# Equivalent Linearization Methods for a Control System with Clutching Inerter Damper
## 1 Introduction  
Recently, the structural vibration control strategy has been widely applied in civil engineering [1; 2; 3; 4; 5; 6]. New passive control devices, such as shape memory alloy dampers [7; 8], bounding dampers [9; 10], and magnetorheological dampers [11; 12], can suppress structural vibration effectively. As an innovative control device, the inertial mass damper (IMD) has gained great popularity in the field of structural vibration control. [PERSON] [13] proposed the concept of inerter, which was first applied in vehicle suspension systems [14]. The mechanical model of the IMD involves an element with two nodes, and the force of the element is positively correlated with the relative acceleration of the two nodes. A study showed that the inerter achieves a damping effect by reducing the load intensity [15], and extends the period of the structure without reducing its stiffness, which is different from base isolation.  
Many inerter-based dampers have been proposed and studied in the field of civil engineering. One possible problem in structures equipped with an IMD is that the IMD or its support may bear a huge force after the structure attains its peak velocity and begins to slow down. This is because of the deceleration delay of the IMD, which results in a faster movement of the damper than the structure. Subsequently, the IMD pushes the structure to move, and the IMD and its support beara huge force in the meantime, which may be destructive for the system. Taking this problem into account, an IMD with a clutch, known as clutching inerter damper (CID), was proposed [15; 16]. The CID works like a bicycle ratchet. In other words, it works when the velocity and acceleration move in the same direction and does not work when they move in the opposite direction. Hence, the CID can reduce the displacement response more effectively after the point when the structure starts to slow down. However, the effectiveness of the CID has not been evaluated clearly in previous studies. Although the working mechanism of the CID is clear, it is difficult for designers to apply it in structures without certain linearization methods, since the equation of the CID is nonlinear, compared with the traditional IMD.  
For a structure with a CID, the time-history analysis can be used for numerical analysis. However, considering the integrated design of structure and CID, it is better to linearize the equation. Therefore, an equivalent linearization method (ELM) needs to be developed. For systems with a bilinear hysteresis model, there have been numerous conventional equivalent linearization methods [17; 18; 19; 20]. For other nonlinear models, the statistical linearization method is widely used [21; 22; 23; 24]. Several new linearization methods were also proposed in recent years [25; 26; 27; 28]. The linearization method for the CID based on the equivalent period and energy was discussed by [PERSON] et al. [16].  
In this paper, the equation of motion of a single-degree-of-freedom (SDOF) system with a traditional IMD is obtained by theoretical analysis; subsequently, the responses of the SDOF system with an IMD and a CID under sinusoidal excitation are compared using numerical simulation. Secondly, three equivalent linearization methods are proposed. The first ELM is based on the adaptation of the frequency response function (FRF), which is denoted as ELM1. The second ELM is based on the equivalent period and the conservation of energy, denoted as ELM2. The third ELM is based on statistical linearization, denoted ELM3. Finally, a numerical example is provided to evaluate the accuracy of all methods under different excitations.