## 1 Introduction  
The passive control of structures subjected to seismic actions through the use of energy dissipating devices (EDD) is a common solution in earthquake-prone areas. A typical building structure with passive control systems consists of a main frame that works in conjunction with special elements called EDDs or simply dampers, whose main role is to dissipate most of the energy input from an earthquake. Among the different types of EDDs available, the so-called hysteretic dampers are probably the most widely used due to their advantages in terms of robustness, reliability and cost. The source of energy dissipation through hysteretic dampers is the plastic deformation of metals (typically steel). In deciding whether or not to replace the EDDs by new ones, a key factor is the non-destructive evaluation of the level of damage to the dampers after a seismic event. This decision has important economic and safety-related consequences, thus making periodic inspection a very relevant issue. Hysteretic dampers do not necessarily need to be replaced after minor/moderate damage, or even after a large earthquake, because they have a high energy dissipation capacity. Hence, the development of reliable Non-Destructive Testing (NDT) techniques for inspection is essential for determining the remaining life of the dampers. Visual inspection is not applicable in such cases, because damage is not visible until the element is on the brim of failure. Past research has shown that the level of damage and the proximity to failure of hysteretic dampers subjected to arbitrarily cyclic loading can be reliably estimated by decomposing the force-displacement curves endured by the damper into the so-called skeleton and Bauschinger parts [1]. This decomposition is the basis of a mechanical damage index called ID that has been extensively validated with static and dynamic tests. However, measuring the force and the displacement of dampers during operation involves very expensive and sophisticated instrumentation, which is hardly justified when the probability of an earthquake occurring is very low. A cheaper and more practical alternative is to conduct simple vibration tests on the dampers by means of low-cost piezoceramic sensors permanently attached to the damper [2; 3; 4; 5]. Damage due to plastic deformations induces changes in the vibration response (resonance frequencies, modes, damping coefficients, etc.) that can be evaluated via proper processing of the signals captured by piezoceramic sensors. Although many papers in the literature deal with time vibration series analysis for damage detection in structures and materials [6; 7; 8; 9; 10; 11; 12; 13], the suggestion of its application to hysteretic dampers is an original contribution by our research group. More concretely, this paper proposes the use of Relative Wavelet Energy Entropy (RWEE) to evaluate damage to hysteretic dampers subjected to seismic events. This index combines wavelet transform with information entropy, taking the advantages of both techniques to obtain a reliable quantification of the damage level of the dampers. Firstly, the wavelet package decomposition of the vibration signals is carried out. Wavelet transform of vibration signals has been widely used for structural damage detection [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30]. Basically, by means of the wavelet decomposition of the time-domain signal on the time-frequency domain, it is possible to observe dominant modes of variability of the signal (frequencies) and how these modes vary over time. Previous results have proven that the energy of the decomposed wavelet vibrational signals using the wavelet packet (wavelet energy) shows higher sensitivity to minor damage [31; 32; 33; 34]. Secondly, Shannon entropy [35; 36; 37; 38] is applied to calculate the wavelet entropy of the vibration signals [39; 40; 41; 42; 43; 44; 45].