# Dynamic and Stability Analysis of Multibolt Plane Joints under Normal Forces
## 1 Introduction  
The contact interfaces between parts are known as mechanical joint surfaces. These interfaces play an important role in the dynamic behavior of mechanisms and mechanical devices. For example, contact stiffness and damping of such systems at machine tools contribute 60% of the overall stiffness and more than 90% of the overall damping [1; 2]. Furthermore, many studies have focused on these joint surfaces. [PERSON] et al. analyzed the dynamic characteristics of the joints in the aero-engine rotor system through FEM simulation and experiment [3]. A virtual gradient material model was proposed by [PERSON] et al. to obtain the contact pressure distribution and dynamic characteristics of the bolted joint [4]. [PERSON] al. researched the dynamic behavior of the bolted joint under heavy loadings utilizing a bolted structure test bed [5]. [PERSON] et al. built a critical two-degree-of-freedom modal coupling model and studied the effect of contact stiffness on the stability and nonlinearity on the basis of fractal surface topography [6]. [PERSON] al. provided a new linear joint model and investigated the influence of preload and surface roughness quality on the identified equivalent stiffness [7]. A new lump model applied in finite element analysis was given by [PERSON] et al. for the nonlinear dynamic response of bolted flanges [8]. With these interfaces withstanding contact pressures and frictions, the contact problems inengineering include geometry errors, microscopic roughness, lubrication, coating, conduction of heat, and electricity, which are nonlinear and complicated. Therefore, studies of the dynamic characteristics of mechanical joint surfacesare of great significance. Research on modeling iscrucial for developing simulations to analyze the dynamic motions of contact interfaces. Conventionally, Hertzian contact theory is utilized to model interacting surfaces. As [PERSON] and [PERSON] consider the fact that all engineering surfaces are not absolutely smooth, they describe the contact interfaces with extended Hertzian theory [9]. Rough surfaces are approximated by a series of spherical summits which have the same radius and Gaussian height distributions. The method was first based on statistical representation of the surface topography and is closer to the actual conditions than previous models, and has an enormous influence on contact theory. In the following, many investigations were focused on the extension of G-W model by [PERSON] et al. [10], [PERSON] [11], [PERSON] [12], and [PERSON] et al. [13]. However, [PERSON] and [PERSON] as well as [PERSON] and [PERSON] suggest that statistical roughness parameters depend on both instrumental resolution and sampling length [14,15]. It was found that the Weierstrass-Mandelbrot (W-M) function satisfies all these properties of a surface profile [14,16]. Therefore, fractal theory has been employed to model engineering surfaces by [PERSON] and [PERSON] (M-B) due to their statistic self-affinity, which is scale-independent [14,17,18,19]. [PERSON] and [PERSON] introduced the domain extension factor for microcontact size distribution to modify the M-B model [20]. Another fractal-based model was proposed by using a multivariate W-M function [21,22]. According to the previous model, [PERSON] et al. deduced the relationship between stiffness and the real area of contact mathematically [23]. [PERSON] et al. analyzed the static force-displacement characteristic of joint surfaces on the basis of a two-variable W-M function using the finite element method and further the dynamic behavior of the contact system [22,24]. The above mentioned studies mostly focused on the experiment test or FE method, which is often considered as a macroscopic system. In addition, the previous studies rarely referred to chaos analysis. In this work, the lumped-mass method is applied for the sake of simplicity of computation and the fractal contact model is employed to the perspective of the surface morphology.