Nanomanufacturing and Metrology

, Volume 2, Issue 4, pp 199–214 | Cite as

An Analytical Model for Determining the Shear Angle in 1D Vibration-Assisted Micro Machining

  • Shamsul Arefin
  • XinQuan ZhangEmail author
  • Senthil Kumar Anantharajan
  • Kui Liu
  • Dennis Wee Keong Neo
Original Articles


In the metal cutting process, an increased shear angle generally leads to lower cutting forces and improved machining performance. It has been reported earlier that 2D vibration-assisted machining (VAM) increases the equivalent shear angle and decreases the chip thickness via an elliptical pulling motion of the tool rake face moving away from the deformed chip. Researchers have conducted several experimental and finite element studies and revealed that an increase of shear angle in 1D VAM as well and this process which started much earlier than 2D VAM is widely used in the machining industries. However, no systematic analysis of chip deformation in 1D VAM has so far been carried out regarding the equivalent shear angle. In this research study, several cutting tests have been conducted initially at a very low frequency (0.5 Hz) and the deformation of chip was found smaller compared to conventional machining. Based on the force analysis, an energy-based analytical model has been derived to evaluate the equivalent shear angle. The model was then verified at ultrasonic 1D VAM against cutting speed ratios. Four different metals have been tested including stainless steel, brass, annealed tool steel, and titanium. The results show that a lower speed ratio in ultrasonic 1D VAM will lead to a decreased chip thickness and as such decreased chip compression ratio. The developed model has been found to be able to well predict the equivalent shear angle for ultrasonic 1D VAM of several types of metal.


Vibration-assisted machining Shear angle Chip compression ratio Ultrasonic vibration Speed ratio 

List of Symbols


Amplitude of vibration along the tangential direction


Frequency of vibration


Frequency of angular vibration


Nominal cutting speed


Maximum tool speed during vibration


Speed ratio


Depth of cut


Formed chip thickness


Undeformed chip area


Contact area at tool flank


Shear stress


Shear strain


Width of cut


Tool flank angle


Friction angle


Instantaneous effective rake angle


Tool negative rake angle


Cutting edge radius


Shear angle


Equivalent shear angle in 1D VAM


Instantaneous effective shear angle in CM


Phase of vibration


Coefficient of friction between the tool rake and the chip


Stress of material beneath the tool flank


Elastic spring-back height of freshly machined workpiece surface


Material hardness


Elastic modulus


Thrust force


Cutting force


Elastic spring force


Elastic deformation force


Plastic deformation force

\(\overline{{F_{{{\text{c}} - {\text{s}}}} }}\)

Average value of elastic spring force

\(\overline{{F_{{{\text{c}} - {\text{e}}}} }}\)

Average value of elastic deformation force


Cutting force in conventional machining


Tool–workpiece engagement step/Tool–workpiece disengagement step


Elastic deformation step


Plastic deformation step


Elastic recovery step

ks, ke, kd

Constants derived from low-frequency 1D VAM

k1, k2

Scaling constant


Volume of material removed per second


Specific energy consumed in 1D VAM


Specific energy consumed in the tool–workpiece engagement step


Specific energy consumed in elastic deformation step


Specific energy consumed in plastic deformation step


Specific energy consumed in CM

1 Introduction

Vibration-assisted machining (VAM) generally applies vibration with specified frequency to the tool or workpiece to create intermittent tool–workpiece motion. In terms of the vibration phase and direction, there are two different types of vibration-assisted machining processes, namely conventional (1D VAM) and elliptical (2D VAM). In the late 1950s, 1D VAM was introduced for traditional macro-scale metal cutting applications. 2D VAM was introduced in the 1990s, and Moriwaki and Shamoto [6] revealed that 2D VAM leads to smaller cutting force, thinner deformed chips, and lower tool wear rate as compared to 1D VAM. This is because the tool rake segregates away from the workpiece, leading to a full separation of the tool and the workpiece, and is known as the reversed frictional motion. The intermittent cutting process in the 1D and 2D VAM depends on the relative tool–workpiece motion in the given vibration and cutting conditions. Nath and Rahman [7] has demonstrated that the tool–workpiece gap during VAM is mainly affected by the speed ratio. There is a threshold value of speed ratio that transforms the intermittent VAM into a continuous cutting process.

The introduction of controlled vibration in the machining process generally lead to smaller machining forces, slower tool wear, reduced surface roughness, and improved machining dynamics. Ma et al. [5] has suggested that the average cutting and thrust force components are smaller using 1D VAM compared to conventional machining (CM) during the cutting of 304 stainless steel. The experimental investigations made by Nath et al. [8] show reduced flank wear in 1D VAM compared to conventional machining while cutting low alloy steel. Zhou et al. [15] reported about the reduced flank wear for 1D VAM of 304 stainless steel by PCD tools. Xiao et al. [12] has made experimental investigation and found reduced chatter suppression in VAM compared to conventional machining that ultimately improved the quality of the machined surface.

In the CM process, the chip is formed along a shear plane due to the compression and friction force applied by the tool on the workpiece and chip. Shear plane angle is one of the essential factors that denotes the metal cutting performance, and its value is affected by all of the cutting conditions, like workpiece material, tool material, cooling condition, tool–chip friction factor, tool edge radius, cutting speed, and so on. A higher shear angle is usually preferred in conventional metal cutting process, due to the resulting better cutting performance characterized by smaller cutting force and less energy consumption.

For the 2D VAM process, Shamoto and Moriwaki [10] have experimentally proven that 2D VAM can significantly increase the equivalent shear angle due to the reverse frictional direction. Later, Shamoto et al. [11] developed a thin shear plane model on 2D VAM to predict the shear angle at various speed ratios. For 1D VAM, as there is no pulling motion to cause the reversed friction, the shear plane model developed in 2D VAM may not apply. Although 1D VAM has been successfully implemented by the machining industry, there are few research studies available on the fundamental material deformation during vibration-assisted cutting as well as the equivalent shear plane angle. Moriwaki and Shamoto [6] reported on a reduction of chip thickness in 1D VAM of copper. Similarly, reduced chip thickness was reported by Zhou et al. [15] in 1D VAM of stainless steel. Xu et al. [13] also reported about chips with thin, smooth, and little distortion in 1D VAM of 304 austenitic stainless steel. Finite element (FE) simulations along with the experimental study of Inconel-718 conducted by Lotfi and Amini [4] showed smaller shear angle in 1D VAM compared to conventional machining. However, a systematic analysis of the chip deformation with a mathematical expression is necessary to substantiate the observed phenomenon.

The tool cuts the workpiece in a cyclic manner and only a part of the cycle is involved in the cutting for 1D VAM. The cutting force gradually increases over the yield limit to start the plastic deformation during each cycle of the tool–workpiece contact. Considering the tool–workpiece contact status and force, the material deformation condition varies significantly in a cutting cycle, and could result in non-conventional cutting energy distribution during the vibration cutting process. Based on the study of specific energy spent in ductile and brittle cutting, Zhang et al. [14] has calculated the transitional uncut chip thickness in the grooving of silicon. Until now, however, there is no predictive model available for the equivalent shear angle in 1D VAM.

In the machining process, shear angle is an important factor in evaluating the cutting mechanism. The shear angle is usually considered to be constant in conventional machining because the material removal process is continuous without interruption mainly from plastic deformation. In micro machining assisted with vibration, the instantaneous shear angle at different tool–workpiece relative position will vary due to the varying tool–workpiece contact status as well as the change of material deformation condition. The tool–workpiece contact status thus determines the energy consumption during cutting process.

In conventional micro machining, the energy consumption mainly happens in three zones, namely primary, secondary, and tertiary zones. The energies in the primary and secondary zones represent the plastic deformation along the shear plane against the tool–workpiece and tool–chip interfaces, respectively. In the tertiary zone, the energy results from the elastic deformation in beneath the tool flank. For every cycle of 1D VAM, the material removal takes place starting from an elastic deformation followed by plastic deformation. The details of the deformation steps can be explained by the proposed force analysis, which is crucial in evaluating the consumed energy during the 1D VAM process. Such unique intermittent cutting process requires a sophistical model to determine various specific cutting energies of removing certain volume of material in each vibration cycle.

With the energy consumption in the VAM cycle, this study aims to derive a theoretical model for evaluating the equivalent shear angle with respect to the variation of speed ratio in 1D VAM. A novel approach has been adopted to understand the elastic and plastic material deformation in the CM and low-frequency 1D VAM by investigating and analyzing the variation of force components. The analysis leads to an energy-based mathematical model to derive the equivalent shear angle in a vibration cutting cycle. Finally, ultrasonic orthogonal 1D VAM tests are conducted on stainless steel, and the thickness variation of formed chips at different speed ratios are evaluated to validate the theoretical model for predicting the equivalent shear angles. To evaluate the robustness of the predicted model, cutting experiments have been conducted on a variety of metals of both soft and hard types at different speed ratios to validate the model on them.

2 Shear Plane Theory in Metal Cutting

In conventional machining of metal or other ductile materials, chip is generated as a result of stress exerted by the tool on chip formation zone of the workpiece material. Figure 1 explains the mechanism of chip formation in an orthogonal cutting process with the force components involved in the chip generation. The resultant force applied by the tool on the workpiece is denoted by R, and it can be decomposed along the cutting direction into two components: the thrust force Ft and cutting force Fc. For zero nominal rake angle, \(\beta\) is used to denote the angle between these two forces (Ft and Fc). R can also be distributed along the tool rake and shear plane into the friction force Ff and the normal force N, and Fs and Fn respectively. \(\gamma\) is used to denote the shear strain. For a tool with 0° rake angle, the angle β turns out to be the friction angle.
Fig. 1

Typical chip generation and force relationship in conventional orthogonal cutting process

The formed chip usually has a larger thickness value (t1) than that of uncut chip (t0). Chip compression ratio (CCR) is used to denote the ratio between formed chip thickness and depth of cut (t1/t0), and it is directly related to the shear angle and can be an important parameter in determining the machining performance. CCR has been used as one of the most important parameters in the methodology proposed by Astakhov and Xiao [3] to determine the cutting force and energy. Their experimental results have shown the relevance of CCR in determining the machining performances. From Fig. 1, the value of CCR can be derived from the geometric relationship as follows:
$$\frac{{t_{1} }}{{t_{0} }} = \frac{{{ \sin }(90^{^\circ } - \varphi - \alpha )}}{\sin (\varphi )}$$
where φ and \(\alpha\) are shear plane angle and absolute value of the tool rake angle, respectively.
Instead of directly observing the shear angle during metal cutting, it is relatively easier to measure the CCR by observing the average value of formed chip thickness. Hence, to derive the shear angle of an orthogonal cutting process, a deliberate approach can be made to obtain CCR by machining a workpiece at the given depth of cuts and measuring the deformed chip thickness. If a zero rake cutting tool is utilized, φ can be derived as below:
$$\varphi = \tan^{ - 1} \left( {\frac{{t_{0} }}{{t_{1} }}} \right)$$
In spite of employing a zero rake tool, there is an instantaneous effective rake angle (ϒne) emerging in an ultraprecision machining process. For a tool with cutting edge radius r, ϒne and effective shear angle (φeff) produced from this instantaneous effective rake angle can be derived from Eqs. (3) and (4) as explained by Zhang et al. [14]:
$$\gamma_{\text{ne}} = \sin^{ - 1} \left( {\frac{{t_{0} }}{r} - 1} \right),{t_{0}} < \, r(1 - \sin \alpha )$$
$$\varphi_{\text{eff}} = \tan^{ - 1} \left( {\frac{{\frac{{t_{0} }}{{t_{1} }}\cos \gamma_{\text{ne}} }}{{1 - \frac{{t_{0} }}{{t_{1} }}\sin \gamma_{\text{ne}} }}} \right)$$

According to Fig. 1 and Eq. (4), an increased thickness of formed chip means a smaller shear angle, which also indicates higher machining forces and more energy required to deform the chips due to the increased shear length (OL). The generated chip thus provides information about the energy consumed and can be used to compare the machining performances.

2.1 Elastic and Plastic Deformation of 1D VAM

Zhang et al. [14] has reported that during a VAM cutting cycle, the workpiece material will undergo elastic deformation, plastic deformation, and elastic recovery for the chip formation process. Figure 2a presents a schematic view of a tool that oscillates along the cutting direction in the 1D VAM. ‘A–B’ represents elastic deformation before the workpiece material goes through the plastic deformation step ‘B–C’, ‘C–D’ represents elastic recovery when the tool segregates from the deformed chip, though there is presence of residual stress that remains in the workpiece material surrounding the primary shear zone.
Fig. 2

a Schematic 1D VAM process with elastic and plastic deformation of workpiece in one cycle with rake angle α. b Tool–workpiece position against vibration phase with zero nominal rake angle in 1D VAM for multiple cycles

Figure 2b schematically depicts the elastic–plastic–elastic regions and also the variation of relative tool position against the vibration phase in 1D VAM, assuming zero nominal rake angle. In each VAM cycle, xi denotes the furthest position where the tool starts reversing and segregates from the chip at ith cycle.

Intermittent contact between the workpiece and tool is an important merit for VAM as compared to the CM process. The amount of tool–workpiece contact in a vibration cycle depends on the nominal cutting speed. Nath and Rahman [7] quantitatively evaluated the effect of cutting speed in VAM, which is termed in another form as speed ratio:
$$R_{\text{s}} = {\raise0.7ex\hbox{$v_{{\text{c}}}$}\mathord{\left/ {\vphantom {v {\left( {v_{\text{t}} } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {v_{\text{t}} } \right)}$}}_{\hbox{max} } = {\raise0.7ex\hbox{$v_{{\text{c}}}$} \!\mathord{\left/ {\vphantom {v {2\pi af}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2\pi af}$}}$$
where vc and (vt)max are the nominal cutting speed and the maximum vibration speed, respectively. (vt)max can be determined with the value of amplitude a and frequency f. Nath et al. [9] recommended that vc should be kept lesser than (vt)max in both 1D and 2D VAM to realize intermittent cutting, i.e., Rs < 1. The increase of speed ratio reduces the intermittent gap between the chip and tool, and the cutting process approaches conventional machining.
In 1D VAM, a relative position of the tool defines as a function of time (t) and other parameters, including amplitude, frequency, and cutting speed:
$$x(t) = a\cos (\omega t) - v_{\text{c}} t$$
where ω is the angular frequency (ω = 2πf).
Figure 3a shows the FE simulation of 1D VAM cutting experiment with relative tool position at different stages of one cutting cycle, while Fig. 3b compares the propagation of the workpiece material’s force components between CM and 1D VAM, which is redrawn based on the data presented by Shamoto and Moriwaki [10]. FE simulation shows different stages of cutting process in 1D VAM with plastic deformation (cutting) and elastic recovery. As illustrated in Fig. 3b, once the cutting tool engages with the workpiece material during the CM and remains non-stop until a complete chip has been formed, they disengage from each other at the end of the entire cutting process. Due to uninterrupted plastic deformation, the force variation in CM is relatively constant.
Fig. 3

a Finite element simulation of the cutting steps in 1D VAM. b Schematic comparison between the force propagation in CM and 1D VAM

In the VAM process, the interrupted cutting causes the variation of the material’s force following the vibration cutting cycle. For any material deformation process under a mechanical load, before the material undergoes the plastic deformation, there will be an elastic region to accumulate the shear stress until it is above a critical value. During the unloading process, the material will undergo elastic recovery and the shear stress will be reduced to residual stress.

2.2 Model Development for the Equivalent Shear Angle

In this section, an energy-based analytical model has been derived to evaluate the equivalent shear angle in 1D VAM. The cutting experiment at low frequency (0.5 Hz) forms the basis for this model. It has been understood that the force dynamometer cannot measure the instantaneous force values precisely during ultrasonic VAM due to its relatively low natural frequency. In order to measure the instantaneous force values precisely in the VAM process, 1D VAM with a low frequency was applied by imitating the vibration cutting motion using programmed tool and workpiece movements.

1D vibration with a relatively low frequency was implemented along the tangential cutting direction, and the locus of tool motion is generated according to Eq. (6) through applying G codes to two linear axes of an ultraprecision machine (Toshiba ULG-100) with a programming resolution of 1 nm. The imitated vibration frequency is set to be 0.5 Hz with a 10-µm amplitude and a 0.63-µm/s nominal cutting speed. As the imitated vibration has a much smaller frequency than the upper limit frequency of a typical dynamometer, the force signal is able to be accurately captured and processed. During the cutting process, a separate computer is utilized to record the force data for further analysis. Figure 4 presents both the schematic and physical of a setup.
Fig. 4

a Procedures for imitating the low-frequency 1D VAM. b Physical experimental setup

To eliminate the effect of round nose on cutting force and simplify the force analysis, an orthogonal cutting is conducted using a flat-nose cutting tool with specified width and front clearance angle, as shown in Fig. 5a. The workpiece is made of stainless steel with a hardness of 303 HV (≈ 32 HRC) and a modulus of elasticity of 196 GPa, and it is prepared by micro milling to have straight rim structure with 500 µm width, as shown in Fig. 5b. The workpiece is mounted on a customized fixture to be held against the vacuum chuck. In this study, the cutting tool is made of CBN, depth of cut and width of cut are 10 µm and 500 µm (same as that of the rim), respectively.
Fig. 5

a Flat-face CBN cutting tool (Grade: BN7000) with specifications. b Customized straight rims on steel workpiece. c Measurement of cutting edge radius using a 3D microscope

2.2.1 Categorization of Deformation Steps Based on Force Analysis

During the low-frequency VAM process, the cutting-directional and thrust-directional force components are recorded, and the ratio between these two measured forces in two directions, Ft/Fc, are also calculated, as shown in Fig. 6. Such ratio is important to analyze the force propagation in VAM process, and it can also denote the important tool–workpiece friction status.
Fig. 6

a Recorded relative tool position, b calculated force ratio, and c measured cutting-directional force and thrust force against the vibration phase (the unit cycle time is 2 s)

From Fig. 6, it can be seen that at the beginning of each cycle of vibration cutting, the tool rake starts contacting the workpiece, and the force ratio is unstable before the tool rake completely engages with the undeformed chip. This tool–workpiece engagement step (Ds) is considered dominated by static friction. During the full engagement of the tool rake with the undeformed chip, the workpiece–material starts undergoing the elastic deformation step (De) and the plastic deformation step (Dp), whose length can be calculated by Eq. (6). Exact values of the specified steps are derived from the graph.

The plastic deformation step will cease when the tool starts moving backward and separating from the chip material, and then the elastic recovery step (Dr) starts. Similar to the elastic deformation step, the stress of undeformed chip material will continuously vary until the start of tool–workpiece disengagement step (Ds), which should have an identical length to the tool–workpiece engagement step. After the tool disengages with the chip material, it will keep in physical contact with the machined surface and plough it using a rounded cutting edge of flank face, as shown in Fig. 6. The values of Ds, Dr, and De can be evaluated based on the position of the friction ratio and cutting force.

2.2.2 Comparison with Conventional Machining Process

As a comparison with the 1D VAM test, a CM test without vibration on the same material has also been conducted, keeping the other machining conditions identical (i.e., cutting tool, depth of cut, etc.). The cutting-directional and thrust-directional force components, as illustrated in Fig. 7a, are measured to be about 15.67 N and 12.36 N, respectively. Besides such a significantly larger cutting force than that of the 1D VAM, the thickness of formed chip (22.19 µm) in CM is also found to be much larger compared to that formed by 1D VAM (16 µm). Figure 7b presents the images of the deformed chips using scanning electron microscope (SEM).
Fig. 7

a Measured cutting and thrust forces in CM. b SEM pictures of the deformed chips in CM without vibration and 1D VAM

2.2.3 Mathematical Formulation

Energy consumed in the cutting process is responsible for deforming the material and generating the chips. In conventional machining process, energy is mainly consumed in shearing the undeformed chip and overcoming the friction between the interfaces. In micro-machining assisted with vibration, the energy is not only spent in plastic deformation in the shearing of material but also has to be consumed in the elastic deformation step before the accumulated stress of undeformed material is sufficient for the plastic deformation to occur. Such elastic energy will be stored in the material and then released during the elastic recovery step and the tool is gradually moved away from the chip. The energy spent at different material deformation steps can be calculated if the instantaneous cutting forces and the respective tool–workpiece positions are given. As stated before, although the cutting force with respect to time can be measured during low-frequency VAM, it cannot be obtained in ultrasonic VAM. Hence, with the analysis of cutting force and force ratio, the force identified in the vibration cutting cycle is categorized and modeled in this section, and a simplified graph of five steps is illustrated in Fig. 8.
Fig. 8

Mechanism of material deformation with respect to tool–workpiece engagement in a VAM cycle against the vibration phase

From Fig. 8, the instantaneous cutting force for five deformation steps defines as below:
$$F_{\text{c}} = \left\{ {\begin{array}{*{20}l} 0 & {x < x_{S} } \\ {F_{{{\text{c}} - {\text{s}}}} } & {x_{S} \, \underline{ < } \, x < x_{A} } \\ {F_{{{\text{c}} - {\text{e}}}} } & {x_{A} \, \underline{ < } \, x < x_{B} } \\ {F_{{{\text{c}} - {\text{p}}}} } & {x_{B} \, \underline{ < } \, x < x_{C} } \\ {F_{{{\text{c}} - {\text{r}}}} } & {x_{C} \, \underline{ < } \, x < x_{D} } \\ {F_{{{\text{c}} - {\text{s}}}} } & {x_{D} \, \underline{ < } \, x < x_{E} } \\ 0 & {x \, \underline{ > } \, x_{E} } \\ \end{array} } \right.$$
where Fc–s represents the elastic spring force in the step of Ds, Fc–e is the elastic deformation force in De, Fc–p is the plastic deformation force in Dp, and Fc–r is the elastic recovery force in Dr.
The cutting force in micro machining is the summation of the force components in the three zones (i.e., primary, secondary, and tertiary). Arcona and Dow [1] derive a model to determine the cutting force in a precision machining, which considers the material hardness (H) as well as the workpiece material’s elastic recovery. In vibration-assisted micro machining, as the material shearing only occurs within the step of plastic deformation, the cutting force during such step is considered to include the same force components and can be written as below:
$$F_{{{\text{c}} - {\text{p}}}} = \frac{H}{3\sqrt 3 }\frac{{A_{\text{c}} }}{{\sin_{{\varphi_{\text{e}} }} }}\cos \varphi_{\text{e}} + \frac{H}{3}A_{\text{c}} + \mu_{\text{f}} \delta_{\text{f}} A_{\text{f}}$$
where Ac and Af are the undeformed chip area and the engagement area between tool flank and workpiece, respectively. In this study, the following two equations by Arif et al. [2] are employed to calculate the values of these two parameters:
$$A_{\text{c}} = wt_{0}$$
$$A_{\text{f}} = w\left( {\frac{s}{{\sin \theta_{\text{f}} }}} \right)$$
where w and θf, are width of cut and tool flank angle, respectively. s is height of elastic recovery, which is derived from this equation:
$$s = k_{1} r\frac{H}{E}$$
where k1, E and r are a scaling constant, elastic modulus of the workpiece and radius of cutting edge, respectively.
Arcona and Dow [1] has derived δf from the material hardness and elastic modulus as follows:
$$\delta_{\text{f}} = k_{2} H\sqrt {\frac{H}{E}}$$
where k2 is a scaling constant. The values of K1 and K2 used by Arcona and Dow [1] are 43 and 4.1, respectively. After substituting the expressions of Af, s, and δf into Eq. (8), the cutting force for Dp is given as a function of the equivalent shear angle φe as follows:
$$F_{{{\text{c}} - {\text{p}}}} = \frac{{Hwt_{0} }}{3}\left( {\frac{{\cot \varphi_{\text{e}} }}{\sqrt 3 } + 1} \right) + \frac{{\mu_{\text{f}} ws}}{{\sin \theta_{\text{f}} }}\left( {k_{2} H\sqrt {\frac{H}{E}} } \right)$$
From Fig. 8, it could be understood that the force at the plastic deformation step remains almost constant during each cutting cycle. In contrast, during the elastic spring and elastic deformation steps, the cutting force continuously increases with the increment of vibration phase from zero to a maximum value, which is equal to Fc–p. For simplicity, the average values of cutting force in these two steps can be derived as follows:
$$\overline{{F_{{{\text{c}} - {\text{s}}}} }} = \int_{{\psi_{S} }}^{{\psi_{A} }} {\frac{{F_{{{\text{c}} - {\text{p}}}} }}{{\psi_{B} - \psi_{S} }}} {\text{d}}\psi \to \overline{{F_{{{\text{c}} - {\text{s}}}} }} = k_{\text{s}} F_{{{\text{c}} - {\text{p}}}}$$
$$\overline{{F_{{{\text{c}} - {\text{e}}}} }} = \int_{{\psi_{A} }}^{{\psi_{B} }} {\frac{{F_{{{\text{c}} - {\text{p}}}} }}{{\psi_{B} - \psi_{S} }}} {\text{d}}\psi \to \overline{{F_{{{\text{c}} - {\text{e}}}} }} = k_{\text{e}} F_{{{\text{c}} - {\text{p}}}}$$
where ψA, ψB, and ψS are the corresponding phase of vibration for the tool–workpiece positions xS, xA, and xB, and their values can be derived from accordingly with the experimental data. ks and ke are two simplified factors to describe the average elastic force with the utilization of the plastic cutting force.

2.2.4 Energies in One Vibration Cutting Cycle

In both CM and VAM processes, for nominal cutting speed of vc, the exact volume of material removed per second in the machining process is:
$$V = v_{\text{c}} t_{0} w$$
In conventional machining without vibration, the cutting energy is mainly contributed by force along the cutting direction. For the same cutting speed, depth, and width, the specific cutting energy in conventional machining is:
$$E_{\textrm{SP-CM}} = \frac{{F_{\textrm{c-CM}} v_{\text{c}} }}{{v_{\text{c}} t_{0} w}}$$
where Fc-CM represents the cutting force in CM.
For a given frequency f, the specific cutting energy in the plastic deformation step can be described as follows:
$$E_{{\textrm{SP-p}}} = \frac{f}{V}\int_{{x_{B} }}^{{x_{B} + D_{\text{p}} }} {F_{{{\text{c}} - {\text{p}}}} } {\text{d}}x \to E_{{\textrm{SP-p}}} = \frac{f}{{v_{\text{c}} t_{0} w}}\int_{{x_{B} }}^{{x_{B} + D_{\text{p}} }} {F_{{{\text{c}} - {\text{p}}}} {\text{d}}x}$$
Similarly, the specific cutting energy in the elastic spring and elastic deformation steps can be described as below, respectively:
$$E_{\textrm{SP-s}} = \frac{f}{{v_{\text{c}} t_{0} w}}\int_{{x_{S} }}^{{x_{S} + D_{\text{s}} }} {\overline{{F_{{{\text{c}} - {\text{s}}}} }} } {\text{d}}x$$
$$E_{{\textrm{SP-e}}} = \frac{f}{{v_{\text{c}} t_{0} w}}\int_{{x_{A} }}^{{x_{A} + D_{\text{e}} }} {\overline{{F_{{{\text{c}} - {\text{e}}}} }} {\text{d}}x}$$
The total specific energy spent for cutting unit volume of material in 1D VAM will be:
$$\begin{aligned} E_{{\textrm{SP-VAM}}} & = E_{\textrm{SP-s}} + E_{\textrm{SP-e}} + E_{\textrm{SP-p}} \\ & \quad \to E_{\textrm{SP-VAM}} = \frac{f}{{v_{\text{c}} t_{0} w}}\left( {\int_{{x_{S} }}^{{x_{S} + D_{\text{s}} }} {\overline{{F_{\text{c - s}} }} {\text{d}}x} + \int_{{x_{A} }}^{{x_{A} + D_{\text{e}} }} {\overline{{F_{{{\text{c}} - {\text{e}}}} }} {\text{d}}x} + \int_{{x_{B} }}^{{x_{B} + D_{\text{p}} }} {F_{{{\text{c}} - {\text{p}}}} {\text{d}}x} } \right) \\ \end{aligned}$$

The value of Ds is considered to be equal to the height of elastic spring back for freshly machined workpiece surface, which is given in Eq. (11). In the elastic recovery and tool–workpiece disengagement step, there is no energy consumed by the tool, and the elastic energy kept inside the material or chip is dissipated. Hence, such disengagement steps are neglected in the calculation of total specific cutting energy for the vibration-assisted cutting cycle.

As the elastic deformation step occurs before the shearing process to accumulate the stress and energy for the impending plastic deformation, the value of De is assumed to be in proportion to the plastic deformation Dp. Their relationship is written as follows:
$$D_{\text{e}} = k_{\text{d}} D_{\text{p}}$$
where kd is assumed to be a factor whose value is proportional to the uncut chip thickness.
The specific cutting energy spent in the region of plastic deformation Dp of low frequency 1D VAM can be denoted as ESP-CM. For identical cutting conditions, the effective cutting energy spent to remove unit volume of material for VAM and CM, the relationship can be drawn as, \(E_{\textrm{SP-VAM}} = \eta *E_{\textrm{SP-CM}}\), where η is a force factor to be evaluated experimentally from the ratio of cutting force between low-frequency 1D VAM and CM. The value of η from the cutting experiment for low-frequency 1D VAM and CM on stainless steel has been found to be 0.45. Hence, the following equation can be derived from Eqs. (7) and (13).
$$\frac{f}{{v_{\text{c}} t_{0} w}}\left[ {\int_{{x_{S} }}^{{x_{S} + D_{\text{s}} }} {k_{\text{s}} F_{{{\text{c}} - {\text{p}}}} {\text{d}}x} + \int_{{x_{A} }}^{{x_{A} + k_{\text{d}} D_{\text{p}} }} {k_{\text{e}} F_{{{\text{c}} - {\text{p}}}} {\text{d}}x} + \int_{{x_{B} }}^{{x_{B} + D_{\text{p}} }} {F_{{{\text{c}} - {\text{p}}}} } {\text{d}}x} \right] = \eta *\frac{{F_{\textrm{c-CM}} v_{\text{c}} }}{{v_{\text{c}} t_{0} w}}$$
After substituting Fc-p of Eq. (13) into Eq. (23), as all the other parameters and factors are known or can be derived from the cutting parameters as well as the workpiece properties, the equivalent shear angle, φe, in a vibration cutting cycle with specific conditions can be calculated from the following generalized form of the equation:
$$\varphi_{\text{e}} = \cot^{ - 1} \left[ {\frac{3\sqrt 3 }{{Hwt_{0} }}\left\{ {\frac{{\eta *F_{\textrm{c-CM}} v_{\text{c}} }}{{k_{\text{s}} D_{\text{s}} f + k_{\text{e}} k_{\text{d}} v_{\text{c}} + v_{\text{c}} }} - \frac{{\mu_{\text{f}} ws}}{{\sin \theta_{\text{f}} }}\left( {k_{2} H\sqrt {\frac{H}{E}} } \right) - \frac{{Hwt_{0} }}{3}} \right\}} \right]$$

3 Experimental Verification Using Ultrasonic 1D VAM

3.1 Ultrasonic 1D VAM Experiment at Varying Speed Ratios on Stainless Steel

The values of experimental and predicted shear angle in low-frequency (0.5 Hz) 1D VAM have been found as 37.23° and 41.52°, respectively. Although the categorization of force and energy variation is derived from the measured low-frequency force, it is necessary to validate the developed model through several ultrasonic 1D VAM tests. The tests were first conducted on the same stainless-steel material which was used in the low frequency cutting tests, and then on other three common metal alloys for further verification.

A tangential tool vibration with 38.87-kHz frequency and 2-µm amplitude were applied, as shown in Fig. 9. The same CBN cutting tool was utilized, and the workpiece was fabricated to be a circular ring with its dimensions shown in the following figure. Speed ratio (Rs) is a critical cutting parameter in VAM. The developed model has been verified with this speed ratio. For a fixed vibration speed (2πaf), the speed ratio is basically a variable of cutting speed (vc) as can be found in Eq. (5). The equivalent shear angle (φe) has been evaluated for a variety of cutting speed from Eq. (24). Six nominal cutting speeds (0.91, 1.46, 2.19, 2.74, 3.29, and 3.65 m/min) were applied and the corresponding speed ratios can be derived as 3.11, 4.95, 7.48, 9.35, 11.22, and 12.46%, respectively. The depth of cut is 4 µm, and the deformed chip thicknesses are inspected and measured using SEM.
Fig. 9

Experimental setup for ultrasonic 1D VAM tests

To predict the equivalent shear angle according to the model developed in the last section, the cutting and vibration parameters are input as given values to the model, together with the material properties and the geometrical parameters of tool. From the previous Fig. 6, the values of the factors (ks, kd, and ke) utilized in Eq. (24) can be derived as 0.137, 1.81, and 0.363, respectively. The value of Fc-CM can be extracted from Fig. 7. By putting all the cutting and vibration parameters and tool/workpiece properties into Eq. (24), the equivalent shear angle φe at different cutting speed can be predicted accordingly, as shown in Table 1.
Table 1

Chip morphology and derived shear angles for ultrasonic 1D VAM of stainless steel

Nominal cutting speed, vc (m/min)

Speed ratio, Rs

(%) @38.87 kHz

Experimental results

Predicted equivalent shear angle, φe

SEM pictures of generated chips

Measured chip thickness (Averaged) t1 (µm)

Experimental shear angle



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Table 1 also lists the SEM photographs of the chips as well as the calculated actual shear angles from the measured chip thickness and CCR against the variation of speed ratios can also be derived as shown in Fig. 10a. It can been observed that CCR increases accordingly when the speed ratio increases, meaning that the material is compressed more by the cutting tool if the length of plastic deformation step (Dp) is increased. In 1D VAM, the amount of compression should be dependent on the type of material and their response on the varying cutting and vibration conditions.
Fig. 10

a Chip compression ratio against the variation of speed ratio. b Comparison between predicted and experimental shear angle against the speed ratio for ultrasonic 1D VAM of stainless steel

Figure 11b compares the shear angles derived from the experimentally measured chip compression ratio and the predicted ones from the proposed model based on specific cutting energy as in Eq. (24). From the graph, it is found that with the increment of speed ratio, the shear angle decreases, resulting in larger deformation of the undeformed chip material. The energy consumption for one vibration cutting cycle will increase as a result of both the increasing deformation of material (i.e., decreasing shear angle) and the increased feeding distance per cycle (vc/f). It is also apparent to observe that there is a gap between the theoretical and experimental values, which is considered to be caused by the estimation of experimental shear angle through Eq. (4), which is mostly used for conventional machining.
Fig. 11

Further experimental verification for three additional common alloys: a force analysis for low-frequency 1D VAM, b variation of CCR, and c comparison between predicted and experimental shear angles for ultrasonic 1D VAM

3.2 Experimental Verification on Three Common Metal Alloys

Although the developed theoretical model based on the force analysis of low-frequency vibration cutting incorporating material properties is found feasible to predict the experimental values of shear angles for ultrasonic 1D VAM of stainless steel, it is also important to provide further verification for the proposed model through cutting tests on other metal alloys. Three common engineering metal alloys (tool steel, titanium alloy, and brass) are tested to study the variation of shear angle under different speed ratios. Table 2 describes the cutting and vibration parameters that are employed for all the experiments.
Table 2

Cutting and vibration parameters for CM, low-frequency, and ultrasonic 1D VAM

Workpiece material

 Stainless steel, 303 HV

 Tool steel, 220 HV

 Titanium alloy, 290 HV

 Brass, 112 HV

Conventional machining of straight rim

 Width of cut (µm): 500

 Depth of cut (µm): 10

 Cutting speed (mm/min): 10

Low-frequency 1D VAM of straight rim

 Width of cut (µm): 500

 Depth of cut (µm): 10

 Nominal cutting speed (µm/min): 37.8

 Frequency of vibration (Hz): 0.5

 Amplitude of vibration (µm): 10

Ultrasonic 1D VAM of circular rim

 Width of cut (µm): 500

 Depth of cut (µm): 5

 Nominal cutting speed (speed ratio) (m/min): 0.91(3.11%), 1.46(4.98%), 2.19(7.48%), 2.74(9.35%), 3.29(11.22%), and 3.65(12.46%)

 Vibration frequency (kHz): 38.87

 Vibration amplitude (µm): 2

Similar to the cutting experiments of stainless steel, 1D VAM experiments with the same low-frequency and cutting conditions were conducted on all the three metal alloys to identify the relevant parameters. Figure 11a illustrates the resulting cutting force against the phase of vibration within a vibration cycle as well as the identified material deformation steps. A series of ultrasonic VAM tests with different speed ratios are also conducted on the same materials, which are prepared in a similar way like stainless steel. Figure 11b, c present the results of CCRs and the comparative results between the predicted and experimental shear angles, respectively. It can be observed for all the three types of metals that the shear angle increases with the decrease of speed ratio.

Both the tool steel and titanium have similar variation trend of CCRs against speed ratio, which is also quite close to that for stainless steel. The developed model predicts well the variation trend of experimental shear angle, although the gap between the two curves still exists for the predicted and experimental values. In comparison, the effect of reduced speed ratio on CCR in ultrasonic VAM of brass is not as obvious as the other three metals, and the experimental variation trend can be well predicted by the developed model, as shown in Fig. 11c. The difference of variation trend between brass and the other three metal alloys is considered to be caused by its relatively low hardness value, which possibly leads to an increased energy consumption during the tool–workpiece engagement step with elastic deformation, as shown in Eq. (20) and Fig. 11a.

3.3 Surface Roughness of the Machined Surfaces

As explained in the previous section, CCR increases with the increase of cutting speed for 1D VAM. The increase of CCR represents more energy is consumed in the cutting process. The effect of this increased energy consumption requires a quantitative analysis of the machined surface quality. The surface roughness of machined surface has been investigated for each of the metals cut at different cutting speeds using a Taylor-Hobson Form Talysurf-120 profilometer. A graph of surface roughness for four different metals is shown in Fig. 12a. The roughness profile is very insignificant for soft metals like brass. For harder metals like tool steel and stainless steel, the graphs are showing slightly increasing though not significant. However, there is clearly an increasing trend for titanium. Titanium is a poor thermal conductor and the heat generated during the machining process cannot be dissipated easily. The highly localized deformation along the shear zone usually causes plastic instability that ultimately damages the surface. This plastic instability increases with the increase of cutting speed and causes more damage to the surface. Notwithstanding the facts that force fluctuation is common in the machining of Ti in CM, 1D VAM interestingly provides a cutting force without any fluctuation. In fact, there is no force fluctuation for any of the other metals, as shown in Fig. 11a. The intermittent nature of cutting with a very small amount of cutting in each vibration cycle suppress the fluctuation.
Fig. 12

a Average roughness of four different types of metals against cutting speed in 1D VAM. b Optical image of workpiece surface at cutting speed of 0.912 m/min (mag: 400×)

From the roughness profile shown in Fig. 12a, it is evident that a lower cutting speed provides the best surface condition in the cutting of hard metals in 1D VAM. Figure 12b shows the images of workpiece surfaces taken by a Keyence optical microscope (VH Z450) under 400× magnification for each of the four metal machines at the lowest cutting speed of 0.912 m/min.

4 Conclusions

An analytical predictive model has been derived with the proposed force analysis of elastic-plastic deformation steps of low-frequency 1D VAM for the equivalent shear angle, which is an important parameter for evaluating the cutting mechanism and the following conclusions are summarized from this research study:
  1. a.

    1D VAM with a very low frequency also produces increased shear angle compared to conventional machining.

  2. b.

    The predicted model, considering various factors including material properties and cutting conditions, provides a shear angle appropriate for 1D VAM.

  3. c.

    The predicted model applied on four different metals shows a similar trend for equivalent shear angle and chip compression ratio (CCR).

  4. d.

    A smaller nominal cutting speed leads to a smaller CCR and a larger equivalent shear angle in ultrasonic 1D VAM.

  5. e.

    For difficult-to-cut metals, a lower cutting speed leads to better machined surface quality.

  6. f.

    The developed model can predict well the variation trend of equivalent shear angle with respect to different speed ratios in ultrasonic 1D VAM.




This project was partially sponsored by Shanghai Pujiang Program (19PJ1404500).


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Copyright information

© International Society for Nanomanufacturing and Tianjin University and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.National University of SingaporeSingaporeSingapore
  2. 2.Shanghai Jiao Tong UniversityShanghaiChina
  3. 3.Singapore Institute of Manufacturing TechnologySingaporeSingapore

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