1 Introduction

Owing to the rapid development of the modern manufacturing process, micromilling has received considerable research attention within the fields of microelectromechanical systems (MEMS), biomedical engineering, aerospace, and other industries due to its high machining efficiency, fabrication accuracy, and low processing cost [1]. Recent studies have demonstrated that applying external energy, such as high-frequency vibration, to the machining process reduces the cutting force, improves machining precision, and suppresses surface damage [2]. This method is referred to as vibration-assisted machining (VAM). The periodic separation between a tool and workpiece can be achieved, which can reduce the cutting heat and remove burs, resulting in smooth surface generation and low temperature [3].

VAM, originally designed to enhance the machinability of difficult-to-machine materials, has recently demonstrated its capability to generate surface textures [4]. Engineered textured surfaces feature a regular texture structure, which allows the component surface to possess superior attributes, such as reduced adhesion friction, improved lubricity, increased wear resistance, varied hydrophilic performance, and enhanced optical properties [5]. MEMS technology, energy beam processing technology, electrical discharge machining, and ultraprecision machining technologies can be used in microsurface texture manufacturing. However, these technologies are limited in their applicability, as they require long fabrication periods for large surface areas and certain materials. VAM has gained extensive attention due to its low cost, high efficiency, and environmental protection [6].

VAM can generally be implemented in two ways: applying vibration to the tool and the workpiece. The milling cutter typically revolves at an extremely high speed during the milling process; thus, the second method of delivering vibration to the workpiece is generally available for the subsequent design. Constructing a flexible stage is necessary to fix the workpiece to implement the aforementioned method. A piezoelectric actuator (PZT) is frequently utilized as a high-frequency excitation source to facilitate the flexible stage during the VAM process. Moreover, considering the PZT operating frequency, VAM can be classified as resonance or non-resonance VAM [7]. Resonance and non-resonance VAM both have advantages and limitations [8]. The resonance VAM can achieve efficient vibration with a large motion displacement. However, its vibration frequency is also fixed around the resonant frequency. The non-resonance VAM can achieve a flexible section of the vibration frequency [9]. However, low vibration efficiency and motion displacement restrict this approach in certain fields [10].

VAM can be commonly classified into three types corresponding to the direction of vibration: 1D, 2D, and 3D VAM. In a 1D VAM, vibration is delivered in either the feed or crossfeed direction, facilitating the movement of the workpiece in a single direction. In 2D VAM, vibration co-occurs in the feed and crossfeed directions, causing an elliptical motion of the workpiece in a plane. Furthermore, in a 3D VAM, the vibration is applied perpendicular to the machining plane, leading to a space ellipse trajectory in the cutting process. Correspondingly, the vibration stages can be classified into 1D, 2D, and 3D stages. 1D VAM can be realized in the 2D and 3D vibration stages; however, severe coupling errors will occur, considerably impacting movement accuracy.

Zhu et al. [11] presented a novel 2D vibration-assisted-compliant cutting system for conventional machines to generate textured surfaces with uniform and accurate topography. However, the method used in the cutting process is highly suitable for turning. Zheng et al. [12] suggested a new 2D PZT-driven vibration stage with a double-parallel four-bar linkage construction with double-layer flexible hinges to guide the motion and reduce displacement coupling between the two directions. However, the accuracy of the vibration stage may be impacted due to the vast size between the PZT and the stage. Börner et al. [13] proposed a 2D resonance VAM system in which the vibrator was designed in the shape of a cross, and the PZT was used to provide high-frequency vibration to the cross converter. The resonance frequency of the vibrator may be altered by enormous mass specimens. Ding et al. [14] described a 2D non-resonant vibration stage in which two pairs of PZTs were symmetrically distributed in flexible hinges around the mechanical oval structure, and a circuit board was integrated into the vibration stage to improve control accuracy. However, coupling error does exist and affects the motion accuracy due to the structural design.

Extensive research efforts have been devoted to designing the vibration stage. The authors believe that most vibration stages for VAM are integrated; therefore, only a single vibration type and performance can be supplied. Consequently, achieving efficient vibration for different requirements is considerably difficult. The separable design proposed in this paper could easily solve this problem. Furthermore, the worn stage could be replaced conveniently, which could reduce production costs and improve productivity.

In this study, the VAM platform is designed to be separable, which implies that the central stage can be replaced with different types, such as 1D and 2D stages with distinct sizes. Compared to the 2D vibration stage, the 1D stage possesses a large stiffness in the cross-vibration direction, which represents a small coupling error in the one-direction vibration application. Flexible applications can be obtained under various situations by changing stages with different types and vibration characteristics. In many cases, due to the limitations in the power and frequency of stacked piezoelectric ceramics, driving the flexible stage at the resonance frequency of the vibration stage is typically required to achieve efficient and reinforced vibrations.

This paper aims to provide a vibration-assisted platform with a separable design and high precision. Different types and dimensions of the stages could be used by applying the separable design approach to satisfy distinct operating requirements. A hybrid decoupling technique is proposed based on a circular-fillet hinge and parallel structure, which can simultaneously achieve high accuracy and output decoupling. The remainder of this article is structured as follows. Section 2 describes the mechanical design and operation of the suggested vibration platform. Section 3 discusses finite element simulations and experimental tests. Finally, Sect. 4 presents the findings.

2 Mechanical Configuration

2.1 Structural Design

The basic structure of the proposed 2-DOF vibration-assisted platform, primarily comprising a pair of PZTs, a changeable flexible stage, and a support base, is illustrated in Fig. 1. The four corners of the flexible stage are positioned close to the support base to improve the accuracy of the assembly. The changeable flexible stage can be commonly divided into 1D and 2D stages, and their schematics are illustrated in Fig. 2. The main factors influencing the motion accuracy of a flexible stage include the coupling error, response hysteresis, and vibration. The coupling error can be defined as the coupling displacement in directions other than the one corresponding to stage vibration, which considerably influences the vibration process [2]. The entire structure is symmetrically designed, and a two-layer mechanism is utilized in the two vibration directions to achieve two decoupling directions. The compression stiffness of the double-parallel flexure hinge is substantially higher than its rotational stiffness. Thus, a novel double-parallel flexure hinge is used as the outside framework of the stage to further reduce the coupling effect. The inner layer mechanism uses a single beam hinge to extend the vibration range of the mechanism. In addition, circular-fillet hinges are employed throughout the construction to increase the motion precision and reduce the stress concentration.

Fig. 1
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Structure of the proposed separable vibration-assisted platform

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Fig. 2
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Schematic of the a 1D and b 2D flexible stages

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The necessary driving force in the VAM is relatively large in the current study. Therefore, stacked piezoelectric ceramics with excellent driving force performance were used in the experiment. The stacked piezoelectric ceramics have a limitation in the lower lateral torque bearing; thus, a PZT with a hemisphere block was selected to ensure protection from undesired shearing stress. The preloading force in the two directions must be identical; thus, cylindrical-encapsulated ceramics with a built-in preload were chosen. Meanwhile, the adjusting modular is designed and positioned between the bolt and PZT to restrict the vibration of PZT primarily in the horizontal direction. Figure 3 shows the cylindrical-encapsulated PZT with its adjustment mechanism, including the adjusting modular and bolt.

Fig. 3
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Cylindrical-encapsulated PZT with its adjustment mechanism

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In the milling process, the resonant frequency and compliance are the most critical parameters because they are directly responsible for the vibration-assisted performance of the flexible stage. If the spindle speed is constant, the experimental frequency determines the amount of vibration assistance in one revolution. When the stage resonant frequency is used as the experimental frequency to improve the driving efficiency of piezoelectric ceramics, a high resonant frequency also implies increased vibration assistance in one milling revolution, which can improve the milling performance. Furthermore, the resonant frequency plays a crucial role in engineered textured surface machining. Simultaneously, the vibration displacement in the vibration-assisted process was determined by compliance. If the driving force remains constant, then the high compliance indicates a highly substantial displacement. Minor vibration-assisted displacements will lead to an unclear experimental outcome. Other parameters such as mass, elastic modulus, and density, also affect the performance of the flexible stage; thus, their impacts are reflected in the resonant frequency and compliance. Therefore, these two factors must receive increased consideration during the flexible stage design.

2.2 Kinematic Analysis

The theoretical analysis of the vibration stage is of considerable importance to the analysis of static and dynamic characteristics. Multiple challenges can be addressed via theoretical analyses, such as calculating the compliance and natural frequency and determining the specific requirements for the PZT stiffness, output force, and stroke range. Several modeling methods, such as pseudorigid body, Castigliano’s second theorem, and finite element methods (FEMs), have been developed for flexible mechanism modeling. However, these methods have their applications and limitations. A matrix-based compliance modeling method built using the linear Hooke’s law for the material was recently developed. In the modeling process, the flexure hinge in the stage was uniformly divided into N pieces, with each piece being treated as a micro Euler–Bernoulli beam. High calculation accuracy and efficiency can easily be obtained by employing matrix operations. Thus, these operations are highly suitable for modeling spatial-compliant mechanisms with complex structures. Therefore, the matrix-based method was selected for modeling the proposed vibration stage. The compliance at the center point of the 2D flexible stage was the same in both directions due to the extreme symmetry design. Therefore, the stage can only be suitably simplified in one direction for computing the compliance of the entire mechanism.

Figure 4a shows a simplified static analysis model of the stage in one direction, and Fig. 4b shows the element of the outer-hinge structure of the simplified model.

Fig. 4
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a Static analysis model of the stage. b Element of the outer-hinge structure of the simplified model. c Transformation matrix. d Dimension and force analyses of the circular-fillet hinge

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The two parallel circular-fillet hinges can be defined as a1 and a2. Based on the matrix method, the outer compliance relative to the global coordinate \({\varvec{C}}_{{{P}}_{{A}}}\) in Fig. 4b can be expressed as:

$${\varvec{C}}_{{P_{A} }} = \left( {{\varvec{k}}_{{P_{A} }} } \right)^{ - 1} = \left[ {({\varvec{C}}_{{a_{1} }}^{o} )^{ - 1} + ({\varvec{C}}_{{a_{2} }}^{o} )^{ - 1} } \right]^{ - 1}$$
(1)

where \({\varvec{k}}_{{P_{A} }}\) is the outer stiffness relative to the global coordinates, and \({\varvec{C}}_{{a_{1} }}^{o} ,{\varvec{C}}_{{a_{2} }}^{o}\) are the global compliance of hinges a1 and a2 relative to the stage center o-point, respectively. The local compliance \({\varvec{C}}_{i}^{r}\) must be transferred from the local coordinate system Or − xryrzr to the global coordinate system Oe − xeyeze, as shown in Fig. 4c, to calculate the outer compliance \({\varvec{C}}_{r}^{e}\). The calculation formula is as follows:

$${\varvec{C}}_{r}^{e} = {\varvec{T}}_{r}^{e} {\varvec{C}}_{i}^{r} \left( {{\varvec{T}}_{r}^{e} } \right)^{\text{T}}$$
(2)

where \({\varvec{T}}_{r}^{e}\) is the coordinate transformation matrix and can be obtained according to Ref. [12].

Figure 4d shows the dimension and force analyses of the circular-fillet hinge. The hinge is assumed to be subjected to the force vector:

$${\varvec{F}} = [F_{x} ,F_{y} ,F_{z} ,M_{x} ,M_{y} ,M_{z} ]^{\text {T}} ,$$
(3)

where Fx, Fy, Fz and Mx, My, Mz are the forces and torques along the x-, y-, and z-axes, respectively. The corresponding deformation was defined as:

$${\varvec{X}} = [\Delta_{x} ,\Delta_{y} ,\Delta_{z} ,\alpha_{x} ,\alpha_{y} ,\alpha_{z} ]$$
(4)

where Δx, Δy, Δz and αx, αy, αz represent the displacements and torques along the direction of the x-, y-, and z-axes, respectively.

Considering the complex shear and torsion effects, the corresponding flexibility matrix Ch of each hinge is established to describe the yield strength. Therefore, the following relationship can be expressed as:

$${\varvec{X}} = {\varvec{C}}_{h} {\varvec{F}}$$
(5)

Then, Ch could be calculated by:

$${\varvec{C}}_{h} = \left\{ \begin{array}{*{20}c} \frac{{\Delta_{x} }}{{F_{x} }} & 0 &0 & 0& 0& 0 \\ 0& \frac{{\Delta_{y} }}{{F_{y} }} & 0 & 0 & 0 & \frac{{\Delta_{y} }}{{M_{z} }} \\ 0 & 0 & \frac{{\Delta_{z} }}{{F_{z} }} & 0 & \frac{{\Delta_{z} }}{{M_{y} }} & 0 \\ 0 & 0& 0 & \frac{{\alpha_{x} }}{{M_{x} }} & 0 & 0 \\ 0 & 0& \frac{{\alpha_{y} }}{{F_{z} }} & 0 & \frac{{\alpha_{y} }}{{M_{y} }} &0 \\ 0 & \frac{{\alpha_{z} }}{{F_{y} }} & 0 & 0 & 0 & \frac{{\alpha_{z} }}{{M_{z} }} \end{array} \right\}$$
(6)

The compliance of the flexure hinge can be obtained in accordance with Ref. [15, 16]. The flexible stage is constructed using 7075 aluminum alloy, and its mechanical properties are outlined in Table 1.

Table 1 Mechanical properties of 7075 aluminum
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The semi-inner structure contains only one flexure hinge due to the symmetrical design. The compliance of this structure corresponding to the local coordinate \({\varvec{C}}_{{o}}^{\text{in}}\) is equal to its compliance \({\varvec{C}}_{{{m}}_{1}}^{{o}}\) corresponding to the global coordinate system Oe − xeyeze. The compliance matrix of the vibration stage in one direction C can generally be expressed as:

$${\varvec{C}} = [({\varvec{C}}_{{P_{A} }} /2)^{ - 1} + ({\varvec{C}}_{{m_{1} }}^{o} /2)^{ - 1} + ({\varvec{C}}_{{P_{A} }} /2)^{ - 1} ]^{ - 1}$$
(7)

The dynamic analyses of the stage in each direction are identical due to the completely symmetrical design. Therefore, the unidirectional dynamics analysis discussion is sufficient. The equivalent stiffness of the flexure hinge in the X and Y directions is expressed in this model by kix and kiy, respectively (i is the label of the flexure hinge; i = a, b, c,…, m). Suppose that the displacement under the action of force F = [Fx, 0, 0]T is s = [x, 0, 0]T; the corresponding kinetic energy Tk and potential energy Uk can be respectively expressed as:

$$\begin{array}{*{20}l} {T_{k} = \frac{1}{2}\dot{x}^{2} M_{1} + 4\dot{x}^{2} M_{2} + 4I_{z} \theta_{z}^{2} } \hfill \\ {U_{k} = \frac{{x^{2} }}{2C}} \hfill \\ \end{array} ,$$
(8)

where M1 and M2 are equivalent masses of the hinge and center stage, respectively, and Iz and θz are the inertia moment and corresponding rotation angle of the flexure hinge around the Z-axis, respectively. By substituting the kinetic and potential energy into the Lagrange equation, the dynamic equation of the vibration stage can be obtained as follows:

$$\frac{{\text{d}}}{{{\text{d}}t}}\left( {\frac{{\partial T_{k} }}{{\partial \dot{x}}}} \right) - \frac{{\partial T_{k} }}{{\partial \dot{x}}} + \frac{{\partial U_{k} }}{\partial x} = F_x$$
(9)

The dynamic equation describing the free motion of the stage can generally be expressed as follows:

$$\left( {M_{1} + 8M_{2} + \frac{{8I_{z} }}{{l^{2} }}} \right)\ddot{x} + \frac{x}{C} = F_x$$
(10)

where l is the hinge length.

The natural frequency of the vibration stage can then be obtained by:

$$\omega = \sqrt {\frac{1}{{\left( {M_{2} + 8M_{1} + \frac{{8I_{z} }}{{l^{2} }}} \right)C}}}$$
(11)

Finally, the theoretical compliance and the first natural frequency of the vibration stage can be calculated by Eqs. (7) and (11), respectively.

3 FEM Simulations and Experimental Tests

3.1 FEM Simulation

The FEM simulation of the vibration stage was conducted using the ANSYS Workbench 19.2. The meshing of the vibration stage is based on the hexahedron-dominated method, while the mesh refinement method was applied at the hinge connection. Figure 5 presents the FEM model setup of the vibration stage; the key parameters (as presented in Fig. 2) are given in Table 2. Considering the mode of the vibration stage in the prestressed state, the natural frequency and compliance of the vibration stage were obtained by constraining the four corner blocks. Figure 6 shows the first two resonance frequencies of the stage, which correspond to 3697.9 and 3701.3 Hz. The first- and second-order natural frequency shapes represent the translational motion of the vibration stage along the y- and x-axes, respectively.

Fig. 5
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Setup of the FEM model

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Table 2 Key parameters of the structure
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Fig. 6
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First two resonance frequencies of the flexible stage: a first-order model and b second-order model

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Figure 7a shows that the axial displacement of the stage is 0.06373 µm for the case of the axial cutting force on the worktable at 1 N. Therefore, the axial stiffness and compliance of the stage are 20.86 N/µm and 0.0479 µm/N, respectively, which satisfies the actual machining requirements. Figure 7b shows the stress distribution in the vibration stage. The findings reveal that the maximum stress value is 192.28 MPa, which is substantially lower than the yield strength of the material (455 MPa).

Fig. 7
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Static simulation: a compliance simulation and b stress simulation

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Meanwhile, the theoretical calculation was also conducted with the key parameters shown in Table 2. The theoretical results were compared with that of the FEM simulation, as shown in Table 3.

Table 3 Comparisons of the static and dynamic results corresponding to the flexible stage
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The difference between the analytical model and FEM is within 10%, which indicates the excellent accuracy of the analytical model. Reasonably, the first natural frequency of FEM is slightly lower than that obtained from the theoretical results. The FEM considers every stage as flexible elements, while the theoretical method only considers the compliances of the flexible hinges in the vibration direction. Moreover, the compliances of the hinges in cross-vibration, as well as structure M1 shown in Fig. 4a, are completely ignored.

3.2 Experimental Tests

Figure 8a shows that the VAM stage is fixed on a precision three-axis micromilling machine. The support base comprised AISI 1045 steel to provide as much stiffness as possible. The 1D and 2D vibration stages, as respectively shown in Fig. 8b and c, were manufactured by utilizing an aluminum alloy 7075-T651. Ensuring a high natural frequency also provides considerable compliance. The two cylindrical-encapsulated PZTs are from CoreMorrow (PSt150/10/20 VS15 model). The driving signals were generated by the Agilent 33500B signal generator and further amplified by the piezoelectric control system (CoreMorrow’s E01.A2). A laser Doppler vibrometer was then fixed on the other side to monitor the real-time output displacement of the vibration stage. An NI 9221 input DAQ card was selected to collect the stage displacement data from the laser Doppler vibrometer, and the data were sent to the LabVIEW software for analysis.

Fig. 8
figure 8

a Experimental setup: b 1D vibration stage, and c 2D vibration stage

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A series of test experiments were designed and conducted to verify the performance and efficiency of the separable VAM system. By exerting a sinusoidal sweep voltage signal of 15 V to the PZT at an interval of 100 Hz over the frequency range of 0–7000 Hz, the frequency amplitude results of the 2D flexible stage can be obtained as shown in Fig. 9. The frequency amplitude result shows that the natural frequency of the 2D stages is approximately 3700 Hz, which verified the simulation results.

Fig. 9
figure 9

Frequency amplitude results of the 2D flexible stage

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Moreover, the coupling displacement in one direction was tested with the application of vibration in the other direction to compare the coupling error between the 1D and 2D stages. The measuring range of the laser vibrometer was set to 100 µm to ensure that the motion displacement of the guideway and the vibration displacement of the platform can be tested simultaneously. Figure 10a–d shows the stage displacement and coupling motion in the experimental process corresponding to the 1D and 2D stages, respectively. The test results are summarized as shown in Table 4, which presents the substantial low-coupling error performance of the 1D stage. Notably, the 1D vibration stage exhibits considerably large stiffness in the cross-vibration direction. Contrastingly, Zheng [12] used the PID (Proportional, Integral, Differential) control method to reduce the coupling displacement of the 2D stage from 0.25 to 0.15 μm, while the coupling error was controlled to within 2.6%, which is also higher than the 1D stage proposed in this paper. Meanwhile, the test results shown in Fig. 10 reveal the remarkable performance of the 1D and 2D stage during the application of large displacement. The motion displacements of the guideway and vibration stage do not impact each other, leading to the remarkable precision of composition displacement.

Fig. 10
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Test results of 1D stage: a stage movement, b coupling motion; 2D stage: c stage movement, d coupling motion

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Table 4 Comparisons of the coupling error between the 1D and 2D stages
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The VAM platform and test devices were arranged as shown in Fig. 11 to evaluate the coupling motion under the vibrations of two directions. Sinusoidal signals of 15 and 7.5 V were sent to the feed and crossfeed directions, respectively. Figure 12 shows the vibration displacement of the stage with 0°, 90°, and 180° vibration phase differences. When the phase difference is 90°, an elliptical trajectory can be obtained by combining the vibration displacements of two directions, as shown in Fig. 12b. The total vibration trajectory errors (including the inherent noise of the capacitive sensors) are 0.21 μm, signifying that the designed vibration-assisted system possesses precise movement capability.

Fig. 11
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Vibration motion trajectory test mechanism

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Fig. 12
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Trajectory of the vibration stage with a phase difference of 0°, b phase difference of 90°, and c phase difference of 180°

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4 Conclusions

A novel separable vibration-assisted platform was designed, analyzed, and tested for micromilling in the current study. A separable design can be realized on the basis of the detachable structure. 1D and 2D flexible stages were designed and utilized in the experiment, which adopted a two-layer mechanism with double-parallel fillet hinges to achieve high motion accuracy and reduce the coupling effect. The static and dynamic characteristics considering the designed flexible stage were obtained using theoretical analysis, FEM, and experimental tests, which formed a contrast verification. The coupling error of the 1D and 2D stages was tested and compared, and the results present a substantial low-coupling error performance of the 1D stage. A motion trajectory test was performed to evaluate the vibration precision of the stage, and an elliptical trajectory with a maximum error of 0.21 μm was obtained. The findings denote that the designed vibration-assisted system possesses precise movement capabilities.