Effect of joint interfacial contact stiffness on structural dynamics of ultra-precision machine tool

Highlights

• Stronge contact model and domain extension factor are adopted to improve the joint interfacial contact stiffness model.

• The rotational contact stiffness is first derived and validated by a representative test structure.

• Transfer matrix method for multibody systems and fractal contact model are used jointly to establish the dynamic model.

• One method to ensure the structural stability is proposed by discovering the sensitive joints and restraining bolt loosening.

Abstract

The contact stiffness of the joint interface significantly affects the dynamic characteristics of a machine tool structure. Herein, an improved theoretical model is proposed to calculate the joint interfacial contact stiffness. This model is established based on fractal theory. To mitigate the plastic and elastic contact problems, the Stronge contact model is adopted to express the contact force in the plastic regime innovatively, whereas the elastic contact force is interpreted using Hertz theory; the domain extension factor is compensated to revise the microcontact size distribution. Furthermore, the rotational contact stiffness is first derived, which renders the contact model more comprehensive. A representative test structure is designed, and the contact model is validated by comparing the simulated contact stiffness with experimental results. The transfer matrix method for multibody systems is incorporated with fractal contact model to establish a dynamic model of an ultra-precision machine tool; this ensures that the dynamic characteristics of the machine tool can be predicted more precisely. Experiments involving a vibration response test and a modal test demonstrate the feasibility of the fractal contact and dynamic models. Based on these two models, the effect of the interfacial contact stiffness on the structural dynamics is investigated. As the preloads at two joints loosen by 15%, the maximum vibration displacements increase by 1.74% and 2.90%, respectively, indicating that the joint interface between the column and lathe bed is more sensitive. The procedure presented herein will facilitate the design and optimization of the complex structures.

Introduction

Ultra-precision machine tools (UPMTs) have become increasingly important for the manufacturing of high-quality products having sub-micron form machining tolerance and nanometer-scale roughness [1]. Widespread applications of UPMTs have been found in fabricating highly precise components in various fields such as aeronautics, astronautics, precision instruments, and military [2]. Researchers have reported that the performance of a machine tool is enormously affected by its dynamic properties [3]. From published literature, it is clear that joints have significant impact on the structural dynamics of the machine tool. More than 60% of vibration problems on machine tools originated from the joint interface, and interfacial contact stiffness constituted approximately 40%–60% of the total stiffness [4]; this suggests that the improvement in structural stability is significantly associated with the joint interfacial contact stiffness.

For the study of joint interfacial contact stiffness, one of the most difficult tasks is to establish a theoretical model that characterizes the contact interface and simulates the contact behaviors accurately. Ample literatures are available regarding modeling using various approaches. The finite element method (FEM) is a general and useful approach, which has been utilized by many researchers. Xu et al. [5] modeled a spindle-holder taper joint to predict the stiffness and stress distribution under different clamping and centrifugal forces with FEM. Qin et al. [6] used three dimensional FEM to investigate the time-varying stiffness of a joint interface caused by bolt loosening. Considering the uneven pressure distribution, Li et al. [7] built a finite element model for bolted joint and calculated the contact stiffness distribution on the joint interface. However, one prominent shortcoming of the FEM is that it reveals the macroscopic deformation behaviors and neglects the surface topography of the joint interface, whereas the real surface is not ideally smooth and contains multiscale geometrical characteristics. A contact model for joint interface that considers the surface topography has been developed by Greenwood and Williamson [8], popularly known as the G-W model. It is a probability-statistical contact model in which two undulated surfaces are assumed to have asperities with the same radius near their spherical summits and Gaussian height distribution. Jackson and Green further proposed an elastoplastic contact model to complement the G-W model [9]. Based on the G-W model and experimental findings, Zhang et al. [10] put forward modified assumptions to improve the contact mechanics for rough surfaces. However, statistical contact models merely account for elastic deformation and neglect the plastic deformation of the microcontact, whereas the statistical roughness parameters depend on the sampling length and measuring resolution.

In view of the above-mentioned shortcomings, fractal theory is appropriate for characterizing joint interfaces owing to its preponderances of scale invariance and self-affinity. Fractal model performs well in describing the stress distribution [11] and analyzing the interaction between particles like surface asperity [12]. Based on fractal theory and the variation of the heat generation power, Zheng et al. [13] developed a complete thermal model for the high-speed press system. Similarly, Liu et al. [14] presented a fractal contact mechanics model to calculate the actual contact area of solid joints in a high-speed spindle system for further thermal contact resistances analysis. Wang et al. [15] and Chen et al. [16] discovered the relationship between contact stiffness and normal pressure based on the Hertz contact theory and fractal theory. However, Wang focused on the asperity interaction, whereas Chen further considered the friction factor. Jiang et al. [17] proposed a fractal model for the translational contact stiffness of machined plane joint, and conducted an experimental validation.

Modeling the joint interface precisely is the fundamental step, whereas applying it to reveal the structural dynamic characteristics has more practical significance. Considering the importance of dynamic parameters in structural modeling, Tian et al. [18] proposed an analytic model for fixed joint interface with virtual material hypothesis-based method; and the analytic solutions of virtual material properties further facilitated the structural dynamic modeling. Liu et al. [19] analyzed the influence of bolt pretightening sequence based on the elastic interaction stiffness theory, and predicted the structural deformation of a machine tool in different pretightening sequences. Inverse receptance coupling substructure analysis method was a well suited experimental method to determine the joint interfacial stiffness and damping, Fu et al. [20] presented a computational model with it to predict the dynamic response of the workpiece, and Ealo et al. [21] developed it in the FEM model of a milling machine to analyze the dynamic behaviors of linear guideways. Compared with other machine components, the vibration of tool tip plays the most critical role in the formation of surface roughness. Deng et al. [22] predicted the position-dependent tool tip response efficiently with the FEM model after identifying the joint parameters through the substructural synthesis method. Different updating methods have been developed to establish dynamics models of UPMTs. However, it is difficult to achieve high computational accuracy and high efficiency at the same time.

Herein, an improved model of the joint interfacial contact stiffness based on fractal theory is proposed. The Stronge contact model was adopted to express the translational contact force in the plastic regime. Meanwhile, domain extension factor was incorporated to revise the microcontact size distribution [23]. Studies regarding the modeling and calculation of rotational contact stiffness have not been performed; hence, this study is the first to present a comprehensive model for the said stiffness. A test instrument was set up to obtain the translational and rotational contact stiffness of the designed structure using different processing methods. The simulated contact stiffness was consistent with the experimental data. Dynamic modeling was performed using the transfer matrix method for multibody systems (MSTMM), which demonstrated a remarkable capability of solving rigid-flexible coupled multibody dynamic problems with high computational speed [24,25]. Lu et al. [26] analyzed the dynamic modeling approach of UPMTs using MSTMM, whereas Ding et al. [27] investigated the effects of its dynamic characteristics on the mid-frequency waviness of the machined surface based Lu et al.'s approach. In this study, MSTMM and fractal theory were used jointly to establish a structural dynamic model of UPMT [[28], [29], [30]], and the dynamic model was validated based on vibration response test and modal test. Finally, comparing the effects of joint interfacial contact stiffness on structural dynamics, the sensitive joint interface was discovered.