Signal complexity approach based investigation of flutter instability in sectional models of bridges in wind tunnel

Abstract

Signal complexity analysis based entropy measure has been used to propose a model-free approach towards assessment of flutter critical wind speed and identification of the type of flutter in wind tunnel testing of long span bridges. The approach is based only on the measured output responses from the structure, which accounts for appropriate effects that stem from the non-linearity of the structure and fluid structure interaction. The usefulness of the proposed approach has been demonstrated with dynamic sectional models of two deck cross-sections in wind tunnel. The gain in regularity of responses on approaching the flutter critical wind speed, demonstrated with time history of responses and recurrence quantification plots, has been quantified using sample entropy. Based on the sensitivity of the sample entropy values on the embedding dimension, it has been suggested to adopt embedding dimension in the range of 2 and 5, to obtain a swift estimate of the state of the dynamic system. The proposed approach has shown potential to provide valuable insight about the flutter instability and distinguish between different types of flutter without requiring prior knowledge about the system under consideration. The study offers a promising complementary methodology to the conventional model-based approaches for evaluating flutter critical wind speed.

Keywords

signal complexity bridge deck section flutter derivatives reduced velocity flutter critical wind speed free oscillation sample entropy

Introduction

Long span bridges are highly flexible with low damping characteristics, making them sensitive to dynamic wind action. Lack of considerations on the dynamic action of wind has resulted in catastrophic failure of bridges in the past (Jurado et al., 2011; Miyata, 2003). Wind tunnel testing continues to be a reliable mode of evaluation of wind-structure interaction effects on the structural stability (Chen et al., 2017, 2018, 2021, 2022). Flutter and galloping are dominant wind induced aeroelastic phenomenon in long span cable stayed/suspension bridges that needs to be accounted for during the early stages of design. Flutter is a self-excited phenomenon, characterized by sustained, high-frequency oscillation of the structure that can grow rapidly and lead to structural failure. Galloping is an aeroelastic instability, characterized by large amplitude, low frequency vibration of the structure caused by the interaction between wind and structure (Hu et al., 2015, 2016). The focus of the present study is flutter phenomenon which is reported to be critical owing to the potential to cause structural failure (Caracoglia, 2013; Cheng et al., 2005).

Conventionally, the motion-induced aeroelastic forces in bridges are expressed through non-dimensional “flutter derivatives” in the frequency domain or “impulse response/indicial function” in time domain (Fung, 1993; Scanlan and Tomko, 1971), among which Scanlan model has been widely accepted among all other alternate formulations (Chowdhury and Sarkar, 2003; Li et al., 2019; Wu et al., 2020). The objective of all the formulations is to identify the flutter critical wind speed (V_cr), which is a supercritical Hopf bifurcation point. Established numerous approaches for evaluation of flutter critical wind speed have been reported in literature (Abbas et al., 2017; Kavrakov and Morgenthal, 2018). All these approaches adopt a model based procedure involving solution of a non-linear eigen value problem, where flutter critical wind speed is identified as the wind speed at which net damping of the system becomes zero. Variation in flutter critical wind speed of the order of 5% to 30% have been reported in literature from comparison of inter-laboratory data (Caracoglia et al., 2009). This variability is attributed to numerous factors like method of extraction of V_cr, accuracy of the mathematical model in description of the physical system, type of deck section, nature of flutter instability, etc., Further, the flow around bridge decks/bluff bodies are highly complex involving turbulent eddies of multitude scales and frequencies (Chen et al., 2023; Gao et al., 2022; Hu et al., 2017; Zhou et al., 2022). The structure hence exhibits a variety of behaviors, viz, nonlinearity, dependence on inflow wind characteristics, geometric shape, etc (Gao et al., 2020). Adopting a universal mathematical model/formulation for characterizing the system behavior may not be accurate. The assessment of critical wind speed through model based approaches could be better supported by a “model free” methodology that is based on the output responses from the structure. The model-free approach takes into account of the appropriate effects that stem out of the non-linearity of the structure and fluid structure interaction. Further, it offers flexibility for application to the full scale structure to monitor occurrence of wind induced excitation from the output only responses. Venkatramani et al. (Venkatramani et al., 2016, 2017, 2018) for the first time had adopted a model free approach towards evaluation of flutter boundary and suggested a set of precursors for flutter of an aerofoil section.

The present study proposes a similar “model-free” approach using signal complexity analysis based entropy measures towards assessment of flutter critical wind speed and identification of the type of flutter. Application of such signal complexity/entropy based approaches for studies on flutter phenomenon of long span bridge deck sections or other civil engineering strucures have not been reported in literature. The usefulness of the proposed approach has been demonstrated from dynamic sectional models of two deck cross-sections, viz, thin plate section (Model-1) and trapezoidal section with overhangs (Model-2) in wind tunnel. The geometric details of the models are shown in Figure 1. In the wind tunnel tests, identification of modal properties of the sectional model becomes challenging at higher wind speeds, especially close to flutter critical wind speed, due to limited length of useful data in vertical degree of freedom (Boonyapinyo and Janesupasaeree, 2010; Janesupasaeree and Boonyapinyo, 2009). This could give rise to uncertainty in the evaluation of flutter critical wind speed, which can be overcome by adopting an entropy based complementary model free methodology.

Figure 1.

Geometric details of the bridge deck cross-sections studied.

At wind speeds less than flutter critical wind speed, transition in time history of responses from random pattern to sinusoidal motion has been observed from the experimental data. Quantification of this reduction in randomness, as occurring during flutter phenomenon has been studied through entropy measures. Such impetuous changes that occur in the states of physical system have been studied only with the aid of measured responses. The results show the potential of the proposed approach to provide valuable insight about the flutter instability and distinguish between different types of flutter without requiring prior knowledge about the system under consideration. The paper is organized as follows: The details of experimental set up and basic results have been presented in the Section 2. A brief mathematical background and formulations of the entropy measures have been presented in Section 3, followed by results and discussions with parametric studies on entropy measures.

Experimental investigations

The wind tunnel investigations have been carried out in the wind tunnel facility of CSIR-Structural Engineering Research Centre (Figure 2). The cross-sectional dimensions of the wind tunnel are 2.5 m (width) × 1.8 m (depth). Model-1 is a rectangular section with a width of 0.445 m and a depth of 0.025 m, thus having cross-sectional aspect ratio (the ratio of width to depth) of 20. This cross-section has been closely regarded as thin plate section in literature (Gu and Qin, 2004). Model-2 is a trapezoidal cross-section with overhangs. Both the sectional models with a length of 2.3 m, occupying nearly full width of the wind tunnel has been fabricated to avoid overestimation of the span wise correlation effects (King, 2003). By virtue of this arrangement, 2-D flow around the model has been ensured.

Figure 2.

Wind tunnel facility of CSIR-SERC. (a) Schematic view, (b) Photo of the facility (viewed from inlet).

Model fabrication and test set-up

The models have been fabricated with a skeletal frame of extruded aluminium sections, with a single core member appropriately stiffened throughout the length of the model. The stiffened core member has been attached to suspension arms at both the ends. Over the skeletal frame, the external geometry of the sectional model has been achieved by means of acrylic sheets with thickness of 3 mm.

Modern day steel/steel concrete composite bridges of widths 30–40 m have been observed to have mass per unit length in the range of 15,000 kg/m to 40,000 kg/m and mass moment of inertia per unit length in the range of 1,500,000 to 6,000,000 kg-m²/m (Zhang and Xu, 2018). A geometric scale of 1:50 is typically adopted for dynamic sectional model testing. Hence, for this geometric scale, these values would be in the range of 6 kg/m to 16 kg/m and 0.24 to 0.96 kg-m²/m for the model. Congruent with these practical values, the properties of the model have been chosen and they are listed in Table 1. The model has been suspended from the two aluminium frames attached to the walls of wind tunnel using eight springs, i.e., four at the top and four at the bottom. The depth of supporting frame is designed to be large towards accommodating the springs of requisite stiffness for mounting the model (Ge et al., 2018). The springs have been attached to the model through the suspension arms. Based on the required frequencies of the sectional model in vertical and torsional degrees of freedom, the stiffness of tension springs has been designed.

Table 1.

Properties of the sectional models.

Property	Model-1A	Model-1B	Model-2
Length	2.3 m	2.3 m	2.3 m
Width (B)	0.445 m	0.445 m	0.45 m
Depth (D)	0.025 m	0.025 m	0.07 m
B/D ratio	20	20	6.36
Mass per unit length	7.15 kg/m	7.15 kg/m	6.9 kg/m
Mass moment of inertia per unit length	0.1511 kg-m²/m	0.1511 kg-m²/m	0.0796 kg-m²/m
Vertical frequency	1.7 Hz	2.6 Hz	2.6 Hz
Torsional frequency	3.1 Hz	5.1 Hz	5.3 Hz
Frequency ratio (torsional frequency/vertical frequency)	1.75	2.01	2.05
Vertical damping ratio	1.4%	1.2%	1.2%
Torsional damping ratio	0.7%	0.65%	0.65%

The combined stiffness of the system of springs under still air condition in both vertical and torsional degrees of freedom has been evaluated from the load displacement plots (Amandolese et al., 2013). For the evaluation of vertical stiffness, the standard weights (load) were applied in vertical direction along the centre line of the model. For the evaluation of torsional stiffness, the weights were applied eccentrically about the centre line of the model. Based on the load displacement relationship, the stiffness of the combined system in vertical and torsional degrees of freedom have been evaluated. It can be observed that the system of springs behaves linearly for the applied range of loads. The vertical and torsional stiffness have been obtained as 4280.5 N/m and 208.5 N-m/rad, respectively (Figure 3).

Figure 3.

(a) Vertical and (b) torsional stiffness of the system.

Ball bearing of rolling element type has been used at both the ends of the model to allow movement of the model in torsional degree of freedom. A linear motion bearing has been employed in conjunction with the ball bearing of rolling element type to allow movement of the sectional model in vertical degree of freedom. This linear motion bearing slides along a polished steel shaft attached to the frame. The friction due to contact of the balls in case of linear motion bearing have been minimized by applying lubricant along the surface of the steel shaft. This end fixture arrangement (close-up view provided Figure 4(b)) arrests the movement of the sectional model in drag/along wind degree of freedom. Schematic view of the test set up has been shown in Figure 4. An enclosure with acrylic sheets with aerodynamically rounded corners has been made to avoid springs from being directly exposed to the oncoming wind flow, and also to act as end plate for the model. Views of the sectional models mounted in wind tunnel have been shown in Figures 5 and 6 for Model-1 and Model-2, respectively.

Figure 4.

Schematic view of the test set-up for mounting sectional models on springs. (a) Overall view, (b) Close up view of the end set up with bearings.

Figure 5.

(a) (b) Views of Model-1 in wind tunnel including (c) close-up view of the instrumentation.

Figure 6.

View of Model-2 in wind tunnel.

For Model-1, experiments corresponding to two different sets of dynamic properties have been carried out, viz, Model-1A and Model-1B (Table 1), by varying the spring stiffness and separation distances. The model has been instrumented with four miniature accelerometers (DeltaTron Accelerometer Type 4507 B 006).

Two accelerometers have been located two at the center of the span of model and the other two at the end of the model to measure the acceleration responses. Further, four non-contact laser displacement sensors with two at the centre and two at one end of span of the model have been used to measure the displacement response. The model has been mounted at a height of 0.8 m from the floor of the wind tunnel, which is nearly half the height of the wind tunnel.

Experimental cases

All the experiments have been carried out under uniform, smooth inflow condition with turbulence intensity less than 1%, as ensured through velocity measurement (Figure 7). Time history of wind velocity at the level where model is mounted is presented in Figure 8(a). Based on the measurements, profiles of mean wind velocity and turbulence intensity with height (z) are obtained and are shown in Figure 8(b) and Figure 8(c). At the level of 0.8 to 0.9 m, where the model has been mounted, smooth flow with turbulence intensity of 0.6% has been observed. This value is consistent with the values reported in literature for smooth inflow condition. The experimental cases carried out has been listed as follows:

(a) Free vibration in vertical degree of freedom under still wind condition.

(b) Free vibration in torsional degree of freedom under still wind condition.

(c) Free decay in combined vertical and torsional degrees of freedom at various wind speeds (with initial excitation provided in both vertical and torsional degrees of freedom) – also termed coupled free decay tests.

(d) Free oscillation flutter tests for various wind speeds.

Figure 7.

Velocity measurement using Cobra probes.

Figure 8.

Time history of wind velocity, profile of mean wind speed and turbulence intensity.

It is to be noted that the model behavior during free oscillation flutter tests closely resemble the kinetic characteristics under natural wind rather than forced oscillation approach (Zhang et al., 2012). Under wind flow, the model has been given an initial displacement/rotation to obtain the free decay responses. For free vibration tests under still wind conditions, initial displacement in vertical degree of freedom (h_io) of 30 mm is provided. A value of h_io of 20 mm has been provided along with an initial rotation (α_io) of +3° and −3° for free decay tests at various wind speeds. A sampling frequency of 1200 Hz has been used for sampling the data from accelerometers and laser displacement sensors for a duration of 120 s.

The vertical (h) and torsional (α) components of the responses have been obtained from accelerometer and displacement sensors as follows (Qin and Gu, 2004):

h = \frac{s 1 + s 2}{2}; α = \frac{s 1 - s 2}{l}

(1)

where s1 and s2 are the output (displacement or acceleration) from two sensors along the measurement axis, separated by a distance of l.

Measured responses

Free vibration tests - still wind condition

From the time history of responses, Welch’s method is used to estimate power spectral density (PSD). The time history is divided to a number of overlapping segments of equal length and periodogram is computed for each segment. The periodograms from various segments are averaged to obtain PSD in frequency domain, also termed as spectrum. The peak in spectrum corresponds to the natural frequency of the model. The damping ratio of the model has been evaluated based on logarithmic decrement technique.

Time history and spectrum of the free vibration tests in vertical and torsional degrees of freedom under still wind for Model-2 have been shown in Figure 9. Similar plots for other models have not been presented for the sake of brevity. The natural frequencies and damping of the models have been evaluated from these plots for all the models.

Figure 9.

Free vibration responses in vertical and torsional degrees of freedom for Model-2.

Free oscillation flutter tests at various wind speeds

Model-1A and Model-1B have been tested from a wind speed of 3 m/s up to wind speed of 12.2 m/s and 19.67 m/s, respectively. The wind speed for wind tunnel testing has been increased from 3 m/s at interval of 1 m/s. The flutter critical wind speed for the considered dynamic sectional model has been initially calculated using Selberg’s formula (Equation (16)), which is empirical in nature. On approaching the wind speed close to that predicted as flutter critical wind speed using Selberg’s formula, finer wind speed interval of 0.3 m/s to 0.4 m/s has been adopted to ensure that the increase in response would be systematically captured.

The models have been tested upto wind speed at which they exhibit divergent amplitude behavior. From the time histories of the vertical and torsional responses, the maximum, mean and root mean square (rms) components have been evaluated. Figure 10 shows the variation of fluctuating components of vertical and torsional response with wind speed and reduced velocity (U/f_hB), calculated using corresponding frequencies at every wind speed.

Figure 10.

Variation of fluctuating components of vertical and torsional responses with reduced velocity for Model-1.

It can be observed from Figure 10 that for Model-1A, the rms value of vertical (h_rms) and torsional (α_rms) displacements show sharp increase by 172% and 305%, respectively on slight increase in wind speed from 11.77 m/s to 12.2 m/s, similar to the studies on rectangular section by Bartoli and Righi (2006). For Model-1B, this nature of sharp increase in fluctuating components of displacements with marginal increase in wind speed could not be carried out owing to rapidly growing oscillations during the tests. A good comparison of fluctuating components of responses and reduced velocity (U/fB) for onset of flutter has been observed for Model-1A and Model-1B. This aids in gaining confidence on the prediction of occurrence of flutter instability in wind tunnel testing and extendibility of the results, for the same geometric model with different set of dynamic properties.

At a wind speed of 11.77 m/s, the model oscillated with same frequency in vertical and torsional degrees of freedom, undergoing behavior similar to Limit Cycle Oscillations (LCO), a manifestation of non-linearity in time domain (Gao et al., 2021). At this flutter boundary, marked by flutter critical wind speed (V_cr), the damping of the dynamics system vanishes. Hence, pure harmonic oscillations (Nieto et al., 2009) are caused and is termed LCO as shown in Figure 11. On crossing wind speeds beyond this speed, divergent nature of responses has been observed. Time history and spectrum of scaled vertical displacement and torsional rotation at this wind speed is shown in Figure 11 for Model-1A.

Figure 11.

Time history and spectrum of free oscillation flutter test based responses at wind speed of 11.77 m/s for Model-1A. (a) Time history of responses, (b) Spectrum of vertical response, (c) Spectrum of torsional response.

The variation of fluctuating component of responses from free decay and free oscillation flutter tests for Model-2 has been shown in Figure 12. Comparison of rms of responses with a geometrically similar model (Mannini, 2006) corresponding to Model-2 has been included in Figure 12. Similarity in the trend of rms of responses with literature has been observed. The differences in magnitude is due to difference in modal properties of natural frequency and damping ratio between the Model-2 and that of literature. Owing to reduced energy of the vortex shedding (Mannini, 2006; Mannini et al., 2016; Zhou et al., 2017) at such low wind speeds, no distinct peak corresponding to vortex shedding has been observed in the plot of rms of responses. The onset of vortex induced vibration of first few modes being less than 2 m/s and relatively small amplitude of vibration from first vibration regime starting at reduced velocity of 4.8 have been reported (Zhou et al., 2017). For positive mean incident angle case of +3°, occurrence of divergent nature of resposes have been observed at lesser wind speed of about 14.5 m/s.

Figure 12.

Variation of fluctuating components of (a) vertical and (b) torsional responses with wind speed for Model-2.

Free decay tests and free oscillation flutter tests - at various wind speeds

The evolution of frequencies and damping ratios with wind speeds for Model-1A and Model-1B, based on free decay tests and free oscillation flutter tests carried out at various wind speeds are presented in Figures 13 and 14 respectively. The frequencies and damping ratios from free oscillation flutter tests have been evaluated based on Data driven Stochastic Subspace system Identification (SSI-DATA) (Boonyapinyo and Janesupasaeree, 2010) algorithm. The difficulty with free oscillation flutter tests were overcome by adopting a methodology of data driven SSI-DATA algorithm, the procedure involved being briefly presented. The simulated responses (y_n) have been organised into block Hankel matrix, which is then divided into past reference $Y_{p}$ and future $Y_{f}$ . An LQ decomposition of the Hankel matrix has been carried out to express it as a product of lower triangular matrix ‘L’ and an orthogonal matrix ‘Q’ with property Q Q^T = Q^TQ = I, as given below

[H_{0 | 2 i - 1}] = \frac{1}{\sqrt{j}} (\frac{\begin{array}{c} y_{0} & y_{1} & . . . & y_{j - 1} \\ . . . & . . . & . . . & . . . \\ y_{i - 1} & y_{i} & . . . & y_{i + j - 2} \\ y_{i} & y_{i + 1} & . . . & y_{i + j - 1} \end{array}}{\begin{array}{c} y_{i + 1} & y_{i + 2} & . . . & y_{i + j} \\ y_{i + 2} & y_{i + 3} & . . . & y_{i + j + 1} \\ . . . & . . . & . . . & . . . \\ y_{2 i - 1} & y_{2 i} & . . . & y_{2 i + j - 2} \end{array}}) = (\frac{Y_{0 | i}}{Y_{i + 1 | 2 i - 1}}) = (\frac{Y_{p}}{Y_{f}}) \frac{↕ l (i)}{↕ l (i)} = [L] [Q]

(2)

Figure 13.

Variation of (a) frequency and (b) damping ratio with wind speed for Model-1A.

Figure 14.

Variation of (a) frequency and (b) damping ratio with wind speed for Model-1B.

The forward state of the system can be identified by projection of row space of future outputs on to the row space of past outputs. Hence, projection matrices have been constructed as given below.

\begin{array}{l} [Π_{i}] = [Y_{f}] / [Y_{p}] \\ [Π_{i - 1}] = [{Y_{f}}^{-}] / [{Y_{p}}^{+}] \end{array}

(3)

The projection matrix can be factorised into observability matrix $[O_{i}]$ and Kalman filter state sequence $[{\hat{S}}_{i}]$ as per the main theorem of stochastic subspace identification, as given by.

[Π_{i}] = [O_{i}] [{\hat{S}}_{i}]

(4)

The Singular Value Decomposition (SVD) of the projection matrix has been carried out and maximum model order ‘n’ of the system has been identified.

[Π_{i}] = [U] [Σ] {[V]}^{T} = [\begin{array}{c} [U_{1}] & [U_{2}] \end{array}] [\begin{array}{c} [Σ_{1}] & [0] \\ [0] & [0] \end{array}] [\begin{array}{c} [V_{1}^{T}] \\ [V_{2}^{T}] \end{array}] \approx [U_{1}] [Σ_{1}] [V_{1}^{T}]

(5)

where

[U]

and

[V]

are orthonormal matrices.

[Σ]

is a matrix comprising of positive singular values in descending order. The observability and Kalman filter state sequence

[{\hat{S}}_{i}]

and shifted state sequence

[{\hat{S}}_{i + 1}]

has been used to construct the system matrices.

[\begin{array}{c} [A] \\ [C^{*}] \end{array}] = [\begin{array}{c} [{\hat{S}}_{i + 1}] \\ [Y_{i | i}] \end{array}] {[{\hat{S}}_{i}]}^{+}

(6)

where,

[Y_{i | i}]

is the (i+1)^th row of the Hankel matrix.

The modal parameters, viz, natural frequencies and damping ratios of the dynamic system have been evaluated by performing eigen value analysis of the state matrix $[A]$ as given below

\begin{array}{c} [A] = {[Ψ] [Λ] [Ψ]}^{- 1} \\ [Φ] = [C^{*}] [Ψ] \end{array}

(7)

where

[Λ]

and

[Ψ]

are the eigen values and eigen vectors matrices

$[Φ]$ is the mode shape matrix.

From the eigen values, the vertical and torsional frequencies (ω_i) and damping ratios (ξ_i) have been evaluated as follows

\frac{1}{Δ t} \ln ([Λ]) = d i a g (a_{i} \pm i b_{i})

ξ_{i} = - \frac{a_{i}}{\sqrt{a_{i}^{2} + b_{i}^{2}}} = - \frac{r e a l (\ln ([Λ]) / Δ t)}{a b s (\ln ([Λ]) / Δ t)}

ω_{i} = - a b s (i m a g (\ln ([Λ]) / Δ t))

(8)

where a_i and b_i are real and imaginary parts of the natural logarithm of the eigen values of state matrix, divided by the sampling time (Δt). The advantage of SSI-DATA is the ability to identify the system matrices and modal parameters with measured responses which are inherently noisy (Boonyapinyo and Janesupasaeree, 2010).

The vertical and torsional frequencies were clearly discerned by corroborating the values from the free decay tests and free oscillation flutter tests. However, it can be observed from the plots that the scatter in variation of damping ratio at higher wind speeds is very high. This could be the reason for variation in the estimated value of critical wind speed of the order of 5% to more than 30% as reported in literature (Caracoglia et al., 2009).

For Model-1A and Model-1B, the torsional frequency has been observed to reduce with increasing wind speed, whereas the vertical frequency is observed to remain constant up to a wind speed (of 10.97 m/s for Model-1A and 19.35 m/s for Model-1B). However, beyond this wind speed, vertical frequency sharply increased and reached the value of torsional frequency as also reported in literature. This is typical behavior observed in case of coupled flutter (Bartoli and Righi, 2006; Glumac et al., 2017; Gao et al., 2020).

Based on the mean trend in the variation of damping ratio, it has been observed that the damping ratio in vertical degree of freedom increased with increase in wind speed reaching values as high as 8% for Model-1A at a wind speed of 10.64 m/s and 14% for Model-1B at a wind speed of 19.67 m/s. The damping ratio in torsional degree of freedom is observed to gradually increase with increase in wind speed and then reduced sharply to a value close to zero value at a wind speed of 11.67 m/s for Model-1A.

For Model-1B, at a wind speed of 19.67 m/s, reduction of damping in the torsional degree of freedom to an extent lesser than that of the damping in still wind condition has been observed. This can be clearly seen by comparing the free decay response at this wind speed with that of still wind in Figure 15. The response in both degrees of freedom has been observed to be dominated by torsional mode as shown in Figure 16. The behavior of flat plates with frequency separation ration, i.e., ratio of torsional natural frequency (f_α0) to vertical natural frequency (f_h0) of two as considered in the present study has been reported to be dominated by torsional mode (Yang et al., 2018).

Figure 15.

Time history and spectrum of responses from free decay tests at wind speed of 19.67 m/s for Model-1B. (a) Time history of responses, (b) Spectrum of vertical response, (c) Spectrum of torsional response.

Figure 16.

Comparison of time history of responses from free decay tests at wind speed of 19.67 m/s and still wind for Model-1B.

For Model-2 (Figure 17), the torsional frequency has been observed to reduce with increasing wind speed, whereas the vertical frequency is observed to remain constant for the entire range of wind speeds. The torsional frequency has been observed to reduce from a value of 5.3 Hz to 4.67 Hz at wind speed of 16 m/s. It has been observed that the damping ratio in vertical degree of freedom increased with increase in wind speed reaching values as high as 13% at a wind speed of 16 m/s. The damping ratio in torsional degree of freedom is observed to gradually increase with increase in wind speed and then reduced sharply to a value close to zero value at a wind speed of 16.1 m/s. It has been observed from the variations of frequency and damping ratio that modal coupling, i.e., vertical frequency reaching torsional frequency has not been observed in case of Model-2. The damping in torsional degree of freedom reaching zero value is the governing parameter for occurrence of 1-DOF torsional flutter instability as reported by Bartoli and Righi (2006) and others (Matsumoto et al., 2008; Tang et al., 2017).

Figure 17.

Variation of (a) frequency and (b) damping ratio with wind speed for Model-2.

The scatter as observed in the damping ratio will be reflected in the flutter derivatives evaluated based on output responses. Hence, flutter derivatives approach based evaluation of flutter critical wind speed involves a level of uncertainty, that need to be assessed through probabilistic approaches (Cheng and Dong, 2017). Further, during experiments, sudden increase in fluctuating component of responses as in Figures 10 and 12 have been observed, which means that the model can get into divergent nature of oscillations without any warning. However, from the time history of vertical and torsional responses, evolution of regularity on approaching flutter critical wind speed has been observed. Similar behavior at higher reduced velocities in case of aeroelastic model of tapered structure has been reported by Chen et al. (2021, 2022). The present study takes advantage of the evolution of regularity to manifest the occurrence of flutter.

Typical plots of vertical and torsional acceleration responses for Model-1A has been presented in Figures 18 and 19, respectively. Beyond wind speed of 8.1 m/s, occurrence of intermittent periodic bursts in the responses has been observed. Similar behavior for testing of aerofoil in wind tunnel has been reported by Venkatramani et al. (2018). At wind speed of 11.5 m/s and above, purely periodic nature of responses have been observed. The amplitudes of response increases with further increase in wind speed.

Figure 18.

Time history of vertical responses showing evolution of regularity on approaching flutter critical wind speed for Model-1A. (a) U = 4.8 m/s, (b) U = 6.5 m/s, (c) U = 8.1 m/s, (d) U = 9.7 m/s, (e) U = 10 m/s, (f) U = 10.3 m/s, (g) U = 11.5 m/s, (h) U = 11.77 m/s.

Figure 19.

Time history of torsional responses showing evolution of regularity on approaching flutter critical wind speed for Model-1A. (a) U = 4.8 m/s, (b) U = 6.5 m/s, (c) U = 8.1 m/s, (d) U = 9.7 m/s, (e) U = 10 m/s, (f) U = 10.3 m/s, (g) U = 11.5 m/s, (h) U = 11.77 m/s.

The response plots of Model-1B were similar to that of Model-1A. Hence, they have not been presented for the sake of brevity. The plots of vertical and torsional acceleration responses for Model-2 has been presented in Figures 20 and 21, respectively. The amplitudes of response increases with increase in wind speed. The occurrence of intermittent periodic bursts is more pronounced in the torsional responses than the vertical responses. This could be attributed to the fact that geometric configuration of Model-2 is susceptible to torsional flutter. At wind speed of 16.2 m/s, purely periodic nature of torsional responses have been observed.

Figure 20.

Time history of vertical responses showing evolution of regularity on approaching flutter critical wind speed for Model-2. (a) U = 3.2 m/s, (b) U = 6.5 m/s, (c) U = 8.1 m/s, (d) U = 11.3 m/s, (e) U = 14.5 m/s, (f) U = 16.2 m/s.

Figure 21.

Time history of torsional responses showing evolution of regularity on approaching flutter critical wind speed for Model-2. (a) U = 3.2 m/s, (b) U = 6.5 m/s, (c) U = 8.1 m/s, (d) U = 11.3 m/s, (e) U = 14.5 m/s, (f) U = 16.2 m/s.

Recurrence Quantification (RQ) plots have been utilized to better identify and visualize the recurrence pattern in the response data from the models (Li et al., 2004). RQ plots are constructed by creating a recurrence matrix that shows the recurrence of a trajectory of measured response data. The recurrence matrix is constructed by setting a threshold value of distance in phase space that represents all possible states of the dynamic system (Marwan et al., 2007). The entries of this matrix include binaries, i.e., zeros (points that fall outside the threshold distance) and ones (points that fall within the threshold distance) and a colour code is given to both of the them for the purpose of plotting. Banded RQ plots for random/Brownian disrupted motion response and checkerboard pattern for periodic/harmonic oscillations have been observed in literature (Marwan et al., 2007).

Windowed recurrence plots (Marwan et al., 2007), reported to be suitable for investigating the effects of external forces on system dynamics has been used. For generation of RQ plots, embedding dimension (m), delay time (τ) and threshold value (ε) of two, one and 0.5 have been adopted. The value of ε has been suggested to be 0.2 times the standard deviation of the observational noise in the data. Since the choice of ε also depends on the dynamic system studied, a number of values of ε from 0.1 to 5 have been tried to arrive at a value of 0.5 that provided better visualisation of the evolution of regularity in responses. RQ plots of torsional responses for Model-1A for various wind speeds have been presented in Figure 22. Random nature of responses at wind speed of 4.8 m/s have resulted in banded nature of RQ plots as in case of Figure 22(a). The spacing between bands have reduced in case of responses with intermittent periodic bursts, as can be seen in Figure 22(b) and (c) at wind speeds of 8.1 m/s and 10.3 m/s, respectively. A fully periodic and recurrent RQ plot has been observed at wind speed of 11.77 m/s. Similar trend of reduction in spacing of bands on approaching flutter critical wind speed and occurrence of fully periodic nature of RQ plots at flutter critical wind speed have been observed for Model-1B and Model-2, as shown in Figures 23 and 24. For Model-1B, the tests were conducted only upto 19.67 m/s as mentioned in Section 2.3.2. At this wind speed, a complete checkerboard pattern of RQ plots could not be observed in the torsional response. This could have been achieved at slightly higher wind speed exposure of the model. Quantification of this change in random pattern of responses to sinusoidal nature of periodic responses, as occurring during flutter phenomenon has not been studied so far in literature with regard to bridge aerodynamics.

Figure 22.

Recurrence quantification plot of torsional responses for various wind speeds for Model-1A. (a) U = 4.8 m/s, (b) U = 8.1 m/s, (c) U = 10.3 m/s, (d) U = 11.77 m/s.

Figure 23.

Recurrence quantification plot of torsional responses for various wind speeds for Model-1B. (a) U = 8.1 m/s, (b) U = 12.9 m/s, (c) U = 17.7 m/s, (d) U = 19.67 m/s.

Figure 24.

Recurrence quantification plot of torsional responses for various wind speeds for Model-2. (a) U = 6.5 m/s, (b) U = 11.3 m/s, (c) U = 14.5 m/s, (d) U = 16.2 m/s.

Entropy measures

Entropy measures are suitable candidate for the quantification of reduction in randomness and also they depend only on the output responses. Adoption of model free approach, like entropy measures offer two-fold advantage:

• To provide sufficient warning before occurrence of flutter phenomenon (Venkatramani et al., 2018) and

• To corroborate or provide realistic estimate the flutter critical wind speed, as evaluated from model based approaches.

The definition of entropy and proof that every new state of a system contains additional information have been presented elsewhere (Delgado-Bonal and Marshak, 2019; Hilborn, 2000; Pincus et al., 1991; Richman and Moorman, 2000). The approach is a coalescence of concepts of information theory and chaos theory.

KS entropy is the measure of the mean rate at which new information is created, and hence, can be used to classify the dynamic systems based on information generation. The issue of underestimation of entropy with increase in order of entropy (n) was overcome by Grassberger and Procassia via calculation of K₂ entropy, which is lower bound value of KS entropy. Considering a time series of length N given by ${X_{i}} = {x_{1}, x_{2}, . ., x_{i}, . ., x_{N}}$ . The template vectors of length ‘m’ is given by

u_{m} (i) = {x_{i}, x_{i + 1}, . . . ., x_{i + m - 1}}, 1 \leq i \leq N - m + 1

(9)

Let

n_{i}^{m} (r)

represent the number of vectors

u_{m} (j)

that are close to

u_{m} (i)

, with their Euclidean distance,

d [u_{m} (i), u_{m} (j)] \leq r

, where r is the tolerance parameter. The probability that any vector

u_{m} (j)

is close to

u_{m} (i)

is given by

C_{i}^{m} (r) = n_{i}^{m} (r) / (N - m + 1)

. The average of

C_{i}^{m}

represents the probability that any two vectors are within r, a defined tolerance parameter. K₂ entropy is defined by

K_{2} = \lim_{N \to \infty} \lim_{m \to \infty} \lim_{r \to 0} - \ln [C^{m + 1} (r) - C^{m} (r)]

(10)

Eckmann and Ruelle defined $Φ^{m} (r) = 1 / (N - m + 1) \sum_{i = 1}^{N - m + 1} \ln C_{i}^{m} (r)$ , by considering difference between the corresponding scalar components of the vectors. They represented $Φ^{m + 1} (r) - Φ^{m} (r) \approx \sum_{i = 1}^{N - m + 1} \ln [C_{i}^{m} (r) / C_{i}^{m + 1} (r)]$ as average conditional probability that sequences that are close to each other for m consecutive data points will still be close to each other when an additional point is known. Applying Eckmann and Ruelle rule, KS entropy ( $H_{K S, E R}$ ) is calculated as

H_{K S, E R} = \lim_{N \to \infty} \lim_{m \to \infty} \lim_{r \to 0} [Φ^{m} (r) - Φ^{m + 1} (r)]

(11)

The above formula, though useful for classification of chaotic systems tend to fail in case of finite noisy data from experiments, the main issue being the strange parameter limits, viz, $r \to 0, m \to \infty a n d N \to \infty$ (Pincus et al., 1991). For short and noisy data, Pincus (Pincus et al., 1991) proposed an entropy measure termed Approximate Entropy (ApEn), which is an approximation of KS entropy measure.

Approximate entropy

For a set of fixed parameters m, r and N, Pincus (Pincus et al., 1991) observed that the value of the term within the bracket in the Equation (11) has intrinsic measure of regularity in the time series data. Approximate Entropy (ApEn) is defined as

\begin{array}{l} A p E n (m, r, N) = \lim_{N \to \infty} [Φ^{m} (r) - Φ^{m + 1} (r)] \\ = \frac{1}{N - m + 1} \sum_{i = 1}^{N - m + 1} \ln [C_{i}^{m} (r)] - \frac{1}{N - m} \sum_{i = 1}^{N - m} \ln [C_{i}^{m + 1} (r)] \end{array}

(12)

For finite data sets, it is estimated as

A p E n (m, r, N) = [Φ^{m} (r) - Φ^{m + 1} (r)]

(13)

It is the conditional probability that two segments of a time series that match at a length ‘m’, also matches at length ‘m + 1’, the next incremental comparison, with the parameter ‘r’ being tolerance for accepting the matches in the time series. The value of the tolerance ‘r’ is defined as a percentage of the standard deviation of the time series considered. The major drawbacks of ApEn are discussed as follows (Richman and Moorman, 2000):

• In order to avoid occurrence of zero probability, self matches are considered in calculation of ApEn. Hence, it becomes a biased statistic and shows higher similarity in the data than actual.

• The evaluated value of ApEn shows high sensitivity to length of data. This can be overcome if the data is large, i.e., ‘N’ tending to infinity, wherein the biased and unbiased estimate would match. In practical cases, availability of large data sets is seldom feasible.

Higher value of ApEn is obtained for unordered, unpredictable data, whilst a value close to zero is obtained for an ordered/regular data. It is based on the premise that an ordered data generates lesser quantum of new information, as it becomes predictable. For smaller value of ‘r’, ApEn value has been observed to be less for irregular data than regular data (Richman and Moorman, 2000), which could be misleading and thus lacks relative consistency. ApEn was shortly followed by introduction of Sample Entropy (SampEn) developed by Richman and Moorman (2000), which overcame the limitations of the former measure.

Sample entropy

Sample Entropy (SampEn) is another improved statistical measure to characterise the regularity in the time series. It has been extensively used for analysis of physiological time series and detection of fault in rotating machinery, etc (Cao et al., 2017; Richman and Moorman, 2000; Udhayakumar et al., 2017). Unlike ApEn, which compares a data with itself, i.e., self-match to avoid zero probability in using template wise approach, SampEn does not use template wise approach. Conceptually, it counts the instances of a template vector, similar to the other vector that is being compared. Sample Entropy is defined as negative natural logarithm of the conditional probability that two such template vectors similar for ‘m’ points remain similar for ‘m + 1’ points

S a m p E n (N, m, r) = - \ln (\frac{A^{m} (r)}{B^{m} (r)})

(14)

\begin{array}{l} A^{m} (r) = \frac{1}{N - m} \sum_{i = 1}^{N - m} C_{i}^{m} (r) \\ B^{m} (r) = \frac{1}{N - m} \sum_{i = 1}^{N - m} C_{i}^{m + 1} (r) \end{array}

(15)

$A^{m} (r)$ is the number of template vectors having $d [u_{m + 1} (i), u_{m + 1} (j)] < r$ .

$B^{m} (r)$ is the number of template vectors having $d [u_{m} (i), u_{m} (j)] < r$ .

For the time series data, the distance function is given by $d [u_{m} (i), u_{m} (j)] w i t h i \neq j$ . Template match is counted if the difference is within the tolerance criteria ‘r’. In the definition of sample entropy, self-matches are not included (i ≠ j) in the calculation of probability which makes the measure independent of the length of signal. The parameters related to calculation of sample entropy, viz, N, m and r have to be fixed prior to the calculation. Similarity in epochs result in lesser value of entropy. Hence, sample entropy value close to zero is indicative of full degree of regularity in the signal. Cross-ApEn or Cross-SampEn can be used to compare two different time series for extent/degree of asynchrony/dissimilarity.

Parametric sensitivity

KS entropy based irregularity measures are estimated for a specific values of embedding dimension (m) and tolerance parameter (r). The evaluated value of entropy is highly dependent on the choice of these parameters. Value of embedding dimension, i.e., the number of subsequent lengths of data that are to be compared is suggested to be one or more, while tolerance parameter is suggested to be in the range of 0.1 and 0.25 times the standard deviation of the data (Pincus et al., 1991; Richman and Moorman, 2000) in order to get stable entropy value. The dependence of r value on the standard deviation of the data also aids in improving the consistency (Huachun et al., 2021). The data length of 10^m–20^m has been reported to be sufficient towards evaluation of ApEn and SampEn for accurate estimation of the conditional probability (Pincus et al., 1991; Richman and Moorman, 2000). However, Venkatramani et al. (2018) have evaluated the entropies of the aerofoil section by adopting embedding dimension of 10, with data length of 1 × 10⁵ points and obtained reasonable results with shorter data length than suggested in literature.

Results on entropy estimates

Tolerance parameter (r) has been fixed as 0.2 times the standard deviation of the signal in consistent with the literature (Cao et al., 2017; Richman and Moorman, 2000). The entropies have been evaluated for embedding dimension in the range of two and 10. The sample entropy values computed from the vertical and torsional responses of Model-1A and Model-1B have been presented in Figures 25 and 26, respectively. It can be observed that for Model-1A, beyond wind speed of 8.1 m/s, the values of sample entropy of both vertical and torsional responses have been observed to reduce, indicating gain in periodicity in the responses.

Figure 25.

Sample entropy values of (a) vertical and (b) torsional responses for Model-1A.

Figure 26.

Sample entropy values of (a) vertical and (b) torsional responses for Model-1B.

At the wind speed of 11.77 m/s, the values of sample entropy of vertical and torsional responses have been observed to be 0.09 and 0.01, respectively, which is indicative of complete periodicity and occurrence of flutter phenomenon (Figure 27).

Figure 27.

Sample entropy values of (a) vertical and (b) torsional responses for Model-2.

Observations on Model-1B indicate that the sample entropy values for both the vertical and torsional components of responses decrease beyond a wind speed of 8.1 m/s. At a wind speed of 20 m/s, the sample entropy values reach minimum values of 0.14 and 0.04 for the vertical and torsional responses, respectively. This reduction in sample entropy values and the occurrence of minima at a specific wind speed indicate the critical wind speed for the system being studied. As observed from experiments, Model-1 undergoes coupled flutter instability, wherein coupling between degrees of freedom takes place during instability, with governing degree of freedom ascertained as torsional degree of freedom (Gao et al., 2020; Yang et al., 2018). Hence, high degree of regularity with least value (close to zero) of sample entropy has been observed in the torsional component of response.

The reduction of the sample entropy value is distinctly observed with significant reduction in magnitudes for embedding dimension (m) in the range of two and five. Further, increase in embedding dimension involves increased computational effort and time, as longer time history records have to be compared to evaluate the probability. Hence, for further studies on the subject, the embedding dimension is suggested to be limited to five.

Figure 28, Figures 29 and 30 demonstrates the inconsistency in estimate of ApEn for the response data from Model-1A, Model-1B and Model-2, respectively. For lesser wind speeds, lesser value of ApEn has been observed, which contradicts the physics of the system behavior with random nature of responses. Further, reduction of ApEn for wind speeds approaching flutter critical wind speed has not been observed for embedding dimension more than three. Hence, SampEn can give a more consistent value of system entropy, rather than ApEn. Further, the computational time required for SampEn is nearly half of that required for ApEn.

Figure 28.

Approximate entropy values of (a) vertical and (b) torsional responses for Model-1A.

Figure 29.

Approximate entropy values of (a) vertical and (b) torsional responses for Model-1B.

Figure 30.

Approximate entropy values of (a) vertical and (b) torsional responses for Model-2.

Model-2 having trapezoidal cross-section, been observed to undergo torsional flutter. Hence, the sample entropy of the torsional response alone has been observed to reach the zero value, while that of vertical response remains almost constant with a value of 0.38. Thus, it is also possible to discern the type of flutter the considered cross-section from the sample entropy values of responses. Table 2 presents the flutter critical wind speed of Model-1 and Model-2 evaluated using empirical expressions available in literature, complex eigen value analysis (Jurado et al., 2011) based on flutter derivatives evaluated from the present study, experimental variation of frequencies and damping ratio with wind speed and based on absolute minimum of sample entropy of responses. Selberg (1961), referred to as Selberg-empirical in Table 2, developed Bleich’s formula, originally introduced for flutter of airfoil and thin plate sections, by putting the shape ratio to apply for various types of bridge sections. Selberg’s formula for critical flutter speed is given by

U_{c r} = 0.44 B \sqrt{(ω_{T}^{2} - ω_{V}^{2}) \frac{\sqrt{υ}}{μ}}

(16)

where

υ = 8 {(r / B)}^{2}

and

μ = π ρ B^{2} / 2 m

; r is the radius of gyration of the cross-section (I = mr²); m is the mass per unit length.

ω_{T} (= 2 π n_{T})

and

ω_{V} (= 2 π n_{V})

are the circular frequencies in the first torsional mode and first vertical bending modes, respectively. Similarly, Rocard gave an empirical formula towards evaluation of flutter critical wind speed given by

U_{c r} = 6.282 f_{α} B \sqrt{(1 - \frac{1}{γ_{ω}^{2}}) \frac{r_{α}^{2} μ^{*}}{1 + 8 r_{α}^{2}}}

(17)

where

r_{α} = \sqrt{I / m B^{2}}; γ_{ω} = ω_{α} / ω_{h}; μ^{*} = μ / 2

Table 2.

Comparison of flutter critical wind speed for both models.

Model	Flutter critical wind speed, V_cr (m/s)
Model	Selberg-empirical	Rocard-empirical	Complex eigen value analysis – Experimental data	Actual observed - exp	Sample entropy based
Model-1A	12.09	12.02	11.43 (free oscillation flutter)	11.77	11.77
Model-1B	21.05	20.93	19.8 (free oscillation flutter)	19.67	19.67
			20.6 (free decay α_io = −3°)
			20.3 (free decay α_io = +3°)
Model-2	20.24	19.55	16.12 (free oscillation flutter)	16.2	16.2
			16 (free decay α_io = −3°)	16.2	16.2
			15.6 (free decay α_io = +3°)	15.7	15.7

The flutter critical wind speed evaluated based on the above formula is referred to as Rocard-empirical in Table 2.

Complex eigen value analysis of experimental data in Table 2 is a two-stage approach involving evaluation of flutter derivatives from free oscillation flutter tests with the aid of a system identification approach (SSI-DATA in the present study) and using flutter derivate based forcing function of self-excited forces to solve the dynamic equation of motion of the sectional model. Hence, the onset of flutter has been evaluated by solving the eigenvalue problem of the considered dynamic system (Jurado et al., 2011). The sample entropy based evaluation of V_cr corroborated with the experimentally observed value of V_cr.

The value of V_cr as evaluated from experimental data through complex eigen value analysis have been observed to vary by about ± 5% with respect to the experimentally observed value of V_cr. Hence, model free approaches like sample entropy based method would aid in evaluation of the actual value of V_cr, that closely resemble the kinematic behavior of the model.

Summary and concluding remarks

Studies on flutter phenomenon based on free oscillation flutter and free decay based dynamic sectional model testing of typical bridge deck sections have been carried out in wind tunnel. Based on the measured responses of the model in considered degrees of freedom, flutter characteristics of the bridge deck sections have been studied. Complex eigen value analysis using flutter derivatives evaluated from responses has been carried out to assess the critical wind speed for occurrence of flutter instability. Quantification of the reduction in randomness, as occurring during flutter phenomenon been studied using entropy based model free approach, on the principle signal complexity analysis. Sample entropy has been proposed to be used towards characterising the regularity in the response. The value of sample entropy evaluated from the response time history approaching a value close to zero can be considered as a potential indicator of occurrence of flutter instability. Further, reduction in sample entropy values are indicative of the gain in regularity, which needs to be duly considered during scaled model testing in wind tunnel as well as during full-scale testing of the structures. From the parametric studies carried out, embedding dimension for estimation of sample entropy has been proposed to be in the range of 2 and 5, to balance computational effort and better discerning of the reduction in value of sample entropy.

Since entropy is a function of probability of occurrence of a template vector in a time series, it hence does not depend on the absolute values/magnitude of responses. This aspect makes it attractive for its applicability for dynamic systems with various types of aerodynamic non-linearities, viz, stiffness and damping related. Further, it does not require apriori knowledge about the characteristics of the system. Hence, this approach can further be extended to smart condition monitoring of full scale long span bridges, which adopt sensors for structural health monitoring. They can be integrated with the active/semi-active control mechanism incorporated in long-span bridges. The response information from these sensors can be processed with a minimal computational cost and time to provide a forewarning before occurrence of flutter phenomenon or vortex induced vibration, under normal and extreme wind events. Hence, safety of the long span cable stayed bridge can be ensured by implementation of this approach.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

M Keerthana

Data availability statement

Data available on request from the authors. The data that support the findings of this study are available from the corresponding author, upon reasonable request. The data given this article is .

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