To explain the effectiveness of the advanced wind tunnel test techniques, unsteady aerodynamics (e.g., aerodynamic forces and aerodynamic damping) and aeroelasticity (e.g., galloping) measured from the techniques are reviewed.

#### 4.1. Characteristics of Unsteady Wind Force

The distributed pressure as well as the overall wind load of a prism can be evaluated from a SMPSS test. Wind pressures on the windward face of a structure are positive as the approaching flow directly acts on the face whereas they are negative on the other three faces (leeward face and side faces) due to suction. Wind forces in the along wind direction are defined as drag force and those in the crosswind direction are defined as lift force (

Figure 8).

For three-dimensional prisms, the local and overall mean, root mean square (RMS) lift and drag force coefficient are defined as follows. It should be noted in the crosswind direction the mean lift force coefficient is close to zero and is therefore neglected. The base moment coefficient can also be evaluated from the observed local pressure coefficients.

The local wind force coefficients of a test model at height

z above the ground are defined as

where

${\text{}\overline{\mathrm{C}}}_{\mathrm{D}}\left(\mathrm{z}\right),{\text{}\tilde{\mathrm{C}}}_{\mathrm{D}}\left(\mathrm{z}\right)$ and

${\tilde{\mathrm{C}}}_{\mathrm{L}}\left(\mathrm{z}\right)$ are the local mean drag force coefficient, the local RMS drag force coefficient and the local RMS lift force coefficient, respectively.

${\mathrm{F}}_{\mathrm{D}}\left(\mathrm{z}\right),{\mathsf{\sigma}}_{{\mathrm{F}}_{\mathrm{D}}\left(\mathrm{z}\right)}{\text{}\mathrm{and}\text{}\mathsf{\sigma}}_{{\mathrm{F}}_{\mathrm{L}}\left(\mathrm{z}\right)}$ are the mean drag force, the RMS drag force and the RMS lift force, respectively.

${A}_{z}$ is the sectional area.

${q}_{H}=\frac{1}{2}\rho {U}^{2}$, where

$\rho $ is air density and

$U$ is wind velocity.

Integrating the obtained local wind force coefficients at each level, the generalized force coefficients are expressed as

where,

${\text{}\overline{\mathrm{C}}}_{\mathrm{D}},{\text{}\overline{\mathrm{C}}}_{\mathrm{D}}$ and

${\tilde{C}}_{L}$ are the generalized mean drag force coefficient, the RMS drag force coefficient and the RMS lift force coefficient, respectively.

$D$ and

$H$ are the width and height;

$\mathsf{\varphi}\left(\mathrm{z}\right)$ denotes the mode.

The base force coefficient is expressed as

where

${\text{}\overline{\mathrm{C}}}_{\mathrm{MD}},{\text{}\tilde{\mathrm{C}}}_{\mathrm{MD}},{\text{}\overline{\mathrm{C}}}_{\mathrm{ML}}{\text{}\mathrm{and}\text{}\tilde{\mathrm{C}}}_{\mathrm{ML}}$ denote the mean drag base moment, the RMS drag base moment, the mean lift base moment and the RMS lift base moment, respectively;

${\overline{\mathrm{F}}}_{\mathrm{MD}},{\text{}\tilde{\mathrm{F}}}_{\mathrm{MD}},{\text{}\overline{\mathrm{F}}}_{\mathrm{ML}}{\text{}\mathrm{and}\text{}\tilde{\mathrm{F}}}_{\mathrm{ML}}$ denote the mean drag force, the RMS drag force, the mean lift force and the RMS lift force, respectively.

Based on observed wind pressures and the above definitions, static wind force characteristics of structures have been comprehensively analyzed [

9,

21,

45,

46]. Most recently, static wind force characteristics of backward or forward inclined prisms were investigated [

47]. The effect of inclination on the force coefficients was analyzed. It was pointed out that the base shear force and moment tend to decrease with increasing the inclinations apart from a small backward inclination case. The possible reasons were illustrated in terms of base force spectra, pressure coherences of the side face pressures and vortex shedding frequencies.

Many studies have focused on unsteady wind forces on bluff bodies. Bearman and Obasaju [

29] have investigated the pressure fluctuation of oscillating two-dimensional circular and square-sectional cylinders. Both the mean and fluctuating crosswind pressures of the cylinders at or away from the lock-in range were observed, and substantial differences between the two cylinders were discovered (

Figure 9). It was found that, at high wind speeds, oscillating pressures are in close agreement with that measured from a static test model, suggesting that the pressures are under quasi-steady state and the effect of structural oscillation is slight. At low wind speeds (around the lock-in range), a significant peak took place (

Figure 10), which was ascribed to the interaction of structural motion and vortex shedding. The vortex shedding process will impart forces on prisms and the prisms may respond to these forces. When the shedding frequency is equal or in close agreement to the natural frequency of the prism, a coupling between the response and the wake exists. In this case, the oscillating frequency of the prism dominates the vortex shedding frequency of wind flow at a certain wind speed range and the range increases as the structural damping decreases.

In steady flow, bluff bodies can undergo vortex-induced-vibrations (VIVs) because of the von Karman vortices generated in wake flow which will create unbalanced forces on the structure. VIVs of prisms occur at the lock-in range where the exciting frequency may close to the natural frequency of the body. In addition, downstream structures behind the bluff bodies generating vortices may also undergo VIV because of the impingement of induced flow and the low pressure core region of the vortices [

48]. Because of the practical and theoretical importance of VIV, it has received considerable attentions [

48,

49,

50,

51,

52,

53]. The phenomena as well as the mechanism about the VIV of bluff bodies with simple configurations were primarily investigated. The commonly accepted interpretation of the mechanism underlying the VIV is that there is a net flux energy from the fluid flow into a structure and, with respect to the structure, negative damping is set up which reduce the total damping of the structure. In the case of low structural damping and mass of a structure, oscillations of the body may be enlarged. It is usually regarded as the lower the structural damping and mass are, the greater the VIV of the body are. However, the VIV limits themselves to amplitudes on the order of the body dimension even for the low structural damping and mass case. This feedback mechanism is implicitly assumed to be nonlinear. The flow and thereby the negative damping are altered by the body motion. The altered flow provides less net energy flux to the structure at some oscillating amplitudes and equilibrium of energies is achieved. The oscillating frequency not always coincides with the natural frequency of the structure. Oscillations may occur at twice or three times the shedding frequency, which may be ascribed to the effect of the airflow added mass and the geometry of the vortex shedding street [

54,

55].

Cooper et al. [

30] and Katagiti et al. [

56,

57] investigated unsteady pressures of bluff bodies. The effect of oscillating amplitude and reduced velocity on the sectional alongwind and crosswind force coefficients were analyzed. It was found that, in the alongwind direction, oscillating amplitude and wind velocity have a slight effect on the observed force coefficients, whereas they have a significant effect on the observed force coefficients in the crosswind direction. An example, local force coefficients in the crosswind direction, is given in

Figure 11. It shows that, at low wind speeds, the local force coefficients increase with oscillating amplitude whereas at high wind speeds, the local force coefficients are in close agreement with that of a stationary case.

Based on observed unsteady pressures, spanwise correlations of cylinders have been analyzed [

32,

58]. It is noteworthy that, at the vortex lock-in range, the spanwise correlation increases considerably with oscillating amplitude and away from the vortex lock-in range, the spanwise correlation changes slightly with increasing oscillating amplitude, suggesting that oscillating amplitude has a significant effect on spanwise correlation around the vortex lock-in range and has a slight effect on spanwise correlation away from the vortex lock-in range. A study [

32] also pointed out that the vibrating pattern, wind incidence and Reynolds number also have considerable effects on spanwise correlation and the effects cannot be neglected (

Figure 12).

The above studies have not only advanced our understanding of characteristics of unsteady wind forces of structures, but also provide a foundation for further analysis. The following section will introduce the identified aerodynamic damping based on the unsteady wind forces obtained from a forced vibration test.

#### 4.2. Aerodynamic Damping Force

The observed unsteady wind force is composed of a random component due to unsteady wake effects and turbulence, and a motion-induced force component. Based on a previous study [

10], it is expressed as

where

$\mathrm{W}\left(\mathrm{t}\right)$ is the observed unsteady wind moment;

${\mathrm{W}}_{\mathrm{l}}\left(\mathrm{t}\right)$ and

${\mathrm{W}}_{\mathrm{m}}\left(\mathrm{t}\right)$ are the random wind moment component and the motion-induced moment component, respectively;

${\mathrm{C}}_{\mathrm{l}}\left(\mathrm{t}\right)$ and

${\mathrm{W}}_{\mathrm{m}}\left(\mathrm{t}\right)$ are the random wind force coefficient and the motion-induced force coefficient, respectively.

The corresponding base force is written as

where

$F\left(t\right)$ is the observed unsteady wind force;

${F}_{l}\left(t\right)$ and

${\mathrm{F}}_{\mathrm{m}}\left(\mathrm{t}\right)$ are the random wind force component and the motion-induced force component, respectively.

A complex aerodynamic impedance,

${K}_{a}$ is defined and expressed as

Then, a dimensionless form of impedance

${G}_{a}$ is defined as

where

${\mathrm{M}}_{\mathrm{s}}\mathsf{\eta}$ denotes the generalized mass of the prism. For a test model with linear mode shape, it is expressed as

As mentioned, the motion-induced force can be divided into an aerodynamic stiffness term and an aerodynamic damping term (

${\mathrm{G}}_{\mathrm{a}}=\mathsf{\lambda}+\mathrm{iu}$). The real part

$\mathsf{\lambda}$ corresponds to the aerodynamic stiffness term and the imaginary part

$\mathrm{u}$ corresponds to the aerodynamic damping term. Then, the base moment is re-written expressed as

In a forced vibration test, the test model oscillates harmonically, and the tip response can be expressed as

where

$\mathrm{y}\left(\mathrm{t}\right)$ is the tip displacement response and

$\widehat{\mathrm{y}}$ is tip oscillating amplitude.

Considering the orthogonality of trigonometric functions, yields,

Then, the aerodynamic stiffness and damping coefficients are expressed as

It should be clarified that

${I}_{1}$ and

${I}_{2}$ can be derived from the time-history base force or the time-history pressure measured from a forced vibration test. By using the identification scheme introduced above, Steckley [

10] identified the aerodynamic stiffness and aerodynamic damping coefficients of bluff bodies. The aerodynamic stiffness and damping coefficients of the prisms under different oscillation amplitudes and turbulence intensities observed in a previous study are presented in

Figure 13.

Figure 13a shows that the aerodynamic stiffness coefficients are negative under all reduced velocities.

Figure 13b shows that the aerodynamic damping coefficients are positive at low wind speeds and are negative at high wind speeds. The peaks of the aerodynamic stiffness coefficients occur at the reduced wind speed around 10 which is in the vortex lock-in range. In this range, the aerodynamic damping coefficients change from positive to negative. Furthermore, the oscillating amplitude has a great effect on the aerodynamic stiffness and damping coefficients. The magnitudes of the aerodynamic stiffness and damping tend to increase with oscillating amplitude. In addition, turbulence intensity also has an impact on the two terms. The magnitudes tend to decrease with increasing oscillating amplitude. The effect of aspect ratio has also been investigated and it was found that the aerodynamic damping and stiffness coefficients are not sensitive to the aspect ratio apart from the aspect ratio of 6.67.

With linear mode shape assumption, based on random vibration theory, the variance of the tip response of a test model is estimated by

where

$\text{}\overline{\mathrm{y}}$ is the variance of the tip response;

${\mathrm{K}}_{\mathrm{s}}$ is the generalized stiffness;

$\mathrm{f}$ is the frequency;

${\mathrm{S}}_{\mathrm{FF}}\left(\mathrm{f}\right)$ is the spectrum of generalized force that is obtained from a HFBB test;

$\mathrm{H}\left(\mathrm{f}\right)$ is the modulus of the mechanical admittance function and is expressed as

where

${f}_{s}$ is the natural frequency;

${\xi}_{s}$ is the structural damping ratio.

It should be emphasized that, for high-rise prisms and buildings, the aerodynamic stiffness force component is small and is often neglected, and only the aerodynamic damping force component is concerned [

59]. The above analytical scheme has been proved to be reasonable and reliable in aerodynamic damping identification. Following the analytical scheme, a few studies have identified the aerodynamic damping of several kinds of prisms [

30,

56,

60]. Substituting the identified results into Equation (16), the mechanical admittance is re-written as

Substituting Equation (16) into (15), the response predictions can be improved.

For the purpose of convenient use, nonlinear mathematical models of the identified aerodynamic damping have been developed. Watanabe et al. [

61] have proposed an empirical mathematical function which is a function of oscillating amplitude and reduced wind velocity, to model the aerodynamic damping. The empirical model is expressed as

where

${F}_{d}$ is aerodynamic damping coefficients;

${\mathrm{U}}_{\mathrm{cr}}$ is the vortex wind speed;

$\chi $,

${\mathrm{H}}_{\mathrm{s}}$ and

${\mathrm{A}}_{\mathrm{p}}$ are parameters of the function, which are functions of oscillating amplitude.

Chen [

59] has revised the mathematical model by a second-order polynomial. Even though the expression is different, it is also a function of oscillating amplitude and reduced wind velocity. It is written as

or

where

${\mathrm{a}}_{1},{\text{}\mathrm{a}}_{2}$ and

${\mathrm{a}}_{3}$ are parameters that are functions of reduced wind speed.

Self-excited vibration (galloping) occurs when the total damping of a system is negative. It is in a steady state when the total damping

${\mathsf{\xi}}_{\mathrm{t}}={\mathsf{\xi}}_{\mathrm{a}}+{\mathsf{\xi}}_{\mathrm{s}}$ becomes zero. The steady state amplitude is determined by

where

${S}_{cr}$ is Scruton number and expressed as

${S}_{cr}={m}_{s}{\xi}_{s}/\left(\rho {D}^{2}\right)$.

${m}_{s}$ is the mass ratio.

Solving Equation (23), yields

By using the developed polynomial, the aerodynamic damping of a test model was evaluated and compared with that estimated by Equation (18) and the quasi-steady theory (

Figure 14). The identified aerodynamic damping has been well used for response predictions and fatigue estimations of structures.

Apart from the forced vibration technique, the aerodynamic damping can also be evaluated based on spectral and time series approaches (i.e., auto-regressive (AR) or auto-regressive and moving-averages (ARMA) techniques, half power bandwidth techniques and random decrement techniques (RDT) [

62,

63,

64]). However, these methods have either mathematical limitations or data record problems, and are incorrect in some cases, as reported in previous studies [

3,

65].

For forward inclined prisms, the generalized aerodynamic damping coefficients are shown in

Figure 15, where

${\sigma}_{y}/y$ means the normalized amplitude of vibration and inclination and

$U/fD$ means reduced wind velocity.

$\alpha $ means the inclination angle. It is clear that the generalized aerodynamic damping coefficients reach the peak when the reduced wind speed is

$1/St$.

#### 4.3. Unsteady Self-Excited Force

Chen and Kareem [

66] pointed out that aerodynamic forces are commonly separated into static, buffeting and self-excited force components. Among them, the self-excited force component which contains unsteady wind effect is used for predictions of the flutter of deck sections and galloping of prisms. A linear self-excited force model of airfoil and bridge deck has been proposed by Scanlan and Tomo [

67]. Due to its simplicity and practicability, it has been widely accepted and used by engineers. Based on the linear model, other self-excited force models with the similar expressions have been proposed [

68,

69]. Despite the fact that the linear model has proven its utility for many applications, it may not be able to address issues aerodynamic nonlinearities and unsteady effect [

66]. Many studies have focused on nonlinear self-excited force models [

70,

71,

72,

73,

74] that are used for amplitude-dependent flutter derivative identification, high-order flutter force spectrum identification, nonlinear hysteresis analysis and low-speed flutter analysis.

Self-excited galloping force is commonly estimated by a quasi-static method. However, the quasi-static force excludes the effect of structural oscillation and is therefore not applicable to predict galloping responses of structures with respect to unsteady effect. Bouclin and Geoola [

75] have proposed a wake-oscillator model which combined the quasi-static force model with the Hartlen-Currie lift model to explain the galloping phenomena of a square cylinder. The classic quasi-static force model was, therefore, improved by introducing the forces induced by the vortex shedding process. Corless and Parkinson [

76,

77] improved the wake-oscillator model to predict the VIV-galloping combined response of cylinders. Tamura [

78] has combined the quasi-static force model with the Birkhoff wake-oscillator model [

79] to evaluate the combined effects of VIV and galloping. Mannini et al. [

80] have investigated the self-excited force models proposed by Corless and Parkinson, and Tamura through wind tunnel studies on VIV-galloping of rectangular cylinders. They pointed out that though the self-excited force models can substantially reflect the interaction of VIV and galloping, they cannot well predict the combined vibrations. Most recently, Gao and Zhu [

41,

81] have identified the unsteady galloping force of two-dimensional cylinders by using the aforementioned hybrid aeroelastic-force experimental system (

Figure 7). Chen et al. [

25] measured the unsteady aerodynamic force and galloping response of a prism from a HAFB wind tunnel test. The unsteady aerodynamic force measured form the hybrid aeroelastic-pressure balance test includes galloping force components and buffeting force components. For the cases of

V_{r} = 28, 36 and 42, unsteady aerodynamic forces on the prism are shown in

Figure 16. For the static measurement case, the only peak was induced by vortex shedding. This means in a static measurement case, the effect of structural motion on aerodynamic force was excluded which will lead to underestimation in the prediction of wind-induced response. The galloping responses of the cylinders predicted by the unsteady galloping force are coincident well with the experimentally observed (

Figure 17), suggesting the identified unsteady galloping force is reasonable and reliable.

The identified unsteady galloping force includes three components: an aerodynamic damping component, an aerodynamic stiffness component and a residual force (buffeting force) component. Among them, only the aerodynamic damping or stiffness component inputs or dissipates energy to an oscillating system. Additionally, it has been validated that the effect of the aerodynamic stiffness is slight. On these considerations, a nonlinear mathematical model that is simplified as a fifth order polynomial was established to model the unsteady galloping force. It is expressed as Equation (25).

where

${\mathrm{P}}_{\mathrm{se}}\left(\mathrm{y},\dot{\mathrm{y}}\right)$ is unsteady self-excited forces acting on a bluff body;

$\mathrm{D}$ the height of the test model;

$\mathsf{\varphi}\left(\mathrm{z}\right)$ is the mode shape;

$y$ is the response of the model;

$\dot{\mathrm{y}}$ is the velocity of the model;

${\mathrm{p}}_{1},{\text{}\mathrm{p}}_{2},{\text{}\mathrm{p}}_{3},{\mathrm{p}}_{4},{\mathrm{p}}_{5}{\text{}\mathrm{and}\text{}\mathrm{p}}_{6}$ are the aerodynamic coefficients of unsteady self-excited forces.

It was found that the classical quasi-steady theory fails to predict the galloping response of a slender prism, as presented in

Figure 18a. By contrast, the galloping response predicted by the developed model is in close agreement with experimental results, as shown in

Figure 18b. This suggests that the shortcomings of the classical quasi-steady theory in predicting galloping instabilities of bluff bodies can be well address by the developed model that is established according to unsteady aerodynamic forces measured from the HAPB test.