Gust Buffeting and Aerodynamic Admittance of Structures with Arbitrary Mode Shapes. II: A POD-Based Interpretation

Consider a structure whose length $l$ is much greater than the reference size $b$ of its cross section. Referring to Fig. 1 of Solari and Martín (2020), $x$, $y$, $z$ is a Cartesian system with origin at $o$; $z$ coincides with the structural axis, $x$ is aligned with the mean wind direction, and $o$ lies at the height $H$ above ground. Furthermore, $X$, $Y$, $Z$ is a Cartesian system with origin at $O$; the $X$, $Y$-plane is coplanar with the ground; the $Y$, $Z$-plane is coplanar with the $y$, $z$-plane; $X$ is parallel to $x$; $Z$ is directed upwards and passes through $o$; and $z$ is rotated $\phi$ with respect to $Z$.

The wind loading $f_{\alpha }(z,t)={\overline{f}}_{\alpha }(z)+f_{\alpha }^{\prime }(z,t)$ is a stationary Gaussian field, where $\alpha =x$, $y$, $\theta$; $t$ is time; $f_x$, $f_y$, and $f_{\theta }$ are alongwind and crosswind force and torsional moment around $z$ per unit length; and ${\overline{f}}_{\alpha }$ and $f_{\alpha }^{\prime }$ are the mean value and the fluctuation of $f_{\alpha }$.

The structure has linear elastic behavior with viscous damping and three uncoupled components of motion: the alongwind and crosswind displacement, $x$ and $y$; and the torsional rotation $\theta$ around axis $z$. Each $\alpha =x,y,\theta$ motion component is a stationary Gaussian process defined as $\alpha (r,t)=\overline{\alpha }(r)+{\alpha }^{\prime }(r,t)$, where $0\le r\le l$, and $\overline{\alpha }$ and ${\alpha }^{\prime }$ are mean value and fluctuation of $\alpha$. Solari and Martín (2020) provided the expressions of the parameters needed to determine the dynamic response for any mode $k$.

The power spectral density (PSD) of the $k$th modal loading in the $\alpha$ direction is given bywhere $n$ = frequency; $\zeta =z/l$ and ${\zeta }^{\prime }=z^{\prime }/l$ = nondimensional coordinates; ${\psi }_{\alpha k}$ = $k$th mode in $\alpha$ direction; and $S_{f\alpha k}$ = cross-PSD (CPSD) of $f_{\alpha }^{\prime }$, expressed aswhere ${\mathrm{\Sigma }}_{\epsilon }$ = sum of three fluctuating loading terms with indices $\epsilon =u,v$, and $w$, related to the turbulence components $u^{\prime },v^{\prime }$, and $w^{\prime }$; $\rho$ = density of air; $\overline{u}$ = mean wind speed; ${\lambda }_x={\lambda }_y=1$; ${\lambda }_{\theta }=b$; ${\delta }_{\epsilon u}$ = Kronecker’s delta; and $c_{\mathrm{\alpha \epsilon }}$ = $\alpha$, $\epsilon$th element of 3×3 aerodynamic matrixwhere $c_d,c_{\ell },$ and $c_m$ = drag, lift, and torsional moment coefficients; $c_d^{\prime },c_{\ell }^{\prime }$, and $c_m^{\prime }$ = prime angular derivatives; $I_u={\sigma }_u/\overline{u},I_v={\sigma }_v/\overline{u}$, and $I_w={\sigma }_w/\overline{u}$ = turbulence intensities, where ${\sigma }_u,{\sigma }_v$, and ${\sigma }_w$ = standard deviations of $u^{\prime },v^{\prime }$, and $w^{\prime }$; $S_{\epsilon }^{*},S_{\epsilon \epsilon }^{*}$, and ${\mathrm{Coh}}_{\epsilon \epsilon }^{*}$ = PSD, CPSD, and coherence function of ${\epsilon }^{*}=u^{*},v^{*},w^{*}$, where $u^{*}=u^{\prime }/{\sigma }_u,v^{*}=v^{\prime }/{\sigma }_v$, and $w^{*}=w^{\prime }/{\sigma }_w$ are reduced turbulence components. Adopting the model of Solari and Piccardo (2001)where $d_u=6.868;d_v=d_w=9.434$; $L_{\epsilon }$ = integral length scale; and $c_{\epsilon }$ = exponential decay coefficient. Eq. (6) may be replaced by any other expression. The use of Eq. (7) is functional to the subsequent developments.

The enhanced equivalent spectrum technique consists of replacing the actual turbulent field with an equivalent monovariate process, assuming ${\mathrm{Coh}}_{\epsilon \epsilon }=1$ and replacing $S_{\epsilon }^{*}$ withwhere $S_{\epsilon ,eq}^{*}$ = PSD of reduced equivalent turbulent fluctuation; ${\overline{\zeta }}_{\alpha k}$ = reference value of ${\zeta }_{\alpha k}$; and ${\chi }_{\alpha k\epsilon }$ = frequency filter that plays the role of aerodynamic admittancewhere $F_{\alpha k}$ = generalized participation coefficient; and ${\kappa }_{\alpha k\epsilon }$ = reduced frequency

Accordingly, Eq. (1) may be rewritten as

Solari and Martín (2020) proved thatwhereas for $n$ tending to infinity, the tail of Eq. (9) is given by

Fig. 2 of Solari and Martín (2020) shows the rigorous solutions of Eqs. (9) and (10) provided by MATLAB version 2019a software for 17 noteworthy modes ${\psi }_{\alpha k}$ classified into three families, referred to as Types A, B, and C. It also provides ${\chi }_0={\chi }_{\alpha k\epsilon }(0)$, the ${\kappa }_m={\kappa }_{\alpha k\epsilon }$ value for which ${\chi }_{\alpha k\epsilon }$ is maximum; the relative maximum ${\chi }_{\max }$ of ${\chi }_{\alpha k\epsilon }$; and the $k^{*}=k_{\alpha k}^{*}$ value defined by Eq. (14).

Solari and Martín (2020) highlighted some noteworthy trends that, to authors’ knowledge, never have been interpreted in an exhaustive way. They concern the growth of aerodynamic admittance ${\chi }_{\alpha k\epsilon }$ with the reduced frequency $\kappa ={\kappa }_{\alpha k\epsilon }$, the achievement of the relative maximum ${\chi }_{\max }$ for $\kappa ={\kappa }_m$, and its decrease as $\kappa$ tends to infinity. In particular, ${\kappa }_m$ increases with the increase of the number $N$ of subdomains in which the piecewise mode ${\psi }_{\alpha k}$ has a constant sign. As $N$ increases, ${\chi }_{\max }$ decreases; for Type C modes, ${\chi }_0$ also decreases as $N$ increases.

This behavior is explained only partially by energy cascades (Solari and Martín 2020) due to the limitation of describing the turbulence at a single point, and therefore lacking any information about the phase shift of its component eddies at different points. POD (Di Paola 1998; Tamura et al. 1999; Solari and Carassale 2000; Chen and Kareem 2005; Solari et al. 2007; Carassale et al. 2007) and DMT (Solari and Carassale 2000; Carassale et al. 2001; Tubino and Solari 2007) provide concepts and rules that contribute greatly to clarify the preceding trends.

Investigating the phase shift of eddies at different points highlights a potential problem, namely the use of a turbulence model [Eqs. (6) and (7)] neglecting the imaginary part of the CPSD, and therefore quad-coherence. According to ESDU (1991), this quantity is negligible for horizontal structures, but not for vertical structures. In this second case, however, ESDU states that quad-coherence can be neglected when the atmosphere is isotropic, i.e., in the inertial subrange and in the high-frequency range. Thus, quad-coherence is relevant only in the low-frequency range. Herein its importance may become relevant when the separation between two points is large, namely when one of them is high above ground level. In this case, however, turbulence is approaching isotropy and the role of quad-coherence again is questionable. In other words, the importance of quad-coherence is restricted to a limited range within which uncertainties are huge, as confirmed by field experiments (Mann 1994). For all these reasons, the authors highlight the importance of research in this field but herein follow the almost totality of researchers who used a real turbulence CPSD to investigate gust buffeting.

Let ${\gamma }_{j\epsilon }(n)$ and ${\theta }_{j\epsilon }(\zeta ,n)(j=1,2,\cdots )$ be the eigenvalues and eigenfunctions of the CPSD of the reduced turbulence component ${\epsilon }^{*}$ [Eq. (4)]. They are the nontrivial solutions of the Fredholm integral equation of the second type (Carassale and Solari 2002)

Because $S_{\epsilon \epsilon }^{*}$ is bounded, real [Eqs. (4) and (7)], and nonnegative definite, its eigenvalues are real and nonnegative; in addition, its eigenfunctions are real and can be made orthonormal as

The completeness of the set of the eigenfunctions enables the spectral decomposition

In the case of infinite eigenvalues, the sum in Eq. (18) has infinite terms; however, sorting the eigenvalues in decreasing order, it can be truncated, retaining a finite number of significant terms.

Assuming that turbulence PSD may be evaluated for $\zeta =\overline{\zeta }={\overline{\zeta }}_{\alpha k}$ and substituting Eqs. (7) and (11) into Eq. (15), the eigenvalue problem assumes the formwhereis the reduced eigenvalue; it depends on $n$ through $\kappa ={\kappa }_{\alpha k\epsilon }(n)$ (Carassale and Solari 2002).

Fig. 1 shows the analytical (solid lines) and numerical (dashed lines) solutions of the reduced eigenvalues and eigenfunctions (Carassale and Solari 2002). Fig. 1(a) highlights the similarity between the reduced eigenvalues and the aerodynamic admittances related to Type A and B modes. The largest detachment caused by the analytical solution concerns the first wind mode (Carassale and Solari 2002) for $\kappa <0.1$ (shaded area). The eigenfunctions [Fig. 1(b)] represent waves or elementary shapes, also called wind modes (Di Paola 1998), which are perfectly coherent along the structural axis and perfectly incoherent with each other (Li and Kareem 1995). The change of sign of the wind modes along $\zeta$ corresponds to counter-phase turbulence coherent structures that energy cascade alone is not able to grasp.

The preceding formulation lends itself to noteworthy conceptual, physical, and engineering interpretations by expanding the turbulence component ${\epsilon }^{*}$ into a series of eigenfunctions, the wind modes, weighted by independent time-histories the PSDs of which are the wind eigenvalues. In other words, eigenfunctions define elementary shapes of the wind loading, whereas eigenvalues specify their power content. Substituting this series expansion into structural modal analysis originates DMT (Solari and Carassale 2000; Carassale et al. 2001), in which a parameter of great expressiveness arises, the cross-modal participation coefficient iswhere $A_{\alpha kj\epsilon }$ = influence of $j$th wind mode (associated with $\epsilon$ turbulence component) on $k$th structural mode in $\alpha$ direction. Using modal truncation, only a subset of wind modes excites a subset of structural modes. In addition, due to the reciprocal shape of wind and structural modes, several cross-modal participation coefficients often are negligible; in this case, the $j$th wind mode is said to be quasi-orthogonal to the $k$th structural mode with respect to ${\overline{f}}_{\alpha \epsilon }$. Thus, only a limited subset of wind modes usually excites a limited subset of structure modes. Furthermore, in some cases, the similar shape of wind and structural modes makes the cross-modal participation coefficients with equal indexes much greater than others; in this case, each wind mode mostly excites the sole corresponding structural mode.

Now, substitute Eqs. (4), (7), (11), and (20) into Eq. (18). Delete the term $S_{\epsilon }^{*}$ that appears in both the left-hand and right-hand members. It follows that

Finally, substitute Eqs. (11) and (22) into Eq. (9). It follows that aerodynamic admittance can be expanded into the following series of the reduced wind eigenvalues:

In particular, unlike Eq. (9), which involves the solution of a double integral, Eq. (23) provides a novel expression of aerodynamic admittance by a single integral. Moreover, it establishes the link among aerodynamic admittance, structural mode, and reduced wind eigenvalues and eigenfunctions. The weight of individual wind eigenvalues depends on the reciprocal shape of structural and wind modes, similarly to double modal transformation, in which Eq. (21) includes the term ${\overline{f}}_{\alpha \epsilon }$ missing in Eq. (23). Similarly, Eq. (23) can be simplified when most wind modes are orthogonal or quasi-orthogonal to the $k$th structural mode; in this case, Eq. (23) reduces itself to a linear combination of a few wind eigenvalues. Furthermore, in some particular cases only one mode, usually the mode with index $j=k$, contributes to reconstruct aerodynamic admittance; in this case, ${\chi }_{\alpha k\epsilon }$ tends to reproduce the shape of the pertinent wind eigenvalue.

This section examines aerodynamic admittance for some noteworthy structural modes and interprets its physical meaning based on cross-modal participation coefficients [Eq. (21)] and the expansion through a series of wind modes [Eq. (23)]. It also provides some preliminary expressions of the parameters needed to construct a CFS.

With the rigorous expression of ${\chi }_{\alpha k\epsilon }$ [Eq. (9)] known, the evaluation of aerodynamic admittance through a series of wind modes [Eq. (23)] makes it possible to recognize which wind modes are necessary to reconstruct this function and thus to supply its physical interpretation. From this point of view, analyses carried out in previous sections are exhaustive. Effective turbulence (Tubino and Solari 2007) contributes to clarifying this issue further.

Effective turbulence is defined as the turbulence reconstructed taking into account the wind modes capable of effectively reproducing a given quantity, in this case aerodynamic admittance. Invoking Eqs. (18), (20), and (22), effective turbulence is characterized here by the PSD and coherence functionwhere $j=E$ denotes a sum over the sole effective wind modes, namely those modes necessary to reconstruct aerodynamic admittance with reasonable approximation.

Focusing on Type A regular modes, Piccardo and Solari (1998) provided a simple and precise CFS of ${\chi }_{\alpha k\epsilon }$. For Type B and C modes, Solari and Martín (2020) developed a CFS that consists of dividing the structural domain into subdomains in which piecewise structural modes are regular and do not change sign. This made it possible to derive a solution based on the application of classical methods to each subdomain. This solution is precise and simple for modes with a few changes of sign, but becomes laborious with increasing mode complexity.

Accordingly, a new CFS was developed here, which is easier and more straightforward, and consists of expressing aerodynamic admittance by an empirical formula that (1) reproduces its shape qualitatively; (2) satisfies Eq. (13) for $n=0$; (3) has a relative maximum ${\chi }_{\max }$ for $\kappa ={\kappa }_m$; and (4) matches the trend of Eq. (14) for $n$ tending to infinity. A preliminary expression that fulfils Points 1, 2, and 4 rigorously and Point 3 approximately is given by (Fig. 13)where ${\chi }_0$ is given by Eq. (13); ${\kappa }_m$ is given by Eq. (26); ${\chi }_{\max }$ is given by Eq. (27) or Eq. (29); and $k^{*}=k_{\alpha k}^{*}$ is given by Eq. (14). Table 1 compares the rigorous and approximate values of these parameters. A better agreement probably is possible through more-refined analyses.

Fig. 14 compares the rigorous expressions of ${\chi }_{\alpha k\epsilon }$ given by Eq. (9) for Modes 6, 10, 14, and 15 (solid lines) with Eq. (33) applied to both the rigorous (dashed lines) and approximate (dashed-dotted lines) ${\chi }_0,{\kappa }_m,{\chi }_{\max }$, and $k^{*}$ values. This figure and other diagrams not reported here show that this method always produces excellent approximations for Type B modes, whereas it is less accurate for Type C modes. The estimation of the approximate parameters of Eq. (33) is coarse, especially for Mode 14, which once again is the most complex to interpret and analyze.

In conclusion, deriving analytical expressions of the aerodynamic admittance for arbitrary modes, independently of the method applied and the precision attained, is a secondary result with respect to the physical interpretation of this issue. At present, numerical analysis generally is better than using simplified formulas. However, such formulas explain the explicit dependence of aerodynamic admittance on its parameters.

The crosswind fluctuating load in the vertical direction $y$ was due to the longitudinal and vertical turbulence components, $u^{\prime }$ and $w^{\prime }$. Fig. 15 shows the aerodynamic admittances and the PSD of the modal force. Solid lines indicate the rigorous solution, and dashed lines indicate the CFS, where $k^{*}=0.405$ [Eq. (14)], $F_{\alpha k}=2/\pi$ [Eq. (10)], ${\kappa }_m=4.00$ [Eq. (26)], ${\chi }_0=0$ [Eq. (13)], and ${\chi }_{\max }=0.25$ [Eq. (27)]. Aerodynamic admittances were null at the zero frequency, and their relative maximum was well below unity. The modal force was due mainly to longitudinal turbulence.

Table 2 lists the main parameters of the vertical displacement for $z=l/4$ due to the first vertical Mode 6, evaluated by the rigorous and approximate solutions. Similar to the method derived by Solari and Martín (2020), the CFS gave almost exact results. The background response was small, and the resonant response prevailed regardless of structural flexibility.

The alongwind fluctuating load was due to the longitudinal turbulence. Fig. 16 shows the aerodynamic admittance and the PSD of the modal force. Solid lines indicate the rigorous solution, and dashed lines indicate the CFS, where $k^{*}=0.344$ [Eq. (14)], $F_{\alpha k}=0.3026$ [Eq. (10)], ${\kappa }_m=4.00$ [Eq. (26)], ${\chi }_0=0.194$ [Eq. (13)], and ${\chi }_{\max }=0.356$ [Eq. (29)]. The aerodynamic admittance remained well below unity, eroding quite uniformly the harmonic content of the modal force. The CFS was quite coarse due to the coarse estimation of ${\chi }_{\max }$ (Table 1).

Table 3 lists the main parameters of the alongwind displacement for $z=l$ due to the first longitudinal Mode 14, evaluated by the rigorous and approximate solutions. The errors due to the latter were greater than those provided by the CFS in Solari and Martín (2020) due to the coarse estimation of ${\chi }_{\max }$ (Table 1) and the limited quality of Eqs. (27) and (29) for Type C modes. These aspects involve ample room for improvement.