Modeling and simulation of surface roughness in ultrasonic assisted magnetic abrasive finishing process

Accepted Manuscript

Modeling and simulation of surface roughness in ultrasonic assisted magnetic abrasive finishing process Aviral Misra , Pulak M. Pandey , U.S. Dixit PII: DOI: Reference:

S0020-7403(17)31961-6 10.1016/j.ijmecsci.2017.08.056 MS 3908

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

18 July 2017 23 August 2017 30 August 2017

Please cite this article as: Aviral Misra , Pulak M. Pandey , U.S. Dixit , Modeling and simulation of surface roughness in ultrasonic assisted magnetic abrasive finishing process, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.08.056

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ACCEPTED MANUSCRIPT Highlights  A novel approach to mathematically model the surface roughness during UAMAF based on process physics has been presented. The model incorporates and recognizes the existence of a critical roughness value; further finishing does not improve the roughness value.



The instantaneous rate of reduction in surface roughness has been considered as a function of the instantaneous value of surface roughness.



The various constants used during the modeling of the surface roughness were determined by inverse estimation from the experimental observations.



The model establishes an exponential correlation between instantaneous roughness value and finishing time during finishing.



The maximum difference in predicted and experimental values of surface roughness was found to be ±7.35%.

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

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ACCEPTED MANUSCRIPT Modeling and simulation of surface roughness in ultrasonic assisted magnetic abrasive finishing process Aviral Misraa, Pulak M. Pandeya*, U.S. Dixitb a b

Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India–110016.

Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India-781039.

Abstract

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An ultrasonic assisted magnetic abrasive finishing (UAMAF) is a hybrid finishing process. In UAMAF, ultrasonic vibrations are introduced into the finishing zone of magnetic abrasive finishing (MAF) process to finish the workpiece surface more efficiently as compared to MAF in the nanometer range. In the present work, a model of surface roughness during UAMAF process has been presented. The model assumes that the instantaneous rate of

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change of surface roughness is proportional to the material removal rate (MRR) and the amount of irregularities present on the surface. A model of MRR was presented, considering it to be attributed to two simultaneous and independent phenomena— a steady state material removal and a transient material removal. The MRR model has further been used to model the instantaneous surface roughness during finishing. The surface roughness model not only

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incorporates the effect of initial surface roughness value but also assimilates a critical surface roughness value below which no reduction in roughness is possible. The constants included

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in the modeling are predicted by using an inverse method. A simulation of 3‒D surface roughness profile has also been presented to visualize the effect of finishing numerically. The

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developed model predicted the surface roughness in UAMAF as a function of supply voltage, working gap, rpm of the electromagnet, amplitude and frequency of ultrasonic vibration,

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hardness and initial surface roughness of the workpiece. The predicted value shows a good agreement with the experimental observation with a maximum deviation of ±7.35%. The model affirms that an exponential correlation exists between instantaneous surface roughness

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value and finishing time during finishing. Keywords: UAMAF; MAF; Modeling; Simulation; Surface roughness; Material removal; Flexible abrasive finishing

*Corresponding author. Department of Mechanical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India. Email: [email protected] Tel.: +9111 2659 6083; Fax: +9111 2658 2053.

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ACCEPTED MANUSCRIPT Nomenclature Amplitude of vibration Area of surface finished by electromagnet Horizontal projected cross−sectional area of indentation Transient coefficient Surface roughness coefficient

Diameter of impression of abrasive particle

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Mean diameter of abrasive particles

Average normal force acting on single abrasive particle Frequency of vibration Brinell hardness number

Constant

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Peak to valley height

Steady state material removal coefficient RPM of the electromagnet

Number of active abrasive particles

Mass material removal rate UAMAF

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̇

Steady state mass material removal rate ̇

Transient mass material removal rate

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̇

Mass of the irregularities available for transient removal

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Centerline average surface roughness value Initial centerline average surface roughness value

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Critical surface roughness (minimum theoretical) value Instantaneous surface roughness value at time .

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Radius of path of abrasive particle Finishing time Depth of the indentation Resultant velocity of an abrasive particle at time Initial volume of irregularities Resultant average velocity of an abrasive particle Instantaneous volume of irregularities Volume of irregularities available after time 3

ACCEPTED MANUSCRIPT Angle subtended by indented portion on center of abrasive particle Correlation lengths Distance from center of abrasive particle to workpiece surface Density of the workpiece material standard deviation (or RMS value) of surface heights Flow stress of workpiece material

1

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Angular velocity of the abrasive particle Introduction

The surface finish of a component plays a significant role in its quality for cases requiring precision fits or application of cyclic loading. The rapid progress in the semiconductor, optical, electronic, atomic energy, aerospace component industry, etc., increased the

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significance of the quality of surface finish and integrity [1]. These advancements lead to a greater demand for fine surface finishing capabilities in a varied range of industrial applications as the finishing operation is a vital and costly segment of the whole production process. Due to the rapid advancements in the field of materials, the materials having properties like high hardness, toughness, high strength to weight ratio and fragility have now

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become popular in industries. Extraordinary features make them demanding for different industrial applications. Finishing of such materials by conventional finishing processes is the

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principal challenge for the manufacturing industry [2]. The traditional finishing processes like grinding, honing and lapping create micro/nano burrs, subsurface damage and residual

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stresses. They are also unable to finish the fragile materials like glass. Achieving the surface roughness values of the order of nanometer by conventional finishing processes is an onerous

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and an uneconomical proposition. In the last few decades, developments have taken place for fine finishing of these materials. The application of flexible abrasive process with the use of

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gentle forces is an effective solution to the problem of obtaining a nano-level surface finish. In some flexible abrasive finishing process, abrasive particles are supported by a carrier, whose properties are controlled by an external magnetic field. These include magnetic abrasive finishing (MAF), magnetic float polishing (MFP) and magnetorheological finishing (MRF). MAF uses magnetic force in the finishing zone to allow the flexible abrasive particles to shear‒off the material in the form of microchips. Recently, ultrasonic assisted magnetic abrasive finishing (UAMAF), an improved version of MAF to enhance process capabilities of MAF had been introduced.

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ACCEPTED MANUSCRIPT UAMAF (Fig. 1) is a hybrid finishing process in which ultrasonic vibrations are being applied in the finishing zone of magnetic abrasive finishing (MAF) process for attaining enhanced surface topology within a reasonably shorter period of time. It was observed experimentally that UAMAF produces better results as compared to MAF for the same set of finishing parameters [3]. Fig. 1 depicts a schematic as well as an actual system of the UAMAF setup. The set up consisted of a specially designed fixture for the workpiece and an ultrasonic vibration generator unit. The electromagnet had four poles arranged alternately as

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north and south poles to generate circular magnetic lines of forces. The circular magnetic line of forces are capable to hold the abrasives along the direction of cutting (tangential direction) [4]. Ultrasonic vibrations were applied to the workpiece fixture by a horn attached to the transducer. Thus, a vibratory motion is provided to the workpiece surface in addition to the rotation of electromagnet. The vibratory motion enhances the interaction of flexible magnetic

(a)

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abrasive brush (FMAB) with the workpiece surface.

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Spindle of CNC milling Electromagnet

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Ultrasonic horn Workpiece

Workpiece fixture

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(b)

Fig. 1: Experimental setup of UAMAF: (a) Schematic diagram; (b) Actual system

In UAMAF, a uniform mechanical mixture of the abrasive and ferromagnetic particle is used and is called unbonded (or loose) magnetic abrasive particles (UMAPs). The influence of magnetic field impels the ferromagnetic particles in UMAPs to position along the magnetic

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lines of force due to magnetic dipole interaction. The non‒magnetic abrasive particles get entrapped in between these chains, thus forming the FMAB that acts as a multi‒point cutting

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tool. During the interaction of FMAB with the workpiece surface, only a fraction of total abrasive particles come in contact with the workpiece surface and take part in the finishing

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action. They are known as ‗active abrasive particles‘. The magnetic levitation force getting transferred on these non‒magnetic active abrasive particles forces them to create an

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indentation onto the workpiece surface. These indented active abrasive particles induce abrasion on the workpiece due to a relative motion by the rotation of electromagnet, which is

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additionally enhanced due to ultrasonic vibrations. Also, the interaction of FMAB with the workpiece surface generates contact stresses at the contact points. If the generated contact stresses are more than a critical value, it shears‒off the irregularities at the contact point. Therefore, in UAMAF the material is removed due to nano‒scratching and micro‒chipping in the form of micro‒chips [5]. The ultrasonic vibrations induce high kinetic energy (or momentum) in abrasive particles, resulting in large shearing of irregularities and thereby increasing the rate of reduction of surface roughness [6].

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ACCEPTED MANUSCRIPT Kim and Choi [7] simulated the cylindrical MAF assuming conical shaped abrasive particles. They used the equation for micro-cutting to develop an expression for the volumetric material removal rate and predicted surface roughness as a function of finishing time. The results for surface roughness agree well with the experimental observations for the lower values of magnetic flux density. Kremen et al. [8] observed experimentally that the material removal rate (MRR) was high in the initial phase of the process due to large out-of-roundness (OOR) of surface profile and decreased with a decrease in unevenness. By their findings, they

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established an empirical expression to evaluate machining time to produce the surface with specified out-of-roundness value. They observed a hyperbolic relationship between OOR error and finishing time. Jayaswal et al. [9] assumed spherical shaped abrasive particles interacting with surface irregularities of triangular profile distributed uniformly over the workpiece surface. They performed a two-dimensional finite element analysis of magnetic

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field to calculate normal magnetic force for the estimation of indentation on the surface of the workpiece. They estimated the volume of material removed by an indented abrasive particle, which was used to estimate the surface roughness with finishing time. Wang and Hu [1] observed that MRR is critically affected by finishing speed, abrasive

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material, abrasive concentration and abrasive size. They also concluded that MRR increases linearly with finishing speed and decreases with finishing time until saturation is reached and

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a steady-state material removal occurs. The study of MAF for finishing cylindrical surface by Jain et al. [10] brings out the working gap and the circumferential speed of the workpiece as

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the critical parameters, which significantly affect the MRR and the percentage change in surface roughness. Chang et al. [11] investigated the effect of concentration, shape, and

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hardness of abrasive particles on MRR and surface roughness for cylindrical MAF. They observed that initial reduction in surface roughness was high that gradually reduced to a

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steady-state value called saturated surface roughness. Yin and Shinmura [12] applied the vertical vibration to magnetic abrasive polishing for the surface and edge finishing and performed deburring of magnesium alloy in three different modes— horizontal (X), vertical (Z) and combined (X) and (Z). On the application of vertical vibrations micro-burrs in magnesium alloy could easily be removed and deburring efficiency was considerably increased. Judal et al. [13] applied vibrations in cylindrical magnetic abrasive finishing. They observed experimentally that the material removal and change in surface roughness were improved approximately by 100% and 150%, respectively, as compared to MAF without vibrations. 7

ACCEPTED MANUSCRIPT Mulik and Pandey [3] experimentally investigated the surface roughness in UAMAF and analyzed the observations using response surface methodology to conclude that voltage is the most dominating parameter followed by percentage weight of abrasives, pulse‒on time of the ultrasonic vibrations and the rotation speed of the electromagnet. They also observed that the percentage change in surface roughness was higher in UAMAF than that in MAF for the similar set of parameters.

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The aforementioned literature survey reveals that the attempts have been made to mathematically model the surface roughness [7‒9] and to experimentally analyze the material removal and surface roughness as a function of different process parameters of MAF process [1,10,11]. Researchers studied the effect of vibrations on the material removal and surface roughness during MAF [12,13]. So far, only one attempt [3] was made to experimentally

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analyze the surface roughness in UAMAF as a function of the various process parameters, and hitherto no attempt has been made to mathematically model the surface roughness in UAMAF considering the physics involved during the process. Therefore, this work aims to mathematically model the surface roughness in UAMAF based on the process physics as a function of different processing parameters. A novel approach is proposed to determine

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instantaneous surface roughness with finishing time during finishing. Firstly, modeling of material removal rate (MRR) was carried out by assuming it to consists of two independent

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and simultaneous phenomenon— (i) steady state material removal and (ii) transient material removal [5]. The modeling of surface roughness is performed by assuming the rate of change

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of surface roughness to be directly proportional to the MRR. Also, change in surface roughness at any instant is assumed to be proportional to the instantaneous amount of

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irregularities available for removal. This work considers surface roughness value as an important factor for the instantaneous rate of change of surface roughness during finishing as observed by various researchers [1,8,14,15] experimentally. This model also includes the

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effect of other important factors in UAMAF process such as workpiece material hardness, the size distribution of magnetic abrasive particles and operating variables, viz, voltage, rpm of the electromagnet, amplitude and frequency of vibrations. The constants used in the modeling were evaluated by the inverse method. The model incorporates a critical roughness value [7] evaluated from finishing parameters. When the critical value is reached, further finishing will not improve the surface roughness. The model confirms that there is always a critical roughness value and it depends on the finishing parameters.

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Mathematical modeling to predict surface roughness during UAMAF

The material removal rate affects the rate of change of surface roughness. Hence, first the modeling of material removal rate is presented and then the model of surface roughness is developed based on the modeling of MRR. The interaction of FMAB with the workpiece surface is a complex system. Thus, for the mathematical modeling of the surface finish during UAMAF, following realistic assumptions are made to simplify the analysis:

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1. All abrasive and ferromagnetic particles in FMAB are considered spherical in shape. 2. The size of ferromagnetic or abrasive particles is a function of their mesh or sieve number. The diameter of abrasive (

) or ferromagnetic (

the mesh number ( ) as

) particles can be related to

. If abrasive particles are specified

in terms of a range of the mesh number then the size distribution is assumed to be

(

)

, where

pass through the sieve and

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normal and is symmetric about the mean particle diameter

given by

is the lower mesh number that allows all the grains to is the upper mesh number that detains most of the grains

in the sieve. It is assumed that the size of the particles remains constant throughout the

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FMAB.

3. In FMAB, the chains of ferromagnetic particles are continuous and remain unaffected

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by the presence of the non‒magnetic abrasive particles. 4. The distribution of magnetic‒flux density over the workpiece surface is uniform and

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steady.

5. The magnetic properties of FMAB do not change during the finishing. As UAMAF has

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a low MRR, the removed material mixed with of unbonded magnetic abrasive particles (UMAPs) does not significantly alter the properties of the FMAB. 6. The FMAB consists of abrasive and ferromagnetic particles without voids.

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7. The material properties of workpiece remain unaffected by the interaction by FMAB. 8. Workpiece material is perfectly plastic and no pile‒up takes place due to ploughing action.

2.1

Calculation of average velocity of abrasive particle

The abrasion by an abrasive particle is influenced by the length of the path followed by it [16]. Hence, an analysis for calculation of the average velocity of the abrasive particles is presented in this section. During UAMAF, the electromagnet is rotated about the Z‒axis (Fig. 1(a)) with

revolutions per minute (RPM) and the ultrasonic vibrations are imposed on the 9

ACCEPTED MANUSCRIPT workpiece in the horizontal direction (i.e. X‒axis) as shown in Fig. 1(a). The displacement of the workpiece at any time

is given by

, where

is amplitude and

is

frequency of ultrasonic vibrations. Thus, the coordinates of an abrasive particle during UAMAF can be expressed as (1) (2)

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It is assumed that Z‒plane corresponds to the workpiece surface, the origin of the co-ordinate system concurs with the center of the electromagnet and initially an abrasive particle lies at X‒axis with radius

away from the origin i.e., at ( , 0). Here

active abrasive particle calculated as

is the angular velocity of the

. The velocity of abrasive particle

in the X‒ and Y‒directions, respectively, can be obtained by differentiating Eqn.

and

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(1) and (2) with respect to time . Thus,

(3)

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(4)

The resultant velocity

of an active abrasive particle at the time is given by }

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√[{

{

} ]

(5)

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Fig. 2 shows the variation of the velocity of an active abrasive particle with time during UAMAF. It is observed from the graph that velocity is periodic in nature and it repeats after a period of time. Hence for simplification of theoretical calculations, average velocity for the

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abrasive particles of the cycle is considered; its value is the average of all the values between

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reference lines (or during one cycle) and is represented by

.

Reference lines

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Resulatnt velocity (m/s)

1.6 1.4 1.2

1 0.8 0.6 0.4 0.2 0 0

50

100

150

200

250

300

350

400

450

500

Time (s)

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ACCEPTED MANUSCRIPT Fig. 2: Variation of resultant velocity with time.

2.2

Modeling of material removal rate

It was observed experimentally by various researchers [5,8,14,15] that during finishing, initially, the MRR is high, thereafter it decreases with finishing time and attains a steady state after a certain period of time. Initially, due to high surface roughness value, a smaller apparent contact area existed in which high contact stresses were generated due to the interaction of FMAB and workpiece surface. The irregularities get sheared-off if some

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combination of generated contact stresses exceeds a critical value. During finishing, the reduction in surface roughness value leads to an increase in the apparent area of contact, which reduces the contact stresses. Once the contact stresses fall below a critical level the shearing-off of irregularities becomes hard to realize, and a steady-state is achieved [17]. Thus, MRR shows a transient nature, which vanishes as the surface roughness value fall

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below a certain value and a steady state is thus observed. Hence, it is concluded that MRR at any instant during UAMAF is a function of the instantaneous value of surface roughness. In the modeling of MRR, it is assumed that MRR is attributed to two independent and simultaneous phenomena and at any instant the sum of their individual contributions determines the magnitude of MRR. The two components of material removal rate are a and a transient material removal component

. The steady state removal component embodies the abrasion/ploughing/scratching

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( ̇

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steady state material removal component ( ̇

action performed by the abrasive particles and transient removal component manifests the shearing-off of irregularities due to interaction of FMAB with the workpiece surface. The

̇

̇

̇

(6)

CE

PT

instantaneous material removal rate ( ̇ at any time is mathematically represented as

Here, it is assumed that while finishing only a part of active abrasive particles shears-off the

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irregularities and another part indents and abrades the workpiece surface. 2.2.1 Steady state removal rate component The wear due to abrasion is directly proportional to the load on the abrasive particles, shape as well as size of abrasive particles, length of contact and probability factor of abrasive particles to take part in material removal and inversely proportional to the hardness of the workpiece [16,18]. The steady state material removal rate ( ̇ ) by an abrasive particle can be written as [5]

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ACCEPTED MANUSCRIPT ̇ where

(7)

is a constant and signifies the probability factor that an active abrasive particle

would take part in material removal,

is the density of workpiece material and

number of active abrasive particles. Here

is

represents the horizontal projected cross-

sectional area of indented portion of an abrasive particle that encompasses the effect of shape and size of the abrasive particle, normal force (or load) acting on it and the workpiece

directly proportional to the average velocity

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hardness. The length of the contact for abrasive particles during the finishing time .

is

(Normal force on active abrasive particle)

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Projected area of indentation 𝐴𝐻𝑃

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Fig. 3: Sectional view of an indented active abrasive particle

The estimation of normal force on the active abrasive particle has been discussed in detail by

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Misra et al. [5]. The normal force (or magnetic levitation force) acting on a non‒magnetic active abrasive particle will force it to indent on to the workpiece surface (Fig. 3). A circular

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impression is created on the workpiece surface like the impression created by the Brinell

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indenter. The diameter of the impression can be given as [5]

where

√

(8)

is hardness of workpiece material and

is the constant, its value is 1 for brittle

material more than 1 for ductile material (for steel

=3.0). The depth of the indentation ( )

of the abrasive particle can be calculated as [5] √

(9)

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ACCEPTED MANUSCRIPT The horizontal projected cross‒sectional area of indentation

(shown by the dotted

portion in Fig. 3) can be derived from the geometry and is given by [5] [

(

)

]

(10)

Thus substituting Eqn. (10) in Eqn. (7), the steady-state mass material removal rate ( ̇ ) in UAMAF can be written as

where

[

(

)

]

(11)

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̇

is the number of active abrasive particles. Its value is found to be

.

The details for the calculation of number of active abrasive particles is given in paper by

2.2.2 Transient material removal rate

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Misra et. al. [5].

Transient material removal refers to the shearing/fracture of irregularities during the interaction of FMAB and workpiece surface due to generated contact stresses. The rate of removal of irregularities due to transient removal phenomenon at any instant is assumed to be a function of the instantaneous mass of irregularities available on the surface for transient

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removal. It is mathematically represented as [5]

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(12)

where

is the instantaneous mass of the irregularities available for transient removal and

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is a constant of proportionality termed as the transient coefficient. The negative sign signifies the reduction of surface irregularities with time. Initially, it is assumed that the mass

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of irregularities available on the workpiece surface is

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reduces to

, and after a finishing time of

it

. Hence, Eqn. (12) can be written as ∫

∫

(13)

Integrating Eqn. (13) and applying the initial conditions, the mass of irregularities available (

) on the workpiece surface after time is given as (14)

Thus, the total mass of irregularities removed (

) due to transient phenomenon after time

is given by

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ACCEPTED MANUSCRIPT (15) where

is the initial mass of irregularities and is a function of initial surface roughness. It is

calculated by assuming the surface irregularities to be a uniform array of triangular ridges (Refer Fig. 4(a)). The justification for this assumption is that UAMAF operation is performed after the grinding operation and researchers [19] proposed that surface obtained after the

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grinding operation has triangular ridges due to abrasion by the grinding wheel.

Area that takes part in finishing (𝐴𝑓 )

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Poles of electromagnet

Fig. 4: (a) Schematic of 2-D triangular profile of ground surface; (b) Illustration of area of surface finished during UAMAF

where

PT

ED

Hence, initial mass of surface irregularities

is peak to valley height and

on the surface is given by (16)

is the area of the surface that comes in contact with

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FMAB during one rotation and actually finished during UAMAF operation as shown in Fig. 4(b). The average surface roughness value related to peak to valley height

is given by

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following relation

∫| |

(17)

Thus from Eqn. (16) and (17), the initial mass of surface irregularities on the surface is given by (18) where

is the initial surface roughness. Substituting Eqn. (18) into (15), the total transient

mass material removal (

) after time is given by 14

ACCEPTED MANUSCRIPT (19) Therefore, the transient material removal rate can be calculated as ̇

(20)

Thus, substituting Eqns. (11) and (20) in Eqn. (6), the material removal rate

̇ at any time

̇

[

(

)

(21)

]

The Eqn. (21) contains two constants

and

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during UAMAF is given by

that are evaluated by the experimental data

by inverse method. The experiments were performed to evaluate total material removal at an interval of 20 s between 0–120 s. The unbonded (or loose) magnetic abrasive particles

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(UMAPs) are used. The concentration of abrasive particles by weight in UMAPs used was 20%, the total weight of UMAPs was 8 grams. The material removed from the workpiece was measured by weighing machine (A&D Instruments India Private Limited Series GR-200 with the smallest measurement size of 0.001 g). The experimental observations for MRR are

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given in the Table 1 and the conditions used during experiments are given in Table 2. Table 1: Experimental observations for MRR [5]

PT

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0‒20 21‒40 41‒60 61‒80 81‒100 101‒120 Finishing time interval (s) Total material removed after 14.358 23.645 30.033 35.152 39.683 44.061 the interval (in mg) Material removal rate (in mg/s) 0.720 0.460 0.320 0.260 0.225 0.220 During the interval 101120 it has been observed that there was an insignificant change in

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MRR compared to interval 81100. Hence, it was assumed that a steady state condition is thus achieved. The value obtained for MRR during the last interval is slightly more than the

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value of steady state MRR. Eqn. (11) is then used to calculate the steady state coefficient where

,

̇ is assumed to equal to 0.22. The substitution predicts the value of steady state

coefficient as

. The transient coefficient

is estimated by fitting all the

experimental observation for MRR in Eqn. (19). A curve fitting was performed by using the MATLAB software utilizing ‗curve fitting tool box‘. The predicted value for

was found to

be 0.034. Fig. 5 exhibits the predicted as well as experimental plot of MRR and the goodness of fit.

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ACCEPTED MANUSCRIPT 0.8

Experimental Theoretical

Coefficients

Parameters Initial surface roughness = 0.3234 m Supply voltage = 60 V Working gap = 1.5 mm RPM of electromagnet = 300 Amplitude of vibration = 8 m Frequency of vibration = 20 kHz

0.6 0.5 0.4 0.3

(with

95%

confidence

bounds): = 0.034 The goodness of fit: SSE: 1.014×10-15

0.2

R-squared: 0.9946

0.1

Adjusted R-squared: 0.9946

0.0 0

20

40

60

80

100

120

Finishing time (s)

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Material removal rate (mg/s)

0.7

RMSE: 1.424×10-8

Fig. 5: The experimental and predicted results of MRR

Modeling of surface roughness

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2.3

The rate of change of surface roughness was initially high that goes on decreasing with the finishing time till a saturation is achieved [1,8,14,15]. Hence, it is concluded that rate of change of surface roughness at any instant depends on the amount of irregularities available at that instant; higher the amount of irregularities present higher is the rate of change of

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surface roughness. The steady state material removal requires the abrasive particle to indent into the workpiece surface. Hence there is a limit to which surface finish can be achieved by

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the process. The theoretical minimum surface roughness obtained by the process is termed as ‗critical surface roughness‘ below which further surface roughness reduction is not possible. is dependent on the depth of indentation by the abrasive particle on the

PT

The value of

workpiece surface and is calculated by the following relation [20]

CE

is evaluated by Eqn. (9). Therefore the maximum change in surface roughness that

AC

where

(22)

can be observed during finishing is the difference between the initial value of surface roughness and critical surface roughness value. The measure of the amount of irregularties available for removal at any instant is the difference between the instantaneous surface roughness

) and critical roughness value

. The MRR at any instant depicts the

instantaneous rate of removal of irregularties. Thus, it is assumed that the rate of change of surface roughness at any instant is directly proportional to i.

Instantaneous material removal rate

̇ and

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ACCEPTED MANUSCRIPT ii.

The instantaneous measure of irregularities available for removal {

}.

The rate of change of surface roughness can be mathematically represented as

} ̇

{ where

(23)

constant of proportionality termed as surface roughness constant. Here negative

time

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sign accounts for the reduction in the rate of change of surface roughness with finishing . Using Eqn. (6), Eqn. (23) can be written as }[ ̇

{ where ̇ remains constant with finishing time and

(24)

̇ ]

̇ is a function of instantaneous surface

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roughness value and changes with finishing time. Thus, substituting

̇ from Eqn. (20), the

rate of change of surface roughness is given by

}[ ̇

{

Assuming initial roughness value of the surface is

]

and after time

(25)

it reduces to

.

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Rearranging and applying the initial conditions, Eqn. (25) can be written as

{

}

∫

( ̇

)

ED

∫

(26)

{

PT

Integrating and applying initial conditions we get }

}

CE

{

(27) ( ̇

[

)]

AC

Rearrangement provides {

*

}

( ̇

(

))+

(28)

Substituting, ̇ from Eqn. (11), the value of instantaneous surface roughness at any time

is

given by {

}

*

(

[

(

)

]

(

))+

(29)

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ACCEPTED MANUSCRIPT Eqn. (29) provides the surface roughness value after finishing time . The unknown constant is determined from the experimental observation using inverse method. The experiments were performed on UAMAF setup as shown in Fig. 1 and all the parameters are stated in Table 2. Table 2: Numerical values for material properties and finishing parameters value

SS 304

Workpiece hardness (

231 BHN

Initial surface roughness

0.3234 µm

Size of abrasive particle (mesh number)

450 (constant)

Size of ferromagnetic particle (mesh number)

300 (constant)

Concentration of abrasive particles by weight

20%

Amplitude of vibrations

8 µm

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Workpiece material

Frequency of vibrations

20 kHz

Supply voltage

60 V

RPM of electromagnet

300

1.5 mm

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Working gap

The surface roughness values are measured for a period of 120 s at an interval of 20 s. The

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value of surface roughness was measured by using Talysurf (Taylor Hobson, Leicester, UK) with a Z−height resolution of 16 nm and a cut-off evaluation length of 0.8 mm. The

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experimental observation for surface roughness is given in Table 3 Table 3: Experimental observations for surface roughness

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Finishing time (s)

0

20

40

60

80

100

120

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Surface roughness (µm) 0.3234 0.2084 0.1573 0.1286 0.1095 0.0952 0.0837 A curve fitting was performed using MATLAB software to fit the experimental data for surface roughness in Eqn. (29) to predict the value of

as 3.289 ×104. Fig. 6 shows the

experimental and predicted plot of surface roughness and the goodness of fit values as obtained by curve fitting.

18

0.35

Simulation Experimental

Coefficients (with 95% confidence

0.30 Constant parameters Initial surface roughness = 0.3234 m Supply voltage = 60 V Working gap = 1.5 mm RPM of electromagnet = 300 Amplitude of vibration = 8 m Frequency of vibration = 20 kHz

0.25

0.20

bounds): = 3.289×104 The goodness of fit: SSE: 2.121×10-16

0.15

R-squared: 0.995

0.10

Adjusted R-squared: 0.995 0.05 0

20

40

60

80

100

120

Finishing time (in sec)

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Instantaneous surface roughness (in m)

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RMSE: 5.945×10-9

Fig. 6: The experimental and predicted results for surface roughness

in Table 4. Table 4: Predicted constants

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The predicted constants for Eqn. (29) for the finishing conditions stated in Table 2 are given

Value

Steady state coefficient

4.600×10-5

Transient coefficient

0.034

Surface roughness constant

3.289×10-4

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Simulation of surface roughness

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3

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Constants

The exact quantification of a surface profile can be done using surface profilometer using a

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stylus, but its mensuration is rather complex. Its measurement is done by evaluating a number of parallel profile that also requires the relocation technique for the alignment of the profiles.

AC

It is achieved through a precise numerically technique using an additional software [21]. However, the numerical generation of surface roughness profile is rather simple and extend definite advantages. It eliminates the requirement for surface measuring instruments. Also, the determination of statistical properties related to surface roughness is rather easy to compute numerically as compared to the measurement from the surface. Hence, a numerical simulation of surface roughness profile obtained during UAMAF is presented in this section. The numerical simulation of three-dimensional surface roughness having a Gaussian profile with specified exponential autocorrelation function has been performed in this section. The two statistical functions of a rough surface— the height distribution (or the frequency density 19

ACCEPTED MANUSCRIPT function) and the auto‒correlation function (ACF) are sufficient to determine the most of its statistical parameters [21]. The digital finite response filter technique proposed by Hu and Tonder [22] for generation of rough surfaces having Gaussian heights distribution and an exponential autocorrelation function has been used for the generation of random 3‒D surface. The exponential autocorrelation function

used for the generation of rough surface is

given by [22]

Here

and

√[(

)

(

(30)

) ]}

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{

are the correlation lengths and are length at which the value of

autocorrelation function reduces to 10% of the value at the origin i.e. assumed as 0.1 µm, where

is the standard deviation (or RMS value) of surface roughness. is used as a measure of surface roughness, hence, in Eqn. (30),

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However in this paper

and their values are

instead of standard deviation, the mean deviation (i.e.

) is used. The total size of the

simulated surface is 50 × 50 µm2. The surface map consists of 50 × 50 data points. The surface generated at different roughness value is shown in Fig. 8 ‒ Fig. 11. Validation

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4

To validate the model the predicted theoretical results calculated from Eqn. (29) were

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compared with experimental observations. The experimental observations for different supply voltage were compared with the theoretical predictions at the same set of parameters. The

PT

other parameters were kept constant and their values are given in Table 2. Fig. 7 shows the plot of the theoretical prediction and experimental observations for surface roughness with finishing time at supply voltage of 40 V and 80 V. The model predicts the surface roughness

AC

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with the maximum deviation of ±7.35%.

20

Parameters Initial surface roughness of sample for 40 V = 0.3076 m Initial surface roughness of sample for 60 V = 0.3088 m Supply voltage = 60 V Working gap = 1.5 mm Simulation at 40 V RPM of electromagnet = 300 Experimental at 40 Amplitude of vibration = 8 m Simulation at 80 V Frequency of vibration = 20 kHz

0.35

0.30

0.25

V

Experimental at 80 V 0.20

0.15

0.10

0.05 0

20

40

60

80

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Instantaneous surface roughness (in m)

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100

Finishing time (in sec)

120

5 5.1

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Fig. 7: Comparison of theoretical and experimental observation for surface roughness at different supply voltages

Results and discussion Surface profile

The initial surface of the workpiece is a ground surface and is subjected to UAMAF under different supply voltage keeping all other parameters constant (for details refer Table 2). Fig.

M

8 ‒ Fig. 11 shows the initial as well as final surface profile observed experimentally and through simulation after finishing for 120 s at a supply voltage of 40 V, 60 V and 80 V,

ED

respectively. Fig. 8(a) shows the measured initial surface roughness profile; it shows a high variation of the maximum peaks to valleys distance. This high unevenness in profile is being represented by the presence of high peaks and low valleys in Fig. 8(b). It leads to a higher

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initial surface roughness value of 0.3234 µm. After finishing the irregularities were sheared‒ off and the variation between peaks and valleys reduces. The part (a) of Fig. 9‒Fig. 11 shows

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the measured profile of the finished surface at different voltages. It shows that the unevenness in profile decreases, this is due to the reduction of peaks from the surface also depicted by the

0.1 mm

𝑅𝑎

0.3234 µm

Micrometers

AC

simulated surfaces in part (b) of Fig. 9‒Fig. 11.

(a)

21

(b)

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Fig. 8: Surface roughness profiles of initial surface; (a) measured (b) simulated

0.1208 µm

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Micrometers

𝑅𝑎

ED

M

(a)

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(b)

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Fig. 9: Surface roughness profiles after finishing at 40 V; (a) measured (b) simulated at parameters level mentioned in Table 2.

Micrometers

AC

𝑅𝑎

0.0837 µm

(a)

22

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(b)

0.0613 µm

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Micrometers

𝑅𝑎

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Fig. 10: Surface roughness profiles after finishing at 60 V; (a) measured (b) simulated at parameters level mentioned in Table 2.

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M

(a)

(b)

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Fig. 11: Surface roughness profiles after finishing at 80 V; (a) measured (b) simulated at parameters level mentioned in Table 2.

The numerically simulated surface profiles provide a better sense of the surface condition as

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compared to measured ones. It can also be concluded from Fig. 8 ‒ Fig. 11 that higher reduction in surface roughness occurred for a higher voltage keeping all the other parameters

AC

constant. 5.2

Supply voltage

Fig. 12 shows the variation of surface roughness with finishing time at the different supply voltage. It can be deduced from the graph that the instantaneous surface roughness during UAMAF decreases with increased supply voltage. This is because increasing supply voltage increases the magnetic‒flux density in the working gap, forming strong chains of ferromagnetic particles; as a result the rigidity of FMAB increases. Higher the strength of FMAB the higher is the indentation force that allows greater indentation by the abrasive 23

ACCEPTED MANUSCRIPT particles onto the workpiece surface. Also, rigid FMAB does not allow the active abrasive particles to roll, and it will efficiently take part in shearing-off of peaks. However, increasing the voltage will limit the quality of surface finish obtained during the process due to higher indentation by the abrasive particles because an increased indentation increases the critical surface roughness value obtained by the process.

Constant parameters Initial surface roughness = 0.3234 m Working gap = 1.5 mm RPM of electromagnet = 300 Amplitude of vibration = 8 m Frequency of vibration = 20 kHz

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0.30 0.25 0.20 0.15

20 V

40 V

0.10

60 V

0.05 0.00 0

20

40

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Instantaneous surface roughness (m)

0.35

60

80

100

80 V 100 V

120

Finishing time (in sec)

5.3 RPM of electromagnet

M

Fig. 12: Variation of surface roughness with finishing time at different supply voltage

ED

The length of the path traveled by an abrasive particle is proportional to the RPM of the electromagnet. Fig. 13 shows the effect of RPM of the electromagnet on the instantaneous surface roughness during UAMAF. It was observed that higher the RPM the lower is the

PT

instantaneous surface roughness. It is because with an increased RPM the contact length of active abrasive particles in FMAB is large for the similar time interval. Hence more amount

AC

CE

of abrasion/ploughing and shearing-off of irregularities takes place.

24

ACCEPTED MANUSCRIPT

0.30

Constant parameters Initial surface roughness = 0.3234 m Supply voltage = 60 V Working gap = 1.5 mm Amplitude of vibration = 8 m Frequency of vibration = 20 kHz

0.25

0.20

0.15

150 rpm 250 rpm 350 rpm

0.10

0.05 0

20

40

60

80

Finishing time (in sec)

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Instantaneous surface roughness (in m)

0.35

100

120

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Fig. 13: Variation of surface roughness with finishing time at different rpm of electromagnet.

Experimentally, it was observed that instantaneous value of surface roughness decreases up to a critical value of RPM and thereafter it increases. It is due to the fact that, on increasing the RPM the centrifugal force acting on abrasive particles in FMAB increases and it tends to

M

pull the abrasive particles out from the FMAB (Fig. 14). A non-deformed chain of abrasive particles has a magnetic force ion energy

[23]. This pullout force is balanced by a

ED

component of magnetic force; hence, the component of magnetic force that acts normal to the workpiece surface decreases. Increasing the RPM beyond a critical limit will reduce the effectiveness of the FMAB and the further increase will cause splashing of the abrasive

PT

particles away from the FMAB. Therefore, increased RPM will tend to reduce instantaneous surface roughness value for similar parametric conditions only when the rigidity of FMAB is

AC

CE

maintained.

(a)

(b)

25

ACCEPTED MANUSCRIPT Fig. 14: Chain of ferromagnetic particle (a) non deformed (b) deformed.

5.4 Working gap The distance from the pole of the electromagnet to the workpiece surface is referred to as working gap. Fig. 15 illustrates the effect of working gap on instantaneous surface roughness during UAMAF; it was observed that instantaneous surface roughness decreases with a reduction in the working gap, keeping all other parameters constant. This is because for the

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smaller gap, the magnetic lines of forces remain straight and concentrated. On the other hand for increased gap they diverge and strength decreases. Therefore, at a lower working gap, a higher strength of FMAB formed, which provides the greater magnitude of forces. A rigid FMAB abrades and shear-off the surface irregularities more efficiently. The indentation of abrasive particle puts a limit on a reduction in working gap; lowering the working gap

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increases the value of the minimum surface roughness obtained by the process.

Constant parameters Initial surface roughness = 0.3234 m Supply voltage = 60 V RPM of electromagnet = 300 Amplitude of vibration = 8 m Frequency of vibration = 20 kHz

0.30

0.25

M

0.20

0.15

0.10

0.05

2.5 mm 2.0 mm

ED

Instantaneous surface roughness (in m)

0.35

PT

0

20

1.5 mm 1.0 mm

40

60

80

100

120

Finishing time (in sec)

CE

Fig. 15: Variation of surface roughness with finishing time at different working gap

5.5 Amplitude and frequency of vibration

AC

In UAMAF, amplitude and frequency of ultrasonic vibrations play a significant role. Fig. 16(a) and (b) shows the variation of the percentage change in surface roughness with a change in amplitude and frequency of vibrations, respectively. An increased amplitude and frequency of vibrations increases the percentage change in surface roughness. An increased amplitude and frequency increases the path traced by the abrasive particle and also the resultant velocity (Eqn.(5)), which brings about enhanced abrasion/ploughing of material and also the increased momentum of FMAB, improves the fracturing/shearing of irregularities from the surface. 26

ACCEPTED MANUSCRIPT 81 Constant parameters Initial surface roughness = 0.3234 m Supply voltage = 60 V Working gap = 1.5 mm RPM of electromagnet = 300 Frequency of vibration = 20 kHz

79 78

Finishing time = 120 s

77 76 75 74 8

9

10

11

12

Vibration amplitude (m)

75.0

Finishing time = 120 s

M

74.5

74.0

73.5

14

Constant parameters Initial surface roughness = 0.3234 m Supply voltage = 60 V Working gap = 1.5 mm RPM of electromagnet = 300 Amplitude of vibration = 8 m

ED

 change in surface roughness

75.5

13

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(a)

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 change in surface roughness

80

20

21

22

Vibration frequency (kz)

(b)

CE

PT

19

Fig. 16: Variation in percentage change in surface roughness for different (a) vibrations amplitude; (b) vibrations frequency

AC

5.6 Hardness of workpiece material Fig. 17(a) illustrates the effect of workpiece hardness on the change of surface roughness during UAMAF. It shows that with an increased hardness of the workpiece surface the change in surface roughness obtained during finishing decreases. Fig. 17(b) depicts that there was a reduction in percentage change in surface roughness with increased hardness of the material. It can be seen that the percentage change in surface roughness was high for the materials of low hardness value. It is because for softer materials indentation depth (or horizontal projected area) obtained for similar conditions was more, as it varies with 27

ACCEPTED MANUSCRIPT hardness. Also, lower strength irregularities can get sheared-off easily. The high hardness of material reduces the role of indentation and reduction in surface roughness occurs mainly due to the removal of irregularities from the surface and rate of reduction of surface roughness decreases. 0.35 Constant parameters Initial surface roughness = 0.3234 m Supply voltage = 60 V Working gap = 1.5 mm RPM of electromagnet = 300 Amplitude of vibration = 8 m Frequency of vibration = 20 kHz

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0.25

0.20

0.15

350 BHN

0.10

250 BHN

0.05

150 BHN

0

20

40

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Surface roughness (m)

0.30

60

80

100

120

Finishing time (s)

(a)

80

75

M ED

85

Constant parameters Initial surface roughness = 0.3234m Supply voltage = 60 V Working gap = 1.5 mm RPM of electromagnet = 300 Amplitude of vibration = 8 m Frequency of vibration = 20 kHz Finishing time = 120 s

PT

 change in surface roughness

90

70

CE

65

60

AC

100

150

200

250

300

350

400

Hardness of workpiece material (BHN)

(b)

Fig. 17: Variation of (a) instantaneous surface roughness with finishing time for different hardness value; (b) percentage change in surface roughness with hardness value

5.7 Initial surface roughness The initial surface roughness of the workpiece surface is an important parameter that affects the rate of change of surface roughness. Fig. 18(a) illustrates the effect of initial surface roughness on the instantaneous surface roughness with finishing time during UAMAF. It 28

ACCEPTED MANUSCRIPT depicts that initially for the surface of high roughness value, the rate of reduction of surface roughness is high. It is because initially, removal of irregularities takes place at a greater rate due to high contact stresses at the interface with FMAB. However, when the surface roughness reaches a critical value, the rate of reduction in surface roughness value becomes steady. Fig. 18(b) shows the percentage reduction in surface roughness after finishing time of 120 s. It demonstrates that for the surface having high roughness value the percentage change

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in surface roughness value is more for similar finishing conditions.

Constant parameters Supply voltage = 60 V Working gap = 1.5 mm RPM of electromagnet = 300 Amplitude of vibration = 8 m Frequency of vibration = 20 kHz

0.5

0.5 m

0.4

0.35 m

0.25 m

0.3

0.15 m

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Instantaneous surface roughness (m)

0.6

0.05 m

0.2

0.1

0.0 0

20

40

60

80

100

120

M

Finishing time (s)

85

PT

 change in surface roughness

80

ED

(a)

75

AC

CE

70 65

Constant parameters Supply voltage = 60 V Working gap = 1.5 mm RPM of electromagnet = 300 Amplitude of vibration = 8 m Frequency of vibration = 20 kHz Finishing time = 120 s

60 55 50 45 0.0

0.1

0.2

0.3

0.4

0.5

Initial surface roughness (m)

(b) Fig. 18: Variation of (a) instantaneous surface roughness value with finishing time for initial surface roughness value; (b) percentage change in surface roughness with initial surface roughness value

29

ACCEPTED MANUSCRIPT 6

Conclusion

In this work, a mathematical model for surface roughness during UAMAF is presented. The proposed model considers instantaneous surface roughness as a function of the instantaneous amount of irregularities on the surface of the workpiece and instantaneous MRR; MRR is assumed as a function of two independent and simultaneous phenomenon— steady state material removal and transient material removal. The experiments were performed on the stainless workpiece to validate the model. The constants used in the modeling are estimated

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by using an inverse method. The following conclusions are deduced from the presented work:  The reduction in surface roughness of the workpiece surface increases with a decrease in working gap and an increased supply voltage. Also, an increased RPM enhances the rate of reduction of surface roughness.

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 The percentage change in surface roughness during finishing increased with an increased amplitude and frequency of vibrations.

 A lesser rate of reduction of surface roughness is realized with an increased hardness in the workpiece surface. The percentage change in surface roughness decreases asymptotically with an increase in the hardness value.

M

 The model established the percentage change in surface roughness during UAMAF as a function of the initial surface roughness of the workpiece; higher the initial surface

ED

roughness the higher is the rate of reduction.  The model confirms that there is a critical surface roughness value that can be

PT

obtained by the given finishing parameters below which no reduction in roughness value occurs.

CE

 The mathematical model for prediction of the surface roughness successfully establishes the presence of steady state and transient material removal phenomena during finishing. The instantaneous surface roughness has a great influence on the

AC

finishing process.

 The developed model is validated with the experimental observation and found to be in good agreement. The model also predicts that there is an exponential relationship between change in surface roughness and finishing time.

Acknowledgements Funding from the Engineering and Physical Sciences Research Council (UK) through grant EP/K028316/1 and Department of Science and Technology (India) through grant DST/RC-

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ACCEPTED MANUSCRIPT UK/14-AM/2012 for project ―Modeling of Advanced Materials for Simulation of Transformative Manufacturing Processes (MAST)‖ is gratefully acknowledged. References [1]

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doi:10.1016/j.ijmachtools.2004.06.014. [2]

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Cylindrical–Magnetic Abrasive Finishing of Aluminum Workpiece. Mater Manuf Process 2013;28:1196–202. doi:10.1080/10426914.2013.811725. [14] Fox M, Agrawal K, Shinmura T, Komanduri R. Magnetic Abrasive Finishing of Rollers.

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ACCEPTED MANUSCRIPT 2013;213:2089–100. doi:10.1016/j.jmatprotec.2013.06.011. [21] Patir N. A numerical procedure for random generation of rough surfaces. Wear 1978;47:263–77. doi:10.1016/0043-1648(78)90157-6. [22] Hu YZ, Tonder K. Simulation of 3-D random rough surface by 2-D digital filter and Fourier analysis. Int J Mach Tools … 1992;32:83–90. doi:10.1016/0890-

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6955(92)90064-N. [23] Mori T, Hirota K, Kawashima Y. Clarification of magnetic abrasive finishing mechanism. J Mater Process Technol 2003;143–144:682–6. doi:10.1016/S0924-

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Graphical abstract

33