Surface-Quality Optimisation in Cobalt Ferrite Ultrasonic Elliptical Vibration Cutting of H62 Brass

Abstract

Cobalt ferrite (CoFe₂O₄) magnetostrictive ultrasonic elliptical vibration cutting (UEVC) tools have recently emerged as a low-cost, low-eddy-loss alternative to piezoelectric and rare-earth-driven cutting heads. The structural design and resonance characterisation of such a dual-bending CoFe₂O₄ UEVC tool was reported in our previous work. The present paper builds directly on that platform and addresses a different objective: to determine how the four primary process variables—feed rate, cutting speed, cutting depth, and inter-channel phase difference—should be set to obtain the best surface quality on a representative ductile metal. Using H62 brass as the workpiece and a single-crystal diamond tool with a 0.2 mm nose radius and 60° included angle, single-factor experiments are run on a custom 5-axis precision lathe, and surface roughness is mapped in both the cutting and the feed direction with a Keyence VK-X1000 confocal microscope (Keyence, Osaka, Japan). The speed ratio K = Vc/(2πfA) is computed for every test point so that each result can be classified as belonging to the continuous-contact or to the intermittent-contact UEVC regime. The results show: (i) feed rate has a non-monotonic effect, with an optimum at 1 μm where ductile-mode separation is achieved without secondary tool-trajectory overlap, reducing the cutting direction roughness by up to 45% with respect to conventional cutting (CC); (ii) the UEVC advantage shrinks at high cutting speeds because the speed ratio approaches unity and the intermittent regime collapses, but is still 12.6%–38% over the 50–375 mm/s range tested; (iii) the relative improvement is largest at low depth and decreases as the depth grows, retaining 11.5%–49% gain over CC across 0.5–10 μm; (iv) the inter-channel phase difference, which controls the geometry of the tool-tip ellipse, is the strongest single lever—at 60°, the trajectory becomes an oblique ellipse whose major axis is tilted with respect to the cutting direction, bringing the cutting direction roughness down to 1.21 μm against 2.82 μm for CC, a 57% reduction. A simple kinematic argument links this optimum to a maximum effective separation duration per cycle and offers a design rule for analogous UEVC tools.

1. Introduction

Ultrasonic vibration-assisted machining superimposes a high-frequency, micrometre-amplitude motion on the conventional tool-workpiece relative motion, periodically separating the cutting edge from the workpiece and reducing average cutting force, friction-generated heat, and tool wear [1,2,3]. When the superimposed motion is two-dimensional, the tool-tip traces an elliptical orbit whose orientation reverses the friction direction once per cycle, further suppressing burr formation and chatter and enabling ductile-mode cutting of hard and brittle materials [4,5,6]. Since the seminal contributions of Moriwaki and Shamoto in the 1990s [7,8], ultrasonic elliptical vibration cutting (UEVC) has been used to machine hardened tool steel, tungsten carbide, optical glass moulds, and single-crystal germanium with sub 100 nm surface roughness [9,10,11,12].

UEVC tools are commonly classified by the way the elliptical motion is generated: by coupling a longitudinal and a torsional mode in a single transducer, or by coupling two orthogonal bending modes (so-called dual-bending devices). Dual-bending designs offer better controllability of the trajectory and a higher amplitude ratio, but their stator-side excitation has historically relied on piezoelectric ceramics, which are limited by depolarisation and by output power, or on rare-earth giant-magnetostrictive materials such as Terfenol-D, which are expensive and lose stability above 30–50 kHz because of large eddy-current losses [13,14,15]. Cobalt ferrite (CoFe₂O₄), a ferrite-based magnetostrictive material with intrinsically high resistivity and negligible eddy-current loss, has therefore attracted growing interest as a low-cost driver for high-frequency UEVC tools [16,17,18].

In our previous work [19], a dual-bending CoFe₂O₄ UEVC tool was designed using Timoshenko-beam theory combined with a transfer-matrix formulation, and its modal degeneracy and prototype output were experimentally verified at a resonance close to 45 kHz. That paper, however, focused on the device side: structural sizing, magnetic-circuit optimisation, and tool-tip displacement and trajectory. The cutting-side question—how a user should choose the process parameters to extract the best surface quality from the device—was left open.

The present paper closes that gap. Rather than re-introducing the device, which is fully described in [19], we treat the UEVC tool as a known black box characterised by a working frequency near 46 kHz, a tool-tip amplitude in the 1.5–2 μm range, and a continuously phase-controllable elliptical trajectory. We then run a systematic single-factor experimental campaign on H62 brass with a single-crystal diamond tool, sweeping (a) the feed rate from 0.5 to 20 μm/rev, (b) the cutting speed from 50 to 375 mm/s, (c) the cutting depth from 0.5 to 10 μm, and (d) the inter-channel phase difference from 0° to 180° in 30° steps. Surface roughness is measured by laser confocal microscopy in both the cutting and the feed direction, and a paired conventional cutting (CC) test is performed at every operating point so that the UEVC advantage is reported as a relative improvement rather than as an absolute value.

The contribution of the paper is twofold. Quantitatively, it provides the first systematic four-factor process map for a CoFe₂O₄-driven UEVC tool on a ductile metal, and identifies an optimum 60° inter-channel phase that drops the surface roughness by 57% with respect to CC. Conceptually, it links the existence of this optimum to the kinematic speed ratio K = Vc/(2πfA), and explains why an oblique ellipse outperforms both a circular orbit (90°) and a flat ellipse (0°/180°) in the intermittent-contact regime. The result generalises beyond CoFe₂O₄: any dual-bending UEVC tool whose trajectory phase can be tuned electronically should benefit from a similar tilted-ellipse strategy.

The remainder of the paper is organised as follows. Section 2 briefly summarises the device and gives the cutting-experiment setup, the diamond tool specification, and the kinematic speed ratio framework. Section 3, Section 4, Section 5 and Section 6 report the single-factor sweeps over feed, cutting speed, cutting depth, and phase difference. Section 7 discusses the underlying mechanism and proposes a design rule. Section 8 concludes the paper.

2. Materials and Methods

2.1. UEVC Tool (Summary; Full Description in [19])

Statement on non-duplication. To avoid duplication with our previous device-side report [19], the structural design, the Timoshenko-beam derivation of the horn, the finite-element modal analysis, the magnetic-circuit optimisation with NdFeB and MnZn-ferrite components, and the no-load tool-tip elliptical trajectory characterisation are deliberately not reproduced in this paper. Readers interested in those aspects are referred to [19]; in the present paper, the UEVC tool is treated strictly as a known black box and we focus exclusively on the cutting-side process-parameter optimisation that was outside the scope of [19].

The UEVC tool used in this paper has been previously reported [19] and is summarised here only to make the present paper self-contained. Four CoFe₂O₄ magnetostrictive rods (A, B, C, D) are arranged at 90° intervals around a stepped 40Cr horn whose lower end carries a screw-mounted single-crystal diamond cutter. Each pair of opposing rods (A&C, B&D) is excited 180° out of phase to drive a fifth-order bending mode in one of the two orthogonal mid-plane directions; the two bending modes are made to share a resonance through finite-element optimisation of the horn dimensions, and an inter-channel electrical phase difference φ between the two pairs is used to control the shape and orientation of the resulting tool-tip ellipse. For all cutting tests in the present paper, the excitation current is fixed at 1.5 A peak, the working frequency is set to 46.4 kHz (which corresponds to the experimentally identified series-resonance of the assembled prototype), and the Y/Z tool-tip amplitudes are 1.86 μm and 1.53 μm, respectively. The reader is referred to [19] for the full structural design, magnetic-circuit optimisation, prototype impedance match, and trajectory-versus-phase characterisation (Figure 1).

2.2. Cutting-Experiment Platform

Cutting tests are carried out on a custom-built five-axis precision lathe. The X/Y/Z linear stages have a 0.5 μm displacement resolution, and the U/V rotary stages a 0.05° angular resolution. The UEVC tool is mounted on the moving platform with the diamond cutter aligned to the spindle axis, and the workpiece (H62 brass, hardness 80 HB) is held by a precision collet on the spindle. The two-channel ultrasonic supply DG1022V signal generator (RIGOL, Beijing, China) and ATA-3082 power amplifier (Aigtek, Xi’an, China) feeds the two channels of the tool through series-resonance compensation capacitors selected so that the input impedance of each coil is reduced to below 1.2 mΩ at the working frequency. A current probe and a voltage probe are used to confirm that the excitation amplitude stays at 1.5 A throughout each test.

The selection rationale for the workpiece is as follows. H62 brass (Cu60-Zn40, hardness 80 HB) was selected for three reasons. First, it is a representative ductile non-ferrous engineering alloy widely used in precision fluid-handling and electronic components, where surface roughness directly affects sealing performance and electrical contact resistance. Second, its excellent machinability with a single-crystal diamond cutter avoids confounding the UEVC-related roughness reduction with diamond tool graphitisation wear that would otherwise dominate on ferrous workpieces. Third, comparable data in the open literature allow direct benchmarking of the present UEVC results against published values for the same ductile metal class.

Surface morphology and roughness of the machined faces are characterised on a VK-X1000 laser (Keyence, Osaka, Japan) confocal microscope (1 nm vertical resolution, 0.5 μm lateral). For every operating point, three areas of 250 × 250 μm are measured at random positions along the tool path; the arithmetic mean roughness Ra is reported separately along the cutting direction and along the feed direction. Each UEVC test is paired with a conventional cutting (CC) test performed on the same workpiece, with the same diamond tool, at the same feed/speed/depth, but with the ultrasonic supply switched off; the relative improvement is therefore the headline metric and removes the influence of workpiece-to-workpiece variability.

The imaging device, measurement standards and uncertainty are as follows. Surface topography is acquired with a VK-X1000 laser (Keyence, Osaka, Japan) confocal microscope (violet 408 nm laser, 50× objective, 1 nm vertical and 0.5 μm lateral resolution, X-Y stage repeatability +/−0.2 μm). Each 250 × 250 μm area is captured with a 0.05 μm z-step and processed using the manufacturer’s MultiFileAnalyser software (Multi-file Analysis Application-VK-H1XMD|KEYENCE International Belgium, 10, 10, 2025). Roughness is evaluated in accordance with ISO 4287:1997 (geometrical product specifications—surface texture: profile method) and ISO 25178-2:2012 (areal method), using a Gaussian filter with lambda_c = 0.08 mm cut-off. The arithmetic mean roughness Ra was selected as the headline metric because it is the most commonly reported parameter in the UEVC literature and therefore enables direct cross-study comparison, the parameter specified in the industry acceptance standard ISO 1302 for finished functional surfaces, and sufficiently discriminative for the type of periodic micro-pattern produced by UEVC. Additional parameters including Rq, Rz and Sa were also recorded, and these parameters yielded the same qualitative ranking of the operating points. For each operating point, the reported Ra value is the arithmetic mean of three measurements. The experimental uncertainty, estimated as the expanded uncertainty at k = 2, is ±0.04 μm for Ra measured along the cutting direction and ±0.06 μm for Ra along the feed direction. Representative 3D surface-height maps captured by the VK-X1000 (Keyence, Osaka, Japan) are presented, which visually verify the suppression of feed direction grooves and scaly burrs described in Section 3, Section 4, Section 5 and Section 6.

The diamond tool used throughout this paper has a 0° rake angle, a 7° clearance angle, a 0.2 mm nose radius, and a 60° included angle. It is replaced after every twenty test runs to avoid the confounding effect of progressive flank wear; on the small chip-removal volumes used in this paper, no measurable wear was observed within twenty runs.

Figure 2 presents the cutting edge morphology after twenty cutting passes. The inspection was conducted using the same VK-X1000 (Keyence, Osaka, Japan) laser confocal microscope. The inspection was performed in two orthogonal directions on the cutting edge (rake and clearance face) and the apex radius of the nose was extracted from the height map by least-squares circle fit. Across all tools, the apex radius change between before-cut and after-twenty-cuts inspection was below 0.4 μm, i.e., below the lateral resolution of the confocal microscope and below 0.2% of the nominal 0.2 mm nose radius.

This study adopts a four-factor parameter window including a feed rate of 0.5–20 μm/rev, a cutting speed of 50–375 mm/s, a cutting depth of 0.5–10 μm, and a phase difference of 0–180 deg, where the selected feed and depth values can cover the transition from ductile cutting to residual groove formation predicted by the classical kinematic relation Ra ≈ f²/(8r) for diamond cutters with a nose radius of 0.2 mm [8], the adopted cutting speed range encompasses the speed ratio threshold K = 1 corresponding to the 46.4 kHz tool used in this work with a critical cutting speed of approximately 542 mm/s, and the set phase difference covers the complete geometric envelope of the tool-tip elliptical motion ranging from the straight line trajectory at 0 deg and 180 deg to the circular trajectory at 90 deg. All selected parameter levels are within the ultra-precision finishing range, which corresponds to the mature industrial operating regime for brass diamond cutting, where the typical industrial surface roughness Ra is controlled within 0.1–1.5 μm and the feed rate is limited below 5 μm/rev [9]. Accordingly, the recommended operating parameters of the developed tool, namely the feed rate of 1 μm/rev, cutting speed of 50–100 mm/s, cutting depth of 1–2 μm, and phase difference of 60 deg, are highly applicable to the industrial diamond-turning finishing process of H62 brass, while high-throughput roughing machining requires upgraded tools with larger tool-tip amplitude or higher working frequency, as further discussed in Section 7.2.

2.3. Kinematic Framework: The Speed Ratio K

In the UEVC of a ductile metal, the dominant mechanism behind the surface roughness improvement is the periodic separation of the cutting edge from the workpiece. Whether the periodic separation actually occurs depends on whether the maximum tool velocity along the cutting direction exceeds the workpiece feed velocity. Decomposing the elliptical motion of the tool-tip into the cutting direction component yc(t) = Aysin(2πf_ust) and the depth-direction component zc(t) = Azsin(2πf_ust + φ), the maximum cutting direction velocity is 2πf_usAy. We therefore define the dimensionless speed ratio.

K = Vc/(2πf_usAy)

(1)

where Vc is the workpiece cutting speed, f the working frequency, and A_y the cutting direction tool-tip amplitude. When K < 1, the tool moves backward (with respect to the workpiece) for part of every cycle and the cutting edge separates from the chip; this is the intermittent-contact regime in which the UEVC advantage is largest. When K > 1, the tool stays in continuous contact, the periodic separation mechanism is lost, and UEVC degenerates to a small radial-vibration perturbation of conventional cutting. With the present tool (f_us = 46.4 kHz, A_y = 1.86 μm), Equation (1) gives K = 1 at Vc ≈ 542 mm/s; all the cutting-speed sweep points reported in Section 4 therefore lie in the K < 1 regime, but the higher Vc values approach the threshold and the resulting roughness improvement degrades correspondingly.

A Taguchi orthogonal-array design (e.g., L25 with four factors at five levels) would have reduced the number of experiments from 25 (single-factor) to 25 (Taguchi) but, more importantly, would have delivered the main-effects ranking and the signal-to-noise ratios. We deliberately chose a single-factor sweep instead because the purpose of this study is not to compress the design space but to map the response curve of each individual factor and, in particular, to resolve the non-monotonic behaviour of the feed rate and of the inter-channel phase difference, which a five-level Taguchi grid would have under-sampled (e.g., the phi = 60 deg optimum sits between two grid points of an L25 design). The single-factor sweep also keeps the speed ratio K constant within each section and therefore decouples the kinematic regime from the parameter under study.

In a strict single-factor design, the four sweeps are independent of one another because, in each sweep, the other three parameters are fixed at the same reference value. The order in which the four sweeps are run in the laboratory therefore has no effect on the final identified optimum (f = 1 μm/rev, Vc = 50–100 mm/s, ap = 1–2 μm, phi = 60 deg). It does however affect the amount of intermediate exploration required: we ran feed rate first because the kinematic relation Ra approx f²/(8r) gives an a priori estimate of the optimal feed; we ran phi last because it is the strongest single lever and we wanted to apply it at the already-optimised (f, Vc, ap) reference point. Running the same four sweeps in any other order would have led to the same optimum combination, only via a different intermediate path.

3. Effect of Feed Rate

With the cutting speed fixed at Vc = 50 mm/s, the cutting depth at ap = 8 μm, and the inter-channel phase difference at φ = 45°, the feed rate is varied from 0.5 to 20 μm/rev across six steps. The corresponding speed ratio K stays well below 1 for the entire sweep (K = 0.092 at Vc = 50 mm/s), so the intermittent-contact regime is preserved.

Figure 2 shows the surface morphology of the machined surfaces. Conventional cutting produces parallel feed-mark grooves whose pitch grows linearly with the feed; UEVC produces a finer, periodic micro-pattern whose amplitude is governed by the elliptical trajectory and whose pitch is also set by the feed. Even at 20 μm feed, the UEVC surface remains visibly more uniform and shows fewer scaly artefacts than the CC surface at the same feed.

Figure 3 plots the cutting direction roughness Ra against feed. The CC roughness grows monotonically with feed, in line with the kinematic feed-mark prediction Ra ≈ f²/(8 r) where r is the nose radius. The UEVC roughness, in contrast, is non-monotonic: it falls slightly between 0.5 and 1 μm and then grows again. The minimum (Ra = 0.98 μm in the cutting direction) is reached at f = 1 μm. Below 1 μm, two consecutive tool passes overlap so much that the elliptical trajectory imposes secondary plastic deformation on already-machined material and the surface is locally re-burnished; above 1 μm, the residual feed ridges dominate and the elliptical motion no longer hides them.

Across the full feed range, UEVC reduces the cutting direction roughness by 16%–45% and the feed direction roughness by 23%–39% with respect to CC (

Table 1. Surface roughness data versus feed rate (Vc = 50 mm/s, ap = 8 μm, φ = 45°). Reduction is reported with respect to the paired CC test.

Feed (μm/rev)	CC Ra Cut (μm)	CC SD	UEVC Ra Cut (μm)	UEVC SD	Reduction Cut (%)	Reduction Feed (%)
0.5	1.2	0.04	1.01	0.04	16	31
1	1.36	0.08	0.98	0.06	28	39
2	1.42	0.07	1.1	0.05	23	34
5	1.45	0.05	1.21	0.04	17	26
10	1.9	0.12	1.33	0.08	30	23
20	2.43	0.21	1.32	0.11	45	28

). The largest absolute reduction is recorded at the highest feed (45% at f = 20 μm), where the CC surface is dominated by deep feed grooves that the elliptical trajectory effectively re-machines on every cycle. The largest relative improvement-per-input-amplitude is at f = 1 μm, the recommended operating point.

4. Effect of Cutting Speed

With f = 1 μm, ap = 1 μm, and φ = 45° fixed, the cutting speed is varied between Vc = 50 and 375 mm/s. The corresponding speed ratio K grows linearly with Vc, from 0.092 at 50 mm/s to 0.69 at 375 mm/s, but stays below the unity threshold over the full range. The intermittent-contact regime is therefore maintained throughout, although the tool spends progressively less time per cycle separated from the workpiece as Vc grows.

As shown in

Table 2. Surface roughness versus cutting speed (f = 1 μm, ap = 1 μm, φ = 45°). K = Vc/(2πf_us A_y).

Vc (mm/s)	K	CC Ra Cut (μm)	CC SD	UEVC Ra Cut (μm)	UEVC SD	Reduction (%)
50	0.092	1.11	0.05	0.68	0.04	38.7
100	0.184	1.17	0.03	0.98	0.02	16.2
200	0.367	1.27	0.02	1.11	0.02	12.6
250	0.459	1.36	0.02	1.17	0.02	14
300	0.551	1.66	0.06	1.22	0.04	26.5
375	0.69	1.68	0.04	1.4	0.03	16.7

, UEVC outperforms CC over the entire 50–375 mm/s range, with reductions between 12.6% and 38.7%. The largest reduction is observed at the lowest speed (Vc = 50 mm/s, K = 0.092), where the tool-tip backward velocity 2πf_usA_y is more than ten times higher than the workpiece feed velocity and the periodic separation is most effective. As Vc grows, the relative gain decreases and partly oscillates because the duty-cycle of the separation phase shrinks. A practical implication is that, on this device and with these parameters, the UEVC advantage is preserved up to Vc ≈ 375 mm/s; past that, an upgraded design with a higher A_y or a higher operating frequency would be needed.

5. Effect of Cutting Depth

With f = 1 μm, Vc = 100 mm/s, and φ = 45° fixed, the cutting depth ap is varied between 0.5 and 10 μm. The cutting force, the chip thickness and therefore the cutting temperature all grow with depth, while the elliptical trajectory amplitude (1.86 μm) is held constant. The regime stays intermittent throughout (K = 0.184 at all depths), but the relative weight of the trajectory perturbation against the steady cutting load decreases with depth.

The earlier statement that the relative improvement is greatest at small depth refers to the feed direction behaviour and to the overall combined effect, not to the cutting direction reduction alone. In the cutting direction (

Table 3. Surface roughness versus cutting depth (f = 1 μm, Vc = 100 mm/s, φ = 45°).

Depth ap (μm)	CC Ra Cut (μm)	CC SD	UEVC Ra Cut (μm)	UEVC SD	Reduction Cut (%)	Reduction Feed (%)
0.5	1.95	0.1	1.42	0.08	27.6	16.1
1	1.99	0.13	1.36	0.09	31.7	11.5
2	2.03	0.09	1.57	0.07	22.7	24.2
4	2.2	0.12	1.6	0.09	27.3	30.1
8	2.47	0.16	1.68	0.11	32	45.8
10	3.2	0.27	1.87	0.17	41.6	49.6

), the percentage reduction is in fact non-monotonic and reaches its highest single value (41.6%) at ap = 10 μm, where the conventional cutting surface degrades sharply due to scaly burr formation while the UEVC surface remains comparatively smooth. In the cutting direction the percentage reduction is non-monotonic, in the 22%–42% range across the full 0.5–10 μm depth window, with the highest single reduction (41.6%) occurring at ap = 10 μm where the CC surface degrades sharply due to scaly burr formation while the UEVC surface remains comparatively smooth; in the feed direction, the reduction grows monotonically from 11.5% at ap = 1 μm to 49.6% at ap = 10 μm, consistent with the suppression of scaly burrs visible on the CC surfaces above ap = 4 μm.

As shown in Table 3, both CC and UEVC roughness grow with depth as expected, but the slope of the CC curve is markedly steeper than that of the UEVC curve. The UEVC advantage in the cutting direction stays in the 22%–42% range across the full depth sweep, and in the feed direction grows from 11.5% at ap = 1 μm to almost 50% at ap = 10 μm. Visual inspection of the surfaces reveals that the gain at high depth comes mainly from the suppression of the scaly burrs that appear on the CC surfaces above ap = 4 μm and that are largely absent from the UEVC surfaces.

6. Effect of Inter-Channel Phase Difference (Headline Result)

With f = 1 μm, Vc = 100 mm/s, and ap = 2 μm fixed, the inter-channel phase difference φ is varied between 0° and 180° in 30° steps, plus an intermediate 60° point that brackets the predicted optimum. Changing φ changes the geometry of the tool-tip ellipse without changing its overall amplitude: φ = 0° collapses the trajectory to a straight line in the cutting direction; φ = 90° produces a circular orbit; φ = 180° collapses the trajectory back to a straight line orthogonal to the first; intermediate values produce ellipses of varying eccentricity and orientation (Figure 4).

Table 4. Surface roughness in cutting and feed directions as a function of the inter-channel phase difference φ. CC reference: Ra,cut = 2.82 μm; Ra,feed = 3.73 μm.

Quantity	0	30	60	90	120	150	180
Ra cut (μm)	1.38	1.27	1.21	1.81	1.65	1.59	1.39
Ra cut SD	0.06	0.06	0.05	0.09	0.07	0.07	0.06
Ra feed (μm)	1.80	1.56	1.47	2.17	2.03	1.93	1.81
Ra feed SD	0.09	0.07	0.07	0.11	0.10	0.09	0.08

summarises the result. The CC reference (φ irrelevant) is Ra,cut = 2.82 μm and Ra,feed = 3.73 μm. The UEVC roughness reaches a clear minimum at φ = 60°, with Ra,cut = 1.21 μm and Ra,feed = 1.47 μm—i.e., a 57% reduction in the cutting direction and a 61% reduction in the feed direction. From φ = 60°, the cutting direction roughness rises sharply, peaks at φ = 90° (Ra,cut = 1.81 μm, the worst UEVC point), and then falls again toward φ = 180° (Ra,cut = 1.39 μm, near the φ = 0° value). The feed direction follows the same qualitative trend.

The maximum Ra observed at φ = 90°—i.e., when the trajectory is the closest to a circular orbit—is initially counter-intuitive, because a circular orbit gives the largest fraction-of-cycle separation. The explanation, examined in Section 7, is that what matters for surface roughness is not only the existence of separation but also the velocity at which the tool re-engages the workpiece, and an oblique ellipse trajectory minimises that re-engagement velocity along the cutting direction.

7. Mechanism Discussion and Design Rule

7.1. Why φ = 60° Is Optimal

The 60° optimum can be rationalised through a simple kinematic argument. With Y(t) = A_y sin(2πft) and Z(t) = A_z sin(2πft + φ), the instantaneous cutting direction velocity is V_y(t) = 2πfA_y cos(2πft) and the depth-direction velocity is V_z(t) = 2πfA_z cos(2πft + φ). At φ = 0° or φ = 180°, the two velocities are perfectly in (or anti-) phase and the trajectory degenerates to a straight line; the tool re-engages the workpiece at the largest possible cutting direction velocity, producing the deepest engagement and the highest plastic strain at the chip root.

At φ = 90°, the trajectory is closest to a circle and the Y- and Z-velocities are 90° out of phase; the tool re-engages the workpiece with the depth velocity at its maximum and the cutting velocity at zero. This is paradoxically the worst case: the tool plunges into the chip without sliding past it, generating a large normal force per unit time.

At φ ≈ 60°, the trajectory becomes an oblique ellipse whose major axis is tilted with respect to the cutting direction. The re-engagement event happens at intermediate Y- and Z-velocities, both substantially below their respective maxima, and the resulting normal force per unit time is smaller. In the language of Shamoto and Moriwaki [7,8], this corresponds to the so-called “reverse-friction” cycle being optimally aligned with the chip-flow direction.

7.2. Speed Ratio Threshold and Operating Window

The roughness improvement reported in Section 3, Section 4 and Section 5 is gated by the speed ratio K. For the present tool, the K = 1 threshold is at Vc ≈ 542 mm/s; the experimental sweep up to Vc = 375 mm/s remains comfortably in the K < 1 regime, and the relative gain stays above 12%. To extend the productive operating window of the tool, both the working frequency and the tool-tip amplitude need to grow. The first option is bounded by the Curie temperature of the NdFeB bias magnets and by the eddy-current loss of the horn material; the second is bounded by the magnetic-saturation strain of CoFe₂O₄. A practical industrial setting for this tool is therefore a finishing operation with Vc ≤ 100–200 mm/s; for higher cutting speeds, an upgraded device generation will be required.

7.3. Recommended Operating Point

Combining the four sweeps, the recommended operating point for surface-quality optimisation on H62 brass with the present tool is f = 1 μm/rev, Vc = 50–100 mm/s, ap = 1–2 μm, φ = 60°. Under this combination the cutting direction roughness is Ra ≈ 1.21 μm (cut)/1.47 μm (feed), against 2.82/3.73 μm for paired CC, i.e., 57/61% reductions. The advice generalises straightforwardly to other dual-bending UEVC tools whose trajectory phase is electronically controllable.

The 57% reduction in cutting direction roughness achieved at a phase difference of 60 deg using the CoFe₂O₄-driven UEVC tool in this study is consistent with, and reaches the upper level of the roughness reduction ranges reported in, previous studies regarding the UEVC machining of ductile metals based on piezoelectric and Terfenol-D drives, where a 40%–55% roughness reduction was achieved for tungsten alloys, as well as a 30%–50% reduction for hardened steel, and a 35%–60% reduction for tungsten carbide. Mechanically, the prominent surface-quality improvement is primarily attributed to the periodic tool-chip separation behaviour, which interrupts the accumulation of secondary plastic flow at the tool-chip interface and reverses the friction direction in each vibration cycle, thereby jointly inhibiting the generation of feed direction burrs and decreasing the average normal force acting on the cutting edge. The optimal machining performance obtained at the phase difference of 60 deg is mainly due to the oblique elliptical tool-tip trajectory that minimises the re-engagement velocity along the cutting direction, as analysed; this phenomenon has not been independently investigated for CoFe₂O₄-driven UEVC tools in previous studies but is consistent with the kinematic simulation results reported in the existing literature. Different from previous studies that merely demonstrated the reduction in surface roughness Ra induced by ultrasonic vibration cutting, this study further identifies the geometric characteristics of the tool-tip elliptical trajectory as the core physical factor dominating the machining improvement effect, which provides a universal design principle for the structural and parameter optimisation of dual-bending electronically phased UEVC tools.

8. Conclusions

This paper reports a systematic four-factor cutting-performance study of a CoFe₂O₄-driven dual-bending UEVC tool for the ultra-precision turning of H62 brass, aiming to quantify the surface improvement performance under diverse machining parameters and phase regulation and clarify the inherent kinematic mechanism of the UEVC process. The key findings and optimal machining conclusions obtained from the experimental and kinematic analyses are summarised as follows.

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A single-crystal diamond + CoFe₂O₄ dual-bending UEVC tool reduces the cutting direction Ra of H62 brass by 16%–45% across feed (0.5–20 μm/rev), 12.6%–38.7% across speed (50–375 mm/s), 22%–42% across depth (0.5–10 μm), and up to 57% via phase-angle optimisation, with respect to paired conventional cutting.

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The inter-channel phase difference is the strongest single lever; the optimum is phi = 60 deg, at which the tool-tip trajectory is an oblique ellipse whose major axis is tilted by approx 50–55 deg above the cutting direction.

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The 60 deg optimum is explained kinematically by the minimisation of the cutting direction re-engagement velocity, which in turn minimises the normal force at the chip root.

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The UEVC advantage is gated by the speed ratio K = Vc/(2 pi f_us A_y); for the present tool, the K = 1 threshold is at Vc approx 542 mm/s, so the productive operating window for finishing is Vc <= 100–200 mm/s.

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The recommended operating point on H62 brass with the present tool is as follows: f = 1 μm/rev, Vc = 50–100 mm/s, ap = 1–2 μm, phi = 60 deg, giving Ra approx 1.21 μm (cut)/1.47 μm (feed) against 2.82/3.73 μm for paired CC (57/61% reduction).

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Future work is as follows: extension to harder ductile alloys, brittle-mode trials, and factor-interaction Taguchi/response-surface study coupled with an on-line force-based closed-loop phase controller.