## 1 Introduction  
Damping technology is one of the important methods to reduce the vibration and improve the performance of many engineering structures. The traditional passive damping method is to attach a viscoelastic damping material and a restraint layer to the controlled object. It has the advantages of a simple structure, easy implementation, wide control frequency, stability, reliability and low cost. It has been paid attention to by many researchers for many years. Partially constrained layer damping (PCLD) consists of a patch bonded to the structure to be damped. The patch is formed of a viscoelastic layer, constrained by a stiff layer covering part of the structure. The mechanical and damping properties of sandwich structures have been studied for decades. [PERSON] [1] proposed a sandwich structure consisting of a middle layer (damping layer) and an upper and lower layer (constrained layer) and investigated the damping factor of a sandwich plate. [PERSON] et al. [2] derived the sixth order differential governing equation of a three-layer sandwich beam and studied the forced vibration of the beam by using the method of [PERSON]. [PERSON] et al. [3] applied the finite element method to predict the modal damping ratios in three-layer laminates. [PERSON] et al. [4] analyzed the damping and vibration characteristics of a partially covered plate with a simple supported based on the Rayleigh-Ritz method. [PERSON] et al. [5] studied the free vibration of a rectangular sandwich plate with a viscoelastic layer by using the first order shear deformation theory (FSDT). [PERSON] et al. [6] proposed the [PERSON] method to analyze a sandwich plate. [PERSON] et al. [7] presented an accurate dynamic stiffness model for a three-layered sandwich beam to study the free vibration characteristics of the beam. [PERSON] et al. [8] investigated the dynamic problem of the sandwich laminated plates based on a layerwise finite element model, and the dynamic behavior of the model was validated by the public literature. [PERSON] et al. [9] dealt with the harmonic and transient responses of a partially constrained layer damping plate. [PERSON] et al. [10] studied the effects of the uncertainties of viscoelastic sandwich structures in passive and hybrid control strategies on their modal parameters. It can be seen from the above statement that many studies have focused on linear models. Some studies on nonlinear models have also been carried out. [PERSON] et al. [11] studied the realization of the variational iteration method in the Korteweg-de-Vries equation for actual and high-order nonlinear equations. [PERSON] et al. [12] proposed the homotopy perturbation method to obtain explicit solutions to nonlinear fourth-order parabolic equations. [PERSON] et al. [13] proposed semi-analytical solutions of nonlinear differential equations with boundary value problems based on general Lagrangian multipliers. [PERSON] et al. [14] presented a nonlinear dynamic response model of a screen machine which was described by nonlinear coupled differential equations and excited by a self-resonant control system. The nonlinear model has higher calculation accuracy, but it will increase the complexity and calculation time. Therefore, the nonlinear model is difficult to apply to engineering. The optimization of the sandwich structures has also been studied in recent years. For such sandwich structures with constrained damping layers, many literatures focus on the optimization of damping loss factors or the dynamic response. The design variables optimized for these sandwich structures include material parameters, coverage area, thickness, location of the constrained damping layer and structure weight. [PERSON] et al. [15] investigated the optimal constrained viscoelastic tape to maximize damping in laminated beams. [PERSON] et al. [16] studied the optimal damping of beams constrained by a damping layer based on an efficient finite element model.