rotating disk subjected to a reciprocating angular movement [11] and suspension-moving load with an initial runout [12]. The response amplitude increases as the spinning speed is close to the resonance speed of the rotating flexible disk with a stationary loading system, and parametric instability can be suppressed by operating the system at low-frequency and small-amplitude reciprocating external forces. [PERSON] et al. [13] investigated the vortex-induced vibration of elastically supported disk, the analysis shows the vibration of the trajectory mode is related to disc vibration. [PERSON] et al. [14] developed a wobbling model of spinning friction discs and pointed out that the threshold of instability is determined by damping and friction-induced circulation. [PERSON] et al. [15] studied dynamics of rotating disc and shaft system, by taking into account the shaft deformation, they investigated the coupling vibration response of the system. [PERSON] et al. [16] modeled the rotating and contact motion of a rigid disc considering the viscous damping on the edge, and the results indicated that edge friction increased the nonlinearity of the coupling motion between rotating and moving. Compared to conventional annular discs, less attention has been given to the dynamics of deformed discs. [PERSON] et al. [17] investigated the effects of eccentricity of the central hole on the frequencies and mode shapes. A similar problem was discussed by [PERSON] [18], who examined the impact of angular speed and eccentricity on the transverse vibration frequencies of the circular plate. [PERSON] [19] analyzed the effect of initial runout on the dynamics of spinning disks; he assumed an initial curvature for the spinning disk and studied its impact on the dynamics of a spinning disk near a rigid surface. [PERSON] and [PERSON] [20; 21] studied the effect of an unsymmetrical initial runout on the oscillation frequencies and the critical speeds of the spinning disks, and the results showed that the natural frequencies of the disks with runout increase in the rotating frame. The discs driven by the spline shaft have different characteristics from those of free and clamped boundaries. [PERSON] et al. [22] investigated the contact mechanics of a spline coupling, and their work showed that during shaft rotation, most friction regimes encountered gross slip, and only two out of nine cases were stick-slip regimes. [PERSON] and [PERSON] [23] analyzed the critical speed for a spinning disc subjected to an external force with spline constraints, and the spline force was modeled as an elastic spring force. They [24] also analyzed the effect of in-plane edge load on spinning discs and found that spline-guided discs did not undergo merge-type effects or flutter and therefore were more stable than clamped discs at a minor critical speed. In addition, this study requires knowledge to model the impact force between the tilting disc and the boundary. [PERSON] et al. [25] established a rubbing-impact model between clutch discs, adopted the Hertz contact model and successfully predicted the experimental drag torque of a wet clutch. The contact model and coefficients of the present work simply refer to the preceding works [26; 27]. The rest of the paper is arranged as follows: Section 2 starts with the coordinate definition of the spinning discs. Then, Section 2.1 analyzes the tilting moment of the flat disc due to centrifugal force. On this basis, Section 2.2 derives the centrifugal moment of  
Figure 1: Common disc deformation types in a wet clutch. the deformed disc by introducing shape functions. The boundary impact and spline friction are introduced in Sections 2.3 and 2.4, respectively. In Section 3, the nonlinear responses are simulated via the state space method, and the dynamic characteristics of the discs with different deformation angles are compared. Moreover, the impacts of the impulse force, boundary distance and rotation speed are also analyzed. The conclusions are drawn in Section 4.