# Vibration Reduction of a Timoshenko Beam with Multiple Parallel Nonlinear Energy Sinks
## 1 Introduction  
Passive vibration reduction has been widely used in civil engineering [1; 2; 3], spacecraft [4; 5; 6], and mechanical systems [7; 8]. Traditional linear dynamic vibration absorbers (DVA) are effective within a specific narrow frequency band [9; 10]. The concept of a nonlinear energy sink (NES) was firstly proposed by [PERSON] [11] in 2001. Compared to the DVA, an NES has the advantages of a broad vibration suppression band, small additional mass, and not changing the natural frequency of the primary system. NES has been widely used for the vibration reduction of beams [12; 13], plates [14; 15; 16], and shells [17; 18; 19]. Many researchers have applied the NESs to continuous beam structures and investigated their vibration reduction performance. [PERSON] and [PERSON] [20] utilized an NES to dissipate the vibration energy of a simply supported linear beam subjected to an impact excitation. [PERSON] et al. [21] investigated a nonlinear simply supported beam coupled with an NES; a harmonic excitation was used. [PERSON] et al. [22] investigated the vibration mitigation of the nonlinear principal resonance response of a beam-like lattice structure with an NES. [PERSON] [23] investigated the nonlinear vibration of a beam with an NES under nonideal boundary supports. [PERSON] et al. [24] analyzed the effectiveness of an NES coupled to an axially moving string that was subjected to transversal wind loading. [PERSON] et al. [25] investigated the effect of an NES attaching to a laminated composited beam subjected to harmonic excitation. It has been proved that NESs are effective for beams with different structures, excitations, and boundary conditions. Recent research shows that applying multiple parallel NESs can further improve vibration reduction performance. [PERSON] et al. [26] studied the vibration response of a main oscillator coupled with multiple parallel NESs and optimized the damping ratio of the NESs to guarantee that energy pumping would occur. The vibration absorption rate of parallel NESs was significantly improved and was better than that of one NES. [PERSON] et al. [27] experimentally studied the vibration reduction performance of a 4-DOF main oscillator coupled with two parallel NESs and demonstrated the efficiency of the parallel NESs. [PERSON] et al. [28] attached the NES to axially moving beams and compared the vibration reduction efficiencies of one NES and parallel NESs. The results indicated that the parallel NESs with less total attached mass could obtain better vibration reduction. [PERSON] et al. [29] compared the vibration absorption of single and two parallel NESs of equal mass and found that by tuning the nonlinear stiffness and damping ratio of NESs, the parallel NESs could eliminate the higher branches of the system more effectively. [PERSON] et al. [30] applied novel lever parallel NESs to a beam, and the results showed that the vibration absorption performance of lever-type parallel nonlinear energy sinks was better than lever-type single nonlinear energy sinks under the same attached mass. Most of the existing research focuses on the vibration reduction of long beams, with the modeling method based on the Euler beam theory. However, the method is unsuitable for short beams, especially with length/thickness ratios below 5. The Euler beam theory, without considering shear deformation, induces large errors in predicting the dynamic responses of short beams with NESs. There is a lack of research on the theoretical analysis of a short beam attached to multiple parallel NESs. The parameters of NESs are essential to vibration reduction performance. [PERSON] and [PERSON] [20] carried out a sensitivity analysis to obtain the optimal nonlinear stiffness of NESs. [PERSON] et al. [25] optimized the parameters of NESs with a sensitivity analysis. [PERSON] et al. [31] numerically investigated the effects of the damping ratio of NESs, as well as the magnitude of the external force to the beam, and obtained the optimized parameters of NESs. [PERSON] et al.