expressions of some necessary parameters are given above,
Fig. 7. Contribution of various force components to total torsional damping ratio. 9
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Fig. 8. Values of sub-terms at critical wind speeds Ucr under various wind AOAs.
flutter derivatives, the phase lag ψ ’ may undergo a dramatic change when the wind AOA is sufficiently large.
Table 3 Back calculated flutter indexes for Group1-3.
6. Flutter index under various wind AOAs With the critical flutter speed obtained from the wind tunnel test, the corresponding flutter index can also be back-calculated by: , γ ’ ¼ U ’cr
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ω2 mr ωα b 1 2h ωα ρb3
AOA ( )
Group
0
Group Group Group Group Group Group Group Group Group Group Group Group
3
(16)
5
where U ’cr is the critical flutter speed obtained from the wind tunnel test; and γ ’ is the flutter index related to testing cases. Table 3 shows the back calculated flutter indexes based on free vibration results. Similar to the trend of flutter wind speed, flutter index also shows a decreasing trend with increasing wind AOA, which shows that the flutter performance of a certain section can be directly characterized using the flutter index. Fig. 9 presents the curves of the flutter index as functions of reduced velocity in terms of flutter derivatives under wind AOAs of 0 , 3 , 5 , and 7 , which were calculated using the simplified coupled flutter theory (i.e., Eq. (8)-Eq. (11)). The back-calculated flutter indexes from the free vibration test are also presented, as shown by the hollow points in Fig. 9. The solid points as shown in Fig. 9(d) denote the flutter index that backcalculated from the critical wind speeds estimated using the bimodal coupled flutter theory. The curves of Groups D–F are plotted here but not defined yet, detail of Groups D-F can refer to Table 4. The back-calculated flutter indexes are also consistent with the
7
1 2 3 1 2 3 1 2 3 1 2 3
r=b
fα (Hz)
fα /fh
Vcr
Flutter index
0.720 0.828 0.916 0.720 0.828 0.916 0.720 0.828 0.916 0.720 0.828 0.916
2.68 3.26 3.77 2.68 3.26 3.77 2.68 3.26 3.77 2.68 3.26 3.77
1.14 1.39 1.61 1.14 1.39 1.61 1.14 1.39 1.61 1.14 1.39 1.61
6.13 10.36 12.64 6.07 10.06 12.52 5.78 7.92 10.55 4.68 6.02 6.57
0.4011 0.4210 0.4202 0.3948 0.4108 0.4153 0.3760 0.3283 0.2973 0.3008 0.2559 0.1966
theoretical curves (Groups 1–3) for wind AOAs of 0 , 3 , and 5 , as shown in Fig. 9(a)–(c). Fig. 9(d) reveals that the flutter index from bimodal coupled flutter calculation is almost identical to those from the test, while the curves from the simplified formulas are apparently different from the hollow points. It is mainly attributed to the fact that the approximation implied in the simplified theory, which is illustrated as H *3 A*1 þ H *2 A*4 H *3 A*1 , may not available due to the dramatic increase in H *2 at 7 wind AOA, as shown in Fig. 3, which may lead to a certain level of error in the simplified formulas, as presented in Fig. 9(d). Table 3 and Fig. 9 show that the flutter indexes are closely spaced when the wind AOA is relatively small, especially at 0 . However, the flutter indexes are spaced with significant discrepancies at large wind AOAs, as shown in Fig. 9. To recheck the stability of the flutter index of 10
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Fig. 9. Curves of flutter index at various wind AOAs.
For the section that tends to be a bluff body (i.e., with increasing wind AOAs), similar patterns of critical wind speed also exist for wind AOAs of 3 , 5 , and 7 , as shown in Fig. 10. However, the corresponding flutter indexes cannot remain stable and have space regions of 0.38–0.42, 0.30–0.38, and 0.2–0.37 for wind AOAs of 3 , 5 , and 7 , respectively. In light of this situation, the critical wind speed cannot directly be estimated using the flutter index anymore because of its uncertainty with respect to bluff bodies. The stability of the flutter index under various wind AOAs can be explained qualitatively using the curves of the flutter index shown in Fig. 9, in which the curves of Groups 1–3 and Groups D–F are presented as examples. As shown, at 0 wind AOA, the curves of the flutter index are similar to each other and reach stability when U=fB is greater than 6, which can easily lead to a stable flutter index even when the critical wind speeds are widely distributed. At 3 wind AOA, although the flutter index increases with increasing reduced speed, and the mean value is around 0.4 which can also lead to an acceptable result of critical flutter speed. At 5 wind AOA, the curves of flutter index show a “funnel-shaped” tendency, i.e., the flutter index reaches the bottom at a certain reduced speed, which implies that the dynamic parameters and flutter derivatives produce a mutual effect on the critical flutter speed. At 7 wind AOA, the flutter index decreases dramatically with increasing reduced wind speed. Thus, the flutter index cannot remain stable because a slight change in reduced velocity can induce a significant change in flutter index. Despite the uncertainty in flutter index under large wind AOAs, the curves in Fig. 9 can be good references for either better tailoring of bridge sections or better modifying of dynamic parameters and further achieve good aerodynamic characteristics, as well as enhanced flutter performance. Finally, the most important enlightenment in this study is that the flutter index of bridge girders may be stable because the dynamic parameters of long span bridges are not fluctuated dramatically, and further study will be conducted in future especially for the streamlined box girders.
Table 4 Parameters for extra flutter analysis. AOAs ( )
Group No.
m
I
r=b
fh (Hz)
fα (Hz)
fα /fh
0/3/5
A B C D E F
4.30 4.30 7.50 7.50 10.00 10.00
0.18 0.25 0.34 0.43 0.80 1.50
0.51 0.60 0.53 0.60 0.71 0.97
1.31 1.23 1.13 1.02 1.68 2.35
2.55 3.78 1.65 3.59 4.23 5.21
1.95 3.07 1.46 3.52 2.52 2.22
the thin flat plate section model under various wind AOAs, six distinct groups of dynamic parameters (as compared with Table 1) were used for an additional flutter analysis, as listed in Table 4. In addition, the width of the thin flat plate was also alternatively assumed to be 0.8 m. The modal damping ratio for all cases was assumed to be 0.0032. Fig. 10 summarizes the calculated results for critical wind speed as well as the corresponding flutter indexes. For comparison purposes, the flutter indexes of Groups 1–3, as shown in Table 3, are also presented. The critical wind speeds are widely spaced with respect to each other with a maximum wind speed of Ucr ¼ 32.7 m/s and a minimum wind speed of Ucr ¼ 5.9 m/s, for Groups A–F under 0 wind AOA, which reveals that different groups of dynamic parameters may dramatically influence the flutter performance of a system. However, the flutter indexes under 0 wind AOA are spaced closely together within a region of 0.401–0.421, which gives a flutter index approaching 0.416, as proposed by Selberg’s formula. These results illustrate that for a relatively streamlined section at small wind AOAs, its corresponding flutter index may remain relatively stable at values close to 0.41 and may be insensitive to the given dynamic parameters which induced the fluctuation of the critical flutter speed. Therefore, the critical wind speed may be directly estimated by setting the flutter index at 0.41 and avoiding complex flutter theoretical calculations.
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104046
Fig. 10. Critical wind speed and flutter index for Groups A–F and Groups 1–3.
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7. Concluding remarks
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The works described in this paper aim at the measurement of flutter derivatives and analysis of flutter instability of a 40:1 thin flat plate section model under various wind AOAs. Stability of flutter index under various wind AOAs were also discussed. Major concluding remarks can then be drawn as follows: (1) The coupled damping related term, i.e., A*2 , shows significant dependency on wind AOAs, which decreasing in absolute value with increasing wind AOAs and changes its sign from negative to positive at wind AOA of 5.5 . (2) A discussion on the mechanism of flutter under different wind AOAs is conducted. It is pointed out that major reason responses for the weakening of flutter performance at large wind AOA is the decreasing of flutter derivative A*2 , while other flutter derivatives have little influence on coupled damping. However, phase angle ψ ’ will has a dramatic change at 7 wind AOA since the dramatic change of A*1 and A*4 . (3) Flutter index is an effective index for evaluating flutter performance of thin flat plate when the wind AOA is less than 3 , and curves of flutter indexes in this study are still good references for either better tailoring bridge sections or changing groups of dynamic parameters, to achieve good aerodynamic characteristics and enhance flutter performance. Moreover, the flutter derivatives of the flat plate model under various wind AOAs may provide alternative choices for the estimations of flutter onset speed of streamlined box girders with large aspect ratios (>10) at the primary design stage when actual flutter derivatives are not experimentally or numerically identified. The future work of this study is to conduct visualized studies by means of computational fluid dynamics (CFD) technique or particle image velocimetry (PIV) technique to check the leading edge separation patterns of a flat plate under various wind AOAs and to figure out the differences and the similarities of flow characteristics between a flat plate and large aspect ratio (>10) streamlined box girders. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The supports for this work provided in part by the Major State Basic Research Development Program (NO. 2013CB036301) from China Ministry of Science and Technology and the National Natural Science Foundation of China (NO.51308478, NO. 51678508, No. 51778547 and NO. U1434205) are greatly acknowledged. References Andersen, M.S., Øiseth, O., Johansson, J., et al., 2018. Flutter derivatives from free decay tests of a rectangular B/D¼10 section estimated by optimized system identification methods. Eng. Struct. 156, 284–293. Chen, A.,R., He, X.F., Xiang, H.F., 2002. Identification of 18 flutter derivatives of bridge decks. J. Wind Eng. Ind. Aerodyn. 90, 2007–2022. Chen, X., Kareem, A., 2001. Nonlinear response analysis of long-span bridges under turbulent winds. J. Wind Eng. Ind. Aerodyn. 89, 1335–1350.
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