## 1 Introduction
In [28], the limit-cycle oscillations exhibited by long-span suspension bridges in post-flutter condition are investigated. The governing aeroelastic partial differential equations are solved by the Faedo-Galerkin method. Most of the studies about the aerodynamic instability of the bridge were devoted to flutter phenomena, according to which the aeroelastic instability involves two coalescent modes, one torsional, the other flexural. This is the case of bridges with low torsional stiffness, for which the natural torsional and flexural frequencies are close each other, prone to coalesce under aerodynamic excitation. On the contrary, bridges with high torsional stiffness (e.g., box-girder bridges) have the fundamental torsional frequency much larger than the flexural one, so that the aerodynamic instability exclusively occurs in the flexural mode (galloping instability). This class of bridges also attracted some interest, justified by the fact that, after the Tacoma collapse, many existing bridges were torsionally stiffened and the new bridges were designed with stiffer trusses (e.g., the Verrazzano bridge in New York, Tago bridge in Lisbon or the Japanese bridge Akashi Kaikyo). Many works in literature dealt with the galloping instability separately in the beam (see, e.g., [29; 30; 31; 32; 33]) and in the cable (see, e.g., [34; 35; 36; 37]). With reference to the class of bridges with high torsional stiffness, the interest was addressed mainly to the galloping control (see, e.g., [38; 39; 40]). In this paper, the interest is focused on galloping of suspension bridges. Differently from the current research trend, according to which fine discretized models are used to analyze particular bridges, here the aeroelastic behavior is investigated via a continuous model, in order to study families of bridges, and to identify the key dimensionless parameters which govern the mechanical behavior. The main innovative aspect, therefore, relies in directly attacking the continuous problem by perturbation methods (similarly to what done by the same authors in [41]), which leads to approximate formulae, suitable for preliminary design. A simple continuous model of suspension bridge, subjected to steady wind flow, is proposed to analyze the aeroelastic in-plain instability (galloping). Linear visco-elastic beam and cable are considered, coupled by vertical suspenders, modeled as axially rigid links uniformly distributed. Both external and internal damping are accounted, both for the beam and cable, according to the Kelvin-Voigt rheological model. A steady and uniform wind flow is applied exclusively to the beam. The aeroelastic effects of the wind are evaluated via the classical quasi-static theory (a discussion about strengths and weaknesses of the method is reported in [42]). First, the linear free dynamic of the undamped model is studied; then, the nonlinear bifurcation problem is tackled by the Multiple Scale Method (MSM) which naturally reduces the infinite-dimensional system to the proper finite-dimensional Center Manifold. From the bifurcation equation, the limit cycles and their stability are evaluated as function of the mean wind velocity. The asymptotic results are validated via numerical time-integration of ordinary differential equations derived by in-space finite differences. A case study of suspension bridge is analyzed. The paper is organized as follows. In Section 2, a continuous visco-elastic model of standard single-span suspension bridge under uniformly distributed steady wind flow, is formulated. In Section 3, the free dynamic of the undamped problem is addressed. In Section 4 the Multiple Scale Method is directly applied to the partial differential equations, to get the bifurcation equation governing the motion close to a Hopf bifurcation. In Section 5, a finite-dimensional model is formulated. In Section 6, the numerical outcomes are presented. Finally, in Section 7 the main findings of the work are summarized. An Appendix A, containing details about damping parameter identification, closes the paper.