1 Introduction

The demand for machining three-dimensional micro/nano-geometries has recently increased in a variety of industries. Although traditional mechanical precision machining can be used to fabricate miniature and micro components/products, but it still remains a big issue in the predictability, producibility and productivity. Contrast with the traditional big precision machine, miniaturized machine has many distinctive merits, such as well matchable machine size with the machined parts, low consumption of energy, space-saving, assembling and transporting conveniently, etc. So the development of miniaturized machine tool has been concerned deeply and widely. Many attention has been paid on the research of the miniaturized machine design and micro machining, thereof [1–5]. However, the design of miniaturized machine now is still based on the principle of the traditional machine. Some of machine layouts can only meet the basic requirements for the micro-processing equipment. But it lacks of in-depth research work in some aspects such as the profile design of the machine structure and the optimization of the key components in the machine, etc. Therefore, this paper presents a dynamics design optimization and simulation approach using for design of a miniaturized machine tool for micro-milling. Preliminary machining trials have been carried out, which makes a further proof and validates that the dynamics design and optimization method used in this research can assure the high performance of the developed miniaturized machine tool for micro-machining.

2 Machine tool structure design and optimization

Regarding machine tools, the structure is one of the critical factors to maintain the machining speed, precision, and productivity. Moreover, it is critical since the mechanical structure not only provides the support and accommodation for all the machine’s components but also contributes to dynamics performance [6, 7]. Therefore, to design a suitable machine tool structure with high static and dynamic features is very essential.

In this paper, two layouts of the machine tool: column structure (CS) and gantry structure (GS) are proposed in conceptual and fundamental design stage as shown in Fig. 1. Type CS1 has two symmetrical triangular reinforcing ribs on both sides; Type CS2 has two symmetrical L-shaped ribs on both sides; Type CS3 has one triangular reinforcing rib on the middle side; Type GS is a typical gantry structure. All of the configurations have the same mass.

Fig. 1
figure 1

Four configurations of the miniaturized machine tool. a CS1; b CS2; c CS3; d GS

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Natural frequency is an evaluation index of structural dynamic stiffness [8]. Therefore, the natural frequencies of the four structures are calculated, and the vibration modes are also obtained. Figure 2a–d) show the first mode of the four configurations, it can be noted that all of the modes are the pitch mode in the processing sensitive direction. The four lowest natural frequencies of those four structures are listed in Fig. 3. The third configuration has a significant advantage in the first 3 order natural frequency. Therefore, this structure is selected as the final structure.

Fig. 2
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The first modes of the four configurations. a CS1; b CS2; c CS3; d GS

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Fig. 3
figure 3

The four lowest natural frequencies of the four configurations

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The robust structure type is presented from the above analysis. In order to further improve the dynamic performance of the machine structure, the parameter optimization is carried out. The structure parameters A, B, C, D, E and F in Fig. 4 are chosen as the design variables, the mass of the machine tool structure is the state variable, the first order natural frequency has the main influence on the machine tool dynamics, therefore, the first order natural frequency is selected as the optimization goal. The objective function (F ob ) is simply defined as follows:

$$ F_{ob} = {\hbox{max}}\,{f\left( {A,B,C,D,E,F} \right)}$$
(1)
Fig. 4
figure 4

Structural optimization parameters of the miniaturized machine tool

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The effective design ranges of the design variables are shown as follows:

$$ \left\{ \begin{gathered} 20\,{\text{mm}} \le A \le 50\,{\text{mm}} \hfill \\ 100\,{\text{mm}} \le B \le 150\,{\text{mm}} \hfill \\ 250\,{\text{mm}} \le C \le 350\,{\text{mm}} \hfill \\ 150\,{\text{mm}} \le D \le 300\,{\text{mm}} \hfill \\ 20\,{\text{mm}} \le E \le 150\,{\text{mm}} \hfill \\ 10\,{\text{mm}} \le F \le 300\,{\text{mm}} \hfill \\ \end{gathered} \right. $$
(2)

The structure parameters of the initial and optimized value are shown in Table 1, from which it can be noted that the thickness and the height of the structure are decreased and the width of the rid is increased with respect to the initial value. After the optimization the first natural frequency of the machine structure improved from 800 Hz to 1246 Hz, which enhanced the dynamic performance of the machine tool greatly.

Table 1 Optimized results of the structural parameters
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After the machine tool structure optimized, the machine tool is designed with determined parameters as shown in Fig. 5. The movements of the workpiece in x and y direction are achieved by two stage stacking together, the movement in z direction is achieved by a stage carried with the micro-diameter cutter. All of the rolling guides are driven by the linear motor and formed the closed-loop control with the grating detection feedback. The air spindle driven by the air turbine can rotate at revolution of 160000 rpm under air pressure of 0.5 MPa. The profile size of the machine tool is 300 mm × 300 mm × 290 mm.

Fig. 5
figure 5

Geometrical model of the miniaturized machine tool

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In order to predict the dynamic performance of the machine tool, a corresponding finite element (FE) model is proposed in terms of its typical characteristics and the determined parameters of the miniaturized machine tool. As shown in the meshing models of Fig. 6, the rolling bearing in the guide and the air bearing in the spindle are substituted by the spring elements.

Fig. 6
figure 6

FE meshing models of the machine tool and the key parts

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3 Dynamic analysis of the miniaturized machine tool

3.1 Modal analysis and test of the miniaturized machine tool

The whole machine tool has the same first mode with the machine structure as shown in Fig. 7a, but the natural frequency is decreased from 1246 to 600 Hz. The natural frequency of the machine tool mainly affect by the machine structure, which confirmed the necessity of optimizing for the machine structure above.

Fig. 7
figure 7

The modes of the whole machine tool. a The first mode of the machine tool; b the first mode of the guide

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The first mode of the guide is the swing around the rolling balls as shown in Fig. 7b. In order to ensure the reading consistency of the grating sensing element, the installation position of the grating was analyzed and verified. From the simulation, it was shown that the grating should be laid parallel to the guide and the same level of ball’s location, where the minimized deformation on the grating can be obtained.

In order to verify the accuracy of the simulation model and obtain the dynamic performance of the machine tool accurately, a series of modal tests are carried out. Figure 8a shows the setup of the system for measuring dynamic characteristics. It consists of an impact hammer, an acceleration pick-up, a charge amplifier and a data acquisition instrument. The test result is shown in Fig. 8b. The tested first order frequency for the whole machine tool is 592 Hz, the result is consistent well with the simulation result 600 Hz, and verified the reliability of the simulation results. It provides a foundation for the further analysis of chatter vibration stability.

Fig. 8
figure 8

The modal tests and results of the machine tool. a Set-up for modal test; b the response of the machine tool

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The miniaturized machine tool has good dynamic performance up to 592 Hz, but weak static stiffness only 8 N/μm, which occurred in the air spindle. The main factor limited the machining accuracy and efficiency of the miniaturized machine tool is the static stiffness that may lead to poor performance of the machine tool.

3.2 The chatter vibration stability prediction

There are many attempts for finding the optimal values of cutting parameters consider different objective functions, including production-time minimization [9], production-cost minimization [10] and a combination thereof [11, 12]. However, the limiting factor for most optimization methods is the instability involved in milling operations [13]. Therefore, the prediction of the milling chatter vibration stability is carried out. The chatter vibration stability law can be described as Eq. (3).

$$ \begin{aligned} b_{\lim } & = \frac{ - 1}{{2K_{s} \cos \left( \beta \right)\text{Re} \left[ \it{FRF} \right]N_{t}^{*} }} \\ \frac{{f_{c} }}{\Upomega } & = K + \frac{\varepsilon }{2\pi } \\ \varepsilon & = 2\pi - 2\tan^{ - 1} \left( {\frac{{\text{Re} \left[ \it{FRF} \right]}}{{\text{Im} \left[ \it{FRF} \right]}}} \right) \\ \end{aligned} $$
(3)

where, b lim is the limiting chip width to avoid chatter, N* is the average number of teeth in the cut, K s is the cutting coefficient, Re[FRF] and Im[FRF] are real and imaginary values of an eigenvalue respectively. f c is the chatter frequency (should it occur), \( \Upomega \) is the spindle rotation speed in rev/s, K is the integer number of waves of vibration imprinted on the workpiece surface in one revolution, and \( \frac{\varepsilon }{2\pi } \) is any additional fraction of a wave, where \( \varepsilon \) is the phase (in rad) between current and previous tool vibrations. Fig. 9 shows the stability lobe diagram when N = 0–3. In this figure, it can be observed limiting chip width under 160,000 rpm is 1.2 mm.

Fig. 9
figure 9

The stability lobe diagram

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4 Machining trials on the miniaturized machine tool

The micro-milling machine was utilized to machine micro components and products with features at micro-levels on engineering materials, PMMA and HPb63-3. The samples shown in Fig. 10 were machined with tungsten carbide micro-diameter cutter. The typical achieved micro-structure machined with complex surface structure and high processing quality that the feature size was in the micron level. The machined specimen from the experiments showed that the designed machine tool presented super dynamic performance. Further, the machining trials also validated that the dynamics design and optimization method used in this article could assure the high performance of the developed miniaturized machine tool for micro-machining.

Fig. 10
figure 10

The samples machined with micro-cutting tools. a PMMA human face; b brass (HPb63-3) logos

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5 Conclusion

In this paper, a 3-axis miniaturized machine tool for micro-milling was presented in terms of dynamics design and structural optimization. The following conclusions can be drawn.

  1. 1.

    Four typical structures were proposed and contrasted in conceptual and fundamental design stage. It was shown that the column structure with one triangular reinforcing rib on the middle side type had the best dynamic performance. The parameter optimization was carried on this structure type. The optimal results improved the first natural frequency from 800 to 1246 Hz, enhancing the dynamic performance of the machine tool greatly.

  2. 2.

    The influence of dynamic performance of the machine tool on the measuring errors was analyzed. According to the analysis and test, it was denoted that laying the grating parallel to the guide and the same level to the ball’s location could ensure the reading consistency of the grating sensing element and improve the detection accuracy.

  3. 3.

    These experimental tests evaluate and validate the proposed methodology and approach was effective and efficient in optimizing the design of miniaturized machine tools leading to improved operational performance.

  4. 4.

    The simulations based on the approach were used as a powerful tool for supporting the full design process from dynamic points of view, which also enabled the machine design to be optimized efficiently and effectively. The results from this paper provide a benchmark and systematic methodology for the future design of miniaturized machine tool for micro-milling.