Analysis of the cutting ratio and investigating its influence on the workpiece’s diametrical error in ultrasonic-vibration assisted turning

Abstract

Due to its numerous advantages such as reduction of machining force and surface roughness, ultrasonic-vibration assisted turning process has been extensively investigated. In the present paper, a new vibration analysis has been done and it has been shown that in the case of rigid workpiece or stable cutting ratio, negligible diametrical error is created by tool vibration in vibration turning which is not present in conventional turning. On the other part of the study, flexible workpiece has been considered and workpiece deformation has been investigated. It has been shown that in this case, the cutting ratio experiences an increasing trend from spindle to free end of one-end fixed workpiece. It has been also shown that the experimental results are in good agreement with analysis. Workpiece diametrical error in conventional turning is about twice in vibration turning.

Keywords

Conventional cutting ultrasonic-vibration assisted turning diametrical error vibration analysis cutting ratio

Introduction

In ultrasonic-vibration assisted machining (UVAM), high-frequency vibrations are added to conventional motion of cutting tool.¹ Reduction in average cutting forces,² tool wear^3,4 and improvement of surface roughness⁵ and heat tool effects⁶ and increasing equivalent stiffness of the machining system,⁷ are some of the advantages of this process. Many researcher studied influence of machining parameters on this process such as Khajehzadeh et al.⁸ who have worked on residual stress in ultrasonic-vibration assisted turning (UVAT) and shown percentage intensity of ultrasonic power effects on generation of residual stress. These advantages have also been seen in other machining processes.⁹ The vibration can be imposed in different directions and each one has its advantages.¹⁰ The cutting ratio is the main parameter in vibration cutting. It should be noted that the underlying reason for reduced cutting forces in UVAT is the alternate separation and reconnection of tool from and to the cutting zone per each cycle of vibration. The cutting ratio is defined as the ratio of the cutting duration in each cycle of the vibration to the total cycle time.¹¹ One of the problems encountered in both the conventional turning and UVAT is the workpiece’s diametrical error being created during the machining process due to the relative movement between tool and workpiece. Different methods have already been employed for the analysis of the workpiece’s elastic deformation and the ensuing diametrical error, in CT. Among them can be named the finite difference¹² and static¹³ method, FEM,^14,15 experimental model¹⁶ and Neural network.¹⁷ The author of the present paper has investigated the diametric error of workpiece in both CT and UVAT.^18,19 However, the effect of tool vibration on the cutting ratio was not investigated. In order to study the vibration of a machining workpiece, the vibration model of a beam under the moving loads can be employed. Quyang in 2011²⁰ has reviewed the research work carried out on this subject and subsequently reported the results of his own study on the dynamic behavior of a Timoshenko beam subjected to a three-dimensional axially moving force. The developed model represents the basic features of CT with the cutting tool moving on the surface and along the axis of the workpiece. In UVAT process, in addition to the elastic deformation resulting from flexible tool-workpiece interaction, the tool vibration is also responsible for the relative movement between workpiece and tool. Therefore, rigid workpiece will be considered first and diametrical error due to tool vibration is obtained. Then, based on theories of vibrations, a deformation model for workpiece ahead of tool caused by machining forces is presented. In the present paper, minimum boundary conditions are considered, that is, workpiece is fixed to spindle (no tailstock has been used) and initial conditions are zero. Since spindle is more rigid than other parts in workpiece-tool system, it has been considered as rigid. By using these assumptions, the effect of workpiece deformation on the cutting ratio is obtained and the equations of workpiece deformation are derived. In order to verify the results of theoretical analysis, experimental tests have been conducted on diametrical error.

Machining and non-machining force and times in UVAT

The relation for cutting force can be obtained by Fourier series in the following form²¹

f (t) = \frac{Δ t_{2}}{Δ t_{total}} P + \frac{2 P}{π} \sum_{n = 1}^{\infty} \frac{1}{n} \sin (n \frac{Δ t_{2}}{Δ t_{total}} π) . \cos (2 π fnt)

(1)

Where P is the maximum of periodic cutting force; $Δ t_{2}$ and $Δ t_{total}$ are cutting time and period of tool vibration, respectively.²² f is vibration frequency in Hertz. Different time slots in cutting zone and distances from workpiece axis in actual case (i.e. applying vibration motion and cutting speed respectively to tool and workpiece) are as shown in Figure 1.

Figure 1.

Absolute values of $y^{-}$ with respect to fixed coordinate of machine at cutting zone.

In this figure $t_{n 1}$ and $t_{n 3}$ are machining interruption and restart instant respectively and $t_{n 2}$ is the moment of maximum distance of the cutting tool from the shear zone in the nth cycle of vibration. Values of $y^{-}$ is absolute displacement from fixed coordinate of machine at cutting zone and has been named according to the time of arrival to that position. Relative displacement of tool at the restart of machining operation in first pass of tool vibration ( $t_{13}$ ) is equal to its relative displacement at interruption moment of the same pass ( $t_{11}$ ) (equation (2)):²³

a \times \sin (2 π f) t_{11} + v \times t_{11} = a \times \sin (2 π f) t_{13} + v \times t_{13}

(2)

Where v is cutting speed and a is tool vibration amplitude. $t_{11}$ is determined from the following relation:

t_{11} = \frac{\sin^{- 1} (\frac{v}{2 π af})}{2 π f}

(3)

The corresponding relation for $t_{21}$ , that is, interruption moment for the next pass of orthogonal machining is as follow:

t_{21} = \frac{\sin^{- 1} (\frac{v}{2 π af})}{2 π f} + \frac{1}{f}

(4)

Therefore, cutting ratio becomes:

\frac{Δ t_{2}}{Δ t_{total}} = \frac{t_{21} - t_{13}}{t_{21} - t_{11}}

(5)

$t_{12}$ is the time at which tool reaches to its maximum distance from cutting zone during separation time and $t_{a max 1}$ is the time at which tool reaches to its maximum first pass course during cutting. Diametrical error is the difference between workpiece actual diameter and nominal diameter:²¹

Δ D (t) = d (t) - (D - 2 a_{p}) t_{n 3} < t < t_{(n + 1) 1}

(6)

D is the initial diameter of workpiece and $a_{p}$ is the nominal cutting depth. Actual cutting depth ( $a'_{p} (t)$ ) is also the difference between nominal depth and diametrical error, in the following form:

a'_{p} (t) = a_{p} - \frac{Δ D (t)}{2} t_{n 3} < t < t_{(n + 1) 1}

(7)

When cutting tool clamped under workpiece center as an amplitude, the extended profile of workpiece and the previously mentioned parameters can be presented (in the exaggerated form) as in Figure 2:

Figure 2.

Extended profile of workpiece surface.

Equation (6) and Figure 2 mean that the cutting tool vibration proper produce diametrical error without even considering the flexibility of the workpiece. This can be a serious disadvantage of vibration assisted machining. However, several examples are presented in the following section which illustrate that the resulting error ( $(a_{p} - a'_{p} (t))_{max} (mm)$ ) is negligible in the order of few microns. This order of error will be shown later in this paper that cannot overshadow the advantage of vibration assisted cutting in decreasing the diametrical error of the machined workpiece in actual conditions where the flexibility of the workpiece is taken into consideration.

Some examples

By calculating $t_{n 3}$ and $t_{a max n}$ and inserting them in equation (6), maximum and minimum of diametrical error can be obtained. Cutting force in cutting speed direction can be defined based on feed ( $f_{r}$ ) and cutting depth ( $a'_{p} (t)$ ) parameters as in equation (8):¹⁷

F_{y} = k_{s} \times a'_{p} (t) \times f_{r}

(8)

In the equation (8), $k_{s} (N / m m^{2})$ is a specific force which depends on workpiece material. In this section, maximum height of workpiece profile error ( $(a_{p} - a'_{p} (t))_{max}$ in Figure 2) and difference of cutting force ( $Δ Force$ ) relative to when tool does not vibrate (and workpiece is considered rigid) for nine machining cases are summarized in Table 1. Specific force for Aluminum 7075 is calculated from previous experiments²⁴ ( $k_{s} = 500 (N / m m^{2})$ ) and feed has been also set as $f_{r} = 0.1 (mm / rev)$ .

Table 1.

Effects of vibratory tool on maximum height of workpiece profile error and cutting force.

$Example$	$D (mm)$	$a_{p} (mm)$	$a (μ m)$	$(a_{p} - a'_{p} (t))_{max} (mm)$	$Δ Force (N)$
1	30	1	8	0.0000	0.00
2	30	1	40	0.0001	0.01
3	3	0.1	8	0.0000	0.00
4	0.3	0.01	8	0.0002	0.03
5	0.3	0.01	16	0.0009	0.14
6	0.3	0.01	32	0.0036	0.54
7	0.3	0.01	4	0.0001	0.01
8	0.3	0.01	2	0.0000	0.00
9	0.3	0.01	50	0.0086	1.30

This table shows that tool vibration slightly increases diametrical error of rigid workpiece and reduces its cutting force.

Workpiece vibration analysis due to machining force

One-end clamped workpiece was considered for machining process. By using Hamilton principle, the equation for lateral motion of Euler-Bernoulli beam is obtained through minimizing the time integral of the difference between kinetic and potential energies which is as follows:²⁵

m \frac{\partial^{2} w}{\partial t^{2}} + \frac{\partial^{2}}{\partial z^{2}} [EI (z) \frac{\partial^{2} w}{\partial z^{2}}] = F (z, t)

(9)

Where E is workpiece modulus of elasticity, m is mass per unit length and $I (z)$ is workpiece moment of inertia at point z which here is considered as constant. w is the lateral displacement for each point of workpiece. $F (z, t)$ is excitation force at time t and point z. Since workpiece as a continuous system has infinite degrees of freedom which should be solved, then modal analysis has been employed. The solution of excited vibration derived from modal analysis is as follows:²⁶

w (z, t) = \sum_{i = 1}^{\infty} ϕ_{i} (z) q_{i} (t)

(10)

where $q_{i} (t)$ is general coordinate and $ϕ_{i} (t)$ is mode shape. At spindle, deformation and slope of deformed workpiece are zero. Free surface of workpiece does not tolerate moment and cutting force. Therefore, if $ϕ_{i} (z)$ is ith mode shape, then boundary conditions of workpiece are as follow:

\begin{matrix} 1) φ_{i} (z) |_{z = 0} = 0 {\frac{d φ_{i} (z)}{dz} |}_{z = 0} = 0 \\ {2) \frac{d^{2} φ_{i} (z)}{d z^{2}} |}_{z = l} = 0 {\frac{d^{3} φ_{i} (z)}{d z^{3}} |}_{z = l} = 0 \end{matrix}

(11)

By applying these boundary conditions, mode shapes $φ_{i} (t)$ and frequency relations can be obtained. The corresponding relation for frequency is as follows:²⁷

\cos (\sqrt[4]{\frac{m}{EI}} \sqrt{ω_{i}}) l \cosh (\sqrt[4]{\frac{m}{EI}} \sqrt{ω_{i}}) l = - 1

(12)

Where $ω_{i}$ is natural frequency of ith mode. It is assumed that tool force on workpiece is like a moving concentrated force. Since tool force is exerted to workpiece at tool position, therefore, the effect of force is only present at tool position, that is, $f_{r} Nt$ and it is zero elsewhere. $f_{r}$ is feed in $mm / rev$ , N is rotational speed in $rev / min$ and l is workpiece length out of spindle. For applying these conditions in relations, vibration turning force (equation (1)) appears in the following form in equation (9):

\begin{matrix} F (z, t) = (\frac{Δ t_{2}}{Δ t_{total}} P + \frac{2 P}{π} \\ \sum_{n = 1}^{\infty} \frac{1}{n} \sin (n \frac{Δ t_{2}}{Δ t_{total}} π) . \cos (2 π fnt)) δ (z - f_{r} Nt) \end{matrix}

(13)

where $δ$ is Kronecker delta. From the above relations and by using Refs.,^26,27 the corresponding relation for the ratio of workpiece elastic deformation of UVAT to maximum vibration turning force (P) with respect to position and time becomes as follows:

\begin{matrix} \frac{w (z, t)}{P} = \\ \sum_{i = 1}^{\infty} ((\frac{(\cos \sqrt[4]{m / EI} \sqrt{ω_{i}} l + \cosh \sqrt[4]{m / EI} \sqrt{ω_{i}} l)}{(\sin \sqrt[4]{m / EI} \sqrt{ω_{i}} l - \sinh \sqrt[4]{m / EI} \sqrt{ω_{i}} l)} \times (\cos \sqrt[4]{m / EI} \sqrt{ω_{i}} z \\ - \cosh \sqrt[4]{m / EI} \sqrt{ω_{i}} z)) + (\sin \sqrt[4]{m / EI} \sqrt{ω_{i}} z - \sinh \sqrt[4]{m / EI} \sqrt{ω_{i}} z)) \times \frac{1}{m_{ii} ω_{i}} \\ \times \int_{0}^{t} ((\frac{(\cos \sqrt[4]{m / EI} \sqrt{ω_{i}} l + \cosh \sqrt[4]{m / EI} \sqrt{ω_{i}} l)}{(\sin \sqrt[4]{m / EI} \sqrt{ω_{i}} l - \sinh \sqrt[4]{m / EI} \sqrt{ω_{i}} l)} \times (\cos \sqrt[4]{m / EI} \sqrt{ω_{i}} f_{r} N τ \\ - \cosh \sqrt[4]{m / EI} \sqrt{ω_{i}} f_{r} N τ)) + (\sin \sqrt[4]{m / EI} \sqrt{ω_{i}} f_{r} N τ - \sinh \sqrt[4]{m / EI} \sqrt{ω_{i}} f_{r} N τ)) \\ \times (\frac{Δ t_{2}}{Δ t_{total}} + \frac{2}{π} \sum_{n = 1}^{\infty} \frac{1}{n} \sin (n \frac{Δ t_{2}}{Δ t_{total}} π) . \cos (n 2 π f τ)) \\ \times \sin ω_{i} (t - τ) d τ \end{matrix}

(14)

In this relation, except maximum vibration turning force (P), all other parameters are known. Therefore, this has been extracted from the rest of the equation. This makes the comparison with CT more easily. This relation is applicable for investigation of workpiece deformation in tangential and radial directions. P is the maximum cutting force and since it cannot be measured from experimental tests (due to high speed and high frequency), finite element analysis has been used instead. The results show that the value of P is very close to that in CT, therefore, here in order to compare UVAT and CT processes and investigate the effect of tool vibrations, the experimental results of CT has been used instead of P. The last natural frequency which is considered in analysis should be larger than tool vibration frequency. This is due to the fact that if tool vibration frequency is larger than all frequencies of natural modes, then one cannot expect that mode of vibration frequency would be excited. On the other hand, if too many modes are considered, computational relations will become several times larger which consequently makes it impossible to solve the problem by available computers. Also, by increasing number of modes, their effect on solution will diminish. In order to solve equation (14), first it is needed to derive natural frequencies from equation (12). There isn’t any analytical solution for equation (12), so, numerical solution should be used. Maple 14 software has been used for that purpose. If it is assumed that workpiece is from Aluminum 7075 (density of 2823 kg/m³ and modulus of elasticity $7.4426 \times 10^{10}$ Pa) of 500 mm length and 30.36 mm initial diameter, the first 10 natural frequencies of this workpiece are given in Table 2.

Table 2.

First 10 natural frequencies (in Hz).

$f_{1}$	$f_{2}$	$f_{3}$	$f_{4}$	$f_{5}$
86.198	540.195	1512.564	2964.022	4899.741
$f_{6}$	$f_{7}$	$f_{8}$	$f_{9}$	$f_{10}$
7319.365	10222.914	13610.389	17481.789	21837.114

If it is assumed that the vibration frequency of tool is in ultrasonic ranges (20 kHz) the frequency of 10th mode is greater than 20 kHz. Therefore these 10 modes have been considered in the analysis. Through numerical solution of equations (4) and (5) in Maple, the graph of $Δ t_{2} / Δ t_{total}$ with respect to tool vibration amplitude at tool vibration frequency of 20 kHz and cutting speed of 28.27 m/min (rotational speed of 300 rpm and workpiece diameter of 30 mm), is obtained and shown in Figure 3.

Figure 3.

$Δ t_{2} / Δ t_{total}$ versus tool vibration amplitude at vibration frequency of 20 kHz and cutting speed of 28.27 m/min.

According to Figure 3, for a vibration amplitude of 8 μm, $Δ t_{2} / Δ t_{total}$ for a rigid workpiece becomes 0.45. From equation (14), the graph of ratio of workpiece deformation to force at tool position and with respect to cutting time ( $w (z, t) / P$ ) can be obtained. If cutting depth be 1 mm, feed 0.1 mm/rev, spindle speed 300 rpm, workpiece length 500 mm and tool vibration frequency 20 kHz be assumed, then the ratio of workpiece deformation to maximum vibration machining force at tool position and at a distance of 12.5 to 420 mm from spindle is shown in the Figure 4.

Figure 4.

Ratio of workpiece deformation to maximum cutting force along workpiece ( $N = 300 rev / min$ $l = 500 mm$ $f = 20000 Hz$ $a = 8 μ m$ ).

In the Figure 4, a part of the main curve (at distance from 206.755 to 206.75505 mm) has been magnified. (Distance from two vibration peaks are $0.00049795 - 0.0004975 = 0.00000045 mm / N$ .) For instance, if it is assumed that machining force in CT is 100 N, then workpiece vibration amplitude becomes $0.00000045 / 2 mm / N \times 100 N = 0.0225 μ m$ therefore, when compared to tool vibration amplitude of $8 μ m$ , workpiece obeys static trend. Workpiece moves upward by an amount $w (t)$ in each machining time. During this condition, if workpiece can be assumed to be fixed, then this can be inferred that tool vibration center moves downward by an amount $w (t)$ . Therefore, based on these statements and also Ref.,¹⁸ if radial deformation of workpiece is $w_{R} (t)$ , then the relation for diametrical error (7) becomes:

\begin{matrix} Δ D (t) = 2 \sqrt{{[\frac{D}{2} - a_{p} + w_{R} (t)]}^{2} + {(w (t) + a \sin (ω t))}^{2}} \\ - D + 2 a_{p} t_{n 3} < t < t_{(n + 1) 1} \end{matrix}

(15)

Therefore, by determining the start time of machining and end of tool movement in each pass and putting these times in equation (15), minimum and maximum of diametrical error can be obtained. From these two times, the time at which tool arrives to its end is when sinusoidal curve of tool vibration has a positive extremum (if each cutting pass is denoted by n, conditions 16 hold).

\begin{matrix} \overset{\cdot}{y} = a ω \cos (ω t_{a max n}) = 0 \\ \overset{\cdot\cdot}{y} = - a ω^{2} \sin (ω t_{a max n}) < 0 \end{matrix}

(16)

From these relations, the times at which tool reaches to its maximum amplitude in each machining pass can be formulated in the following form:

t_{a max n} = ((n - 1) \times 3 + n) \frac{π}{2 ω}

(17)

Thus, only machining start time ( $t_{n 3}$ ) needs to be determined.

The relation 15 shows the Increasing amplitude (a) cause increase in diametrical error and increasing ultrasonic frequency (more than 20 kHz), didn’t change workpiece deformation ( $w (t)$ )²¹ and so it can be concluded that frequency parameter in ultrasonic rage cannot effect on diametrical error.

Determining cutting times by considering workpiece deformation

If the time for the start of the separation process for a flexible workpiece in first pass and second pass are respectively $t_{11 d}$ and $t_{21 d}$ , then the start of connection for flexible workpiece ( $t_{13 d}$ ) is the time when its relative displacement is equal to relative displacement at separation time. Therefore, the following relation is obtained:

\begin{matrix} w (t_{11 d}) + v t_{11 d} + a \sin ω t_{11 d} = w (t_{13 d}) + v t_{13 d} \\ + a \sin ω t_{13 d} \end{matrix}

(18)

The times of start of Separation (The $t_{11 d}$ and $t_{21 d}$ ) are obtained by applying motions to tool and setting the relative speed to zero From following

\begin{matrix} \overset{\cdot}{w} (t_{11 d}) + v + a ω \cos ω t_{11 d} = 0 \\ \overset{\cdot}{w} (t_{21 d}) + v + a ω \cos ω t_{21 d} = 0 \end{matrix}

(19)

Similar to equation (5), cutting ratio ( $Δ t_{2 d} / Δ t_{totald}$ ) for a flexible workpiece is in the form of equation (20):

\frac{Δ t_{2 d}}{Δ t_{totald}} = \frac{t_{21 d} - t_{13 d}}{t_{21 d} - t_{11 d}}

(20)

From a distance of 12.5 mm to 420 mm along workpiece and in space intervals of 2.5 mm, and by using equation (20) and considering workpiece deformation due to machining forces, the ratio of cutting time to period along workpiece has been obtained and shown in Figure 5.

Figure 5.

Cutting ratio by taking into account the workpiece deformation ( $N = 300 rev / min$ $l = 500 mm$ $f = 20000 Hz$ $a = 8 μ m$ ).

As it is clear from the figure, as the distance from spindle and subsequently deformation increases, the distance between maximum and minimum values of $Δ t_{2 d} / Δ t_{totald}$ increases.

The most important parameter in this process is this value which influences machining conditions. (Even small) changes in this parameter can lead to significant effects on stability of the process and as it has been mentioned earlier, it directly influences on force. Therefore, the schematic of cutting force can be illustrated in the Figure 6:

Figure 6.

Effect of workpiece springback on cutting ratio.

Experimental test

In order to experimentally investigate the spring-back in vibration turning, a setup has been prepared. The necessary vibration has been provided by a Steckmann GmbH generator and transducer and has been applied on tool in cutting direction. For force measurement, a Kistler dynamometer (model 9257BA) has been employed. A TME40 CNC turning machine has been used for the turning process of AL7075 cylindrical workpiece. The vibration amplitude and frequency were 8 micrometers and 20 kHz respectively. A titanium horn designed and fabricated for concentering and transferring mechanical vibration to cutting tool. A Sandvik VBMT 160404 tool has been also used. Clearance angle was 5°, rake angle was 4°. Cutting depth 1 mm, feedrate 0.1 mm/rev and spindle speed 300 rpm have been adjusted. Workpiece length was 600 mm from which 500 mm has been rest out of the spindle.

For the workpieces used in CT and UVAT to have the same mechanical properties and initial deformations, they have been chosen from a 6 mbar. The two ends of this bar has been cut 0.5 m, therefore, the initial workpiece diameter along its length has been found to be $30.34 \pm 0.01$ in vibration turning and $30.16 \pm 0.015$ in CT. when the force is applied for the first time to workpiece, dynamometer needs to reach a steady state for recording correct values. Two factors should be taken into consideration: first is backlash which causes a delay in processing time of dynamometer. And second is overshoot due to dynamic of the process which has an oscillatory mode before acquiring steady state. For stability of machining process, 13 sections 10 mm apart from each other have been considered along workpiece length. From strength of materials science, the following analytical-geometrical relation for diametrical error in CT¹⁸ holds:

\begin{matrix} Δ D (z) = 2 {{[\frac{D}{2} - a_{p} + \frac{F_{x} {(f_{r} \times N \times t)}^{3}}{3 EI}]}^{2} \\ {+ {(\frac{F_{y} {(f_{r} \times N \times t)}^{3}}{3 EI})}^{2}}}^{0.5} - D + 2 a_{p} \end{matrix}

(21)

Where $F_{y}$ is cutting force in cutting speed direction and $F_{x}$ is cutting force in radial direction. The function raised to power 0.5 is a sixth degree polynomial which finally gives a third order polynomial after being raised to power 0.5. As it has been mentioned earlier, in UVAT process, each natural mode plays a role in workpiece deformation (equation (10)). However, the most important and main mode in vibration turning is fundamental mode which has most energy and its shape is generally similar to static deformation curve for workpiece²⁷ and in most approximate methods of natural frequency such as Rayleigh method, is considered as the first mode shape. In the present paper, it has been analytically proven that curve of workpiece deformation due to cutting force obeys a static behavior in vibration turning which is like a third-order curve. In fact, first mode is dominant. Therefore, it can be stated that function of diametrical error in UVAT is similar to that in CT and can be approximated by a third-order function. For comparing theoretical and experimental results, a curve fit should be performed on experimental data. For curve fitting a set of data, some physical points should be taken into account: first, the first derivative of curve fitted function does not change sign at point zero, that is, at spindle position because, from the physical point of view, the trend of workpiece deformation from its free surface to spindle does not change. Second, the second derivative of the curve fitted function does not change sign along its length since again from the physical point of view; there is not a turning point in the deformation curve of an isotropic workpiece subjected to a concentrated load.

Comparison of experimental and analytical results

In order to compare the experimental data with analytical results, the final diameter of workpiece and real cutting depth have been used in the present paper.

Deriving diametrical error from analytical relations and comparing CT and UVAT processes

In the performed analysis, diametrical error can be obtained from calculating elastic deformation of workpiece in cutting and radial directions. In order to provide a comparison between analytical and experimental results, the obtained forces from CT experiments in each workpiece section have been replaced in equation (14) equation. If $F_{CExp}$ , is the experimentally obtained cutting force for CT in cutting direction and $F_{RExp}$ , is the experimentally obtained radial force in CT, then elastic deformations of workpiece in UVAT in both radial and cutting directions can be obtained from equation (22). $w (t) / P$ can be obtained from equation (14) for each section of workpiece.

\begin{matrix} w_{R} (t) = (\frac{w (t)}{P}) \times F_{RExp} \\ w (t) = (\frac{w (t)}{P}) \times F_{CExp} \end{matrix}

(22)

Diametrical error in CT is calculated from equation (21) and maximum of diametrical error in UVAT can be obtained by substituting elastic deformations of workpiece (equation (22)) into equations (15), (17), and (18). The percentage of reduction of diametrical error in UVAT process in comparison with CT process with respect to distance from spindle has been shown in Figure 7.

Figure 7.

Percentage of reduction of diametrical error in UVAT process in comparison with CT process with respect to distance from spindle ( $N = 300 rev / min$ $l = 500 mm$ $f = 20000 Hz$ $a = 8 μ m$ ).

Figure 7, shows the percentage of reduction of diametrical error in UVAT process in comparison with CT process increases with increasing the distance from spindle.

Comparison of experimental and analytical results by taking diametrical error into account

In order to compare experimental and analytical results, final diameter of workpiece has been used. By using the data of the authors’ previous article,²¹ Figure 8 has been drawn. This figure shows the percentage of error between experiments and analysis for both CT and UVAT processes when final diameter of workpiece ( $d (t)$ ) has been taken into account.

Figure 8.

Comparison of percentage of diametrical error between analytical solution and experimental data by taking final diameter of workpiece into account ( $N = 300 rev / min$ $l = 500 mm$ $f = 20000 Hz$ $a = 8 μ m$ ).

As it is obvious from the figure, the discrepancies between analytical and experimental results are small and it can be said that this discrepancy becomes less and less as we move toward spindle. Also, the error between analytical and experimental results in UVAT is less than in CT. Therefore, in order to avoid the error due to deformation of workpiece, cutting depth should change along workpiece and a depth equal to ( $(a_{p} - a'_{p} (z)) + a_{p} = 2 a_{p} - a'_{p} (z)$ ) should be given to CNC.

Conclusion

In the present study, it has been shown that tool vibration causes negligible diametrical error in ultrasonic vibration assisted turning of a rigid workpiece. It has been shown that vibration amplitude of workpiece in UVAT process is much less than tool vibration amplitude and is close to static value. The contact and separation time of tool and workpiece and also machining time has been determined by taking into account the workpiece deformation. And it has been also shown that as workpiece deformation increases, the difference between maximum and minimum cutting time along workpiece increases which changes cutting force in each pass and finally leads to instability. Therefore, changes in the cutting ratio and its effects on stability of machining, can little affect the diametrical error. But both theoretical and experimental results show the average diametrical error of workpiece in the ultrasonic vibration assisted turning process is smaller than conventional turning.

Footnotes

Appendix

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

H. Soleimanimehr

References

Dixit

Pandey

Verma

. Ultrasonic-assisted machining processes: a review. Int J Mechatron Manuf Syst 2019; 12(3/4): 227–254.

Soleimanimehr

Nategh

Jamshidi

. Mechanistic model of work-piece diametrical error in conventional and ultrasonic assisted turning. Adv Mater Res 2012; 445: 911–917.

Liu

Wang

, et al. Experimental study on tool wear in ultrasonic vibration–assisted milling of C/SiC composites. Int J Adv Manuf Technol 2020; 107(1): 425–436.

Khajehzadeh

Boostanipour

Amiri

, et al. The influence of ultrasonic elliptical vibration amplitude on cutting tool flank wear. Proc IMechE, Part B: J Engineering Manufacture 2020; 234(12): 1499–1512.

Puga

Grilo

Carneiro

. Ultrasonic assisted turning of Al alloys: influence of material processing to improve surface roughness. Surfaces 2019; 2(2): 326–335.

Gholamzadeh

Soleimanimehr

. Finite element modeling of ultrasonic assisted turning: cutting force and heat generation. Mach Sci Technol 2019; 23(6): 869–885.

Chen

Cheng

, et al. Investigation of strengthening effect on the machining rigidity in longitudinal torsional ultrasonic milling of thin-plate structures. Proc IMechE, Part B: J Engineering Manufacture 2019; 234(3): 665–670.

Khajehzadeh

Boostanipour

Razfar

. Finite element simulation and experimental investigation of residual stresses in ultrasonic assisted turning. Ultrasonics 2020; 108: 106208.

Unyanin

Khusainov

. Study of forces during ultrasonic vibration assisted grinding. Procedia Eng 2016; 150: 1000–1006.

10.

Zhang

Guo

Ehmann

, et al. Effects of ultrasonic vibrations in micro-groove turning. Ultrasonics 2016; 67: 30–40.

11.

Brehl

Dow

. Review of vibration-assisted machining. Precis Eng 2008; 32(3): 153–172.

12.

Qiang

. Finite difference calculations of the deformations of multi-diameter workpieces during turning. J Mater Process Technol 2000; 98(3): 310–316.

13.

Mayer

JRR

VũPhan

Cloutier

. Prediction of diameter errors in bar turning: a computationally effective model. App Math Model 2000; 24(12): 943–956.

14.

Malluck

Melkote

. Modeling of deformation of ring shaped workpieces due to chucking and cutting forces. J Manuf Sci Eng 2004; 126(1): 141–147.

15.

Phan

Baron

Mayer

JRR

, et al. Finite element and experimental studies of diametral errors in cantilever bar turning. App Math Model 2003; 27(3): 221–232.

16.

Topal

Çoğun

. A cutting force induced error elimination method for turning operations. J Mater Process Technol 2005; 170(1–2): 192–203.

17.

Benardos

Mosialos

Vosniakos

. Prediction of workpiece elastic deflections under cutting forces in turning. Robot Comput Integr Manuf 2006; 22(5–6): 505–514.

18.

Soleimanimehr

Nategh

Amini

. Analysis of diametrical error of machined workpieces in ultrasonic vibration assisted turning. Adv Mater Res 2011; 264–265: 1079–1084.

19.

Soleimanimehr

Nategh

. An investigation on the influence of cutting-force’s components on the work-piece diametrical error in ultrasonic-vibration-assisted turning. AIP Conf Proc 2011; 1315(1): 1145–1150.

20.

Ouyang

. Moving-load dynamic problems: a tutorial (with a brief overview). Mech Syst Signal Process 2011; 25(6): 2039–2060.

21.

Soleimanimehr

Nategh

Najafabadi

, et al. The analysis of the timoshenko transverse vibrations of workpiece in the ultrasonic vibration-assisted turning process and investigation of the machining error caused by this vibration. Precis Eng 2018; 54: 99–106.

22.

Sharma

Pandey

. Mechanistic based cutting force model during ultrasonic assisted turning with self-lubricating cutting inserts. Int J Adv Manuf Syst 2019; 18(01): 133–155.

23.

Razavi

Nategh

Abdullah

. Analytical modeling and experimental investigation of ultrasonic-vibration assisted oblique turning, part II: dynamics analysis. Int J Mech Sci 2012; 63(1): 12–25.

24.

Nategh

Amini

Soleimanimehr

. Modeling the force, surface roughness and cutting temperature in ultrasonic vibration-assisted turning of al7075. Adv Mat Res 2010; 83–86: 315–325.

25.

Han

Benaroya

Wei

. Dynamics of transversely vibrating beams using four engineering theories. J Sound Vib 1999; 225(5): 935–988.

26.

Meirovitch

. Fundamentals of vibrations. International ed. Long Grove, IL: McGraw-Hill, 2001.

27.

Rao

. Vibration of continuous systems. 4th ed. New York: John Wiley & Sons, 2007.