# Effect of Vinyl Flooring on the Modal Properties of a Steel Footbridge
## 3 Viscous Damping Mechanisms for Vibrating Footbridges  
Real footbridges dissipate energy when they vibrate according to different mechanisms. The energy dissipated during the vibratory process may be determined as the sum of the simultaneous actuation of all these mechanisms [29]. According to their fundamental mode of energy dissipation, damping mechanisms may be classified in three groups [30]: (i) dissipation within a solid (material damping); (ii) dissipation within or to a fluid medium (fluid damping); and (iii) dissipation at the interfaces between solids (friction damping). The material damping mechanism refers to the inherent energy dissipation process which converts its kinetic energy to thermal energy during each cyclic deformation. The fluid damping mechanism refers to both the energy lost from a vibrating surface through wave radiation (radiation damping) and the energy lost through drag forces on a solid body moving in a fluid (drag forces damping). Finally, the friction damping mechanism refers to the dissipation process in which mechanical energy is transformed into heat during each cyclic motion [30]. In order to model mathematically these dissipative mechanisms, several damping models have been proposed [31]. In general, for linear systems, these damping models may be classified into two groups: (i) viscous damping models and (ii) non-viscous damping models. In the viscous damping models, the damping forces do not depend on the past history of the motion of the structure, whereas in the non-viscous damping models, the damping forces are strongly dependent of the previous behavior of the structure. Thus, due to its simplicity and better computational behavior, viscous damping models have been widely used. In facts, viscous damping models have been adopted by the most advanced design guidelines [13; 14]. Such viscous damping models are characterized by the damping ratio, \(\zeta_{j}\) (being \(j\) the number of the considered vibration mode), which is defined as the ratio between the damping coefficient and the critical damping coefficient [23]. In this manner, the overall damping ratio of civil engineering structures, \(\zeta_{j}\), may be expressed in terms of the different dissipative elements involved as follows [32]:  
\[\zeta_{j}=\zeta_{s,j}+\zeta_{ns,j}+\zeta_{a,j}+\zeta_{d,j} \tag{3}\]  
where  
\(\zeta_{s,j}\): is the structural damping ratio (associated with the structural elements) [%]. \(\zeta_{ns,j}\): is the non-structural damping ratio (associated with the non-structural elements) [%]. \(\zeta_{a,j}\): is the aerodynamic damping ratio (associated with the air-structure interaction) [%]. \(\zeta_{d,j}\): is the special devices damping ratio (associated with installed external damping devices as tuned mass dampers, sloshing tanks, etc.) [%]. Although there are some design guidelines--as for instance the European standard [32] to check the vibration serviceability limit state of building under wind action, which allow an accurate estimation of the damping ratio in terms of its different dissipative elements--in the case of footbridges, only overall damping ratios are provided by the most advanced design guidelines [13; 14]. Table 2 shows the values recommended by these design guidelines in terms of the construction type and the level of vibrations. The values provided by Table 2 are conservative and do not allow estimating the value of the damping ratio of a particular footbridge in terms of its actual design. In order to overcome this limitation, in this study the effect of a non-structural element, the vinyl flooring, on the damping ratios of steel footbridges is analyzed experimentally. As it was not possible to conduct experimental tests on the Jorge Manrique footbridge, given that its service had to be maintained, a laboratory steel footbridge was built for this purpose.