## 1 Introduction  
A bladed disk is the assembly of the disk and the mounted blades in aero-engines. The interaction between the rotating bladed disk and the flow field often induces severe vibration which might cause high cycle fatigue or structural failure. Currently, frictional damping (generated by contact interfaces [1; 2; 3]) is applied to control the vibration of bladed disks. In recent years, in order to improve the thrust-weight ratio and reliability, the integrally bladed disk as single-piece-made component is developed and implemented industrially [4]. However, due to the lack of friction contacts, the damping decreases dramatically and vibration problem becomes more prominent. For these reasons, new damping strategies for bladed disks are still in demand. The dampers for bladed disks should be light-weight due to strict limit on the added mass in aerospace engineering. The dampers should also be efficient in broad frequency band due to the nature of fluid excitation. On these aspects, piezoelectric shunt damping is found to be a good candidate among others [5; 6]. The basic idea is to embed or paste the piezoelectric materials on the host structure and shunt the electrodes by an electric circuit. Owing to the two-way energy conversion capability of piezoelectric materials, some proportion of the mechanical energy is converted into the electric field when the structure vibrates. The electric energy is dissipated by the electric circuit (e.g., a resistor), and this generates the damping to the mechanical field thus reduces vibration. Since proposed in 1980s [7; 8], piezoelectric shunt damping has been rapidly developed and has already been considered applying to blades [9; 10] and bladed disks [11; 12; 13; 14; 15; 16] in aero-engines. Good single-mode damping can be achieved when we only use passive electric components in the shunting circuit. For example, the shunt with a resistor and an inductor connected in series performs like a dynamic vibration absorber [7; 8]. The resistance and the inductance should be tuned with the intrinsic capacitance of piezoelectric materials and the targeting resonance frequency in a manner analogous to a mechanical vibration absorber. This often leads to the problem of implementing a rather large inductance, but it can be solved by synthetic circuits [17] or digital circuits [18]. Band-pass filters and multiple inductor-resistor branch should be introduced to achieve multi-mode/broadband damping [19; 20]. Nevertheless, the performance of the passive shunts is normally sensitive to the electric parameters or working frequency due to the underlying need of frequency tuning. There have been many attempts of applying piezoelectric shunt damping to aero-engine blades in the literature. Many positive experimental results can be found, including on a rotational test rig [9] and on a real-scale composite fan blade [10]. In modern aero-engines, the disk is becoming thinner to reduce the mass. Consequently, the stiffness of the disk closer to the blade stiffness, and the disk and blades should be considered as a whole mechanical system when developing dampers [21]. Ideally, a bladed disk is a perfect cyclic-periodic structure. However, in reality, there are always small and random deviations in blade properties due to manufacturing, material inhomogeneity, in-operation wear, etc. These small blade-to-blade discrepancies are called mistuning, and it can lead to vibration concentrated in few blade sectors (known as vibration localization [22]). Moreover, due to the break of cyclic symmetry, the vibration modes no longer has a single nodal diameter and thus can be excited by travelling wave excitations with various engine-orders [23]. As a result, it is more favorable to achieve multi-mode damping for vibration control of bladed disks. Multi-mode damping can be achieved if a synchronized switching damping (SSD) circuit [24; 25] is introduced to the piezoelectric shunt. External power is required to drive the switching action, thus makes it a semi-active approach.