1 Introduction

The study of the aeroelastic behavior of slender structures is a fascinating topic, of high scientific and technical value. The literature is rich of studies concerning specific flexible structures (cables [1,2,3,4,5,6,7], beams [8,9,10,11,12], plates [13,14,15,16]) and general aeroelastic phenomena (galloping [17,18,19,20,21], flutter [22,23,24,25], vortex-induced vibrations (VIV) [26,27,28,29,30]). The instability and bifurcation events can be related to different kinds of excitation. Self-excited autonomous systems, as structures subjected to steady wind, are prone to Hopf bifurcations, caused by zeroing of the total damping. Non-autonomous systems, as structures subjected to turbulent wind, are prone to parametric excitation phenomena, leading to divergence, flip and Neimark–Sacker bifurcations. Depending on the nature of the loads, the different kinds of excitation can interact. Some attention has been devoted in literature to interactive aeroelastic phenomena, as galloping parametric excitation [31,32,33,34,35,36,37,38,39,40] or galloping vortex-induced vibrations [41,42,43,44,45]. In particular, as regard the former class, the principal resonance of a single-degree-of-freedom system with two-frequency parametric and self-excitation is investigated in [31, 34]; the method of multiple scales is used to determine the equations of modulation of amplitude and phase, and qualitative analyses are performed out to study steady state, limit cycle, and torus responses. In [32], analytical investigations of the system under parametric, self-excited, and inertial excitations were carried out. In [33], the galloping of tall prismatic cantilevered structures, due to unsteady wind, is analytically studied. The unsteady wind is considered by adding a time varying wind speed component to the mean wind speed. Consequently, the structure is subjected to multi-harmonic external and parametric excitation. The multiple scale method is used to study the effect of primary and secondary resonances on the galloping response of the structure. Starting from this last work, the parametric, external, and self-excitation of one/two towers under turbulent wind flow is studied in [35, 36, 39] and in [37, 38], focusing the attention to the periodic and quasi-periodic galloping motions. It was shown that the unsteady component of wind can cause a significant decrease in the critical speed; however, the contribution of the unsteady component is less prominent at high wind velocities, at which large amplitude oscillations occur. It was detected, moreover, that periodic and quasi-periodic motions exist according to suitable combinations of the steady and unsteady wind parameters.

Here, the effect of a wind flow acting on a slender prismatic structure is investigated. A class of cross sections is considered, suffering sub-critical bifurcation followed by a regain of stability at high amplitudes, known in the literature as the hard-loss of stability [8, 46, 47]. When the turbulence frequency is nearly double a natural frequency of the structure, this latter is parametrically excited at the primary resonance. Therefore, self-excitation and parametric excitation interact, both in linear and nonlinear fields. The investigation, mainly carried out by perturbation methods, is aimed at evaluating: (a) as the turbulent component reduces the galloping critical velocity, and vice versa, (b) as the steady wind component modifies the parametric excitation instability domain. Moreover, (c) the analysis of the limit cycles generated by the flip bifurcations, and that of tori, generated by the Neimark–Sacker bifurcations, are also of interest.

The paper is organized as follows. In Sect. 2, a continuous aeroelastic model of a slender prismatic visco-elastic structure is reduced to a single-D.O.F system via the Galerkin method. In Sect. 3, the bifurcation equations, ruling the slow flow of the amplitude and phase of the involved mode, are derived via the multiple scale method. In Sect. 4, the linear stability analysis is carried out, and a three-dimensional stability domain is built-up. In Sect. 5, a nonlinear analysis of the bifurcation equation is performed; in particular, the bifurcation equations governing the slow-slow-flow on a torus, are derived. In Sect. 6, numerical results are reported for a sample structure; they are compared with direct numerical integration of the original partial integro-differential equation of motion. In Sect. 7, some conclusions are drawn. In “Appendix A”, the sample system is described. Two other “Appendixes”, supplying details, close the paper.

2 Aeroelastic model

A continuous model of a slender prismatic visco-elastic structure, of length l, restrained to belong to the \((\mathbf {i},\mathbf {j})\)-plane and constrained at its ends A, B, subjected to a turbulent wind flow of velocity U(t)k, acting normally to the plane, is formulated (Fig. 1). The structural model is assumed to be linear (since, as it is well-known, elastic nonlinearities mainly affect the frequency, but not the amplitude of the aeroelastic response); the aerodynamic model is instead nonlinear. The generic equation of motion and the boundary conditions are:

$$\begin{aligned}&m\ddot{v}+\mathscr {L}_{e}\left( v\right) +\mathscr {L}_{v}\left( \dot{v}\right) +c_{e}\dot{v}+p_{a}=0\nonumber \\&\quad \mathscr {B}_{eH}\left( v\right) +\mathscr {B}_{vH}\left( \dot{v}\right) =0\qquad H=A,B \end{aligned}$$
(1)

In these equations: \(v\left( s,t\right) \) is the deflection of the prism in the \(\mathbf {j}\)-direction at the abscissa \(s\in \left( 0,l\right) \) and time \(t\in \left( 0,\infty \right) \); m is the mass per unit length; \(\mathscr {L}_{e}\) and \(\mathscr {L}_{v}\) are elastic and viscous linear operators, respectively; \(\mathscr {B}_{eH}\) and \(\mathscr {B}_{vH}\) are elastic and viscous linear operators acting at the boundaries; \(c_{e}\) is an external damping coefficient; \(p_{a}\left( s,t\right) \) are aerodynamic loads in the \(\mathbf {j}\)-direction; a dot indicate partial differentiation with respect to the time. Equation (1) can be referred as a metamodel of prismatic structures, describing a class of real important structures, as columns, towers, chimneys, and bridges.

Fig. 1
figure 1

Prismatic structure under transverse turbulent wind flow

Full size image

2.1 Aerodynamic model

The aerodynamic forces are determined according to the quasi-static theory [8]. This theory holds when the vortex-shedding phenomenon is well-separated from galloping. Only the vertical component of the aerodynamic forces is considered.

By assuming the wind velocity \(U\left( t\right) \) uniform in space, the quasi-static theory supplies the forces as nonlinearly depending on the structural velocity \(\dot{v}\left( s,t\right) \) (which is responsible for modifications of the attack angle). By assuming that \(\dfrac{\dot{v}}{U}\ll 1\), and the structure cross section is symmetric with respect to the horizontal axis, they are given by the odd-power series:

$$\begin{aligned} p_{a}= & {} -\dfrac{1}{2}\varrho _{a}{DU}^{2}\left[ A_{1}\left( \dfrac{\dot{v}}{U}\right) +A_{3}\left( \dfrac{\dot{v}}{U}\right) ^{3}\right. \nonumber \\&\left. +A_{5}\left( \dfrac{\dot{v}}{U}\right) ^{5}+\cdots \right] \end{aligned}$$
(2)

where: \(\varrho _{a}\) is the air mass density and \(A_{i}\) are dimensionless aerodynamic coefficients, depending on the cross-section shape, of which D is a characteristic length. The aerodynamic forces include all the nonlinearities of the problem.

By letting:

$$\begin{aligned} U(t)=\bar{U}+u(t) \end{aligned}$$
(3)

in which \(\bar{U}\) is the (leading) steady-state wind velocity, and u(t) the (small) turbulent component, and linearizing Eq. (2) in the ratio \({\displaystyle \frac{u}{\bar{U}}}\ll 1\), it follows:

$$\begin{aligned} p_{a}= & {} -\bar{U}^{2}\left[ b_{1}\left( 1+\frac{u(t)}{\bar{U}}\right) \left( \dfrac{\dot{v}}{\bar{U}}\right) \right. \nonumber \\&\quad \left. +\,b_{3}\left( 1-\frac{u(t)}{\bar{U}}\right) \left( \dfrac{\dot{v}}{\bar{U}}\right) ^{3}\right. \nonumber \\&\quad \left. +\,b_{5}\left( 1-\frac{3u(t)}{\bar{U}}\right) \left( \dfrac{\dot{v}}{\bar{U}}\right) ^{5}\right] \end{aligned}$$
(4)

where \(b_{i}:=\dfrac{1}{2}\varrho _{a}DA_{i}\) (for \(i=1,3,5\)) has been introduced to simplify the notation.

In the following, it will be assumed that the excitation is harmonic, namely:

$$\begin{aligned} u(t)=\hat{u}\cos \varOmega t \end{aligned}$$
(5)

with \(\hat{u}\ll \bar{U}\) the amplitude and \(\varOmega \) the frequency of the turbulent component.

2.2 Single degree-of-freedom system

The equations of motion and boundary conditions Eq. (1), by accounting for the aerodynamic forces (4) and the turbulence law (5), are recast in the following nondimensional form:

$$\begin{aligned} {\displaystyle \begin{array}{l} \ddot{v}^{*}+\mathscr {L}_{e}^{*}\left( v^{*}\right) +\mathscr {L}_{v}^{*}\left( \dot{v}^{*}\right) +c_{e}^{*}\dot{v}^{*}=p_{a}^{*}\\ \mathscr {B}_{eH}^{*}\left( v^{*}\right) +\mathscr {B}_{vH}^{*}\left( \dot{v}^{*}\right) =0\qquad H=A,B \end{array}} \end{aligned}$$
(6)

where:

$$\begin{aligned} p_{a}^{*}:= & {} -b_{1}^{*} \left( \bar{U}^{*}+\hat{u}^{*}\cos \varOmega ^{*}t^{*}\right) \dot{v^{*}}\nonumber \\&-\,b_{3}^{*}\left( \frac{1}{\bar{U}^{*}}-\frac{\hat{u}^{*}}{\bar{U}^{*}{}^{2}}\cos \varOmega ^{*}t^{*}\right) \dot{v}^{*}{}^{3}\nonumber \\&-\,b_{5}^{*}\left( \frac{1}{\bar{U}^{*}{}^{3}}-\frac{3\hat{u}^{*}}{\bar{U}^{*}{}^{4}}\cos \varOmega ^{*}t^{*}\right) \dot{v}^{*}{}^{5}, \end{aligned}$$
(7)

and the following positions have been introduced:

$$\begin{aligned}&v^{*}:=\frac{v}{l},\quad s^{*}:=\frac{s}{l},\quad t^{*}:=t\omega _{r},\nonumber \\&\varOmega ^{*}:=\frac{\varOmega }{\omega _{r}},\quad \omega _{r}:=\frac{1}{l}\sqrt{\frac{T_{r}}{m}},\nonumber \\&c_{e}^{*}:=c_{e}\frac{\omega _{r}l^{2}}{T_{0}},\quad b_{i}^{*}:=b_{i}\frac{\omega _{r}^{2}l^{3}}{T_{r}},\quad \bar{U}^{*}:={\displaystyle \frac{\bar{U}}{\omega _{r}l},}\nonumber \\&{\displaystyle \hat{u}^{*}:=\frac{\hat{u}}{\omega _{r}l}} \end{aligned}$$
(8)

Moreover, \(\mathscr {L}_{e}^{*}\left( v^{*}\right) ,\mathscr {L}_{v}^{*}\left( \dot{v^{*}}\right) ,\mathscr {B}_{eH}^{*}\left( v^{*}\right) ,\mathscr {B}_{vH}^{*}\left( \dot{v^{*}}\right) \) are nondimensional operators; \(\omega _{r}\) is a reference frequency and \(T_{r}\) a reference mechanical characteristic, having the dimension of a force; finally a dot denotes differentiation with respect \(t^{*}\).

The continuous system (6) is discretized as a single-D.O.F. system via the Galerkin method, by assuming:

$$\begin{aligned} v^{*}\left( s^{*},t^{*}\right) =q\left( t^{*}\right) \phi \left( s^{*}\right) \end{aligned}$$
(9)

Here \(q\left( t^{*}\right) \) is a Lagrangian parameter and \(\phi \left( s^{*}\right) \) is a natural mode of the undamped system, normalized according to \(\max \left| \phi \right| =1\), of frequency \(\omega ^{*}\), satisfying the boundary value problem:

$$\begin{aligned} \begin{array}{l} \mathscr {L}_{e}^{*}\left( \phi \right) -\omega ^{*}{}^{2}\phi =0\\ \mathscr {B}_{eH}^{*}\left( \phi \right) =0\qquad H=A,B \end{array} \end{aligned}$$
(10)

By weighting the residuals in the domain and at the boundaries, the following condition is enforced (Virtual Work Equation):

$$\begin{aligned}&\int _{0}^{1}\left[ \ddot{q}\phi +q\mathscr {L}_{e}^{*}\left( \phi \right) +\dot{q}\left( \mathscr {L}_{v}^{*}\left( \phi \right) +c_{e}^{*}\phi \right) -p_{a}^{*}\right] \phi \,\mathrm{d}s^{*}\nonumber \\&\quad +\,\dot{q}\left[ {\displaystyle \mathscr {B}_{vA}^{*}\left( \phi _{A}\right) }{\displaystyle \phi _{A}+{\displaystyle \mathscr {B}_{vB}^{*}\left( \phi _{B}\right) }{\displaystyle \phi _{B}}}\right] =0 \end{aligned}$$
(11)

from which a nondimensional ordinary differential equation is drawn:

$$\begin{aligned}&\ddot{q}+\omega ^{*}{}^{2}q+D_{0}\dot{q}+D_{1}\left( \bar{U}^{*}+\hat{u}^{*}\cos \varOmega ^{*}t^{*}\right) \dot{q}\nonumber \\&\quad +\,D_{3}\left( {\displaystyle \frac{1}{\bar{U}^{*}}}-{\displaystyle \frac{\hat{u}^{*}\cos \varOmega ^{*}t^{*}}{\bar{U}^{*2}}}\right) \dot{q}^{3}\nonumber \\&\quad +\,D_{5}\left( {\displaystyle \frac{1}{\bar{U}^{*2}}}-3{\displaystyle \frac{\hat{u}^{*}\cos \varOmega ^{*}t^{*}}{\bar{U}^{*4}}}\right) \dot{q}^{5}=0 \end{aligned}$$
(12)

In this equation, the structural and wind coefficients are defined as:

$$\begin{aligned}&\omega ^{*}{}^{2}:={\displaystyle \frac{1}{I_{m}}}{\displaystyle {\displaystyle \int _{0}^{1}}}{\displaystyle \mathscr {L}_{e}^{*}\left( \phi \right) \phi \,\mathrm{d}s^{*}},\nonumber \\&D_{0}:=c_{e}^{*}+{\displaystyle \frac{1}{I_{m}}}\displaystyle \left( {\displaystyle {\displaystyle \int _{0}^{1}\mathscr {L}_{v}^{*}\left( \phi \right) \phi \,\mathrm{d}s^{*}}}\right. \nonumber \\&\qquad \left. +{\displaystyle \mathscr {B}_{vA}^{*}\left( \phi _{A}\right) }{\displaystyle \phi _{A}+{\displaystyle \mathscr {B}_{vB}^{*}\left( \phi _{B}\right) }{\displaystyle \phi _{B}}}\right) ,\nonumber \\&D_{1}:=b_{1}^{*},\quad D_{3}={\displaystyle \frac{b_{3}^{*}}{I_{m}}}{\displaystyle {\displaystyle {\displaystyle \int _{0}^{1}}}}{\displaystyle \phi {}^{4}\,\mathrm{d}s^{*}},\nonumber \\&D_{5}:={\displaystyle \frac{b_{5}^{*}}{I_{m}}}{\displaystyle {\displaystyle \int _{0}^{1}}}{\phi }{}^{6}\,\mathrm{d}s^{*} \end{aligned}$$
(13)

with \({\displaystyle I_{m}:={\displaystyle \int _{0}^{1}}}{\displaystyle \phi {}^{2}\,\mathrm{d}s^{*}}\). To simplify the notation, the star will be dropped ahead.

3 Bifurcation equation

The one-D.O.F. system (12) is prone to a Hopf bifurcation (galloping, in the technical literature) when \(b_{1}<0\) (i.e., \(A_{1}<0\)) and the steady wind \(\bar{U}\) attains a critical value \(\bar{U}_{c}\). On the other hand, it is also susceptible of parametric excitation when the turbulence has a suitable frequency \(\varOmega \) and the intensity \(\hat{u}\) overcomes a threshold, which depends on \(\varOmega \). Here attention is devoted to the case of primary parametric resonance, for which:

$$\begin{aligned} \varOmega =2\omega +\sigma \end{aligned}$$
(14)

where \(\omega \) is a natural frequency of the aeroelastic system, and \(\sigma \) a small detuning. The two types of bifurcation can interact in the nonlinear field, giving rise to a multiple bifurcation, described by the three bifurcation parameters \(\bar{U},\,\hat{u},\,\sigma \). It is worth noticing, however, that, while self-excitation can occur alone (when the turbulence is absent), the reverse does not hold, since \(\bar{U}\) cannot be zero nor small in the model (due to the hypothesis \(\dfrac{\dot{v}}{U}\ll 1\)).

To analyze the nonlinear phenomenon, the Multiple Scale Method (MSM) is applied here [48] to capture the essential aspects of bifurcation (see, e.g., [49] for several applications of the MSM in bifurcation theory). Internal resonance cases, in principle possible, are excluded in this paper.

According to the MSM, a small dimensionless parameter \(\epsilon \) is introduced by a suitable ordering of variables and parameters. The Lagrangian parameter is rescaled as \(q\left( t\right) \rightarrow \epsilon q\left( t\right) \). Physical coefficients are also rescaled as: \(D_{0}\rightarrow \epsilon D_{0}\), \(D_{1}\rightarrow \epsilon D_{1}\), \(D_{3}\rightarrow \epsilon ^{-1}D_{3}\), \(D_{5}\rightarrow \epsilon ^{-3}D_{5}\). Independent timescales \(t_{0}:=t\), \(t_{1}:=\epsilon t\), ..., are introduced, so that \(\partial _{t}=\partial _{0}+\epsilon \partial _{1}+\cdots \), \(\partial _{t}^{2}=\left( \partial _{0}+\epsilon \partial _{1}+\cdots \right) ^{2}\), where \(\partial _{j}:=\partial /\partial t_{j}\) (\(j=0,1,\ldots \)). Finally, by letting \(q\left( t\right) =\begin{array}{c} q_{0}\left( t_{0},t_{1},\ldots \right) \end{array}+\epsilon q_{1}\left( t_{0},t_{1},\ldots \right) +\cdots \), the following perturbation equations are derived:

$$\begin{aligned}&\epsilon ^{0}: \partial _{0}^{2}q_{0}+\omega ^{2}q_{0}= 0 \nonumber \\&\epsilon ^{1}: \partial _{0}^{2}q_{1}+\omega ^{2}q_{1}= -\left( 2\partial _{0}\partial _{1}q_{0}+D_{0}\partial _{0}q_{0}\right. \nonumber \\&\quad \left. +D_{1}\left( \bar{U}+\hat{u}\cos \varOmega t\right) \partial _{0}q_{0}\right. \nonumber \\&\quad \left. +D_{3}\left( {\displaystyle \frac{1}{\bar{U}}}-{\displaystyle \frac{\hat{u}\cos \varOmega t}{\bar{U}^{2}}}\right) \left( \partial _{0}q_{0}\right) ^{3}\right. \nonumber \\&\quad \left. +D_{5}\left( {\displaystyle \frac{1}{\bar{U}^{2}}}-3{\displaystyle \frac{\hat{u}\cos \varOmega t}{\bar{U}^{4}}}\right) \left( \partial _{0}q_{0}\right) ^{5}\right) \end{aligned}$$
(15)

Equation (15) admits the following solution:

$$\begin{aligned} q_{0}\left( t_{0},t_{1}\right) =A\left( t_{1}\right) e^{i\omega t_{0}}+\text {c.c.} \end{aligned}$$
(16)

where \(A\left( t_{1}\right) \) is a complex modulating amplitude, varying on a (slow) timescale, whose characteristic length is much larger than \(2\pi /\omega \); c.c. stands for complex conjugate terms. By substituting Eq. (16) in the second of numbered Eq. (15) and by vanishing the secular terms, the following complex bifurcation equation is drawn:

$$\begin{aligned} \dot{A}= & {} \left( d_{0}+d_{1}\bar{U}\right) A-{ \frac{d_{1}}{2}}\bar{A}\hat{u}e^{i\sigma t}+\frac{d_{3}}{\bar{U}}A^{2}\bar{A}\nonumber \\&+\frac{d_{3}}{2\bar{U}^{2}}\hat{u}e^{i\sigma t}A\bar{A}^{2}+\frac{d_{3}}{6\bar{U}^{2}}\hat{u}e^{-i\sigma t}A^{3}\nonumber \\&{\displaystyle +\frac{d_{5}}{\bar{U}^{3}}A^{3}\bar{A}^{2}+\frac{3d_{5}}{2\bar{U}^{4}}\hat{u}e^{i\sigma t}A^{2}\bar{A}^{3}}+{\displaystyle \frac{3d_{5}}{4\bar{U}^{4}}\hat{u}e^{-i\sigma t}}A^{4}\bar{A}\nonumber \\ \end{aligned}$$
(17)

where coming back to the unrescaled quantities has been performed, and its coefficients assume the following form:

$$\begin{aligned}&d_{0}=-\frac{D_{0}}{2},\qquad d_{1}=-\frac{D_{1}}{2},\qquad d_{3}=-\frac{3}{2}\omega ^{2}D_{3},\nonumber \\&\quad d_{5}=-5\omega ^{4}D_{5} \end{aligned}$$
(18)

For further analyses, both the Cartesian and polar real forms of Eq. (17) are needed (see [50] for a discussion), as derived ahead.

3.1 Cartesian form

To obtain the Cartesian form of the bifurcation equation, a new variable \(Z\left( t\right) \) is defined via:

$$\begin{aligned} A\left( t\right) =Z\left( t\right) {\displaystyle e^{i\frac{\sigma }{2}t}} \end{aligned}$$
(19)

which transforms the non-autonomous Eq. (17) in an autonomous system. Then,

$$\begin{aligned} Z\left( t\right) =X\left( t\right) +iY\left( t\right) \end{aligned}$$
(20)

is posed, and the real and imaginary parts of the equation are separated. The following two-dimensional dynamical system is obtained:

$$\begin{aligned} \left( \begin{array}{c} \dot{X}\\ \dot{Y} \end{array}\right) =\mathbf {J}_{0}\left( \begin{array}{c} X\\ Y \end{array}\right) +\mathbf {n}\left( X,Y\right) \end{aligned}$$
(21)

where:

$$\begin{aligned} \mathbf {J}_{0}:=\left[ \begin{array}{cc} {\displaystyle d_{0}+\bar{U}d_{1}{\displaystyle -\frac{\hat{u}d_{1}}{2}}} &{} {\displaystyle \frac{\sigma }{2}}\\ {\displaystyle -\frac{\sigma }{2}} &{} d_{0}+\bar{U}d_{1}{\displaystyle +\frac{\hat{u}d_{1}}{2}} \end{array}\right] \end{aligned}$$
(22)

is the Jacobian matrix at the origin, and \(\mathbf {n}=\left( n_{x},\,n_{y}\right) ^{T}\) is the vector of the nonlinear terms, defined as follows:

$$\begin{aligned} n_{x}= & {} \frac{d_{3}XY^{2}}{\bar{U}}+X^{3}\left( \frac{2d_{3}\hat{u}}{3\bar{U}^{2}}+\frac{d_{3}}{\bar{U}}\right) \nonumber \\&+XY^{4}\left( \frac{d_{5}}{\bar{U}^{3}}-\frac{3d_{5}\hat{u}}{4\bar{U}^{4}}\right) \nonumber \\= & {} +{\displaystyle X^{3}Y^{2}\left( \frac{3d_{5}\hat{u}}{2\bar{U}^{4}}+\frac{2d_{5}}{\bar{U}^{3}}\right) +X^{5}\left( \frac{9d_{5}\hat{u}}{4\bar{U}^{4}}+\frac{d_{5}}{\bar{U}^{3}}\right) }\nonumber \\ n_{y}= & {} \frac{d_{3}X^{2}Y}{\bar{U}}+Y^{3}\left( \frac{d_{3}}{\bar{U}}-\frac{2d_{3}\hat{u}}{3\bar{U}^{2}}\right) \nonumber \\&+X^{4}Y\left( \frac{3d_{5}\hat{u}}{4\bar{U}^{4}}+\frac{d_{5}}{\bar{U}^{3}}\right) \nonumber \\= & {} {\displaystyle +X^{2}Y^{3}\left( \frac{2d_{5}}{\bar{U}^{3}}-\frac{3d_{5}\hat{u}}{2\bar{U}^{4}}\right) +Y^{5}\left( \frac{d_{5}}{\bar{U}^{3}}-\frac{9d_{5}\hat{u}}{4\bar{U}^{4}}\right) }\nonumber \\ \end{aligned}$$
(23)

Accordingly, the motion law (9) is expressed, at the leading order, by:

$$\begin{aligned} v\left( s,t\right)= & {} 2\left[ X\left( t\right) \cos \left( \frac{\varOmega }{2}t\right) \right. \nonumber \\&\left. -Y\left( t\right) \sin \left( \frac{\varOmega }{2}t\right) \right] \phi \left( s\right) \end{aligned}$$
(24)

where use has been made of Eqs. (16) and (14).

3.2 Polar form

To derive the polar form of the bifurcation equation (17), the complex amplitude is expressed as:

$$\begin{aligned} A\left( t\right) =\frac{1}{2}a\left( t\right) e^{i\varphi \left( t\right) } \end{aligned}$$
(25)

with \(a\left( t\right) \) the modulus of the amplitude and \(\varphi \left( t\right) \) the phase. Here, according to Eq. (9), a(t) represents the motion of the material point at which \(\phi \) attains its maximum modulus. Consequently, Eq. (17) splits into two real equations:

$$\begin{aligned} \dot{a}= & {} a\left( +d_{0}+d_{1}\bar{U}-\frac{1}{2}d_{1}\hat{u}\cos \gamma \right) \nonumber \\&+\,a^{3}\left( \frac{d_{3}}{4\bar{U}}+\frac{d_{3}\hat{u}\cos \gamma }{6\bar{U}^{2}}\right) \nonumber \\&+\,a^{5}\left( \frac{d_{5}}{16\bar{U}^{3}}+\frac{9d_{5}\hat{u}\cos \gamma }{64\bar{U}^{4}}\right) \nonumber \\ a\dot{\gamma }= & {} a\left( d_{1}\hat{u}\sin \gamma -\sigma \right) -a^{3}\frac{d_{3}\hat{u}\sin \gamma }{6\bar{U}^{2}}\nonumber \\&-\,a^{5}\frac{3d_{5}\hat{u}\sin \gamma }{32\bar{U}^{4}} \end{aligned}$$
(26)

in which \(\gamma :=2\varphi \left( t\right) -\sigma \,t\) is the phase difference. The motion law (9), at the leading order, reads:

$$\begin{aligned} v\left( s,t\right) =a\left( t\right) \phi \left( s\right) \cos \left( \frac{\varOmega }{2}t+\frac{\gamma \left( t\right) }{2}\right) \end{aligned}$$
(27)

Amplitudes and phases in the two representations are related by \(a=2\sqrt{X^{2}+Y^{2}}\), \(\gamma =2\arg \left( X+iY\right) \).

4 Linear stability analysis

The rest position of the structure \(v\left( s,t\right) =0\) \(\forall \left( s,t\right) \), is described by the trivial solution \(A=0\) of the bifurcation equation (17). Since the polar form (26) is singular at \(a=0\), the Cartesian form (21) is used. Stability of the origin is governed by the eigenvalues of the Jacobian matrix \(\mathbf {J}_{0}\), which read:

$$\begin{aligned} \lambda _{\pm }=d_{0}+d_{1}\bar{U}\pm \frac{1}{2}\sqrt{\triangle \left( \hat{u},\sigma \right) } \end{aligned}$$
(28)

where the discriminant is:

$$\begin{aligned} \varDelta \left( \hat{u},\sigma \right) :=d_{1}^{2}\hat{u}^{2}-\sigma ^{2} \end{aligned}$$
(29)

For later purposes, the associated (right) eigenvectors, solving \(\left( \mathbf {J}_{0}-\lambda _{\pm }\mathbf {I}\right) \mathbf {x}=\mathbf {0}\), together with the left eigenvectors, solving the transpose conjugate eigenvalue problem \(\left( \mathbf {J}_{0}-\bar{\lambda }_{\pm }\mathbf {I}\right) \mathbf {y}=\mathbf {0}\), are evaluated as:

$$\begin{aligned}&\mathbf {x}_{\pm }=\left( \begin{array}{c} d_{1}\hat{u}{\displaystyle \mp \sqrt{\triangle \left( \hat{u},\sigma \right) }}\\ \sigma \end{array}\right) ,\nonumber \\&\quad \mathbf {y}_{\pm }=\left( \begin{array}{c} -d_{1}\hat{u}{\displaystyle \pm \sqrt{\triangle \left( \hat{u},\sigma \right) }}\\ \sigma \end{array}\right) \end{aligned}$$
(30)

In the absence of turbulence (\(\hat{u}=0\) , \(\sigma =0\)), the two eigenvalues Eq. (28) are real and coincident. A Hopf bifurcation occurs when \(\lambda _{+}=\lambda _{-}=0\), at the critical galloping velocity \(\bar{U}=\bar{U}_{c}\):

$$\begin{aligned} \bar{U}_{c}=-\dfrac{d_{0}}{d_{1}} \end{aligned}$$
(31)

The coefficients \(d_{0}\) and \(d_{1}\) (see Eq. (18)) depend on the the cross-section shape. Here, the interesting case \(d_{0}<0\) (section prone to galloping) and \(d_{1}>0\) (sub-critical Hopf bifurcation) is considered.

The turbulent component of wind modifies the bifurcation condition, according to the nature of the eigenvalues (28). To investigate its effects, linear stability is studied in the \(\left( \sigma ,\hat{u},\bar{U}\right) \)-space, in which: (i) \(\bar{U}\) is the distinguished parameter, and, (ii) \(\hat{u}\) and \(\sigma \) are the splitting parameters (since they rule the coalescence, as it will appear clear soon). Results are qualitatively summarized in Fig. 2, where the linear stability domain is represented (a) by a 3D view, or (b,c,d) by cross sections parallel to the coordinate planes (here the eigenvalues are also sketched in the different regions).

In the \(\left( \hat{u},\,\sigma \right) \)-plane, the straight lines \(\sigma =\pm d_{1}\hat{u}\) are the locus at which \(\varDelta =0\). These lines separate the plane in an internal \(\mathcal {R}_{P}\) and an external \(\mathcal {R}_{Q}\) region, in which periodic and quasi-periodic motions, respectively, arise when \(\bar{U}\) takes a suitable value, as discussed ahead.

  • In the internal region \(\mathcal {R}_{P}\), it is \(\triangle \left( \hat{u},\sigma \right) >0\), so that the eigenvalues are real and distinct. Here, when \(\bar{U}\) is increased from zero, and it is such that \(\lambda _{\pm }=0\), two successive flip bifurcations occur, at which a periodic response, of period double of that of excitation, is triggered. This happen on the geometrical locus:

    $$\begin{aligned} d_{0}+d_{1}\bar{U}\pm \frac{1}{2}\sqrt{d_{1}^{2}\hat{u}^{2}-\sigma ^{2}}=0 \end{aligned}$$
    (32)

    which is a cone in the parameter space. The lowest bifurcation (\(\lambda _{+}=0\)) manifests at a steady wind velocity \(\bar{U}\left( \hat{u},\sigma \right) <\bar{U}_{c}\), so that turbulence reduces the galloping velocity.

  • In the external region \(\mathcal {R}_{Q}\), it is \(\triangle \left( \hat{u},\sigma \right) <0,\) so that the eigenvalues are complex conjugates. Here, when \(\bar{U}\) is increased from zero, and it is such that \(\mathrm {\text {Re}}[\lambda _{+}]=\text {Re}[\lambda _{-}]=0\), a Neimark–Sacker bifurcation occurs, at which a quasi-periodic response is triggered. This happen on the geometrical locus:

    $$\begin{aligned} d_{0}+d_{1}\bar{U}=0 \end{aligned}$$
    (33)

    i.e., at \(\bar{U}=\bar{U}_{c}\) \(\forall \left( \hat{u},\sigma \right) \).

  • At the interface between \(\mathcal {R}_{P}\) and \(\mathcal {R}_{Q}\), it is \(\triangle \left( \hat{u},\sigma \right) =0\), so that the eigenvalues are real and coincident. They vanish at \(\bar{U}=\bar{U}_{c}\), where the Neimark–Sacker bifurcation degenerates in a flip bifurcation. The splitting parameters decide about the coalescence.

It is worth noticing that, referring to the slow flow ruled by the bifurcation equation (17), the flip bifurcation appears as a divergence (simple zero eigenvalue) and the Neimark–Sacker as a Hopf bifurcation. At the intersections of the relevant manifolds, a double-zero (Bogdanov–Takens) bifurcation occurs, where the Hopf bifurcation degenerates; here, the Hopf manifold dies (see, e.g., [51]). This mechanisms clearly appears in Fig. 2b, c.

A mechanical interpretation of the instability domain can be given. When \(\bar{U}=\bar{U}_{c},\) the aerodynamic damping balances the structural damping, so that the monomodal response of the structure is undamped. Due to the parametric excitation, there exists a sector of the \(\bar{U}=\bar{U}_{c}\) plane, in which the system is unstable, according to the well-known Mathieu equation (see Fig. 2a). When \(\bar{U}<\bar{U}_{c}\), the structural damping prevails over the aerodynamic damping, so that the unstable zone moves apart from the \(\sigma \)-axis. Consequently, a larger turbulence, or a smaller detuning, are needed to make the system unstable, namely: (i) \(\hat{u}>\hat{u}_{m}\left( \sigma \right) \), with \(\hat{u}_{m}:={\displaystyle \frac{\sigma }{d_{1}}}\) the minimum turbulence amplitude (see 2b), or, (ii) \(\left| \sigma \right| <\sigma _{M}\left( \hat{u}\right) \), with \(\sigma _{M}:=d_{1}\hat{u}\) the maximum detuning (see 2c). When (ideal case, beyond the validity limits of the theory) \(\bar{U}=0\), damping is only of structural type, so that \(\hat{u}\) takes its maximum value \(\hat{u}_{c}=\hat{u}_{c}\left( \sigma \right) \) (see Fig. 2b). Out of the parametric excitation zone, at \(\bar{U}=\bar{U}_{c}\), the phenomenon is lead by the self-excitation. However, the periodic motion experienced by the system under non-turbulent wind, is modified by the turbulence, which introduces a second frequency, \(\mathrm {Im}[\lambda _{\pm }]\), into the response, thus transforming the limit cycle in a torus.

Fig. 2
figure 2

Linear instability domain: a 3D representation in the space of the wind parameters; coordinate sections: b \(\hat{u}-\bar{U}\), c \(\sigma -\bar{U}\), d \(\sigma -\hat{u}\). Black solid lines are loci at which periodic (\(\mathcal {P}\)) and quasi-periodic (\(\mathcal {Q}\)) motions arise. Gray solid lines are successive bifurcation loci. The gray filled areas indicate the stable regions. (Color figure online)

Full size image

5 Nonlinear analysis

Bifurcation analysis calls for determining: (a) the limit cycles (periodic solutions) generated by the flip bifurcation; (b) the tori (quasi-periodic solutions) generated by the Neimark–Sacker bifurcations. The periodic solutions, which are equilibrium points for the bifurcation equations, are found by analytically or numerically solving nonlinear algebraic equations. The quasi-periodic solutions, which are periodic orbits for the bifurcation equations, are found (i) via numerical integration of the ordinary differential equations, and (ii) asymptotically, by carrying out a new perturbation analysis (as, e.g., done in [52], and, then, in [37, 38] for a different engineering application).

5.1 Limit cycles

Periodic motions are the equilibrium points \(a=\mathrm {const}\), \(\gamma =\mathrm {const}\) of the polar Eq. (26). By requiring \(\dot{a}=0\) and \(\dot{\gamma }=0\) and by zeroing the right-hand side, \(\sin \gamma \) and \(\cos \gamma \) are obtained and the variable \(\gamma \) condensed using the relation \(\sin ^{2}\gamma +\cos ^{2}\gamma =1\). The resulting equation for the unknown a is the following:

$$\begin{aligned}&\frac{\bar{U}^{2}\left( 16d_{0}\bar{U}^{3}+16d_{1}\bar{U}^{4}+4a^{2}d_{3}\bar{U}^{2}+a^{4}d_{5}\right) {}^{2}}{\hat{u}^{2}\left( \frac{9}{4}a^{4}d_{5} +\frac{8}{3}a^{2}d_{3}\bar{U}^{2}-8d_{1}\bar{U}^{4}\right) {}^{2}}\nonumber \\&\quad +\frac{64\bar{U}^{8}\sigma ^{2}}{\hat{u}^{2}\left( \frac{3}{4}a^{4}d_{5}+\frac{4}{3}a^{2}d_{3}\bar{U}^{2}-8d_{1}\bar{U}^{4}\right) {}^{2}}=1 \end{aligned}$$
(34)

In general, the roots of the Eq. (34) cannot be determined analytically. However, this is viable in the particular case of perfect resonance (i.e., \(\sigma =0\)), for which the solutions read:

$$\begin{aligned}&a_{1_{\pm }}=4\bar{U}\sqrt{{\displaystyle \frac{1}{d_{5}\left( 9\hat{u}+4\bar{U}\right) }}\left( -d_{3}\left( \bar{U}+\frac{2}{3}\hat{u}\right) \pm \sqrt{\varDelta _{+}}\right) }\nonumber \\&a_{2_{\pm }}=4\bar{U}\sqrt{{\displaystyle \frac{1}{d_{5}\left( 9\hat{u}-4\bar{U}\right) }}\left( -d_{3}\left( \bar{U}-\frac{2}{3}\hat{u}\right) \pm \sqrt{\varDelta _{-}}\right) } \end{aligned}$$

with

$$\begin{aligned} \varDelta _{\pm }:= & {} d_{3}^{2}\left( \bar{U}\pm \frac{2}{3}\hat{u}\right) {}^{2}\nonumber \\&\quad {\displaystyle +\frac{d_{5}\left( 9\hat{u}\pm 4\bar{U}\right) \left( d_{1}\left( \hat{u}\mp 2\bar{U}\right) \mp 2d_{0}\right) }{2}} \end{aligned}$$
(35)

together with \(\gamma _{1}=0\) and \(\gamma _{2}=\pi \), respectively. The Eq. (35) consist in multi-valued functions, whose existence depends on the wind parameters and on the nature of the aerodynamics coefficients (i.e., by the cross-section geometry).

To detect stability of the periodic solution, the dynamical system Eq. (26) is perturbed by letting \(a\left( t\right) =a_{e}+\delta a\left( t\right) \) and \(\gamma \left( t\right) =\gamma _{e}+\delta \gamma \left( t\right) \), where \(\delta a,\delta \gamma \) are small deviations from the equilibrium solution \(\left( a_{e},\gamma _{e}\right) \). Substituting in the equation and linearizing in the perturbation, leads to the following variational equation:

$$\begin{aligned} \begin{array}{cccc} \begin{pmatrix}\delta \dot{a}\\ \delta \dot{\gamma } \end{pmatrix} &{} = &{} \mathbf {J}_{e} &{} \begin{pmatrix}\delta a\\ \delta \gamma \end{pmatrix}\end{array} \end{aligned}$$
(36)

where \(\mathbf {J}_{e}:=\left[ J_{ij}\left( a_{e},\gamma _{e}\right) \right] \) is the \(2\times 2\) Jacobian matrix at the equilibrium, whose coefficients are:

$$\begin{aligned} J_{11}\left( a_{e},\gamma _{e}\right)= & {} \displaystyle d_{0}+\bar{U}d_{1}+\frac{3d_{3}a_{e}^{2}}{4\bar{U}}+\frac{5d_{5}a_{e}^{4}}{16\bar{U}^{3}}\nonumber \\&+\left( -\frac{1}{2}\hat{u}d_{1}+\frac{45\hat{u}d_{5}a_{e}^{4}}{64\bar{U}^{4}}+\frac{\hat{u}d_{3}a_{e}^{2}}{2\bar{U}^{2}}\right) \cos \gamma _{e}\nonumber \\ J_{12}\left( a_{e},\gamma _{e}\right)= & {} {\displaystyle \left( \frac{1}{2}\hat{u}d_{1}a_{e}-\frac{\hat{u}d_{3}a_{e}^{3}}{6\bar{U}^{2}}-\frac{9\hat{u}d_{5}a_{e}^{5}}{64\bar{U}^{4}}\right) \sin \gamma _{e}}\nonumber \\ J_{21}\left( a_{e},\gamma _{e}\right)= & {} {\displaystyle \left( -\frac{\hat{u}d_{3}a_{e}}{3\bar{U}^{2}}-\frac{3\hat{u}d_{5}a_{e}^{3}}{8\bar{U}^{4}}\right) \sin }\gamma _{e}\nonumber \\ J_{22}\left( a_{e},\gamma _{e}\right)= & {} \left( {\displaystyle \hat{u}d_{1}-\frac{\hat{u}d_{3}a_{e}^{2}}{6\bar{U}^{2}}-\frac{3\hat{u}d_{5}a_{e}^{4}}{32\bar{U}^{4}}}\right) \cos \gamma _{e} \end{aligned}$$
(37)

By denoting with \(\mu \) the eigenvalues of \(\mathbf {J}_{e}\), the periodic solution is stable if \(\mathrm {Re}\left( \mu \right) <0\) for all \(\mu \), and unstable if \(\mathrm {Re}\left( \mu \right) >0\) for at least one \(\mu \). In the resonant case, the Jacobian matrix is diagonal, with eigenvalues \(J_{11}\left( a_{i\pm },\gamma _{i}\right) \) and \(J_{22}\left( a_{i\pm },\gamma _{i}\right) \).

5.2 Tori

Quasi-periodic motions are found as limit cycles for the bifurcation equations, via a perturbation analysis. Use is made of the Cartesian form (21) of the bifurcation equation and the MSM is newly applied to build-up periodic solutions arising at Neimark–Sacker points. Here, the eigenvalues of the Jacobian matrix \(\mathbf {J}_{0}\) are purely imaginary, i.e., \(\lambda _{\pm }=\pm i\varpi \), with \(\varpi :={\displaystyle \frac{1}{2}}\sqrt{-\varDelta }\), and where \(\bar{U}=\bar{U}_{c}\). At the bifurcation, the motion is:

$$\begin{aligned} \left( \begin{array}{c} X\\ Y \end{array}\right) =R\left( t\right) \left( \begin{array}{c} d_{1}\hat{u}-2i\varpi \\ \sigma \end{array}\right) e^{i\varpi t}+\text {c.c.}+\text {h.o.t.} \end{aligned}$$
(38)

where \(R\left( t\right) ={\displaystyle \frac{1}{2}}r\left( t\right) e^{i\theta \left( t\right) }\) is a complex amplitude, and where use has been made of Eq. (30). By letting \(\bar{U}=\bar{U}_{c}+\epsilon \varDelta U\) to explore the neighborhood of the bifurcation point, and newly applying the MSM (see “Appendix B”), the following polar form of the bifurcation equation (sometimes referred to as ruling the slow-slow-motion), is drawn:

$$\begin{aligned} \dot{r}= & {} {\displaystyle rd_{1}\varDelta U+\frac{c_{3r}}{4}r^{3}+\frac{c_{5r}}{16}r^{5}}\nonumber \\ r\dot{\theta }= & {} {\displaystyle \frac{c_{3i}}{4}r^{3}+\frac{c_{5i}}{16}r^{5}} \end{aligned}$$
(39)

Quasi-periodic solutions are found as equilibrium points \(r=\text {const}\) of Eq. (39), i.e.,

$$\begin{aligned} \begin{array}{l} {\displaystyle r_{\pm }=\sqrt{-\frac{2c_{3r}}{c_{5r}}\pm \frac{2\sqrt{c_{3r}{}^{2}-4c_{5r}d_{1}\varDelta U}}{c_{5r}}}}\\ \theta _{\pm }\left( t\right) =\beta _{\pm }t+\theta _{0} \end{array} \end{aligned}$$
(40)

where \(\theta _{0}\) is an initial phase and:

$$\begin{aligned} \beta _{\pm }:={\displaystyle \frac{c_{3i}}{4}r_{\pm }^{3}+\frac{c_{5i}}{16}r_{\pm }^{5}} \end{aligned}$$
(41)

is a frequency correction.

To detect stability, \(r\left( t\right) =r_{\pm }+\delta r\left( t\right) \) is put in Eq. (39), and this latter linearized:

$$\begin{aligned} \delta \dot{r}=\left( d_{1}\varDelta U+\frac{3}{4}c_{3r}r_{\pm }^{2}+\frac{5}{16}c_{5r}r_{\pm }{}^{4}\right) \delta r \end{aligned}$$
(42)

According to Eq. (9), (16), (20), (38), the quasi-periodic motion of the original system is described, at the leading order, by:

$$\begin{aligned} \begin{aligned} v\left( s,t\right)&=2r\sigma \left( \cos \left( \frac{\varOmega }{2}t\right) \cos (\varpi _{*}t+\theta _{0}+\alpha )\right. \\&\quad \left. -\sin \left( \frac{\varOmega }{2}t\right) \cos (\varpi _{*}t+\theta _{0})\right) \end{aligned} \phi \left( s\right) \end{aligned}$$
(43)

where \(\varpi _{*}:=\varpi +\beta \) is the amplitude-dependent nonlinear modulating frequency, and \(\alpha :=\arctan ({\displaystyle \frac{2\varpi }{\sigma _{M}\left( \hat{u}\right) }})\). The motion has therefore two frequencies, the driving one \({\displaystyle \frac{\varOmega }{2}}\), and the modulating one \(\varpi _{*}\). At any given \(s=\bar{s}\), the modulating amplitude:

$$\begin{aligned} a\left( t\right) =\left( 2r\sigma \sqrt{1-\frac{\sigma _{M}\left( \hat{u}\right) }{\sigma }\cos \left( 2\varpi _{*}t+2\theta _{0}+\alpha \right) }\right) \end{aligned}$$
(44)

exists only when \(\sigma >\sigma _{M}\left( \hat{u}\right) \) and it spans the range:

$$\begin{aligned} a\left( t\right) \in \left[ 2r\sigma \sqrt{{\displaystyle 1-\frac{\sigma _{M}\left( \hat{u}\right) }{\sigma }}},\,2r\sigma \sqrt{{\displaystyle 1+\frac{\sigma _{M}\left( \hat{u}\right) }{\sigma }}}\right] \end{aligned}$$
(45)

whose extreme values represent the lengths of the axes of an elliptical trajectory traveled in the (X,Y) state-space (see the later Fig. 13).

Fig. 3
figure 3

Linear instability domain: a 3D representation; contour plots in the planes: b \(\left( \hat{u},\bar{U}\right) \), when \(\sigma \) spans \(\left( 0,0.07\right) \) by steps 0.014; c \(\left( \sigma ,\bar{U}\right) \), when \(\hat{u}\) spans \(\left( 0.05,0.3\right) \) by steps 0.05; d \(\left( \sigma ,\hat{u}\right) \), when \(\bar{U}\) spans the range \(\left( 0.525,0.7\right) \) by steps 0.035

Full size image

It should be noticed, however, that the perturbation solution carried out here (local bifurcation analysis) is unable to capture possible interactions (global bifurcations analysis) between tori and limit cycles. Therefore, it is expected that it loses validity at high amplitudes of motion. The question will be investigated via numerical analyses.

6 Numerical results

Here, by referring to a sample structure described in “Appendix A”, the nondimensional coefficients appearing in Eq. (18) are taken to assume the following values: \(d_{0}=-0.14\), \(d_{1}=0.19\), \(d_{3}=4717.47\), \(d_{5}=-9.19\times 10^{7}\). These numerical values are consistent with the ordering performed in the perturbation analyses.

The asymptotic solutions, previously determined, are validated ahead against numerical results. These latter consist of: (i) numerical integrations of the nonlinear partial differential equation of motion Eq. (6), discretized by finite-differences (see “Appendix C”), and, (ii) numerical integrations of the bifurcation equations Eq. (26) and (21), for given initial conditions.

6.1 Linear analysis

In the case study, the first natural frequency is \(\omega =2.12\) rad/s (\(\omega ^{*}=8.77\) in dimensionless form) and the corresponding galloping velocity (in the absence of turbulence) is \(\bar{U}_{c}=34.13\) m/s (\(U_{c}^{*}=0.7\) in dimensionless form). The influence of the turbulent component of the wind is illustrated in the (quantitative) linear instability domain in Fig. 3, where (a) a 3D-plot and (b–d) planar contour plots are shown. Plots are extended beyond the limits of validity of the theory, which requires \(\hat{u}\ll \bar{U}\). It is seen that the critical wind velocity decreases when the turbulence increases and/or the detuning decreases. For example, when \(\hat{u}=0.15\) and \(\sigma =0.02\), it is \(\bar{U}=0.64\), i.e., \(\bar{U}_{c}\) is reduced of \(7.7\%\); when \(\hat{u}=0.15\) and \(\sigma =0\) it is \(\bar{U}=0.62\), i.e., \(\bar{U}_{c}\) is reduced by 10.7%.

6.2 Limit cycle analysis in the resonant case

The nonlinear behavior of the system at the perfect resonance \(\left( \sigma =0\right) \) is first analyzed in the \((\bar{U},\hat{u})\)-plane. The relevant bifurcation analysis calls for determining limit cycles and analyzing their stability. Amplitude and phase of the limit cycles are defined by the explicit expressions \(a_{i\pm },\gamma _{i}\) (\(i=1,2\)) in Eq. (35).

First, the existence domain of each solution branch in Eq. (35) is studied in the parameter plane (see Fig. 4). In the same figure, the bifurcation locus, made by two generatrices of the cone, are marked in red. The plot displays the existence of a complex scenario, in which from zero to four limit cycles coexist (the circled numbers denote the number of solutions in each region). To detect the stability of the periodic solutions, the sign of the corresponding eigenvalues \(J_{11}\) and \(J_{22}\) of the Jacobian matrix Eq. (36) is evaluated: They are found to be both negative only when evaluated at the \(\left( a_{2+},\gamma _{2}\right) \) limit cycle, which is therefore the unique stable bifurcated branch.

Fig. 4
figure 4

Existence domains, according to the wind parameters (\(\bar{U}\),\(\hat{u}\)), of: \(a_{1+}\) (blue region), \(a_{1-}\) (gray region), \(a_{2+}\) (black region) and \(a_{2-}\) (pink region). The straight red lines are the bifurcation loci. The circled numbers indicate the number of solutions coexisting in the same region. (Color figure online)

Full size image

Planar bifurcation diagrams are built-up and shown in Figs. 5 and 6, namely: (i) a versus \(\bar{U}\), for selected values of \(\hat{u}\) (ticks I, II in Fig. 4), and, (ii) a versus \(\hat{u}\), for selected values of \(\bar{U}\) (ticks III to VI in Fig. 4).

The \(\left( a,\bar{U}\right) \) diagrams are commented first. In case (I) there exist three families of limit cycles, all branching from the trivial path. Two of them are close to the galloping curve (\(\hat{u}=0\)) (one on the left, the other on the right), generated, when \(\bar{U}\) is increased, at entering and coming out the cone of Fig. 3. A third family bifurcates from \(\bar{U}=0\), at which the theory loses validity; however, the existence of this branch has been confirmed by numerical results, as it will be discussed later. The dangerous phenomenon of hard loss of stability, already manifesting in absence of turbulence, is observed. It is generated by an (unstable) subcritical bifurcation, followed by a regain of stability, which entails a finite jump of the amplitude, just passed the critical velocity. In the numerical example, this amplitude is about 1% of the length (i.e., double of the cross-section height). Moreover, the existence of a stable portion of the lowest branch, creates a potentially dangerous competitive attractor at high amplitude, although existing in a very narrow range of the velocity (0.25, 0.26). In case (II), in which the turbulence is higher, the scenario changes, since the stable and unstable portions of the two lowest curves weld together, the highest remaining undisturbed. Here, once the trivial path loses stability, the system jumps to an even higher-amplitude limit cycle, equal to about 1.5% of the length. The dangerous effect of the turbulence is thus highlighted.

Fig. 5
figure 5

Bifurcation diagrams. Steady amplitude of the limit cycle versus the steady wind velocity, for assigned turbulence amplitudes: (I) \(\hat{u}=0.1\); (II) \(\hat{u}=0.2\). Blue, gray, black and pink lines refer to the non-trivial solutions \(a_{1+}\), \(a_{1-}\), \(a_{2+}\), \(a_{2-}\) , respectively. The thin green curve is the bifurcation diagram when \(\hat{u}=0\); the green markers are numerical results. (Color figure online)

Full size image

When the bifurcation diagrams \(\left( a,\hat{u}\right) \) are plotted, Fig. 6 is obtained. In all cases, just one family of limit cycles exists, generated, for \(\hat{u}\) increasing, at the (unique) crossing of the cone. When \(\bar{U}<\bar{U}_{c}\) (cases (III) and (IV)), a hard loss of stability is observed. However, in case (III), in which \(\bar{U}\) is far from the galloping value, the stable branch fails to overlap the sub-critical region. This aspect could be interpreted as due to the prevailing influence of the parametric excitation with respect the galloping phenomenon. In contrast, in case (IV), the stable branch extends on a small portion of the super-critical region. When \(\bar{U}\ge \bar{U}_{c}\) (cases (V) e (VI)), the trivial solution is unstable, so that motion stabilizes on limit cycles of great amplitude, which increases with the turbulence.

Fig. 6
figure 6

Bifurcation diagrams. Steady amplitude of the limit cycle versus the turbulence amplitude for assigned steady wind velocity: (III) \(\bar{U}=0.52\); (IV) \(\bar{U}=0.59\); (V) \(\bar{U}=\bar{U}_{c}=0.7\); (VI) \(\bar{U}=0.8\). Blue, gray, black and pink lines refer to the non-trivial solutions \(a_{1+}\), \(a_{1-}\), \(a_{2+}\), \(a_{2-}\), respectively; the green markers are numerical results. (Color figure online)

Full size image

To sketch the evolution of the system in presence of one or more attractors, the transients asymptotic solutions, obtained by direct numerical integrations of the bifurcation equation (26), are investigated in the state-space at varying of the initial conditions. By referring to the bifurcation diagram in Fig. 5I, the \(\left( a,\gamma \right) \) trajectories are shown in Fig. 7 for the steady wind velocities: (a) \(\bar{U}=0.5\), at which two stable steady solutions co-exist; (b) \(\bar{U}=0.84\), at which only one stable steady solution exists. It is seen, in Fig. 7a, that trajectories originating at low amplitudes (i.e., below that of the unstable solution) are attracted by the trivial solution, irrespectively of \(\gamma \); in contrast, trajectories originating at higher amplitudes can be attracted by one of the two stable solutions, according to the initial gamma. In Fig. 7b it appears that there exist a trajectory leading to the stable solution which attracts all the surrounding orbits. Finally, the steady-state and transients asymptotic solutions, are validated against exact finite-difference solutions of the partial differential equations (6). In particular, for given initial conditions (\(a=0.01\), \(\gamma =0.05\) in Fig. 7b), the exact response and the asymptotic \(a\left( t\right) \) time-histories are compared in Fig. 7c. After a transient has been exhausted, the motion stabilizes on a limit cycle of amplitude a, depending on the wind parameters. When the limit cycle amplitudes are extracted by the recorded responses, relevant to different wind parameters, the (green) bullets reported in Figs. 5I and 6IV are found. In all cases, the agreement between exact and asymptotic solutions is excellent.

Fig. 7
figure 7

Trajectories in the state space for different initial conditions (black markers) and for the steady wind velocities: a \(\bar{U}=0.5\); b \(\bar{U}=0.84\); unstable and stable steady solutions are denoted by red and green markers, respectively. c Comparison between exact response (green line) and asymptotic a(t) (black line) time histories. Turbulent wind parameters \(\sigma =0\), \(\hat{u}=0.1\). (Color figure online)

Full size image

6.3 Limit cycle analysis in the quasi-resonant case

The nonlinear periodic motions in the quasi-resonant case are analyzed. Differently from the resonant case, explicit forms of amplitude and phase of the limit cycles are not available, so that numerical solutions are obtained. The same values of the turbulence, \(\hat{u}=0.1,\,0.2\), already considered in Fig. 5a, b, are considered in Figs. 8 and 9, respectively. In each of them, three different values of detuning \(\sigma \) have been considered, such that the \(\left( \hat{u},\sigma \right) \) point in the parameter space is (a) inside the region \(\mathcal {R}_{P}\), (b) at the interface between \(\mathcal {R}_{P}\) and \(\mathcal {R}_{Q}\), (c) inside the region \(\mathcal {R}_{Q}\). For comparison, in addition to the galloping curve (thin green line), the curves relevant to \(\sigma =0\) (thin red lines) have also been reported. At the bottom of the figures, the phases are plotted. Referring to Fig. 8a, it is seen that, in the periodic region \(\mathcal {R}_{P}\), the limit cycle are only slightly affected by the small detuning. In contrast, when the detuning is large (Fig. 8b, c), due to the disappearance of the flip bifurcation points \(B_{2},B_{3}\) from the trivial path, limit cycles not crossing this path arise, generated by welding of the former curves, which, however, leaves almost unaltered their stable parts. Concerning the phase, it appears that the detuning alters the constant values \(\gamma _{1,2}=0,\pi \) of the perfect resonance, so that the phase changes with the parameter. The phases, as the amplitudes, show a welding mechanism. A very similar behavior is exhibited in Fig. 9.

Fig. 8
figure 8

Bifurcation diagrams: steady amplitude and phase of the limit cycle versus the steady wind velocity for turbulence amplitude \(\hat{u}=0.1\) and assigned values of detuning: a \(\sigma =0.016\), such that \(\left( \hat{u},\sigma \right) \in \mathcal {R}_{P}\); b \(\sigma =\sigma _{M}=0.019\), such that \(\left( \hat{u},\sigma \right) \) is at the interface; c \(\sigma =0.025\) such that \(\left( \hat{u},\sigma \right) \in \mathcal {R}_{Q}\). Analytical quasi-resonant (black lines), resonant (thin red lines), galloping (thin green lines); the green markers are exact results. (Color figure online)

Full size image
Fig. 9
figure 9

Bifurcation diagrams: steady amplitude and phase of the limit cycle versus the steady wind velocity for turbulence amplitude \(\hat{u}=0.2\) and assigned values of detuning: (a) \(\sigma =0.03\), such that \(\left( \hat{u},\sigma \right) \in \mathcal {R}_{P}\); (b) \(\sigma =\sigma _{M}=0.038\) , such that \(\left( \hat{u},\sigma \right) \) is at the interface; (c) \(\sigma =0.05\) such that \(\left( \hat{u},\sigma \right) \in \mathcal {R}_{Q}\). Analytical quasi-resonant (black lines), resonant (thin red lines), galloping (thin green lines); the green markers are exact results. (Color figure online)

Full size image

Some bifurcation diagrams, plotting the steady amplitude and phase of the limit cycle versus the turbulence amplitude \(\hat{u}\) are shown in Fig. 10 for an assigned detuning \(\sigma =0.05\) and steady wind velocity: (a) \(\bar{U}=0.85\bar{U_{c}}\); (b) \(\bar{U}=\bar{U_{c}}\); (c) \(\bar{U}=1.15\bar{U_{c}}\). A comparison between quasi-resonant (black lines) and resonant (thin red lines) cases is made. It is observed, with respect to the resonant case, that the detuning rounds the cusps of the amplitude bifurcation diagrams, but does not involve qualitative changes. It mainly modifies the unstable branches of the bifurcation diagrams, pushing them forward.

Fig. 10
figure 10

Bifurcation diagrams: steady amplitude and phase of the limit cycle versus the turbulence amplitude for \(\sigma =0.05\) and assigned values of steady wind velocity: a \(\bar{U}=0.85\bar{U_{c}}\); b \(\bar{U}=\bar{U_{c}}\); c \(\bar{U}=1.15\bar{U_{c}}\). Analytical quasi-resonant (black lines), resonant (thin red lines); the green markers are exact results. (Color figure online)

Full size image

The plots of the steady amplitude and phase of the limit cycle versus the detuning parameter \(\sigma \) are shown in Fig. 11 for an assigned turbulence amplitude \(\hat{u}=0.1\) and steady wind velocity: (a) \(\bar{U}=0.85\bar{U_{c}}\); (b) \(\bar{U}=\bar{U_{c}}\); (c) \(\bar{U}=1.15\bar{U_{c}}\). In these bifurcation diagrams it emerges that the detuning strongly influences the amplitude and phase of the limit cycles. In particular, there exist two closed curves (for amplitudes and phases) in the subcritical range and just one curve in the supercritical range. This circumstance proves the existence of a tubular surface in the \(\left( \bar{U},\sigma ,a\right) \)-space, emerging from the plane of the trivial solution (\(a=0\)), whose transverse and longitudinal sections are the bifurcation diagrams in Figs. 8, 9 and 11.

Fig. 11
figure 11

Bifurcation diagrams: steady amplitude and phase of the limit cycle versus the detuning parameter for a turbulence amplitude \(\hat{u}=0.1\) and assigned steady wind velocity: a \(\bar{U}=0.85\bar{U_{c}}\); b \(\bar{U}=\bar{U_{c}}\); c \(\bar{U}=1.15\bar{U_{c}}\). Analytical (solid lines) and exact (green point bullets) results. (Color figure online)

Full size image

The asymptotic solutions are validated against numerical results: green bullets are superimposed to the asymptotic bifurcation diagrams in Figs. 8c, 9c, 10c and 11c. The exact results are found to be in excellent accordance with the analytical predictions.

6.4 Torus analysis

The nonlinear quasi-periodic motions, occurring on tori of the state-space, are studied. These solutions are generated by Neimark–Sacker bifurcations, at \(\bar{U}=\bar{U}_{c}\) and splitting parameters in the \(\mathcal {R}_{Q}\) region. The study is carried out, both numerically (via integration of the bifurcation equations (21)), and asymptotically, as described in Sect. 5.2. Since the bifurcation is of codimension-1, the bifurcation diagram in Fig. 12a is sufficient to explain the phenomenon. Here, in addition to the quasi-periodic solution (exact and asymptotic), represented via the interval \(\left[ a_{\text {min}},a_{\text {max}}\right] \), also the periodic solutions are plotted. At \(\bar{U}=\bar{U}_{c}\), a torus bifurcates from the trivial solution. It is of subcritical type, and manifests again the hard loss of stability phenomenon. After regaining stability, a collision of the torus with a limit cycle occurs, leading to the disappearance of the torus. Exact results (green segments and green bullets) clearly show the stable part of both torus and limit cycle branches. The asymptotic solution displays the qualitative mechanism.

The torus branch plotted in Fig. 12a is just that corresponding to a \(\left( \hat{u},\sigma \right) \) pair; if this latter is changed, other branches of similar type are found. If a cross-section at \(\bar{U}=\bar{U}_{c}\) of all these curves is made, together with \(\sigma =\mathrm {const}\) or \(\hat{u}=\mathrm {const}\), the plots in Fig. 12b, c are obtained. They, in addition to the collision already described, show the existence of stable tori external to the limit cycle tube (see Fig. 12c).

Fig. 12
figure 12

Bifurcation diagrams: (a) a versus \(\bar{U}\) when \(\hat{u}=0.1\) and \(\sigma =0.05\); (b) a versus \(\hat{u}\) when \(\bar{U}=\bar{U}_{c}\) and \(\sigma =0.055\); (c) a versus \(\sigma \) when \(\bar{U}=\bar{U}_{c}\) and \(\hat{u}=0.1\). Exact (green markers) and asymptotic (black lines) periodic solutions. Exact (green segments) and asymptotic (gray regions) quasi-periodic solutions. (Color figure online)

Full size image

The transients asymptotic solutions, drawn by the bifurcation equation (39) through direct numerical integration, are investigated in the \(\left( X,Y\right) \) state space, at varying of the initial conditions. Although the motion essentially occurs in the one-dimensional space r(t), as the first of numbered Eq. (39) proves, the two-dimensional representation adopted here is more clear, since it allows distinguishing closed orbits from equilibria. In particular, the periodic (stable and unstable) solutions are represented by inclined ellipsis, whose axis lengths are the maximum and minimum value of the amplitude, according Eq. (45). By referring to the bifurcation diagram in Fig. 12a, the \(\left( X,Y\right) \) trajectories are shown in Fig. 13 for the steady wind velocities: (a) \(\bar{U}=0.59\), at which two periodic stable solutions co-exist; (b) \(\bar{U}=0.75\), at which only one periodic stable solution exists. Figure 13a shows that, when the initial conditions are inside the unstable closed orbit, the motion is attracted by the trivial solution. In contrast, when they are external to the unstable orbit, the trajectories are attracted by the stable orbit. In Fig. 13b, it appears that all the trajectories dye on the stable orbit.

Fig. 13
figure 13

Trajectories in the state space for different initial conditions (black markers) and for the steady wind velocities: a \(\bar{U}=0.59\); b \(\bar{U}=0.75\). Stable and unstable periodic solutions are represented by green and red ellipsis, respectively. Turbulent wind parameters \(\sigma =0\), \(\hat{u}=0.1\). (Color figure online)

Full size image
Fig. 14
figure 14

Time histories of: a X; b Y; c a. Comparison between exact (green lines) and asymptotic (black lines) results. Wind parameters \(\sigma =0.05\), \(\bar{U}=0.72\), \(\hat{u}=0.1\). (Color figure online)

Full size image
Fig. 15
figure 15

Modulating frequency \(\varpi _{*}\) versus: a \(\bar{U}\) for \(\hat{u}=0.1\) and \(\sigma =0.05\); b \(\hat{u}\) for \(\bar{U}=\bar{U}_{c}\) and \(\sigma =0.055\); c \(\sigma \) for \(\bar{U}=\bar{U}_{c}\) and \(\hat{u}=0.1\). Exact (green point bullets) versus asymptotic (gray lines) results. (Color figure online)

Full size image

Moreover, concerning the asymptotic solution, when compared with the exact one, it is observed that: (i) it captures the qualitative trend, but it is not very accurate for steady wind velocities far from the galloping value; (ii) it is unable to describe the interaction with limit cycles (as a matter of fact, it exists after the collision). Item (i) clearly appears in Fig. 14, where a comparison between the time-histories of the exact (green lines) and analytical (black lines) amplitudes X, Y, a are shown for given \(X=0\), Y=0.0025 initial conditions, represented in Fig. 13a. The rough approximation of the asymptotic solution is mainly due to the fact it has been limited to the first-order. It is guessed that pushing forward the perturbation expansion one step further would give better results. With reference to item (ii), the exact and asymptotic (Eq. (43)) slow frequency \(\varpi _{*}\) is plotted in Fig. 15 versus the wind parameters. It appears that, although an ordering violation occurs in the perturbation scheme, which leads the frequency \(\varpi _{*}\) to vanish, the asymptotic solutions in Fig. 15b, c, capture with reasonably accuracy the zero-frequency occurrence, denoting the collision of torus and limit -cycle. Of course, negative frequencies, beyond the collision, have no meaning. In conclusion, even a low-order perturbation solution, as expected, gives useful information.

7 Conclusions

The aeroelastic stability of a slender prismatic visco-elastic structure under turbulent wind flow has been analyzed by a one-D.O.F. dynamical system, drawn by a Galerkin projection of the continuous model. Aerodynamic forces have been evaluated via the quasi-static theory. The unsteady wind has been split in a steady part and in a fluctuating part, assumed harmonic, of frequency close to the double of the fundamental frequency of the structure. The steady part is responsible for galloping; the harmonic part for principal parametric excitation. The two bifurcation mechanisms interact in the nonlinear field. The multiple scale method has been applied to find bifurcation equations. These have been solved for analyzing linear stability, limit cycles (periodic motions) and tori (quasi-periodic motions), bifurcating from the trivial path. The analytical solutions have been validated against numerical integration of the finite-difference discretized equations. The following main results have been found.

  1. 1.

    The parameter space is made of two regions: In each of them, periodic and quasi-periodic solutions bifurcate from the equilibrium. In the periodic region, the parametric excitation mechanism prevails over galloping. Here, however, the steady wind influences the parametric excitation instability domain, since it modifies the resultant (structural plus aerodynamic) damping. The higher the wind velocity, the wider the instability domain. On the other side, turbulence reduces the galloping velocity of the steady wind case. In the quasi-periodic region, the galloping mechanism prevails over parametric excitation. However, the turbulent component changes the Hopf bifurcation of the autonomous system into a Neimark–Sacker bifurcation for the non-autonomous system, but it does not affect the critical velocity.

  2. 2.

    In the perfect resonant case, there are several limit cycle branches. They are multi-valued, and exhibit the hard-loss of stability phenomenon. Just one branch is stable. When a detuning is introduced, either it weakly affects the resonant solutions (small detuning), or it dramatically changes them (sufficiently large detuning), explaining as unstable bifurcated paths merge and move away from the trivial path. The agreement between exact and asymptotic periodic solutions is excellent.

  3. 3.

    Torus branches bifurcate from the trivial path. They also manifest the hard loss of stability phenomenon. After having regained stability, they collide with a limit cycle and disappear. Meantime, their slow frequency tends to zero. There exist, however, regions of the parameter space where stable quasi-periodic motion take place.

  4. 4.

    A first-order asymptotic solution for quasi-periodic motions has been built-up. It qualitatively captures the features of the numerical solutions, although is not very accurate at high steady wind velocities. On the other hand, it gives a satisfactory estimate of the collision point, at which the slow frequency goes to zero.