Dynamic Characteristics Analysis of a Micro Grinding Machine Tool

Developing miniaturizedultraprecision machine tools (MUMTs) is a rational approach to manufacture micro/mesoscale mechanical components. The dynamic characteristics of MUMTs can be different from that of conventional machine tools, because of its miniaturized structure, downsized components and more flexible assembly. Their dynamics behaviorneeds to be studied. In this paper, the dynamic characteristics of a verticalultraprecisionmicro grinding machine tool were analyzed. An analysis model was established based on finite element method(FEM).Modal analysis was then conducted to obtain vibration mode characteristics of the machine tool, and then the harmonic response analysis was applied to machine tool for verifying its mechanical behaviour. This work helps MUMT developers make a better product at the early design stage with lower cost and development time


Introduction
Currently, most of the micro/mesoscale mechanical components are machined by conventional ultra-precision machine tools and MEMS manufacturing techniques. A rational approach for micro/mesoscale manufacturing is to develop miniaturized ultra-precision machine tools(MUMTs). The advantages of MUMTs include: small footprint, low cost of energy, low operational costs, quick response to demands,etc. [1].
Dynamic characteristics of machine tools structure have an important influence on cutting efficiency and machining accuracy. They are important performance indicator of machine tool and have been widely studied for design and optimization of conventional machine tools to enhance their machining performance [2][3][4][5]. Dynamic characteristics of miniaturized ultra-precision machine tools also need to be analyzed for structure optimization and performance enhancement. In this paper, the dynamic characteristics of a verticalultraprecisionmicro grinding machine tool were researched based on finite element method(FEM).The natural frequency, vibration modes and displacements of dynamic response were analyzed.

Theory of dynamic analysis
Dynamiccharacteristics analysis of the machine tools incl udes modal analysis and harmonic response analysis.

Theory of modal analysis
Modal analysis is used to determine the vibration characteristics of the structure, it is also the basis for the analysis of the harmonic response [6]. The Eigen frequency equation for modal analysis is given by: where K is symmetric stiffness matrix of the structure, Mis symmetric mass matrix, Z is the natural frequency, andM is the vibration mode.iis the number of modes, and it is equal to the number of degrees of freedom.

Theory of harmonic response analysis
Harmonic response analysis predicts the response of a structure subjected to continuous harmonic excitation. There are many forced vibration in engineering applications, the vibration is caused by disturbance force or displacement. The dynamic equations of motion can be expressed as: where M is symmetric mass matrix of the structure, I is internal forces, P is external forces, x is acceleration. By solving equation (2),the steady state response of the structure generated by excitation can be obtained.

FEM model
The finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for partial differential equations. FEM cuts a structure into several elements (pieces of the structure), then reconnects elements at "nodes", as if nodes were pins or drops of glue that hold elements together. This process results in a set of simultaneous algebraic equations. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function. Abaqus is one of the outstanding commercial FEM softwares applied to a wide variety of engineering problems. Its accuracy of calculation results have been verified by many experiments.
The main components of the vertical micro grinding machine tool are machine bed, column, spindle, stages. All the stages are set in their initial position. The 3D model of the machine tool was established in Solidworks, and then imported into Abaqus to get the finite element model. In the FEM simulations, tetrahedron elements defined by 10 nodes are suitable for irregular geometry. Meshing were used to mesh the virtual machine structure. The bottom surface of the 3D structure was constrained during the whole computations. Most of the components are connected by bolts, such as the connection between machine bed and column, spindle and stage in z direction. These connection are simulated by bolt load model in Abaqus. The relationship between carriage and base of the stage was simulated with linear spring elements and an example is shown in Fig. 1.

Modal analysis
Modal analysis can obtain the fundamental vibration mode shapes and corresponding frequencies of the machine tool. Lanzosalgorithm in Abaqus for solving eigenvalue has the most general capabilities and high speed. In this paper, lanczos algorithm is selected to extract the first 60 natural frequencies of the machine tool. The total effective mass of the model is 1.165T, the excited effective mass in x, y, z directions of the first 60 modes are 1.0625T, 1.0587T, and 1.0733T respectively. The proportion of the effective mass of x, y, z direction in total effective mass is 91.2%, 90.8%, 92.1%. It can be considered enough modal frequencies are extracted [7]. The first 60 natural frequencies of the machine tool are listed as below:   Complex vibration shapes 497 8 Complex vibration shapes 557 9 Complex vibration shapes 604 10 Complex vibration shapes 623 Detailed pictorial descriptions of first 6 natural frequencies and vibration modes are showed in Fig. 3.  The excited effectivemass of each mode is listed in Fig. 4.
It can be found the 2 nd ,5 th ,10 th and 13 th mode are main vibration mode ,the excited mass which corresponds to these mode is much more than other modals excited.
The 2 nd mode has the most extracted mass, its corresponding frequency is 239Hz.The 5 th mode has the second most extracted mass, its corresponding frequency is 413Hz. The mass extracted by the other mode are much less than the 2 nd and 5 th mode. It can be considered they have the most influence on dynamic behavior. Figure 3 (b) shows the 2 nd mode of the machine tool, and figure 3 (e) shows the 5 th mode of the machine tool. With the deformation of column, the displacement of spindle isobviously large, and therefore the column is the weak part in the machine tool configuration. It should be paid more attention to improve the structure of column, which could be achieved by increasing the thickness of the column or mounting bolts in the back of the column.  The first natural frequency has been recognized it should be higher than the machine operation frequency in the design of conventional precision machine, for example, the spindle rotational frequency. However, this principle cannot be used in the design of miniaturized ultra-precision micro grinding machine tool, because the maximum spindle speed of the micro grinding machine tool can be up to 2000 Hz (120,000 rpm). Modal analysis results provide a frequency selection guide for machine operation rather than a threshold for maximum spindle speed [8].

Harmonic response analysis
When the natural frequencies and their modes of vibration are obtained, harmonic response analyses were performed to determine the steady-state response of the machine to loads that vary harmonically with time. The harmonic response analysis verify whether or not the structure is able to overcome resonance and harmful effects caused by forced vibrations.
In this analysis, the machine structure was excited by a series of harmonic forces acting between work piece and cutting tool. To analysis the dynamic response of the cutting tool under heavy load, 5N force in 3 directions is applied to cutting tool. Frequency sweep range is set to 25-2100Hzwith 25 Hz intervals, it covers the main frequency range of the structure. Figure5.(a)-(c) show the harmonic response of relative displacement in 3 directions respectively.
The maximum dynamic response displacement of about 2.8μm(y direction) occurs at 350Hz, which corresponds to a dynamic loop stiffness of 1.78N/μm. In the actual operation frequency range(1200-2000 Hz), the dynamic displacement in 3 directions is much less than 0.5μm, and the dynamic loop stiffness is higher than 10N/μm. It can be considered the machine occurs few reasonce when machining.

Conclusion
A series of simulations were performed on the miniaturized ultra-precision micro grinding machine tool, including modal analysis and harmonic analyses. The simulations based on FEM are used as a powerful and effective tool for supporting the full design process in an iterative manner, which also enable the machine design to be optimized efficiently and effectively.
1) The first 60 mode were extracted, the excited effective mass in 3 directions are obtained, and the main vibration shape in x, y,z directions were found. 2) The cutting tool was applied with force to simulate the harmonic response of displacement, and the response curve was obtained. The max displacement is 2.38μm which occurs at 350Hz.In the actual operation frequency range, the machine tool is hardly to occur reasonce. This work contributes to the development of a vertical miniaturized ultra-precision micro grinding machine tool. Simulation method is mainly used in this work. Therefore, the physical experiments are needed to verify the results after building the prototype machine.