ARTICLE IN PRESS
Journal of Wind Engineering and Industrial Aerodynamics 95 (2007) 183–207 www.elsevier.com/locate/jweia
Dynamics of suspended cables under turbulence loading: Reduced models of wind field and mechanical system Vincenzo Gattullia,, Luca Martinellib, Federico Perottib, Fabrizio Vestronic a
Dipartimento di Ingegneria delle Strutture, delle Acque e del Terreno, Universita` di L’Aquila, 67040 Monteluco di Roio, L’Aquila, Italy b Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133 Italy c Dipartimento di Ingegneria Strutturale e Geotecnica, Universita` di Roma ‘‘La Sapienza’’, Via Eudossiana 18, 00184 Roma, Italy Received 12 October 2005; received in revised form 6 May 2006; accepted 16 May 2006 Available online 1 August 2006
Abstract In cables, turbulent wind may cause large amplitude oscillations. The prediction of cable response under wind action requires the use of high-dimensional numerical models either to describe the spatial wind field or to model the expected large cable oscillations. The paper discusses the ability of reduction techniques, for loading and cable descriptions, in reproducing accurately the dynamic response of a suspended cable excited by an artificially generated 3D turbulent wind field. Both the mechanical system and the spatially varying wind velocities are projected on the basis of cable eigenfunctions, retaining in the reduced models few degrees-of-freedom associated with the lowfrequency modes. A numerical investigation performed by a refined finite element model provides novel findings on the cable response to wind and permits to demonstrate the effectiveness of the reduced models in the description of cable dynamics. r 2006 Elsevier Ltd. All rights reserved. Keywords: Cables; Wind turbulence; Aerodynamic damping; Finite element models; Reduced order models; Nonlinear oscillations
Corresponding author. Tel.: +39 0862 434511; fax +39 862 434548.
E-mail address:
[email protected] (V. Gattulli). URL: http://www.ing.univaq.it/webdisat/. 0167-6105/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2006.05.009
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1. Introduction The cable is a simple but important structural element; indeed cables are frequently used to sustain themselves, as in high-voltage transmission lines, or within tall or wide structures, as in the case of guyed towers and suspended/stayed bridges. It is characterized by high resistance, high flexibility and a very small damping, which leads to the fact that a cable is often prone to undergo large amplitude oscillations mainly due to wind loading. The dynamic response of cables has been thoroughly investigated in linear and nonlinear regime under deterministic and stochastic excitations. Since the behavior is inherently nonlinear and the nonlinearities of this system are very rich, a large amount of papers have been devoted to the study of nonlinear dynamic phenomena in the stationary oscillations ([17,27,22,23]), typically in the presence of some resonance conditions, when these phenomena become relevant ([14,1]). In these conditions it has been observed that low-dimensional models, obtained by expanding the displacement functions in a suitable bases—like the eigenfunctions of the modes involved in the resonances—are able to capture most of the nonlinear response. Even though a new generation of numerical methods facilitates the general description of solutions for large-dimensional ODEs, efforts are still necessary to select more refined bases able to minimize the nonlinear coupling terms. Within this framework, it has been shown in a recent paper ([11]) that even when the classical approach based on the use of eigenfunctions of the linearized equations of motion is followed, a reduced model with a low number of modes is usually sufficient; indeed such models are able to describe accurately the response of cables, both qualitatively and quantitatively, predicting all the main bifurcated stable branches highlighted by means of a refined nonlinear finite element model (FEM). Less experience exists when the cable is excited by non-periodic force with varying spatial distribution, such as the case of wind turbulence loading. In this case the degree of approximation of reduced models depends on two conditions: the ability of lowdimensional mechanical system in reproducing the main characteristic of the response and the representation of the loading features through a simple description. This item is tackled diffusely in the literature devoted to the response of linear systems excited by a stochastic process and is, for example, discussed in [2]. In the present work the description of the dynamical response of a non-resonant suspended cable under wind excitation is considered. The aim of the study is mainly to investigate the effect of adopting reduced order models, for both dynamical system and excitation loadings, in the computation of the cable response to wind. In this light, the influence of the characteristics of the turbulent wind field on the cable response was preliminarily analyzed by means of a finite element (FE) based numerical procedure ([19]); more precisely, the effects of the coupled in-plane and out-of-plane turbulence, of the dimensionality of the wind field (1D vs 3D) and of the turbulence coherence were characterized. Based on this investigation, the accuracy of reduced models was analyzed comparing the results of these models to those of a refined FEM, in describing the cable nonlinear response to turbulent loading. To investigate the reduced order modeling of the excitation, the artificially generated wind velocity fields is compared to a reduced one which is represented by a few components obtained projecting the complete wind onto the cable eigenfunctions basis. The paper is organized as follows. The wind model is discussed first (Section 2) and, on the basis of a few numerical results, wind complete and reduced descriptions are compared
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(Section 2.1). In the following Section 3 the governing equations for cable dynamics under turbulent wind are presented for both analytical (Section 3.1) and FE (Section 3.2) models. The FE response analysis (Section 4) is then exploited to the aim of discussing the responses to in-plane and out-of-plane turbulence (Section 4.1), to 1D and 3D wind turbulence fields (Section 4.2) and the effects of turbulence coherence (Section 4.3). Finally, for some selected cases, the previous results are compared to those obtained by direct integration of a low dimensional analytical model (Section 5). Concluding remarks summarize the research findings (Section 6). 2. Wind model When the adequacy of a reduced dynamic model is under consideration, the properties of perturbation forces must be carefully studied by the point of view of both frequency content and spatial variation. The latter aspect deserves particular care when, as in the case of suspended cables, the structural eigenspectrum is such that a number of modes fall in the low-frequency interval, where wind turbulence shows its most significant harmonic components. Turbulent components of wind velocity are often defined, in engineering applications, as realizations of a zero-mean stochastic process which can be assumed as stationary in time but non-homogeneous in space when the boundary layer is considered; the process is usually described, in the frequency domain, in terms of its cross power spectral density function (CPSD). A question arises how the properties of the loading process affects the degree of approximation of a reduced wind model. For a preliminary discussion we shall refer to the classical theory of linearized stochastic wind loading (see [4]); as an illustrative example we shall deal with the cable out-of-plane oscillations due to a single-component turbulent wind directed as the normal to the vertical plane containing the cable under self-weight. In this setting, if the out-of-plane displacement component wðx; tÞ is expanded (see Section 3.1) as P aw linear combination of the eigenfunctions cðxÞ of the linearized problem, i.e. wðx; tÞ ¼ ni¼1 ci ðxÞqiw ðtÞ, the forcing term appearing in the ith discretized equation of motion (generalized load component) can be written in the form Z piw ðtÞ ¼ ci ðxÞgw ðxÞw1 ðx; tÞ dx. (1) D
In Eq. (1) D is the structural domain, w1 ðx; tÞ is the transverse component of turbulent velocity and gw ðxÞ is the positive function gw ðxÞ ¼ G w cD ra bðxÞW ðxÞ,
(2)
i.e. the product of a trigonometric factor Gw times air density ra , section dimension b, aerodynamic coefficient cD and mean wind velocity W ; in Eq. (1) the drag force has been linearized according to the hypothesis that W bw1 . To characterize the dynamic loading intensity at mode i the variance of the generalized load (1), which is a zero-mean process as well, can be computed as its mean-square value. With standard manipulation and by denoting with E½: the expectation operator, we obtain Z Z 2 E½piw ðtÞ ¼ E½w1 ðx1 ; tÞw1 ðx2 ; tÞci ðx1 Þgðx1 Þci ðx2 Þgðx2 Þ dx1 dx2 . (3) D
D
The cross correlation function E½w1 ðx1 ; tÞw1 ðx2 ; tÞ in turn can be decomposed into its frequency components, represented through the CPSD, S w1 ðx1 ; x2 ; f Þ. Doing this and
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interchanging the order of integration, the power spectral density function Spiw ðf Þ of the modal load is implicitly defined according to the following: Z 1Z Z E½p2iw ðtÞ ¼ Sw1 ðx1 ; x2 ; f Þci ðx1 Þgðx1 Þci ðx2 Þgðx2 Þ dx1 dx2 df Z0 1 D D ¼ S piw ðf Þ df . ð4Þ 0
For investigating how the modal loads are affected by the wind distribution, the CPSD of turbulent velocity can be factored as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S w1 ðx1 ; x2 ; f Þ ¼ C w1 ðx1 ; x2 ; f Þ S¯ w1 ðx1 ; f Þ; S¯ w1 ðx2 ; f Þ, (5) where C w1 ðx1 ; x2 ; f Þ is the coherence function and S¯ w1 ðx1 ; f Þ is the power spectrum (PSD) of wind velocity for x ¼ x1 . Moreover, in most of wind engineering applications the coherence is assumed real and characterized by an exponential decay with distance and frequency. By denoting with aðx1 ; x2 Þ the decay coefficient, the PSD of the modal load can be expressed, from Eqs. (4) and (5) as Z Z expfaðx1 ; x2 Þf jx1 x2 jg S piw ðf Þ ¼ D D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S¯ w1 ðx1 ; f Þ; S¯ w1 ðx2 ; f Þci ðx1 Þgðx1 Þci ðx2 Þgðx2 Þ dx1 dx2 . ð6Þ The evaluation of the integral (6) in the limiting conditions of full coherence ðaf ¼ 0; C w ¼ 1Þ and null coherence ðaf ¼ 1; C w ¼ dðx1 x2 ÞÞ suggests interesting considerations. In the first case Eq. (6) gives Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 S¯ w1 ðx1 ; f Þci ðx1 Þgðx1 Þ dx1 (7) S piw ðf Þ ¼ D
which, for a homogeneous process ðS¯ w1 ðx1 ; f Þ ¼ S¯ w1 ðf ÞÞ, leads to a well-known result in random vibration theory. For the delta-correlated process, by exploiting the properties of the Dirac d we get, for Eq. (6), the expression Z S¯ w1 ðx1 ; f Þ½ci ðx1 Þgðx1 Þ2 dx1 (8) S piw ðf Þ ¼ D
which is again a standard result for uncorrelated loading. Expression (7) is the square of the weighted average of the ith eigenfunction; therefore, for high-coherence values ‘‘first mode shaped’’ eigenfunctions, with no sign changes, are clearly dominant in the excitation. Expression (8), on the other hand, delivers the weighted mean-square of the eigenfunction; according to (8), i.e. for low-coherence loading, all modes are excited, in principle, with the same intensity. To gain further insight into this matter, it can be also shown how this behavior can be interpreted in terms of wind reduced models. In this light the description of the turbulence structure can be simplified using normal modes, i.e. the so-called ‘‘wind modes’’ ([24,25,5,7]); following [3] the turbulence CPSD can be decomposed, within the structural domain D, into its eigenfunctions (modal shapes) yj ðx1 ; f Þ and eigenvalues (modal spectral
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densities) gj ðf Þ, i.e. Sw1 ðx1 ; x2 ; f Þ ¼
X
yj ðx1 ; f Þyj ðx2 ; f Þgj ðf Þ,
187
(9)
j
yj being the complex conjugates of yj . By substituting expression (9) into (6) we can write Z Z X Spiw ðf Þ ¼ gj ðf Þ ci ðx1 Þgðx1 Þyj ðx1 ; f Þ dx1 ci ðx2 Þgðx2 Þyj ðx2 ; f Þ dx2 j
X
¼
D
gj ðf ÞjE ij ðf Þj2 ,
D
ð10Þ
j
where Z E ij ðf Þ ¼
ci ðxÞgðxÞyj ðx; f Þ dx.
(11)
D
In [26] it is observed that, in many cases of practical interest, the structural and wind eigenfunctions tend to be similar in shape; in such case the ith modal load tends to be entirely related to the corresponding wind mode. In the same paper the eigenfunctions and eigenvalues of the CPSD (5), are given, in analytical terms, as functions of the decay parameter af . It is also shown how, for high-coherence values (low af ), the first eigenvalue is dominant and tends to the value of the process PSD, i.e. S¯ w1 ðf Þ; for fast coherence decay (high af ) the turbulence power is spread over a large number of wind modes. Therefore, a highly correlated wind structure is inherently suitable for a reduced model of its spatial variation. The most efficient reduction technique should be the projection on the wind modes; however, given the similarity between wind and structural modes (see again [26,3]), it can be easily understood how projection on the first structural modes delivers a very close approximation of the turbulence field. In the numerical experiments here shown, wind fields with different coherence were considered, selecting as common occurrence an average wind coherence (AWC) in comparison to the case of a very low wind coherence (LWC). In the first case, consistently with the data reported in [26] and with the cable setting, a constant value of a ¼ 0:3 s=m was chosen (see Eq. (5)): note that this leads to values of the correlation length parameter 1=ðaf Þ respectively equal to 66.7 and 6.67 m at f ¼ 0:05 and 0.5 Hz. In the LWC case the correlation length was fictitiously reduced of two orders of magnitude, so that a null-coherence case was practically simulated. 2.1. Complete and reduced representation In the numerical procedure here adopted, 3D turbulence was modeled by generating artificial velocity time histories in all nodes of the FE mesh. Generation was based on the probabilistic model proposed in [26]; this model is completely defined when the average velocity W , the terrain factor kr , the roughness length z0 and the minimum height zmin are given. The values here adopted are the following: W ¼ 30 m=s, kr ¼ 0:22, z0 ¼ 0:3 m, zmin ¼ 5 m and correspond to a suburban or industrial area (type III) according to the Eurocode 1 classification ([8]). The artificial generation was performed by wave superposition, according to the classical Shinozuka method ([24,25]): for a gradual build-up of the excitation, the first part (500 s) of the generated turbulence histories, which are 2000 s long, was multiplied by a cosine taper.
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In order to investigate the effect of the accuracy in modeling the spatial variation of the wind field, starting from the generated time histories (complete wind model, CWM) a reduced model (RWM) was derived, which is consistent to the cable reduced analytical model. To this aim, it was assumed that the cable has the supports at the same elevation and that the wind velocity field has its mean component orthogonal to the vertical plane passing through the supports; in this setting the static equilibrium configuration of the cable, which is the reference configuration for the associated eigenproblem, belongs to a rotated plane. Consequently (see Fig. 4c), the fluctuation of the wind velocity can be expressed in terms of its in-plane and out-of-plane components v¯ and w ¯ and then expanded in terms of the corresponding in-plane (ji ) and out-of-plane (ci ) cable eigenfunctions, i.e. v¯ ðx; tÞ ¼
nv X
ji ðxÞZiv ðtÞ;
wðx; ¯ tÞ ¼
i¼1
nw X
ci ðxÞZiw ðtÞ,
(12)
i¼1
where the generalized components Ziv and Ziw of the turbulence can be easily expressed, exploiting orthogonality as shown in the Appendix. By retaining the components of the first in-plane and out-of-plane modes only, the reduced wind model (RWM) was obtained. In Figs. 1a,b the time histories of the horizontal wind turbulence, obtained with CWM and RWM, are depicted over a given time interval at the cable mid-point and at a quarter point of the cable length, respectively. Figs. 1c–f represent the wind turbulence frequency content of complete and reduced wind fields in the range 0–2 Hz only, since the wind spectrum shows a negligible contribution above 2 Hz. Consequently, modal components with the natural frequency above such range are not directly excited by the wind turbulence. In Fig. 2 the spatial mean square value (msv) of the horizontal turbulent components of the wind velocity, defined as the time variation of the quantity R‘ 1=‘ 0 wT ðs; tÞwðs; tÞ ds, is represented for the two models (CWM and RWM). The spatial mean square value is numerically evaluated; the results show how in the AWC case (a) the reduced field still represents most of the wind power, while for LWC (b) significant differences can be appreciated. To highlight the response contribution of the different modes, the modal loads of the first four out-plane modal components due to the horizontal wind turbulence were evaluated and reported in Fig. 3; these were computed according to expressions (1) and (2), in which the trigonometric factor G w is here equal to cos y, where y is the angle of rotation of the cable plane under average (static) wind loading (see Fig. 4c). In Fig. 3 the in-plane modal loads are omitted since the static configuration under mean wind lays in 45 inclined plane and in reason of the similar shape for in- and out-plane modes, their absolute values are very similar to the out-of-plane ones. It results that the modal loads have a similar frequency content and show a notable reduction of the higher modes intensity for the AWC, while this reduction for the LWC is less pronounced. 3. Cable models under turbulent wind Two modelization approaches to derive the equations of motion of a suspended cable loaded by a turbulent wind are briefly described. Firstly, a continuum model is introduced, fully described by three partial differential equations that are simplified by classically proved assumptions. Secondly, the equations of motion of a cable and its aerodynamic interaction are presented in the classical framework of the FE method.
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w1(m/s)
20
(a)
189
CWM (l/4) RWM (l/4)
0
-20
w1(m/s)
26
(b)
CWM (l/2) RWM (l/2)
0
-26 1540
1560
1580
1600
1620
1640
t (sec) 1.8
(c)
(d)
CWM (l/2)
(e)
RWM (l/2)
(f)
CWM (l/4)
RWM (l/4)
1.2 0.6 0
0
1 f (Hz)
20
1 f (Hz)
20
1 f (Hz)
20
1 f (Hz)
2
Fig. 1. Horizontal wind turbulence (AWC), complete and reduced models (CWM–RWM): time histories of wind velocity (a) at ‘=2, (b) at ‘=4, Fourier spectrum of wind turbulence (c) and (d) at ‘=2, (e) and (f) at ‘=4.
3.1. Analytical model The static equilibrium configuration of a cable suspended between two fixed horizontal supports can be described by a curve that lies in the vertical plane (Ox^ y^ in Fig. 4). By referring to taut and shallow cables (i.e. d=‘p1=8, where d is the cable sag and ‘ is the distance between the two supports), the configuration under self-weight can be described ^ ^ ðx=‘Þ ^ 2 , with constant horizontal tension H, by a parabolic function, yðxÞ ¼ 4d½x=‘ assumed to describe the initial tension N 0 , (N 0 ðsÞ ’ H). The interaction between the cable with circular cross-section and the blowing wind is analyzed by referring to a wind action that considers a mean flow acting orthogonally to the cable plane with superimposed turbulence, as described in the previous section. Apart from aeroelastic interaction the wind mean component exerts a constant out-of-plane transversal action. The varied equilibrium configuration under the static wind action can be searched solving the linearized ([21]) or nonlinear ([16]) static problem depending on the mean component intensity.
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5000 (a)
msv (m2/s2)
complete CWM reduced RWM
2500
0 2000 (b)
msv (m2/s2)
complete CWM reduced RWM
0 1540
1560
1580
1600
1620
1640
t (sec) Fig. 2. Spatial msv of the 1D wind turbulence: (a) AWC, (b) LWC.
The evaluation of the static reference configuration C 0 described by the parabolic function yðxÞ lying on a rotated plane from the vertical one, permits to derive the dynamic varied configuration C 1 through the displacement components uðs; tÞ, vðs; tÞ and wðs; tÞ along the new co-ordinate axes x, y, z, respectively, aligned and orthogonal to the rotated plane (Fig. 4c). Following [17], the Lagrangian measure of strain is assumed eðx; tÞ ¼ u0 þ y0 v0 þ 12ðv02 þ w02 Þ
(13)
and the equations of motion of the system are obtained by the Hamilton’s extended principle, mu€ þ mu u_ ½EAe0 ¼ 0,
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8
(a)
p1w p2w p3w p4w
4 piw (kN)
191
AWC
0
-4
-8 5
p1w p2w LWC p3w p4w
(b)
piw (kN)
2.5
0
-2.5
-5 1540
1560
1580
1600
1620
1640
t Fig. 3. Out-of-plane modal loadings due to wind turbulence: (a) AWC, (b) LWC.
m€v þ mv v_ ½Hv0 þ EAðy0 þ v0 Þe0 ¼ f v , mw€ þ mw w_ ½Hw0 þ EAw0 e0 ¼ f w ,
(14)
where E is the modulus of elasticity, A is the area of the cross section, m, mu , mv and mw are the mass and damping coefficients of the cable for unit length; a dot and a prime indicate derivatives with respect to time t and the abscissa x, respectively. The assumption that the gradient of the horizontal component of the dynamic displacement is smaller with respect to the gradient of the transversal components (u0 5v0 ; w0 ), and y0 51, H=EA51, has been introduced and the problem is completed by homogeneous boundary conditions in 0; ‘. _ x; tÞ and f w ð_v; w; _ x; tÞ are the time-dependent components The force components f v ð_v; w; in the y- and z-axes directions, respectively, representing the wind–structure interaction. These forces are acting on the circular cross-section instantaneously, neglecting the timevariation of the aerodynamic coefficients. Besides that, given the central symmetry of the cable section, the drag force f D is independent of the wind angle of attack remaining aligned with the wind relative velocity direction V r , yielding f D ¼ 12cD ra bjV r jV r ,
(15)
where cD is the drag coefficient, ra is the air density and b the section diameter. Furthermore, it is assumed that the wind is described by the horizontal mean component
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192
l ∧
z
∧ ∧
θ
z, w Fdw
∧
x d
u y∧
Fdv
W
C0
∧
z, w
v C1
w
W
∧
∧ y, v
(a)
y, v
(c) x2 ,v2 z
y x W
l
(b)
w1 γr
w2 . -v1
W1 Vr -v. 2
. v2 . v1
x1 ,v1
(d)
Fig. 4. Suspended cable configuration and section: (a) and (c) analytical model, (b) and (d) finite element model.
W enriched by the two time varying components v¯ , w¯ aligned to the in- and out-of-plane displacements in the varied configuration. Thus, the linearized components of the drag force f D in the in- and out-of-plane direction rotated of y with respect the along flow horizontal direction (see Fig. 4c) are written as _ tÞ av v¯ ðx; tÞ avw wðx; f v ðx; tÞ ¼ cD ra bW ½av v_ðx; tÞ þ avw wðx; ¯ tÞ, _ tÞ þ awv v_ðx; tÞ aw wðx; f w ðx; tÞ ¼ cD ra bW ½aw wðx; ¯ tÞ awv v¯ ðx; tÞ,
(16)
where the coefficients aij are given in Appendix as functions of y. Under the previous assumptions, uðx; tÞ can be eliminated by a standard condensation procedure leading to the definition of a constant elongation e¯ as Z ‘ v02 þ w02 0 0 e¯ ðtÞ ¼ 1=‘ yv þ dx. (17) 2 0 Two integral-differential equations of motion in the transverse displacements vðx; tÞ and wðx; tÞ are thus obtained: mv€ þ mv v_ ½Hv0 þ EAðy0 þ v0 Þ¯e0 ¼ f v , mw€ þ mw w_ ½Hw0 þ EAw0 e¯ 0 ¼ f w .
(18)
A non-dimensional form of the equations is introduced by normalizing the variables with respect to the length of the cable and its first in-plane frequency o1v . Hence, in the following the reported quantities are dimensionless, according to the positions given in the Appendix and the tilde is dropped for simplicity. The displacements are described by
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the expansions vðx; tÞ ¼
nv X
wðx; tÞ ¼
ji ðxÞqiv ðtÞ;
nw X
i¼1
ci ðxÞqiw ðtÞ,
(19)
i¼1
where ji and ci are the eigenfunctions of the Hamiltonian linearized equations of motion (18). The associated eigenvalues give the in-plane and out-of-plane frequencies oiv and oiw , which depend on the mechanical cable characteristics through the non-dimensional Irvine parameter, l2 ([13]). The use of the expansions (19) leads to the following expression for the constant strain e¯ ðtÞ ¼
nv X
nv X
b1j qjv þ
j¼1
b2ij qiv qjv þ
i¼1;j¼1
nw X
b3i q2iw .
(20)
i¼1
Thus, the following nonlinear ordinary differential equations describe the motion ! nw nv nv nv X X X X zijv q_ jv þ zijvw q_ jw þ a0ij qjv þ a1i þ a2ij qjv e¯ ¼ piv , q€ iv þ j¼1
q€ iw þ
nw X j¼1
j¼1
zijw q_ jw þ
nv X
j¼1
j¼1
zijwv q_ jv þ o2iw qiw þ a3i qiw e¯ ¼ piw ,
(21)
j¼1
where the in-plane cable frequencies are obtained as o2iv ¼ ða0ii þ a1i b1i Þ, while oiw are the out-of-plane frequencies; on the other hand, structural viscous damping has been assumed as proportional. The linear part of system (21) is composed by diagonal stiffness coefficients, since the off-diagonal terms ða0ij þ a1i b1j Þ vanish due to the orthogonality of the eigenfunctions, while aerodynamic damping coefficients, introduced by the expressions (16) are coupled due to the y-angle between the mean wind and the out-of-plane directions. Furthermore, a rich variety of coupling arises in the nonlinear part, due to quadratic and cubic nonlinear terms. The expressions of the coefficients in Eqs. (20) and (21), as well as the definitions of the pi -functions, can be found in Appendix. 3.2. Finite element model A three-node isoparametric FE is developed and coded following [6]; the element is directly formulated in the coordinates of the global dynamic model, so that no transformation is necessary to form the tangent stiffness and the generalized components of restoring forces. Linear elastic behavior of the cable is assumed in the large displacement and small deformation range to derive the restoring forces increment during the time step Dt; following the classical ‘updated Lagrangian’ formulation, a linear estimate of the restoring forces Re ðt þ DtÞ as functions of the nodal displacement increments Due of the FE is sought in the form t
t
ðgÞ Re ðt þ DtÞ ¼ Re ðtÞ þ ½kðeÞ e þ ke Due .
(22)
In Eqs. (22), the contribution to the predicted restoring force is given in terms of elastic t t and geometric tangent stiffness matrices kðeÞ and kðeÞ e g , as well as by the vector of the restoring forces at time t; these are respectively defined, along with the mass matrix me , by
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the following expressions: Z L ðeÞt ke ¼ EAH T;s H ;s xðtÞxT ðtÞH T;s H ;s ds;
t kðgÞ e
0
Z Re ðtÞ ¼ 0
Z
L
Nðs; tÞH T;s
H ;s xðtÞ ds;
Z
L
¼ 0
Nðs; tÞH T;s H ;s ds,
L
mðsÞH T ðsÞHðsÞ ds,
me ¼
(23)
0
where HðsÞ is the shape function matrix, H ;s ðsÞ its derivative, Nðs; tÞ the axial force and xðtÞ the nodal coordinate vector. When wind loading of the element is considered, ([19]), local axes x1 (parallel to the average wind velocity component W ) and x2 are introduced in the cross-section (Fig. 4d); accordingly, local components of wind turbulence (w1 and w2 ) and cable velocity (_v1 and v_2 ) are defined. Within the context of quasi-steady aerodynamic modeling and for a circular section, local forces per unit length acting along x1 , x2 are cast into the following form: ( ) 2 1 ra bV r cD ðgr Þ cosðgr Þ ðaÞ F ¼ , (24) 2 ra bV 2r cD ðgr Þ sinðgr Þ where V r , gr are the instantaneous relative velocity and angle of attack: w2 v_2 V r ¼ ðW þ w1 v_1 Þe1 þ ðw2 v_2 Þe2 ; gr ¼ arctg . W 1 þ w1 v_1
(25)
To compute the generalized elemental nodal components QðaÞ e of the aerodynamic local forces F ðaÞ , virtual work analysis can be applied leading to the following expression: Z L QðaÞ ¼ F ðaÞ ðt; sÞCe ðs; uÞ ds, (26) e 0
where Ce ðs; uÞ ¼ ae ðs; uÞHðsÞ and ae ðs; uÞ is a transformation matrix depending on the direction of the average wind velocity and on the direction of the section normal, which is in turn defined by nodal displacements u and by shape functions. Consequently, the equation of motion at time t can be written in the form M u€ þ C u_ þ Rðu; u_ Þ ¼ QðaÞ ðt; u; u_ Þ þ QðtÞ. (27) P In Eq. (27), M ¼ k mek is the mass matrix, obtained by assembling elemental mass P matrices; C is the structural damping matrix; R ¼ k Rek is the vector of the generalized P components of restoring forces, obtained by assembling elemental vector; QðaÞ ¼ k QðaÞ ek is the vector listing the generalized components of the aerodynamic forces; Q is the vector of the generalized components of other forces (static or dynamic) acting upon the system and u, u_ are the Lagrangian nodal displacements and velocities. In the numerical implementation of the procedure, the elastic stiffness and the mass matrices are evaluated in closed form; two point Gauss quadrature was instead used to evaluate the integrals involving the variation in the axial force. The equations of motion (27) of the FE model were integrated by means of a numerical procedure based on the Newmark method, as modified by Hilber, Hughes and Taylor for controlling numerical damping of high-frequency oscillations ([12]). To ensure dynamic equilibrium at the end of the time step, the modified Newton–Raphson method was
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adopted; aerodynamic stiffness and damping matrices, leading to non-symmetric terms, were disregarded in building the iteration matrix. 4. FEM response analysis The response of a cable having a section diameter of 0.0281 m and subjected to an artificial turbulent wind field was studied by the FE approach. Under self-weight and an average wind velocity of 30 m/s, the cable lays on an inclined plane and is characterized by the following non-dimensional parameters: m ¼ EA=H ¼ 486, n ¼ d=‘ ¼ 1=45, l2 ¼ 15:36. In such conditions, assuming m ¼ 1:8 kg=m and a distance between the supports ‘ ¼ 266:948 m, the above stated values of m, n lead to EA ¼ 17956:980 kN, H ¼ 36:938 kN and an initial (undeformed) cable length L ¼ 266:749 m. In the FE analysis the cable was subdivided into eight three-node FE of equal length, leading to a system of 45 dofs (Fig. 4b); consistent weight forces were first applied to the nodes, together with drag forces due to average wind velocity, for computing the static equilibrium configuration. The modal analysis, performed upon linearization in this configuration, shows perfect agreement with the analytical results both for natural frequencies (see Table 1) and for modal shapes (for details see [11]). The first in-plane natural frequency was found at 0.4 Hz ðo1v ¼ 2:519 rad=sÞ. Based on the system eigenpairs, a viscous damping matrix C leading to constant damping ratio zi ¼ 0:0044 for the first 35 normal modes was subsequently computed and used in the time history analyses (see Eq. (27)). The FEM was subjected to a turbulent wind field with mean velocity acting along the yaxis of Fig. 4b (transverse direction); the wind forces evaluated at each node according to Eqs. (24) and (25) are directly dependent on the wind velocity field as discussed in the previous sections. The cable response is described through the displacement components vi ði ¼ 1; 2Þ in the along- and across-wind local reference axes that in the studied cases coincide with horizontal and vertical global reference axes. The displacement components and the time are defined as for the non-dimensional variables in Eq. (31) in Appendix. 4.1. In-plane and out-of-plane turbulence Cable dynamic response has been deeply investigated in technical literature, evidencing nonlinear coupling, complex dynamics and bifurcations. These dynamic features are well visible in the frequency response functions to in-plane and out-of-plane harmonic loads. On this respect, a good agreement has been already found between analytical prediction and FEM under periodic loads ([10,11,20]). Table 1 In-plane and out-of-plane non-dimensional cable frequencies In-plane
An.
FE
Out-of-plane
An.
FE
1vs 2va 3vs 4va 5vs
1.000 1.339 2.026 2.677 3.350
1.000 1.337 2.020 2.688 3.382
1ws 2wa 3ws 4wa 5ws
0.669 1.339 2.008 2.677 3.347
0.667 1.339 2.010 2.687 3.372
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To better frame the present analysis in the previous investigations, the cable response to wind was studied projecting the random load along the in- and out-of-plane directions that are rotated with respect to the natural cable configuration depending on the horizontal mean wind intensity. Consequently, the cable static configuration under horizontal mean wind velocity was first determined. In Fig. 5 the static equilibrium positions assumed by the cable mid-point under increasing wind mean velocity are reported as a reference curve showing that the cable configuration under the considered steady wind of 30 m/s belongs to a plane inclined almost 45 . Starting from this nonlinear equilibrium state, cable responses to turbulence applied in different directions was studied. First, the separate effects of in-plane and out-of-plane components of the horizontal turbulence was considered. In the in-plane case, the loading excites directly the in-plane modes while the out-of-plane components are participating due to both linear and nonlinear coupling ascribed, respectively, to aerodynamic damping and large displacements. Indeed, Fig. 5a shows the cable mid-point response under in-plane turbulence, where both displacement components are present. In the absence of aerodynamic forces, the cable oscillates in the plane of static configuration up to a critical value of amplitude, when bifurcation occurs and also out-ofplane component arises ([21,9]). This bifurcation phenomena of in-plane oscillations in a spatial motion is destroyed by the aerodynamic coupling, while the cable nonlinear coupling affects only the high amplitude response. When out-of-plane turbulence is 0 (a)
(b)
v2
v2
-0.01
-0.02
-0.03
v1
v1
0 (c)
(d)
v2
v2
-0.01
-0.02
-0.03
0
0.01
0.02 v1
0.03
0
0.01
0.02
0.03
v1
Fig. 5. Cable mid-point response to wind by FEM under: (a) in-plane turbulence, (b) out-of-plane turbulence, (c) 3D turbulence, (d) reconstructed response. Reference curve: static equilibrium under an increasing mean wind (0–45 m/s).
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considered, see Fig. 5b, both linear and nonlinear coupling are present at any level of excitation intensity, but due to the larger flexibility in the out-of-plane motion, the time histories of the two components are quite different and out-of-plane component is prevailing; the ratio between the two amplitudes is significantly larger than two, a value foreseen by means of the 2 dof analytical model in [21]. Then, the cable response to the simultaneous in-plane and out-of-plane turbulence has been simulated and response with larger amplitudes than the ones obtained superimposing the previous results are obtained, as it is evident comparing the Figs. 5c and d. This difference can be ascribed to a slightly non-linear behavior which is compatible with the cable relatively small displacement range. Looking at a direct comparison of the two time histories it is clearly visible that the overall aspect and amplitude of the cable motion are quite similar; examination of the Fourier spectra (not shown for brevity) points out that the most significant differences are in the background response, at frequencies lying below the cable eigenspectrum. Further analyses have been performed to identify the main factors that are significant in the complex phenomena under study. Among these, the influence of the load description, such as turbulence dimensionality and coherence, and the role of cable model dimension are tackled. 4.2. 1D and 3D turbulence The influence on cable response of the turbulence dimensionality was evaluated considering both 1D and 3D turbulent wind fields. In the 1D case, the turbulence was superimposed to the horizontal mean wind W in the same direction; in the 3D case, three turbulent components at each point were superimposed to the horizontal mean wind. The parameters selected in the wind model are such that the coherence expected for the turbulence components at different points along the cable is within average ([26]) (AWC). Horizontal and vertical cable displacements at ‘=2 and ‘=4 under 1D and 3D turbulent wind are compared in Fig. 6. The differences between the displacement time histories are not much sensitive to the observation point while the richer turbulence description affects mainly the vertical (across-wind) components. Indeed, a 3D turbulent wind field encompasses a description of the across-wind turbulence that contributes mainly to the differences in the vertical cable displacements. However, the maximum of the cable vertical displacement is not significantly affected by the presence of the across-wind turbulence. Consequently, deeper analysis regarding the response prediction by reduced order models were conducted primarily with a 1D wind turbulence description. In this respect, Fig. 7 shows the time histories of the ðv1 ; v2 Þ displacement components of the model at ‘=4, comparing the responses to complete (CWM) and reduced (RWM) wind models for the average coherence (AWC) case. The cable response is still well predicted when a wind turbulence is described by the first in-plane and out-of-plane modes (RWM). Indeed, the effects of CWM higher modes is appreciable only in the horizontal response (see Fig. 7b). The cable mid-point response shows even smaller differences with respect to ‘=4. 4.3. Effects of turbulence coherence The conjecture regarding the reduced representation of the wind in the cable eigenfunction basis should be supported through accurate analyses of the response to three-dimensional turbulent wind even in the case of low coherence (LWC).
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v2 (l/4)
-0.006
(a)
v1 (l/4)
-0.016 0.02
(b) 0.004 3700
3800
3900
4000
3900
4000
τ
v2 (l/2)
-0.008
(c)
v1 (l/2)
-0.024 0.028
(d) 0.004 3700
3800 τ
Fig. 6. Comparisons of FEM displacements under 1D (solid) and 3D (dashed) wind turbulence (AWC): (a) vertical and (b) horizontal displacements at ‘=4; vertical (c) and horizontal (d) displacements at ‘=2.
v2 (l/4)
-0.006
-0.016
(a)
v1 (l/4)
0.02
(b) 0 3700
3800
τ
3900
4000
Fig. 7. Comparisons of FEM displacements at ‘=4 under complete (solid) and reduced (dashed) horizontal 1D wind turbulence (AWC): (a) vertical, (b) horizontal displacements.
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The comparison between the cable response to complete and reduced wind shows that the low turbulence coherence makes a little worse the approximation (Figs. 8). The results confirm that a LWC turbulence excites the high-frequency modes more than in the AWC case. Indeed, significant differences are observed between the cable response to turbulent wind fields with different coherence (AWC and LWC) (see Fig. 9). To explain these differences it may be useful to evaluate an energy measure of the response associated to each mode, selected as the ratio of the elastic strain energy associated at the ith mode normalized to that of the first out-of-plane mode. The elastic strain energy for the ith mode was computed as the modal stiffness times the msv value of the ith modal amplitude. The obtained results for both in-plane and out-of-plane modes are shown in Fig. 10, where the normalized energy can be easily compared (in logarithmic scale) for the two different wind coherence, namely AWC and LWC. The comparison evidences that the
v2 (l/4)
-0.009
(a) -0.015
v1 (l/4)
0.016
(b) 0.008 3700
3800
3900
4000
τ Fig. 8. Comparisons of FEM displacements at ‘=4 under complete (solid) and reduced (dashed) 1D wind turbulence (LWC): (a) vertical, (b) horizontal displacements.
Fig. 9. FEM displacements at ‘=4 under 3D wind turbulence with AWC (solid) and LWC (dashed): (a) vertical, (b) horizontal displacements.
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1
1
0.1
0.1 Ew
Ev
200
0.01
0.001
0.001 0.0001 1
0.01
2
3
(a)
4 Nm
5
6
0.0001 1
7 (b)
2
3
4 Nm
5
6
7
Fig. 10. Energy distribution vs mode number: average (solid) and low (dashed) wind coherences: (a) in-plane, (b) out-of-plane modes.
out-of-plane modes are excited more than the in-plane ones. Furthermore, the AWC and LWC cases clearly show that the wind coherence has an effect on the excited modes; indeed the fourth mode, either for in-plane and out-of-plane case, determines a switch on the relative importance of the involved modal energy by the AWC and LWC loads. However, even though a wind with low coherence has a larger influence on the higher cable modes, the difference is not significant and the involved energy at higher frequencies is two- or three-order smaller than the energy brought at low frequency by the main modes. 5. Cable reduced models In order to investigate the effect of model dimensions a 8 dof analytical model was built through the use of the first four in- and out-of-plane modes. The model, described completely by Eq. (21) and the relative coefficients reported in Appendix, was numerically integrated in order to evaluate the cable response to turbulent wind. During the analyses, special attention was devoted to the linear viscous terms to which contribute the cable material damping and more considerably the aerodynamic damping due to the fluid–structure interaction. The relevance of the aerodynamic damping on suspended cables has been qualitatively described ([5]) and experimentally confirmed ([15,18]). Here, a description of the aerodynamic damping is pursued through the analytical model projected on the eigenfunction basis. On this respect, the expressions of the coefficients in the linear viscous damping matrix as function of the inclination y, that depends on the mean wind velocity, are given in Eqs. (31), (36) and (37) of the Appendix. For the examined case ðW ¼ 30 m=sÞ the coefficients zaij of the aerodynamic damping matrix are reported in Table 2, showing that a relevant coupling between symmetric inplane and out-of-plane modes is present, while material damping was simply simulated as mass proportional, leading to an increase of the diagonal terms according to the expression m in the Appendix with a constant coefficient for each mode ðzm ijv ¼ zijw ¼ 0:0044Þ. The damping matrix highlights the coupling between in-plane and out-of-plane modes due to aerodynamic damping predicted in the continuous model, see Eq. (16). There is no coupling, however, in terms of velocity components in the same direction (in-plane or outof-plane); this is because the aerodynamic damping effects in the same direction can be regarded as mass-proportional, since both the cable properties and the mean wind velocity
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Table 2 Aerodynamic damping matrix for a selected mean wind ðW ¼ 30 m=sÞ
1vs 2va 3vs 4va 1ws 2wa 3ws 4wa
1vs
2va
3vs
4va
1ws
2wa
3ws
4wa
0.187 — — — 0.060 — 0.004 —
— 0.187 — — — 0.063 — —
— — 0.187 — 0.004 — 0.067 —
— — — 0.187 — — — 0.063
0.066 — 0.004 — 0.191 — — —
— 0.063 — — — 0.191 — —
0.004 — 0.059 — — — 0.191 —
— — — 0.063 — — — 0.191
are uniform along the cable. Note that the aerodynamic damping would be fully mass-proportional if damping coefficients were the same in the y and z directions, since in this case coupling terms would cancel out. Unfortunately, this is not the case and coupling terms appear in Eq. (16). In the discretized setting of Eq. (21) the degree of coupling depends on the trigonometric factors av ; aw ; avw and on the magnitude of the cross modal coefficients I aij (see Appendix), which, in turn, is influenced by the cable properties. The effectiveness of the analytical model description of the cable response is tested through direct numerical integration in time of the nonlinear modal equations (21). In particular, the 8 dof analytical model coordinates, obtained numerically, are compared with the response of the FEM model projected at each time instant on the modal basis. Looking at modes associated to high frequency, it can be easily observed that the reduction of the modal loading factors (see Fig. 3) together with the inherent smaller flexibility is sufficient to contrast the decrease of the aerodynamic damping that is roughly inversely proportional to the frequency. The accuracy of the response predicted by the 8 dof analytical model is shown in Figs. 11 and 12 with reference to the first in-plane and out-of-plane modes, and for the two wind coherence values considered. A general excellent agreement is observed between the modal response obtained by analytical and FEM in time domain. Indeed, an exam of the frequency content of the cable response pursued through FFT analysis confirms the optimal matching both for the low-frequency ‘‘quasi-static’’ range and for the resonant component. The contribution of higher modes than the first in-plane and the first out-of-plane is negligible; it is confirmed by the very small difference occurring between the responses of 2 dof and 8 dof models which is not here reported for sake of brevity. The performed comparisons between cable responses obtained by reduced and FEM models evidence that small nonlinear modal coupling with higher modes is involved in the dynamic behavior even if geometric (in the analytical and FEM models) and aerodynamic (in the FEM model) nonlinearities are taken into account in the analyses. These results are in accordance with the fact that high level of damping is contributed by the aerodynamics at the studied wind velocity; moreover, relatively small dynamic amplitudes are involved in the cable response which is strongly influenced by quasi-static motion at very low frequency.
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q1v
0.01
(a)
FEM An
0
a) -0.01 0.01
(b)
q1w
FEM An
-0.01 4000
4050
4100
4150
4200
4250
τ (c)
(d)
(e)
FEM
(f)
An
FEM
Cq1v
Cq1w
An
0
0.5
1 f
1.5 0
0.5
1 f
1.5 0
0.5
1 f
1.5 0
0.5
1
1.5
f
Fig. 11. Comparisons of analytical (dashed) and FEM (solid) results in terms of modal amplitude under 1D wind turbulence (AWC): (a) first in-plane, (b) first out-of-plane, (c) and (e) FFT of analytical response, (d) and (f) FFT of FEM response.
6. Conclusions The cable response to turbulent wind has been analyzed by means of a large dimensional model making use of the FE method and compared to the response predicted using reduced models for wind and mechanical system. Before analyzing the ability of reduced models, the refined FEM was used to enlighten the main features of the cable response to turbulent wind, useful in the subsequent comparison. Under an horizontal mean wind with a superimposed turbulence the vertical cable static configuration rotates on a plane inclined with an angle depending on the mean wind velocity. In the new static configuration the effects of the fluid-structure interaction is such that the linear aerodynamic damping, even for radially symmetric sections, couples the in-plane and out-plane cable oscillations. This coupling increases with the mean wind velocity and in general dominates the response also with respect to the involved geometric and aerodynamic nonlinearities. Two effects of the wind turbulence have been investigated:
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q1v
0.01 (a)
FEM An
(b)
FEM An
0
-0.01
q1w
0.01
-0.01 4000
4050
4100
4150
4200
4250
τ (c)
An
An
FEM
Cq1v
Cq1w
FEM
0
0.5
1 f
1.5 0
0.5
1 f
1.5 0
0.5
1 f
1.5 0
0.5
1
1.5
f
Fig. 12. Comparisons of analytical (dashed) and FEM (solid) results in terms of modal amplitude under 1D wind turbulence (LWC): (a) first in-plane, (b) first out-of-plane, (c) and (e) FFT of analytical response, (d) and (f) FFT of FEM response.
the occurrence of across-wind components and the coherence. Wind turbulence has been described through both 1D and 3D velocity fields, and the cable response has been evaluated in both cases, evidencing that vertical cable response is mainly affected by the wind description, although with small differences. These differences augment when the wind coherence decreases with respect to an average case. The main interest of the study has been devoted to the evaluation of reduced order models in preserving the description of the main features of the cable dynamic response. On this respect, both the mechanical system and the spatially varying wind velocities are projected on the basis of cable eigenfunctions retaining in the reduced models few dof associated with the low frequency modes. In the wind case, the dynamic characteristics of RWM have been compared to a complete description (CWM), both based on artificial generation. The analysis of the FEM response to complete and reduced wind models shows that most of the responses are given by the low modes and thus it is possible to strongly reduce the problem dimensionality. In this respect, to confirm this prediction for the cable mechanical system, the dynamic response delivered by the nonlinear finite element model
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of a suspended cable excited by wind has been compared to the response of an analytical model using few low-frequency modes. The results show that the prediction of the cable response is well described by using reduced models for both wind and cable, in the case of both AWC and LWC. Finally, weak nonlinear modal coupling has been found in the experienced dynamic responses of the investigated cable; this is mainly due to the high level of aerodynamic damping introduced by the fluid-structure interaction. Acknowledgments This work was partially supported by MIUR (Ministry of Education, University and Research) under the project: Dynamic behavior of structures: theory and experiments (COFIN 01-02 and 03-04, http://www.disg.uniroma1.it/fendis). Appendix A A.1. Wind turbulence model The expressions of the generalized components Ziv and Ziw of turbulence introduced in Eq. (12) were easily obtained exploiting the orthogonality, with respect to the mass distribution, of the cable eigenfunctions such as R R ji ðxÞmðxÞ¯vðx; tÞ dx c ðxÞmðxÞwðx; ¯ tÞ dx DR Ziv ðtÞ ¼ ; Ziw ðtÞ ¼ D R i 2 . (28) 2 D ji ðxÞmðxÞ dx D ci ðxÞmðxÞ dx In order to define the complete relation between the generalized components of the turbulence and the modal loading in Eq. (16) the mixed generalized components should be introduced as R R c ðxÞmðxÞ¯vðx; tÞ dx ¯ tÞ dx i ðxÞmðxÞwðx; Dj R 2 . (29) Zivw ðtÞ ¼ ; Ziwv ðtÞ ¼ D R i 2 D ji ðxÞmðxÞ dx D ci ðxÞmðxÞ dx Consequently, the non-dimensional modal loading defined in Eq. (21) due to the turbulence, can be obtained introducing a nondimensional expression for the generalized components of the turbulence as Z~ i ðtÞ ¼ o1v Zi ðtÞ and coherently with the expressions in Eq. (21), omitting the tilde for the sake of simplicity, the expression of the modal loading can be related with the turbulence component as follows: piv ðtÞ ¼ cD w½av Ziv ðtÞ þ avw Zivw ðtÞ;
piw ðtÞ ¼ cD w½aw Ziw ðtÞ þ awv Ziwv ðtÞ.
(30)
A.2. Cable model A non-dimensional form of Eq. (18) can be obtained introducing the following quantities: x x~ ¼ ; l
y~ ¼
y ; d
v v~ ¼ ; ‘
w~ ¼
w ; ‘
~ iv ¼ o
oiv ; o1v
~ iw ¼ o
oiw , o1v
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t ¼ o1v t;
av ¼
m¼
EA ; H
ð1 þ sin2 yÞ ; 2
n¼
aw ¼
d ; ‘
w¼
205
ra bW , mo1v
ð1 þ cos2 yÞ ; 2
avw ¼ awv ¼
sin y cos y . 2
(31)
The modal shapes of a suspended cable are given by the following expressions: b ji ðxÞ ¼ 1 tan i sin bi x cos bi x bi ; i ¼ 1; 3; . . . ðsymmetricÞ, 2 ji ðxÞ ¼ sin ipx; ck ðxÞ ¼ sin kpx;
i ¼ 2; 4; . . . ðantisymmetricÞ, k ¼ 1; 2; 3; . . . ,
(32)
where bi ¼ cosðbi =2Þð1=ðcos bi þ ð1Þh ÞÞ is a normalization constant (h ¼ i for i ¼ 1; 5; . . . and h ¼ 2i for i ¼ 3; 7; . . .) and the spatial frequencies bi are given by the roots of the characteristic equation " # bi bi 4 bi 3 tan ¼0 (33) 2 2 l2 2 which depend on Irvine parameter l2 ¼ 64mn2 , with m and n given by Eqs. (32), ([13]) while the related dimensional natural frequencies can be obtained as oiv ¼
bi os ; p
i ¼ 1; 3; . . . ðsymmetricÞ
oiw ¼ ios ; i ¼ 1; 2; 3; . . . ;
oiv ¼ ios ; i ¼ 2; 4; . . . ðantisymmetricÞ,
p os ¼ ‘
where
rffiffiffiffiffi H . m
(34)
In order to define the coefficients of Eqs. (20), (21), let us introduce the following integrals: Im i
Z ¼
I yi ¼
Z
1 0
Z
f 2i ðxÞ dx;
I eij
1 0
y0 ðxÞf 0i ðxÞ dx;
Z
1
¼ 0
f 0i ðxÞf 0j ðxÞ dx;
I pi ðtÞ ¼
Z
I aij
1
f i ðxÞf j ðxÞ dx,
¼ 0
1
hðx; tÞf i ðxÞ dx,
(35)
0
where f i and f 0i are the placeholders of ji ðxÞ, or ci ðxÞ and their derivatives where appropriate, and yðxÞ is the parabolic function lying in the y rotated plane from the vertical one while hðx; tÞ is the placeholder of gust wind velocity distributions v¯ ðx; tÞ and wðx; ¯ tÞ. a Consequently, the coefficients aij and bij , the damping coefficients zij ¼ zm ij þ zij of the damping matrix, including material and aerodynamic dampings, and the gust loading pi in Eqs. (20) and (21) are defined as follows.
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A.3. In-plane equations I av I avw mv ij ij dij þ av mv cD w; zijvw ¼ avw mv cD w, mo1v Ii Ii I ev I ev I yv 1 ij 1 ij 1 i ¼ mv 2 ; a1i ¼ mn mv ; a2ij ¼ m mv 2 , 2 I i b1 I i b1 I i b1
zijv ¼ a0ij
b1j ¼ nI yv j ;
b2ij ¼ 12 I ev ij ;
b3i ¼ 12 I ew ii , ¯
piv ðtÞ ¼ cD w
vv pwv av I p¯ i ðtÞ þ avw I i ðtÞ . mv Ii
ð36Þ
A.4. Out-of-plane equations I aw I awv mw ij ij dij þ aw mw cD w; zijwv ¼ awv mw cD w, mo1v Ii Ii I ew 1 I ew 1 ii ii ¼ mw ; a3i ¼ m mw , 2 I i b1 I i b21
zijw ¼ o2wi
b1i ¼ nI yv i ;
b2ij ¼ 12 I ev ij ;
piw ðtÞ ¼ cD w
vw aw I pi ww ðtÞ þ awv I p¯ i ðtÞ , I mw i
b3i ¼ 12 I ew ii ,
¯
ð37Þ
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