Mechanics, Materials Science & Engineering, July 2016 – ISSN 2412-5954

MMSE Journal. Open Access www.mmse.xyz

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Mechanics, Materials Sciences & Engineering Journal

, Austria, Sankt Lorenzen, 2016

Mechanics, Materials Science & Engineering Journal (MMSE Journal) is journal that deals in peer-

reviewed, open access publishing, focusing on wide range of subject areas, including economics,

business, social sciences, engineering etc.

MMSE Journal is dedicated to knowledge-based products and services for the academic, scientific,

professional, research and student communities worldwide.

Open Access model of the publications promotes research by allowing unrestricted availability of

high quality articles.

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The Journal Policy declares the acceptance of the scientific papers worldwide, if they passed the

peer-review procedure.

Editor-in-Chief Mr. Peter Zisser

Dr. Zheng Li, University of Bridgeport, USA

Prof. Kravets Victor, Ukraine

Ph.D., Shuming Chen, College of Automotive Engineering, China

Dr. Yang Yu, University of Technology Sydney, Australia

Prof. Amelia Carolina Sparavigna, Politecnico di Torino, Italy

ISSN 2412-5954

e-ISSN 2414-6935

Design and layout: Mechanics, Materials

Science & Engineering Journal, www.mmse.xyz

Technical support: hotmail@mmse.xyz

©2016, Magnolithe GmbH

© Copyright, by the authors

Mechanics, Materials Science & Engineering, July 2016 – ISSN 2412-5954

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CONTENT

I. Materials Science MMSE Journal Vol. 5 ..................................................................................... 6

Detection of Iron Oxide Layer in Quenched and Tempered Gear Steel Using Magnetic

Barkhausen Noise. M. M. Blao, M. M. Sawalem ............................................................................... 7

Development and Determination of the Age Hardening Characteristics of Al-2.00Mg-2.66Si

Wrought Alloy. Ihom A.P., Aniekan Offiong .................................................................................... 13

Effect of Textures on Tensile Properties of Extruded Ti64/VGCF Composite by Powder

Metallurgy Route. Patchara Pripanapong, Shu-feng Li, Junko Umeda, Katsuyoshi Kondoh ........ 22

II. MECHANICAL ENGINEERING & PHYSICS MMSE JOURNAL VOL. 5 .......................................... 33

Strength Analysis of Flat Spring of the Resonant Vibro-Impact Module. Volodymyr Gursky,

Igor Kuzio .......................................................................................................................................... 34

Modelling of Station of Pumping by Variable Speed. Benretem A. ......................................... 49

A Thermal Force Drifting Particles along a Temperature Gradient. Amelia Carolina

Sparavigna ........................................................................................................................................ 58

Error Analysis of Method for Calculation of Non-Contact Impact on Space Debris from

Ion Thruster. Alpatov A.P., Fokov A.A., Khoroshylov S.V., Savchuk A.P. ...................................... 64

Increased Wear Resistance of Surfaces of Rotation Bearings Methods Strengthening-

Smoothing Processing. A.A. Tkachuk, V.U. Zablotskyi, T.V.Terletskyi, O.L. Kaidyk, S.A. Moroz ........ 77

Kinetostatics of Wheel Vehicle in the Category of Spiral-Screw Routes. Kravets V.V.,

Bas K.M., Kravets T.V., Zubariev M.S., Tokar L.A. .......................................................................... 86

An Instrumented Macro-Indentation Method for Determining the Mechanical Properties

of Coconut Shell (Coco Nucifera of Cameroon). E. Njeugna, M.B.K. Ganou, D. Ndapeu,

J.N.T. Foba, N.R.T. Sikame, P.W.M. Huisken ................................................................................... 99

Macroscopic Geometrical Modelling of Oil Palm Mesocarp Fibers of Three Varieties of

Palm Nut. E. Njeugna, P. W. M. Huisken, D. Ndapeu, N. R. T. Sikame, J. Y. Dréan ..................... 107

Comparison of Assemblies of Four-Link Structural Groups of 3

rd

Class on the

Transmission Angle . Matsyuk I.N., Morozova Т.I., Shlyakhov E.М. ............................................ 124

The Mechanics of a Cantilever Beam with an Embedded Horizontal Crack Subjected to an

End Transverse Force, Part A: Modelling. Panos G. Charalambides,

Xiaomin Fang .............. 131

The Mechanics of a Cantilever Beam with an Embedded Horizontal Crack Subjected to an

End Transverse Force, Part B: Results and Discussion. Panos G. Charalambides, Xiaomin Fang ... 157

Lagrangian Representations and Solutions of Modified Emden-Type Equations. Aparna

Saha, Benoy Talukdar ...................................................................................................................... 176

Improvement of Fourier Series Convergence on the Basis of Splines and Its Application for

Numerical Inversion of Laplaсe Transform. Tanya Solyar ....................................................... 188

On the Boltzmann Equation of Thermal Transport for Interacting Phonons and Electrons.

Amelia Carolina Sparavigna ........................................................................................................... 203

Dynamical Analyses of Piston Machines Used in Oil Industry. V.I. Bakhshaliev ................. 216

VII. ENVIRONMENTAL SAFETY MMSE JOURNAL VOL. 5 ............................................................. 223

Study of Ground Treatment on Improvement of Pile Foundation Response in Liquefiable

Soils. Chen Yulong ........................................................................................................................... 224

Mechanics, Materials Science & Engineering, July 2016 – ISSN 2412-5954

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Noise-Induced Hearing Loss in Relation With Vibration Disease and Exposure to

Vibration Among Employees in Latvia. Kristina Karganova, Jelena Reste ............................... 234

IX. ECONOMICS & MANAGEMENT MMSE JOURNAL VOL. 5........................................................ 246

ANFIS and Multi Linear Regression to Estimate the LTLF for the Kingdom of Bahrain.

Mohamed Y. Al-Hamad, Isa S. Qamber ........................................................................................... 247

Taxation of Public Owned Land for Real Estate Reconstruction in Kiev, Ukraine.

М.А. Malashevskyy, I.M. Ciobanu .................................................................................................. 262

The Benchmark Survey Methods of the Lecturers and Chairs Work in the Higher

Educational Establishments, with Using the Cumulative Ranking Index. Protsiv I.V.,

Shevchenko O.V., Protsiv V.V. ........................................................................................................ 270

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Detection of Iron Oxide Layer in Quenched and Tempered Gear Steel Using

Magnetic Barkhausen Noise

M. M. Blaow

1, a

and M. M. Sawalem

2, b

1

– Department of Materials Science and Engineering, Faculty of Engineering, University of Misurata, Aljazera Street,

Misurata, Libya

2

– Department of Mechanical Engineering, Faculty of Engineering, University of Misurata, The Coase Road, Misurata,

Libya

a – mblaow@yahoo.co.uk

b – ferretti355@gmail.com

DOI 10.13140/RG.2.1.1027.5444

Keywods: magnetic Barkhausen noise profile, tempering, oxidation, hysteresis.

ABSTRACT. This paper deals with the non-destructive evaluation of surface oxidation of gear steel using magnetic

Barkhausen noise profiles analysis. Martensitic specimens are subjected to tempering at various temperatures in a

muffle furnace. Tempering induced changes result in Barkhausen profile height increase and peak centers shift to lower

fields. Single peak profiles are associated with specimens tempered up to 500 °C. Double peak profiles are seen with

specimens tempered at 600 °C and 700 °C. Single peak profiles are observed after removing the oxide layer. The

utilization of MBN method for this purpose is based on the difference in the inherent magnetic properties between the

degraded surface layer and the sub-surface unaffected bulk. The observations are discussed in the light of established

models of Barkhausen noise.

Introduction. The manufacturing processes for gear components include heat treatment operations

to achieve surface characteristics for components to increase wear resistance. Heat treatments

include carburizing and induction hardening which introduce a hard layer at the surface and

maintain a soft interior. Recently Ovako 667 steel was developed for low cost manufacture of

wear-resistant elements. This type of steel could be fully hardened by air-cooling from the austenite

region. Another feature is that the material is resistant to over-tempering.

When a ferromagnetic material is magnetized by a varying magnetic field, the local changes in the

magnetization induces voltage pulses in a search coil placed on the surface which are known as

magnetic Barkhausen noise [1]. Magnetic Barkhausen noise (MBN) is mainly associated with the

irreversible domain wall movement and refers to the abrupt discontinues changes in the

magnetization rate that result from domain walls overcoming various types of obstacles in their

path. Obstacles include grain boundaries, voids and precipitates [2]. The sensitivity of MBN to

microstructural inhomogeneities makes it potentially useful as a non-destructive testing technique.

Also, the assessment of microstructure and mechanical properties after initial heat treatment as a

quality control measurement and their subsequent degradation during service such as exposure to

high temperature [3]. In steels, microstructural defects like grain boundaries, inclusions and

dislocations promote both mechanical and magnetic hardening, increasing the area of hysteresis

curve and reducing permeability. This happens because the same defects which pin dislocations

also may “pin” domain walls while moving under the effect of a time varying excitation field [4].

Mechanics, Materials Science & Engineering, July 2016 – ISSN 2412-5954

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In case carburized and decarburized steels a hardness gradient is present [5-9]. This can cause two

MBN intensity peaks to appear at different field strengths. The reason is that each peak originates in

a material layer of different hardness. The variation in ferromagnetic material properties can be

correlated to different parameters derived from the MBN signal profile generated during the

magnetization cycle [10]. The aim of the study is to investigate the magnetic Barkhausen noise

response from iron oxide layer formed at the surface of martensite tempered at elevated

temperatures as a function of weeping magnetic field.

Materials and Method. The composition of the stock material is shown in Table 1. In the present

work, bars (10×10×120 mm) machined from the stock were austenitized in vacuum to prevent

surface oxidation and decarburization of the specimens. Martensitic specimens were tempered in a

muffle furnace at 500, 600 and 700

o

C for 1 hour to produce different degrees of oxidation. The

details of the MBN apparatus is shown elsewhere [13]. In this experiment, the magnetizing

frequency used was 1 Hz to enhance the MBN signal to noise ration and maintain a low

magnetization rate.

Table 1. Composition of Ovako 677 steel

Element

C

Mn

Ni

Cr

Mo

Si

S

P

Wt %

0.67

1.48

0.11

1.03

0.25

1.46

0.007

0.016

Results. Figure 1 shows half-cycles MBN profiles of a quenched specimen and quenched and

tempered at 500°C specimen.

Fig. 1. Half-magnetizing cycle Barkhausen profile from specimen tempered at 500

o

C.

0

10

20

30

40

50

60

-0.25 0 0.25 0.5 0.75 1

MBN ( (Arbitrary units)

Magnetizing current, A

Martensite

tempered at

500

o

C

Air cooled

Mechanics, Materials Science & Engineering, July 2016 – ISSN 2412-5954

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Fig. 2. Half-magnetizing Barkhausen profile from the specimen tempered at 600

o

C.

The specimens oxidized at 600 and 700 °C show MBN profiles appear to consist of two

overlapping peaks (Figs. 2 and 3). It seems reasonable to assume that the MBN profiles reflect the

composition gradient at the skin depth of the specimens, which are the iron oxide at the surface and

the steel at the subsurface. Figure 4 shows that the second peak at higher field from the oxidized

specimen fits with MBN profile of the cleaned specimen which implies that the oxide layer is thin

and does not attenuate the Barkhausen emission from the bulk material. After removal of the oxide

layer, the specimen tempered at 700

°C shows a remarkable increase in the MBN emission (Fig. 3)

compared to that from the oxidized surface.

Fig. 3. Half-magnetizing Barkhausen profile from the specimen tempered at 700

°

C.

0

10

20

30

40

50

60

70

80

90

-0.25 0.25 0.75

MBN (Arbitrary units)

Magnetizing current, A

Martensite

tempered at

600

o

C

Oxidized

surface

Clean

surface

0

50

100

150

200

250

300

350

-0.25 0.25 0.75

MBN (Arbitrary units)

Magnetizing current, A

Oxidized

surface

Clean

surface

Martensite

tempered at 700

o

C

Mechanics, Materials Science & Engineering, July 2016 – ISSN 2412-5954

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This indicates that the oxide layer is able to attenuate the MBN signals from the bulk material.

Barkhausen profiles of the quenched and tempered specimens without oxides are shown in Fig. 4.

Fig. 4. MBN profiles showing tempering induced changes.

Discussion. Although MBN has been attributed to a number of mechanisms, most current thinking

associates it with the irreversible movement of domain walls. Theoretical models highlighting the

connection MBN and the irreversible component of magnetisation M

irr

have been reviewed by Jiles

14

. A basic assumption is that the intensity of emission is proportional to the differential

susceptibility χ

irr

= dM

irr

/dH, where H is the magnetic field. This is illustrated schematically in Fig.

5, where the M

irr

–H hysteresis loop is shown in relation to MBN emission for a complete

magnetisation cycle. The amplitude of emission is greatest when the slope of the M

irr

–H curve is a

maximum, and smallest at points approaching saturation. The MBN characteristics observed for the

different microstructures (Fig. 4) are consistent with the theory. If the hysteresis loop becomes

narrower, with steeper sides, the peak intensity of emission will increase and the position of the

peak will shift towards zero field. The converse will occur if the hysteresis loop becomes broader

with a smaller maximum slope. In the experiments, peak position (Fig. 4) shifted to lower values as

the microstructure changes from martensite to ferrite-cementite microstructure.

0

50

100

150

200

250

300

-0.25 0.25 0.75

MBN (Arbitrary units)

Magnetizing current, A

Quenche

Quenced and

tempered at

600

o

C

Quenced and

tempered at

700

o

C

Mechanics, Materials Science & Engineering, July 2016 – ISSN 2412-5954

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Fig. 5. Magnetic hysteresis loops and the corresponding MBN signals [14].

It is widely accepted that the Barkhausen noise signal is strongly dependent on the number of

pinning obstacles met by domain walls during the magnetization process. The peak amplitude is

thus strongly sensitive to the phase proportion, whereas the position of the peak is usually linked to

the nature and the strength of the obstacles. The discontinuous jumps of Bloch walls are due to their

local pinning by different obstacles such as inclusions, precipitates, grain boundaries, and

dislocation tangles. The MBN increases with the number of these pinning obstacles. A shift of the

peak to the high value of the magnetic field is also observed when the influence of these obstacles

on Bloch walls increases. Because the magnetic structure is directly linked to the nature of

metallurgical state and hence its differential susceptibility, each phase has its inherent magnetisation

saturation and hence a distinctive MBN response [15, 16]. This is consistent with the present

observations on the oxidized and non oxidized specimens. The differential susceptibility of iron

oxide layer is different from that of ferrite and cementite structure and hence this results in two

magnetization responses (MBN) relative to the applied sweeping field .

Summary

1. Iron oxide layer results in a split of the profile into two peak’s profiles revealing the iron oxide

and the bulk material.

2. Iron oxide layer at the surface of the specimens attenuates the MBN signals.

3. Magnetic Barkhausen noise technique is very sensitive to microstructural gradient at the surface

and the subsurface bulk.

References

[1] E. Gorknov, Yu. Dragoshanskii, and M. Mikhovki. Barkhausen noise and its utilization in

structural analysis of ferromagnetic materials (review article v) 5. Effects of volume and surface

thermal processing, Russian Russian Journal of Nondestructive Testing, 36 (6) 389 (2000).

[2] D.C. Jiles, The influence size and morphology of eutectoid carbides on the magnetic properties

of carbon steels, Journal of Applied Physics, 63, 2980, 1988.

[3] D. D'Amato, C. Verdu, X. Kleber, G. Regheere, and A. Vicent, Characterization of

Austempered Ductile Iron Through Barkhausen Noise Measurements Journal of Nondestructive

Evaluation, Vol 22 (4), pp 127-139, 2004, doi: 10.1023/B:JONE.0000022032.66648.c5.

[4] D. C. Jiles, "Dynamics of domain magnetization and the Barkhausen effect, Czechoslovak

Journal of Physics, 50, 893, 2000.

Mechanics, Materials Science & Engineering, July 2016 – ISSN 2412-5954

MMSE Journal. Open Access www.mmse.xyz

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[5] J. Alessandra, , L. Gunther, and F. Gerhardt, Magnetic Barkhausen Noise Profile Analysis:

Effect of Excitation Field Strength and Detection Coil Sensitivity in Case Carburized Steel,

Materials Sciences and Applications, Vol. 16, pp.1015-1019, 2013, doi: 10.1590/S1516-

14392013005000095.

[6] Hao, X. J., Yin, W., Strangwood, M., Peyton, A. J., Morris, P. F., & Davis, C. L. (2008). Off-

line measurement of decarburization of steels using a multifrequency electromagnetic sensor.

Scripta Materialia, 58(11), 1033-1036. DOI:10.1016/j.scriptamat.2008.01.042.

[7] Lo, C.C.H., Kinser, E.R. and Jiles, D.C. Analysis of Barkhausen Effect Signals in Surface

Modified Magnetic Materials Using a Hysteretic Stochastic Model. Journal of Applied Physics,

2006.

[8] M. Blaow, J. Evans and B. Shaw, Magnetic Barkhausen noise: the influence of microstructure

and deformation in bending, Acta Materialia, Vol 53, pp. 279-287, 2005,

doi:10.1016/j.actamat.2004.09.021.

[9] X. Kleber, A. Hug-Amalric, and J. Merlin, Evaluation of the Proportion of Phases and

Mechanical Strength of Two-Phase Steels Using Barkhausen Noise Measurements: Application to

Commercial Dual-Phase Steel, Metallurgical and Materials Transactions A, Vol. 39, pp.1308-1318,

2008, doi: 10.1007/s11661-008-9508-3.

[10] D. C. Jiles, L. B. Sipahi, and G. J. Williams, J. App. Phys Vol. 73, p. 5830, 1993.

[11] M. Kaplan C. Gür and M. Erdogan, J, Characterization of Dual-Phase Steels Using Magnetic

Barkhausen Noise Technique, Journal of Nondestructive Evaluation, Vol. 26, pp.79-87, 2007, doi:

10.1007/s10921-007-0022-0.

[12] M. J. Sablik, J Appl Phys, Vol. 74, pp.5898, 1993

[13] M. M. Blaow, J. T. Evans and B. A. Shaw, J. Magn. Magn. Mat., Vol.303, pp.153, 2006.

[14] D.C. Jiles, Czechoslovak J. Phy. Vol. 50, 893 (2000).

[15]O. Saquet, J. Chicois, and A. Vincent, Mat. Sci. Eng. A, Vol. 269, pp. 73-82, 1999.

[16] M. M. Blaow and B.A. Shaw, Magnetic Barkhausen Noise Profile Analysis: Effect of

Excitation Field Strength and Detection Coil Sensitivity in Case Carburized Steel, Materials

Sciences and Applications, Vol. 5 pp.258-266, 2014, doi: 10.4236/msa.2014.55030.

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13

Development and Determination of the Age Hardening Characteristics of Al-

2.00Mg-2.66Si Wrought Alloy

Ihom A.P.

1, a

and Aniekan Offiong

1

1 – Department of Mechanical Engineering, University of Uyo, Uyo , PMB 1017 Uyo-Nigeria

a – ihomaondona@gmail.com, ihomaondona@uniuyo.edu.ng

DOI 10.13140/RG.2.1.1965.7207

Keywords: precipitation hardening, characterization, solutionising, wrought alloy, development, determination.

ABSTRACT. The study, was carried out by developing the alloy using the foundry route of melting, alloying, and

casting. The produced test samples were machined to produce test specimens which were subjected to precipitation

hardening treatment. The test specimens were for impact and hardness test to inference the response of the developed

alloy to age hardening treatment. The ageing temperature was 190

o

C, and the ageing time was from 1 -5 hrs. The

control specimen was not age hardened and when compared with the age hardened specimens, the tested mechanical

properties of the age hardened specimens were better than the control specimen. The hardness was seen to increase,

with ageing time peaking at 3hrs of ageing to a value of 38.34 HRB, dropped and rose after 4hrs of ageing and

continued to increase, thereby prompting curiosity. The toughness had a steady increase as the ageing time was

increased, which clearly showed that the developed alloy responded to age hardening treatment.

Introduction. A lot of aluminium–based alloys have been developed and characterized as can be

seen in B.S. Aerospace Series, section L (aluminium and light alloys). Specification of the

aluminium-based alloys is clearly stated unfortunately the specification for this research work could

not be sighted however, close compositions were seen. According to the wrought aluminum alloy

designation system, alloys of these series (Al-Si-Mg) are designated 6xxx. Aluminum – Magnesium

– Silicon alloys are heat treatable [2, 5, 11]. Solution treatment followed by either artificial or

natural ageing allows considerable increase in yield-strength (3-5 times). Ductility of the alloy

decreases as a result of the heat treatment. Hardening of the alloys from this group is achieved due

to precipitation of the phase Mg-Si occurring during ageing [5]. The phase has a fixed ratio between

the elements content (valence compound), therefore amount of magnesium and silicon in 6xxx

alloys is balance according to this ratio or with an excess of silicon. Alloys of this series possess

high mechanical strength combined with good formability and corrosion resistance. Excess of

silicon enhances effect of precipitation of the alloys but decreases their ductility because of

segregation of silicon in the regions of grain boundaries. This adverse effect of silicon may be

diminished by addition of chromium and manganese depressing recrystalization during solution

treatment. Temperature of artificial ageing of 6xxx alloys is 320 – 360

o

F (160 – 182

o

C).

Aluminum – Magnesium - Silicon alloys (6xxx series) are used in aircraft and automotive

applications, in architectural applications and as structural materials [2].

Aluminium alloys used in both cast and wrought forms may be precipitation hardened if of suitable

composition. Age-hardening as it was then called, was infact discovered in some aluminium-based

alloys at the beginning of the century and subsequently developed for use in military aircraft during

the First World War. The extent of the formation of coherent precipitates at ordinary temperatures is

limited so that strength attains a fairly low maximum value in a few days and this process used to be

called ‘age-hardening’. At higher temperatures the formation of coherent precipitates proceeds

further and so the strength continues to increase. However, a point is reached where the thermal

activation is such that tiny non-coherent particles of begin to form in accordance with phase

equilibrium. Other wrought aluminium alloys which can be precipitation hardened are those

Mechanics, Materials Science & Engineering, July 2016 – ISSN 2412-5954

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14

containing small amounts of magnesium and silicon. These form the compound Mg

2

Si the

solubility of which, like that of CuAl

2

, increases considerably with temperature [11].

Pure aluminium is relatively soft and weak, it has a tensile strength of no more than 90N/ mm

2

in

the annealed condition-and for most engineering purposes is used in the alloyed form. The strengths

of many aluminium-base alloys can be further increased by precipitation hardening to produce a

strength / mass ratio of the same order as for high-tensile steels. The greater relative volume of

aluminium alloy involved for a specific force-bearing capacity means that greater flexibility in

design is possible. The objective of this research is to develop another aluminium alloy from the

6xxx series (Al-Mg-Si) and determine its ageing characteristics in terms of effects on its hardness

and toughness properties. It’s already established that there is a relationship between hardness and

tensile strength for most metallic materials although it may not be a direct proportional relationship

[1, 11]. In like manner there is a relationship between tensile strength and toughness of a material,

this can be observed in stress-strain curve where the area under the curve is proportional to the

energy required to fail the metal. This energy is equal to the energy required to fail the same

material under toughness or impact test. By implication the toughness of a material can be inferred

from the area under the curve of its stress-strain plot [5].

Materials and Method.

Materials. The materials used for this project included the following: aluminium cables which were

procured from an electrical company in Uyo, pure silicon and magnesium were acquired from Zaria

and other materials like, salt (NaCl), sand, clay and water were locally sourced in the University of

Uyo and Uyo town.

Equipment. The equipment used during the research included; melting furnace, round metal

mould, impact tester, Hardness tester, centre lathe, hack saw, Bench vice, crucible pot, electric

furnace with 1200

o

C peak capacity, furnace pan and tong, venier caliper, electronic measuring

scale, and stop watch.

Method. Aluminum cables known to posses 99.8% purity, silicon powder, industrial salt (NaCl),

and magnesium were used to produce Al-2.00Mg-2.66Si wrought alloy. The Foundry route was

used for the production. Metal moulds, charge preparation and calculation were all carried out

before melting in the crucible furnace where it was possible to stir the melt for homogeneity. The

melt was poured to produce four test samples of 20.6 mm dia x 302 mm. These were used to

produce test specimens for solutionising and ageing treatments before they were subjected to

hardness and toughness tests. Details of the work procedure is as presented below:

Charge Calculations. Total quantity of the developed alloy required for characterisation = 1.4kg

(1400g).

But percentage of the alloying element required for the production of the wrought alloy was

calculated as follows:

The percentage of silicon =

= 2.66%

The percentage of Mg used =

= 2.0%

The total weight of the cast samples was 1400g and the amount of Si and Mg, used were 37.2g and

28g respectively.

Therefore 37.2+28.0 = 65.2g

Subtracting 65.2g from total cast material the balance will be 1334.8 g i.e (1400-65.2)

Total amount in grammes of Al used = 1334.8g.

Preparation of the mould:

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15

Fig. 1. Side view and bottom view of the steel pipe used in the construction of the metal moulds.

Fig. 1 shows a 302 mm length x 22.6 mm diameter pipe before the pipe was divided into half.

Round and flat file was used to remove built up and sharp edges from the surface of the cut to

maintain a smooth surface. This pipe was divided in this manner so as to enable the easy removal of

the solidified alloy after pouring. Flat sheet metal was welded to each half of the pipe using electric

welding. The two halves were brought together again to make one piece, binding wire was used to

hold it together and clay was used to seal the opening at the joint. The moulds prepared in this way

were dried and ready for use.

Melting and Casting.

The charge as calculated above was transferred to the crucible furnace, where the aluminium was

first charged and allowed to melt before the addition of the magnesium and silicon alloying

elements. Industrial salt (NaCl) was sprinkled to the melt and vigorously stirred to obtain a

homogeneous melt before pouring into the already prepared moulds. The castings were allowed to

cool before they were removed from the moulds.

Specimen Preparation.

The sample test bars were as shown in Plate 1, they were machined to produce standard test

specimens for hardness test and toughness test. The hardness test specimens were 20mm x20mm

while the toughness test specimens were prepared according to ISO Standard for V-Charpy impact

test. Plate II shows the prepared specimens.

Plate I. Sample test bars of Al-2.00Mg-2.66Si Alloy.

302 mm

22.6m

m

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Plate II. Impact and hardness test specimens before being subjected to ageing process.

Ageing Treatment. The test specimens shown in Plate II were subjected to ageing treatment with

the exception of the control specimens. The specimens for ageing treatment were first solutionised

at 500

o

C using the furnace shown in plate III. The test specimens were then quenched in warm

water. The quenched specimens were removed and dried before precipitation treatment at 190

o

C in

the same furnace shown in plate III. The ageing time of the test specimens ranged from 1 hr to 5 hrs

in steps of 1hr. After the ageing treatment the test specimens were ready for mechanical properties

testing.

Plate III. The furnace used for precipitation treatment.

Mechanics, Materials Science & Engineering, July 2016 – ISSN 2412-5954

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Plate IV. Removal of specimen after each hour of ageing time.

Hardness Testing Procedure. In this experiment test pieces were in round shape as shown in the

plate IV below:

Plate V. Hardness specimen before undergoing hardness test.

The test pieces were placed on the table of the testing machine, the wheel was rolled to bring into

contact the test piece and the indenter under a minor load of 9.8 kg, which took up the “slack” in the

system while the dial indicator was set to zero. The major load was then applied; the indicator made

about 2 revolutions before becoming steady, and the hardness value was directly read on the

indicator. The machine used had an indenter steel ball (1.6 mm) and Rockwell Hardness B-Scale

with minor load 98N (9.8kgf) and major load 980N (100kgf) were selected. The hardness machine

identity was Karl Frank 6MBH, WEINHEM BIRKENAH, type – 38506 and werk-Nr-21289.

Impact Testing Procedure. A Notch-bar test piece of standard geometry was mounted horizontally

on the anvil of the machine which was struck by a fast moving weighted pendulum with a velocity

of 5.24 m/s, while the energy absorbed in breaking the specimen was measured in joules and read

directly from a dial indicator. It measured the relative toughness of the material, which indicated the

material capacity to absorb energy and deform plastically before fracture. The machine capacity

was 300J scale-Charpy with test temperature being room temperature; velocity of pendulum was

5.24 m/s, and the equipment was made by Avery-Denison, England.

Results and Discussion.

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18

Results. The results of the research are displayed in Tables 1-3 below. The variation of the hardness

and toughness of the material with ageing time is also shown in figs. 2-3.

Table 1. Result of the hardness test.

Ageing Time

Average Hardness Value

0 hr (unaged)

36.50

1 hr (aged)

36.00

2 hrs (aged)

35.95

3 hrs (aged)

38.34

4 hrs (aged)

34.00

5 hrs (aged)

39.33

Fig. 2. Hardness variation of the developed Al-2Mg-2.66Si alloy with ageing time.

Table 2. Result of impact test.

Ageing Time

Energy absorbed (J)

Type of specimen

1hr

8j note: blow hole noticed in the

sample

Standard v- note (square shape)

2 hrs

28j

Standard v- note (square shape)

3 hrs

21j

Standard v- note (square shape)

4 hrs

28j

Standard v- note (square shape)

5 hrs

38j

Standard v- note (square shape)

Control (Unaged

specimen)

9.5j

Standard v- note (square shape)

Table 3. Ageing time versus toughness.

31

32

33

34

35

36

37

38

39

40

0 1 2 3 4 5

Hardness

Ageing Time (hrs)

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19

Ageing Time

Toughness

0 hr (unaged)

9.5j

1 hr (aged)

8j

2 hrs (aged)

20j

3 hrs (aged)

21j

4 hrs (aged)

28j

5 hrs (aged)

38j

Fig. 3. Toughness variation of the developed Al-2Mg-2.66Si alloy with ageing time.

Discussion. The objective of this research is to develop a wrought aluminium alloy of the

composition Al-2Mg-2.66Si and to determine whether it has ageing characteristics. This objective

has been implemented and the burden of proof lies with the results generated from the mechanical

tests carried out on the developed alloy. The results are interesting as discussed below and should

elicit more research work on the alloy from the materials science and engineering research world.

Analysis of Hardness Test Result. Table 1 shows the result of the hardness test and Fig.2 shows a

graphical representation of the variation of hardness and the ageing time at constant temperature

(190

0

C). The hardness value of the unaged specimen was 36.50HRB this value dropped gradually

to 35.95HRB at 2hrs of ageing. This softening is normal because of the precipitates being formed

and rearrangement of atoms. Immediately after 2 hrs of ageing the hardness started increasing and

peaked at 38.34HRB this pattern fits in with previous works which showed the curve peaking at

3hrs and then the hardness droping with further ageing. The hardness in this case actually dropped

to 34.00HRB at 4hrs of ageing but suddenly started rising and rose to 39.33 at 5hrs of ageing. It is

possible that the hardness will continue to increase beyond the 5hrs of ageing. Unfortunately the

work terminated at 5hrs it is suggested that further research work should be carriedout which should

go beyond the 5hrs used in this work to actually establish the ageing time relationship with the

hardness of the developed alloy. The line trend of hardness-ageing time relationship is very

interesting and could mean that at longer ageing time; higher hardness values may be obtained.

Essien and Udo [2] have observed that the behaviour of the hardness-ageing time curve to rise, fall

and start rising again is not uncommon with ageing, it is possible for the hardness to decrease, and

is normally linked to the nature of precipitate that has been formed at a particular time. The

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5

Toughness (joules)

Ageing Time (hours)

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20

formation of Mg

2

Si precipitates is normally associated with the alloy system under consideration.

Coherent and non-coherent precipitates will exhibit different hardness values. Non-coherent

precipitates are normally associated with discontinuities at the interface with the matrix [9]. Their

formation therefore normally results in decrease of hardness of the material. Ihom, et al. [7] agreed

with the above explaination but further added that the ageing process is a difussion controlled

process and is controlled by this equation.

D = D

O

e

Q/RT

(1)

Where D is the diffusion rate;

D

0

is the diffusion co-efficient;

Q is the activation energy required to move an atom;

R is the gas constant and T is the temperature in Kelvin.

At higher temperatures, the movement of solutes is faster because the activation energy required is

met quickly. Also time is required for the atoms to diffuse to new position X; that is why the

distance travelled by the atoms is a function of diffusion coeficient and resident time of the aged

alloy (X =

. According to the equation the extent of interdiffusion X increases with the square

root of time t. Therefore if a longer ageing time is applied it may lead to over-ageing, which means

the hardness cannot continue to increase indefinately but must reduce after some time [7, 9, 10].

Impact Test. The essence of the impact test was to determine the effect of age hardening on the

developed alloy as would manifest in the toughness of the material with increased energy

absorption of the specimens before failure (fracture). The result and the pattern as the ageing time is

increased is shown in Tables 2-3. Fig. 3 shows the toughness variation of the developed Al-2Mg-

2.66Si alloy with ageing time. The graph clearly shows that as the ageing time was increasing the

material toughness was also increasing. There was a drop in toughness value at 1hr of ageing and

the reason was as explained in Table 2. There was a blowhole in the test specimen which reduced

the true value of the toughness of the material, because of the presence of the blowhole underneath

the specimen the energy absorbed before fracture was drastically reduced. The other specimens

were defect free and so gave more realistic results. The toughness kept increasing with ageing time,

however, previous works have shown that there comes a time when over ageing occurs and

thereafter both the hardness and toughness reduces [5, 8]. It therefore means that toughness cannot

continue to increase indefinately with ageing time. The developed alloy from the toughness-ageing

time curve has clearly demonstrated that it has responded to precipitation hardening treatment, and

it is therefore an age hardening alloy. longer ageing periods may be employed to see the extent of

improvement of this property. There is a correlation between toughness of a material and its tensile

strength that is why the area under a stress-strain curve can be used to inference the toughness of a

material. When the area is small it means the toughness of the material is low the energy absorbed

by the material before failure is low, if however, the area under the curve is large it means the

material is a tough material and must absorb a substantial amount of energy before failure [5, 8].

From the preceeding it therefore means that the tensile strength of the developed material was

improved alongside with the toughness of the material as the material was aged at different time

intervals.

Summary. The research work titled “Development and Determination of the Age Hardening

Characteristics of Al-2Mg-2.66Si Wrought Alloy” has been extensively considered and the

following conclusions drawn from the work:

i) the work developed an alloy of the composition Al-2Mg-2.66Si a wrought alloy that was

precipitation hardened at various intervals

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21

ii) the hardness test results of the age hardened alloy at various periods revealed that the developed

alloy responded to age hardening

iii) the impact results of the age hardened alloy specimens clearly showed that the developed alloy

responded to age hardening, as the ageing time was increasing the toughness of the developed alloy

was also increasing, and

iv) the developed alloy exhibited an interesting characteristics as the tested mechanical properties

kept increasing with increase in ageing time necessitating the suggestion that further work should be

carried out on the developed alloy with ageing treatment covering twelve hours.

Acknowledgement. The authors of this work wish to sincerely acknowledge the contributions of

our undergraduate students Mr Essien, E.V., and Mr Udo, U.E. We do hope that the exposure they

had during the course of this work will broaden their horizon and help them to appreciate

metallurgical engineering.

References

[1] Alo O.A., Umolu L.E., Ajao J.A. and Oluwasegun K.M. (2012) Thermal, Hardness and

Microstructural Characterisation of Al-Si-SiC

p

Composites, JMMCE, 11(2), 159-168, doi:

10.4236/jmmce.2012.112013.

[2] Essien E.V and Udo U.E. (2016) Development and Determination of the Age Hardening

Characteristics of Al–Mg-Si based Alloy, B.Eng. Degree project submitted to the Department of

mechanical Engineering University of Uyo, Uyo-Nigeria.

[3] Hassan S.B., Aponbiede O. and Aigbodion V.S. (2008) Precipitation Hardening Characteristics

of Al-Si-Fe-/ SiC Particulate Composite, Journal of Alloys and Compounds, 466(1-2), 268-222.

[4] Hassan S.B. and Aigbodion V.S. (2009) The Effect of Thermal Ageing on Microstructure and

Mechanical Properties of Al-Si-Fe/Mg Alloys, Journal of Alloys and Compounds, 486 (1-2), 309-

314.

[5] Higgins R.A. (1985) Properties of Engineering Materials, 6

th

Edition: Great Britain, Hodder and

Stoughton Educational.

[6] Ihom A.P., Nyior G.B., and Zamanni I.G. (2012) The Effect of Thermal Ageing on

Microstructure and some Mechanical Properties of Al/2.0% Glass Reinforced Composite, IJRRAS

12(3), 1-6.

[7] Ihom A.P., Nyior G.B., Nor, I.J., Ogbodo N.J. (2013) Investigation of Egg Shell Waste as an

Enhancer in the Carburization of Mild Steel, AJMSE, 1(2), 29-33, doi: 10.12691/ajmse-1-2-3.

[8] Markson I.E. (2016) The Effects of Pressure on Squeeze Cast Aluminium Alloys, PhD Degree

Project Submitted to the Department of Mechanical Engineering, University of Uyo, Uyo-Nigeria.

[9] Martins J.W. (1998), Precipitation Hardening, 2

nd

Edition, Butterworth Heinemann, Oxford, pp.

1-20.

[10] Matthews F.L. and Rawlings R.D. (2005) Composite Materials: Engineering and Science, 4

th

Edition, England: Woodhead Publishing Limited, pp. 80-90.

[11] Shragger A.M. (2008), Elementary Metallurgy and Metallography, Second Revised Edition

Reprinted, Dover Publications Inc, Newyork, pp. 136-140.

Mechanics, Materials Science & Engineering, July 2016 – ISSN 2412-5954

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22

Effect of Textures on Tensile Properties of Extruded Ti64/VGCF Composite by

Powder Metallurgy Route

Patchara Pripanapong

1

, Shu-feng Li

2

, Junko Umeda

2

, Katsuyoshi Kondoh

2

1 – Graduated School of Engineering, Osaka University, 2-1, Yamadaoka, Suita, Osaka, Japan

2 – Joining and Welding Research Institute, Osaka University, 11-1 Mihogaoka, Ibaraki, Osaka, Japan

DOI 10.13140/RG.2.1.1120.1525

Keywords: Ti-6Al-4V, VGCFs, composite materials, hot extrusion, dynamic recrystallization.

ABSTRACT. Monolithic Ti-6Al-4V and Ti-6Al-4V composited with vapor grown carbon fibers (VGCFs) were

fabricated by powder metallurgy (P/M) route in this research. Spark plasma sintering (SPS) subsequent by hot extrusion

was applied in order to fabricate a full-density and high strength composite materials. A severe plastic deformation

during hot extrusion resulted in a dynamic recrystallization (DRX) in α-Ti grains. Dynamic recrystallization was

observed in a low deformation temperature region, which yield point of material was also observed in the stress-strain

curve. Furthermore, the addition of VGCFs encouraged the dynamic recrystallization during hot extrusion. Ti64+0.4

wt-% VGCFs shows the highest tensile strength of 1192 MPa at the end part of the extruded rod where the temperature

of material was lower compared to the tip and middle part during extrusion. Additionally, the improvement in tensile

strength was contributed by solid-solution strengthening of carbon element originated from VGCFs in α-Ti matrix.

Introduction. Ti-6Al-4V alloy (Ti64) is the most well-known among Ti alloys, and used in many

industries. High specific strength, good corrosion resistance and biocompatibility promoted a

widely use of Ti and its alloys such as in aerospace and automobile industries, or medical devices

and prosthesis [1, 2]. Many researchers studied the effect of hot working on microstructure and

mechanical properties of wrought or cast Ti64. A. Momeni et al. studied the effect of deformation

temperature and strain rate on microstructure and flow stress of wrought Ti64 under hot

compression test [3]. Ti64 specimens, which experienced a hot compression test at 1273 and 1323

K, exhibited a large recrystallized α-grain with low flow stress on the microstructure. This

correlated with the results proposed by T. Seshacharyulu et al. and R. Ding et al. for the cast Ti64

[4, 5]. G.Z. Quan et al. studied the modelling for dynamic recrystallization in Ti-6Al-4Al by hot

compression test. The result shows that a flow stress decreases with the increasing of deformation

temperature. The high deformation temperature promotes the mobility at the boundaries for

annihilation of dislocation, and the nucleation and growth of dynamic recrystallization [6]. H.Z. Niu

et al. studied the phase transformation and dynamic recrystallization (DRX) behaviour of Ti-45Al-

4Nb-2Mo-B (at-%) alloy. The results show that the DRX modes were strongly depends on

deformation temperature, and a decomposition of lamellar structure along with the DRX of γ and

B2/β grain occurred at low forging temperature [7]. D.L. Ouyang et al. studied the recrystallization

behaviour of Ti-10V-2Fe-3Al alloy after hot compression test. They reported that a partial grain

refinement related to incomplete DRX was observed even after a large strain of 1.6, and an

increment of strain resulted in an increasing of volume fraction of recrystallized grain. The full

grain refinement accompanied by the completely DRX was developed at lower temperature of 1223

K by severe deformation [8]. The dynamic recrystallization behaviour of Ti-5Al-5Mo-5V-1Cr-1Fe

alloy was reported by H. Liang et al. The DRX always occur when the store energy in a material

reaching the critical value. During hot deformation, the increase of flow stress caused by dislocation

generation and interaction resulted in an improvement of strength of Ti alloys. The sample

deformed at 1073 K exhibited higher tensile strength compared to sample deformed at 1153 K

because more dislocations were generated [9]. There are many reports related to dynamic

recrystallization of Ti alloys but no report in dynamic recrystallization behaviour of Ti64 composite

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23

with VGCF, and its effect on tensile properties of composite material was found yet [10, 12]. The

objective of this research is to study the texture of Ti64 and Ti64 composites fabricated by spark

plasma sintering, and subsequently hot extruded. The effect of dynamic recrystallization on tensile

properties of monolithic Ti64 and Ti64/VGCF composite materials were investigated through this

research. The samples cut from the end and middle part of extruded rod (extruded temperature

lower than 1243 K) show a DRX structure which many of small nucleated grains were observed.

Tensile sample obtained from the part that experienced extrusion at low temperature exhibited a

high tensile strength because a large amount of defects was generated in a sample.

Experimental procedure. Ti-6Al-4V atomization powder (Osaka Titanium Technology Co., Ltd.

TILOP64-45) with a spherical shape and diameter of 45 μm (fig. 1a), and vapor grown carbon

fibers (Showa Denko K.K.,VGCFs

TM

) with 8 μm in length and diameter of 150 nm (fig. 1b) were

applied in this research. The chemical composition of Ti64 powder was listed in Table 1. Ti64

powder was mixed with cleasafe oil (0.15 g) by table milling equipment for 3.6 ks with rotation

speed of 90 rpm and subsequently mixed with VGCFs by rocking milling for 7.2 ks. Rocking

milling was performed for long time in order to provide a uniform distribution of VGCFs on

powder surface. The Ti64 composite materials were fabricated in three compositions such as 0, 0.1

and 0.4 wt-% of VGCFs. The Ti64/VGCFs mixed powder was poured in the carbon container,

which has an inner diameter of 42 mm. The inner wall of container was sprayed with boron nitride

to prevent a reaction between Ti64 powder and carbon container during SPS. The monolithic Ti64

and mixed Ti64/VGCFs powder were pre-compacted by hydraulic hand press under pressure of 20

MPa before sintering. The green compacts were consolidated by spark plasma sintering (Syntech

CO. SPS103S) at 1273 K with heating rate of 0.54 K/s for 1.8 ks, and the pressure of 30 MPa was

introduced to a green compact under vacuum atmosphere of 5 Pa. Afterwards, the sintered billets

were preheated in a horizontal image furnace at 1323 K for 0.6 ks under argon atmosphere before

hot extruded into 12 mm in diameter rod by 200 ton press machine (THK Slidepack FBW3950R

1200L) with an extrusion speed of 6 mm/s. The sintered billets and extruded specimens were cut for

microstructure observation. For extruded rod, the samples were cut from three positions which are

shown in fig. 2. The samples were ground with SiC abrasive paper, polished with Al

2

O

3

colloidal

and etched by Kroll etchant (H

2

O:HNO

3

:HF = 100:5:1) for microstructure observation. For EBSD

analysis, the specimens were polished with SiO

2

colloidal by vibratory polisher. The

microstructures observation and phase analysis of sintered and extruded specimens were performed

by optical microscope and scanning electron microscope (JEOL JSM6500F). The grain size and

texture were analysed by electron back-scatter diffraction (EBSD) method. The tensile samples

were machined from three positions in extruded rod (fig. 2) with 20 mm in gauge length and 3 mm

in diameter. The universal tensile test machine (Autograph AGX 50 kN, Shidmazu) was applied for

tensile test with strain rate and cross head speed of 5×10

-4

s

-1

and 6 mm/min, respectively.

Fig. 1. SEM micrograph of (a) Ti-6Al-4V atomization raw powder, (b) Vapor grown carbon fibers

(VGCFs).

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Table 1. Chemical composition of Ti-6Al-4V atomization raw powder (wt-%).

Material

Al

V

Fe

C

O

N

H

Ti

Ti-6Al-4V

6.12

4.48

0.03

0.01

0.13

0.014

0.006

Bal

Fig. 2. Microstructure observation positions in extruded rod.

Results and discussion. The microstructures of Ti64 and Ti64+0.1wt-%VGCFs after SPSed

including texture information such as pole figure (PF), inverse pole figure (IPF) and intensity of

each plane direction are shown in fig.3. The monolithic Ti64 that was SPSed at 1273 K, which is

above β transus temperature (1243 K) shows a large prior β grains of 170 μm (separated by yellow

dash line) and α-lamellar colonies inside β grain (fig. 3a). The microstructure evolution was

explained by the change of allotropy at 1243 K from α (hcp) to β (bcc) during heating followed by

precipitation of α-lamellar phase inside grain and along the grain boundaries of prior β during

cooling. On the other hand, Ti64+0.1wt-%VGCFs shows a different microstructure structures

compared to monolithic Ti64 (fig. 3b). The microstructure of Ti64+0.1-wt%VGCFs consisted of α-

lamellar and α-equiaxed which formed during cooling from β region to α+β region, and α+β region

to α region, respectively [13, 14]. This microstructure evolution was also observed in Ti64+0.4-

wt%VGCFs. The difference in microstructure between monolithic Ti64 and Ti64/VGCFs

composite material was explained by an effect of α stabilizer of carbon that increased β transus

temperature of composite material. Crystal orientation of Ti64 and Ti64+0.1wt-%VGCFs after

SPSed was shown in fig. 3c and 3d, respectively. The random crystal orientation was observed in

both materials including Ti64+0.4wt-%VGCFs, which was similar to cast alloy [15].

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Fig. 3. SEM micrograph of (a) Ti64 and (b) Ti64+0.1wt-%VGCFs with attached pole figure (PF)

and inverse pole figure (IPF) of (c) Ti64 and (d) Ti64+0.1wt-%VGCFs.

The microstructure of Ti64 and Ti64+0.4-wt%VGCFs observed from different positions in extruded

rod in transversal direction by optical microscope were shown in fig. 4. Henceforth, monolithic

Ti64, Ti+0.1-wt%VGCFs and Ti+0.4-wt%VGCFs will be name as T0, T0.1 and T0.4, respectively.

Figure 4 shows a microstructure of T0-1 and T0.4-1 cut from a tip of extruded rod (A-X represents

a sample name, in which A is a material and X is a position in extruded rod showed in fig. 2). A

fine α-lamellar and α-equiaxed structure compared to SPSed sample was observed in extruded

sample due to an effect of plastic deformation. The size of α-lamellar colony and α-equiaxed

observed in T0 and T0.4 at position 2 and 3 (fig. 4b-4c and 4e-4f) was finer compared to sample

obtained from position 1 (fig. 4a and 4d). This phenomenon was also observed in Ti0.1 as well. A

large additional amount of VGCF in Ti0.4 resulted in a carbide phase formation, its periodic

formulae was evaluated by EDS as Ti

6

C

3.75

.

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Fig. 4. Optical microstructure of Ti64 (a) T0-1; (b) T0-2; (c) T0-3 and Ti64+0.4-wt%VGCFs (d)

Ti0.4-1; (e) Ti0.4-2; (f) Ti0.4-3 in transversal direction.

Figure 5 shows crystal orientation, pole figure (PF) and inverse pole figure (IPF) from three

positions in Ti0 rod analyzed by EBSD. Figure 5a shows a crystal orientation of Ti0-1, which was

extruded at temperature higher than β transus (1243 K). This sample exhibited a strong intensity in

[0001] direction which parallel to extrusion direction, and a grain size of 16 μm was observed. Ti0-

2, which was extruded at temperature lower than β transus (1173 K), shows some small nucleated

grains on microstructure with a decreasing in intensity in [0001] direction. This sample shows a

grain size of 10 μm, which was smaller, compared to Ti0-1 (fig. 5b). For Ti0-3, this position was

extruded at temperature lower than 1173 K. A very small grain size of 1.3 μm and a large amount of

nucleated grain was observed, grain morphology was changes from α-lamellar to α-equiaxed. The

intensity in [0001] direction was decreased compared to Ti0-2. These results indicated that a

dynamic recrystallization (DRX) was occurred in Ti0-3 sample (fig. 5c). The important factor that

induced a DRX during hot extrusion was a deformation temperature, which different at each

position in extruded rod. The deformation temperature will control a stored energy in extruded

material [16]. Ti0-1 was extruded at highest temperature compared to other position (over β transus

temperature), deformation at high temperature in a β phase region lead to an insufficient stored

energy for DRX. The stored energy in Ti0-2 (fig. 4b) was higher than Ti0-1 because more defects

such as dislocation and stress was generated in a material deformed at low temperature compared to

high temperature [17]. Similarly, a large amount of dislocations was generated in a material after

cold deformation. This phenomenon was clearly observed in Ti0-3, a high degree of DRX resulted

in a random crystal orientation.

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Fig. 5. Crystal orientation, inverse pole figure (IPF) and pole figure (PF) in transversal direction

of (a) Ti0-1; (b) Ti0-2; (c) Ti0-3.

Figure 6 shows a crystal orientation, PF and IPF of extruded Ti composited with 0.1 wt-%VGCFs.

Ti64 with 0.1wt-%VGCFs shows a nucleation of new grain in a high extruded temperature position

(fig. 6a). The grain morphology of Ti0.1-1 was different from Ti0-1 because microstructure of

Ti+0.1wt-%VGCFs was α-equiaxed after SPSed. An average grain size of Ti0.1-1 was 2 μm after

extrusion but not uniform. The microstructure of Ti0.1-1 composed of prior α-equiaxed grain and a

small nucleated grain, which indicated that DRX was not completed due to low stored energy. The

small nucleated grain was ceased to growth during cooling, and a final microstructure was non-

uniform. A DRX was easier to observe in samples mixed with VGCFs compared to monolithic Ti64

because an interstitial solid solution of carbon in Ti64 matrix acted as a defect, which increased a

stored energy when material was deformed [16]. Ti0.1-2 shows a similar grain morphology, grain

size and intensity in [0001] direction to Ti0.1-1 (fig. 6b). However, Ti0.1-3 shows a uniform grain

size of 1.5 μm, and intensity in [0001] direction was decreased compared to Ti0.1-2 (fig. 6c). The

uniform grain size and a random crystal orientation indicated that Ti0.1-3 exhibited higher DRX

degree compared to Ti0.1-2 [18].

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Fig. 6. Crystal orientation, inverse pole figure (IPF) and pole figure (PF) in transversal direction

of (a) Ti0.1-1; (b) Ti0.1-2; (c) Ti0.1-3.

Figure 7 shows a crystal orientation, PF and IPF of extruded Ti composited with 0.4wt-%VGCFs.

Ti64+0.4wt-%VGCF shows a similar microstructure to Ti64+0.1wt-%VGCF that a small nucleated

grain and prior α-grain with a size of 2 μm was observed in Ti0.4-1 and Ti0.4-2 (fig. 7a and 7b).

For Ti0.4-3, a uniform grain size of 1.5 μm was observed after nucleated grain growth, and a lowest

intensity in [0001] direction among three samples was detected (fig. 7c). From crystal orientation,

PF and IPF results, a DRX was strongly depended on a stored energy in a material. The stored

energy was increased by decreasing of deformation temperature or an effect of solid solution of

carbon. A. LUCCI et al. reported that an addition of various substitution alloying elements to Cu in

a low content enable to induced a dynamic recrystallization after deformation by increasing a stored

energy in a material [19].

Tensile properties of Ti64, Ti64+0.1wt-%VGCF and Ti64+0.4-wt%VGCF obtained from different

position in extruded rod were listed in table 2. For all materials, the lowest and highest of 0.2%YS

and UTS was obtained from tip and end of extruded rod, respectively. During hot extrusion,

position 1 was extruded at highest temperature (the first part to be extruded) compared to other

position resulted in a low shear stress, and small amount of dislocation was formed in material. The

flow stress was decreased by increasing of deformation temperature [3]. Samples that were obtained

from position 2 and 3 show an increasing in tensile strength and yield strength compared to position

1 instead of decreasing because of dislocation removal by an effect of DRX. This was implied that

shear stress generated by hot extrusion induced a DRX, and simultaneously generated a new

dislocations at grain boundary during DRX. The size of new grains were very fine (fig. 4c and 4f)

then a considerable amount of dislocations was formed in material resulted in an improvement of

tensile strength with a traded off of elongation. Ti64 composite with VGCFs exhibited higher

tensile strength compared to monolithic Ti64 obtained from same position because an effect of solid

solution strengthening of carbon [20]. 0.2%YS and UTS of Ti64+0.4wt-%VGCFs was small

decreased compared to Ti64+0.1wt-%VGCFs for sample obtained from same position due to a

large amount of brittle carbide phase formation (fig. 4).

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Fig. 7. Crystal orientation, inverse pole figure (IPF) and pole figure (PF) in transversal direction

of (a) Ti0.4-1; (b) Ti0.4-2; (c) Ti0.4-3.

Table 2. Tensile properties of Ti0, Ti0.1 and Ti0.4 obtained from different position in extrusion rod.

Position

Ti64

Ti64+0.1wt-%VGCF

Ti64+0.4wt-%VGCF

UTS

(MPa)

0.2%YS

(MPa)

Elongation

(%)

UTS

(MPa)

0.2%YS

(MPa)

Elongation

(%)

UTS

(MPa)

0.2%YS

(MPa)

Elongation

(%)

1

990

943

15.2

1127

1087.0

17.7

1191

1069

10.9

2

1121

1002

12.9

1166

1165

18.1

1150

1150

9.1

3

1131

1092

9.4

1190

1170

12.0

1192

1192

7.4

Stress-strain curve of Ti0, Ti0.1 and Ti0.4 extruded material obtained from different positions in

extruded rod was shown in fig.8. For monolithic Ti64, Ti0-1 and Ti0-2 shows ductile behaviour that

stress-strain curve exhibited no yield point. On the other hand, Ti0-3 obtained from DRX region

exhibited a yield point in stress-strain curve with highest UTS of 1130.8 MPa (fig. 8a). This is

because a considerable amount of dislocation formed in a part that extruded at low temperature. In

the case of Ti0.1, yield point was observed in Ti0.1-2 and Ti0.1-3 (fig. 8b and c). From a principle,

yield point was developed in a material that gained a sufficient shear stress, resulted in a permanent

displacement of atom [9]. This was occurred in sample which experienced a deformation at low

temperature such as the end part in extruded rod. For Ti64/VGCFs composite material, yield point

was observed in samples obtained from position 2 because interstitial solid solution of carbon

provided an additional shear stress in material [20]. Ductility of Ti0.4 was much lower than

monolithic Ti0 and Ti0.1 due to a formation of large amount of Ti

6

C

3.75

brittle intermetallic phase.

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Fig. 8. Stress-strain curve of extruded materials obtained from different positions (a) Ti0; (b) Ti0.1;

(c) Ti0.4.

Summary. The high stored energy in monolithic Ti or Ti composites was obtained when material

experienced a deformation at low temperature or by an effect of solid solution of carbon, resulted in

a DRX. The improvement of tensile strength of Ti0, Ti0.1 and Ti0.4 was obtained by DRX

followed by dislocation formation during hot extrusion. The details are mentioned below.

1. The dynamic recrystallization (DRX) was occurred in sample that extruded at low

temperature resulted in high stored energy in material. The small recrystallized grain was observed

as evidence.

2. The solute carbon atom from vapor grown carbon fibers (VGCFs) in Ti64/VGCFs

composite materials acted as a defect that provided additional stored energy in material during hot

extrusion, and facilitate a DRXed.

3. The formation of dislocations at grain boundary during DRX at low deformation

temperature which many small nucleated grains were formed resulted in improvement in yield and

tensile strength. The increased value was small but systematically occurred for all samples.

References

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[2] D. Mareci, R. Cheraliu, D.M. Gordin and T. Gloriant, Comparative corrosion study of Ti-Ta

alloys for dental applications, Acta Biomater., 2009, 5, 3625-3639, DOI:

10.1016/j.actbio.2009.05.037.

[3] A. Momeni and S.M. Abbasi, Effect of hot working on flow behaviour of Ti-6Al-4V alloy in

single phase and two phase regions, Mater. Des., 2010, 31, 3599-3604, DOI:

10.1016/j.matdes.2010.01.060.

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[4] T. Seshacharyulu, S.C. Medeiros, W.G. Frazier and Y.V.R.K. Prasad, Hot working of

commercial Ti-6Al-4V with an equiaxed α-β microstructure: materials modeling considerations,

Mater. Sci. Eng. A, 2000, 284, 184-194, DOI: 10.1016/S0921-5093(00)00741-3.

[5] R. Ding, Z.X. Guo and A. Wilson, Microstructural evolution of a Ti-6Al-4V alloy during

thermomechanical processing, Mater. Sci. Eng. A, 2002, 327, 233-245, DOI: 10.1016/S0921-

5093(01)01531-3.

[6] G.Z. Quan, G.C. Lua, J.T. Liang, D.S. Wu, A. Mao and Q. Liu, Modelling for the dynamic

recrystallization evolution of Ti-6Al-4V alloy in two-phase temperature range and a wide strain rate

range, Comput. Mater. Sci., 2015, 97, 136-147, DOI: 10.1016/j.commatsci.2014.10.009.

[7] H.Z. Niu, Y.F. Chen, Y.S. Zhang, J.W. Lu, W. Zhang and P.X. Zhang, Phase transformation and

dynamic recrystallization behaviour of a β-solidifying γ-TiAl alloy and its wrought microstructure

control, Mater. Des., 2016, 90, 196-203, DOI: 10.1016/j.matdes.2015.10.133.

[8] D.L. Ouyang, M.W. Fu and S.Q. Lu, Study on the dynamic recrystallization behaviour of Ti-

alloy Ti-10V-2Fe-3V in β processing via experiment and simulation, Mater. Sci. Eng. A, 2014, 619,

26-34, DOI: 10.1016/j.msea.2014.09.067.

[9] H. Liang, H. Guo, Y. Ning, X. Peng, C. Qin, Z. Shi and Y. Nan, Dynamic recrystallization

behaviour of Ti-5Al-5Mo-5V-1Cr-1Fe alloy, Mater. Des., 2014, 63, 798-804, DOI:

10.1016/j.matdes.2014.06.064.

[10] C.H. Park, J.H. Kim, J.T. Yeom, C.S. Oh, S.L. Semiatin and C.S. Lee, Formation of a

submicrocrystalline structure in a two-phase titanium alloy without severe plastic deformation, Scr.

Mater., 2013, 68, 996-999, DOI: 10.1016/j.scriptamat.2013.02.055.

[11] Y.Q. Ning, X. Luo, H.Q. Liang, H.Z. Guo, J.L. Zhang and K. Tan, Competition between

dynamic recovery and recrystallization during hot deformation for TC18 titanium alloy, Mater. Sci.

Eng. A, 2015, 635, 77-85, DOI: 10.1016/j.msea.2015.03.071.

[12] Y. Chen, J. Li, B. Tang, H. Kou, X. Xue and Y. Cui, Texture evolution and dynamic

recrystallization in a beta titanium alloy during hot-rolling process, J Alloys Compd., 2015, 618,

146-152, DOI: 10.1016/j.jallcom.2014.08.129.

[13] G.C. Obasi, OM. Ferri, T. Ebel and R. Bormann, Influence of processing parameters on

mechanical properties of Ti-6Al-4V alloy fabricated by MIM, Mater. Sci. Eng. A, 2010, 527, 3929-

3935, DOI: 10.1016/j.msea.2010.02.070.

[14] G.G. Yapici, I. Karaman, Z.P. Luo and H. Rack, Microstructure and mechanical properties of

severely deformed powder processed Ti-6Al-4V using equal channel angular extrusion, Scr. Mater.,

2003, 49, 1021-1027, DOI: 10.1016/S1359-6462(03)00484-6.

[15] S. Roy, S. Suwas, S. Tamirisakandala, R. Srinivasan and D.B. Miracle, Microstructure and

texture evolution during extrusion of boron modified Ti-6Al-4V alloy, Mater. Sci. Eng. A, 2012,

540, 152-163, DOI: 10.1016/j.msea.2012.01.120.

[16] J.D. Verhoeven, Fundamentals of Physical Metallurgy, 1st edn, 55-74; 1975, Canada, John

Wiley and sons, Inc, ISBN: 978-0-471-90616-2.

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10.1016/j.msea.2010.06.044.

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[19] A. Lucci, G. Riontino, M.C. Tabasso, M. Tamanini and G. Venturello, Recrystallization and

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[20] F. Pedix, M.-F. Trichet, J.-L. Bonnentian, M. Cornet and J. Bigot, Relationships between

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8.

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Strength Analysis of Flat Spring of the Resonant Vibro-Impact Module

Volodymyr Gursky

1

, Igor Kuzio

1

1 – Lviv Politechnic National University, Ukraine

DOI 10.13140/RG.2.1.2504.0240

Keywords: flat spring, vibro-impact, resonance, finite element method (FEM), eigenfrequencies, contact stress.

ABSTRACT. The rod model of the resonant vibro-impact module with an electromagnetic drive is considered.

Construction’s design implemented an asymmetrical elastic characteristic by one flat spring with two absolutely rigid

intermediate supports. Eigenfrequency is defined for corresponding location intermediate supports based on the finite

element method. Stress-strain state of the elastic element is graphically represented at the expense of static displacement

of local mass. Contact task is considered and contact force between the flat spring and cylindrical support is calculated.

Also, contact stiffness is determinate. The parameters of volumetric stress state of the contact, calculated analytically

and by modeling in SolidWorks Simulation are shown. The dynamics of the vibro-impact rod system with kinematic’s

disturbance is modeled. Contact and equivalent stresses during operation of the vibro-impact rod system are determined.

1. Introduction. Vibro-impact systems are the basis of technological machines and processes of

environments with challenging physical and mechanical characteristics. This is due the presence

challenging modes as imposing the main and sub-harmonic oscillations, acceleration polyfrequency

range by asymmetry of displacement of the operating mass. The resulting polyfrequency system

generates wave processes, resonance phenomena and power conversion in environments more

effective than harmonic movement law of worker mass. So, the use of frequency vibro-impact

processes and systems is almost justified and promising. However, this requires challenging

practical solutions for the implementation of nonlinear systems and special theoretical analysis to

evaluation the relevant modes.

2. Analysis of recent research and literature. The application of structural nonlinearities is most

appropriate among to the known practical solutions for the implementation of vibro-impact systems

[1, 2]. A few practical solutions are in patents [3–6]. The feature of such systems functioning is the

emergence of sub- and superresonanсe modes with different multiplicities orders [7, 8].

Implementation of such is possible by matching of the system parameters and power options [3].

3. Aim of the article. The main objective for developing the new vibration module is providing

regulated resonant harmonic and vibro-impact modes. Proposed to implement it by resetting the

stiffness properties of a flat spring relative to its base design. The task of reducing the complexity of

operations for the implementation of the various modes as a practical matter is pursued. This is not

enough in presented solutions [3-6].

Adjustable asymmetrical resilient characteristic (fig. 1) will be implemented by the flat spring with

presented functional condition:

.

(1)

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35

Fig. 1. Renewable characteristic of stiffness element.

Implementation of the regulated modes is carried out by a constructive influence on the stiffness

characteristic. It is supposed that the following conditions:

, а

, ,

.

4. Problem statement. Implementation of vibro-impact systems by movement limiters is

accompanied by contact phenomena. That is confirmed in subsequent works [9-12]. It should be

noted that the elastic elements are an important hub of the resonant vibration machines. Strength

characteristics along with the task of rigidity must be complying. Therefore, the lack of integrated

techniques for analyzing stress-strain states in the implementation of the vibro-impact modes is a

problematic.

5. Research materials. Resonant vibration module designed by the two-mass oscillation system

with active 1and reactive 2 masses, which are connected by one flat spring 3. Active mass 1 is

designed as a frame with an horizontal grooves for hard fixing brackets 4. Brackets 4 have vertical

slots for adjustable cylindrical rods 5. Two core 6 are placed on the active mass 1. The reactive

mass 2 contains anchors 7. The resonance module can be the basis for creating of the vibratory

technological machine. Module also can be used as a stand for dynamic testing.

Fig. 2. General view of the vibro-impact module.

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36

This structural implementation allows to use one a flat spring 3 to implement the basic harmonic

mode in the lower frequency region by stiffness coefficient

. Cylindrical shafts 5 are mounted in

vertical grooves of brackets 4 and are used as adjustable intermediate stops for flat springs 3. Using

the horizontal grooves changes the location of the intermediate support along the flat springs.

Accordingly is changes stiffness coefficient

. The appropriate gap between flat spring and

intermediate stops set by using the vertical grooves. So, adjustable asymmetric piecewise linear

characteristics of a flat spring and adjustable oscillations modes with different amplitude and

frequency spectrum will be realized.

With appropriate settlement schemes, which describe asymmetric transverse stiffness change

according to (1), also taking , the stiffness coefficient

is a necessary to define with the

stress-strain state analysis.

For the analysis of rod systems appropriate to use finite element method [13]. Finite-element

scheme of the vibration system in vibro-impact mode based on flat spring implemented on fig. 2.

The standard matrix of nodal reactions (transverse forces and bending moments) of one finite

element rod has the form [13]:

(2)

where – is a bending stiffness of the rod.

Fig. 3. Operation scheme of the vibro-impact system.

Bending rigidity of a flat spring is calculated to realization of the necessary natural frequency

of the harmonic oscillation mode for scheme I:

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37

(3)

where

– is reduced mass of vibrating system.

Finite-element scheme during movement in a negative direction is based on these boundary

conditions:

(4)

The system with 10 degrees of freedom obtained in general form by using the division by 4 rods

finite elements:

,

і

,

– according vertical movement and angles of rotation of the rod

nodal’s points (). According to general stiffness matrix of a rod (2) are formed stiffness

matrix with dimension 10×10 of finite elements rods in the global coordinate system in accordance

with the boundary conditions (4).

,

(5)

,

,

,

The resulting stiffness matrix of finite-element rod system:

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38

.

(6)

Free oscillations equation of a rod system is written in matrix form:

,

(7)

where,

– nodal’s displacements vector of the finite element;

– diagonal matrix of local masses inertial parameters;

– system’s stiffness matrix.

Based on the determinant for the frequency equation

,

(8)

is defined natural frequency of bending vibrations:

,

(9)

Reduced stiffness coefficients of rod systems according to the formulas (1) and (9) have the next

forms:

,

,

(10)

Thus, obtained parametric dependence of the system’s stiffness coefficient as function of

.

Stiffness coefficients ratio, taking into account

has a view:

,

(11)

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39

The eigenfrequencies ratio is

.

By the condition of multiplicities of natural frequencies of the rod systems

,

(12)

formulas for positioning intermediate supports are the next

,

.

(13)

For verification must be

.

Stress-strain state analysis. The vector of nodal displacements for statically loaded rod system

type II can be determined from the relationship:

,

(14)

where – vector of external loads reduced to nodes.

Nodal’s displacement vectors each finite element-rod:

,

,

,

.

(15)

The vectors of nodal reactions each finite element defines by matrix expression (2). The hogging

line of each finite element-rod is based on the known forms functions at nodes single displacement

[13]:

,

,

(16)

,

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40

.

The resulting displacement and flex curve of a rod system ІІ are determined by the following

dependency:

(17)

Resilient condition of the rod evaluated by the bending moment, transverse force, normal and

tangential stresses:

,

,

,

,

(18)

where

,

,

– geometric characteristics of the flat spring’s

cross-section;

і – width and thickness of the flat spring.

The vector of nodal loads is presented with concentrated force with peak value , acting at the local

oscillating mass. Diagrams of deflection line, angle of rotation, transverse forces and bending

moments will be obtained by solving of the equations (14)-(18). The value of the maximum bending

stresses are determined by the known formulas and given for appropriate settlement schemes:

,

,

(19)

where

,

are maximum bending moments in dangerous

sections for consideration rod systems.

The maximum shear stress is acting on the central section of the rod:

,

(20)

The equivalent (von Mises) stress:

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41

(21)

Research results. Analysis of stress-strain state of the rod systems offered to implement by static

displacement of local mass

. The value of renewable forces is set by effected displacement for

the corresponding settlement schemes with formulas:

,

. System parameters

for analysis:

,

, ,

,

, ,

,

,

,

,

Taking

received:

,

. Diagrams of stress-strain state for half flat springs in scheme ІІ (

) shown on the fig. 4.

a

c

b

d

Fig. 4. Theoretical dependence of stress-strain state of elastic half-length rod.

Determination of contact stress. Determination of contact stress is exercise on Hertz formulas by

value of reactions at intermediate supports (transverse force) [11, 14]. The cylindrical support

reaction is the sum of transverse forces in the rod’s nodes:

.

(22)

The pressure in contact zone is calculated by the formula [14]:

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42

,

(23)

where,

,

, – length of contact (width of flat spring);

– radius of intermediate support;

та

– Poisson's ratio of materials in contact.

Vectors of volumetric stress state determined as [14]:

,

,

.

(24)

where,

The main stresses are identical accordingly to axial components:

,

,

.

(25)

The maximum shear and equivalent von Misses stress presented graphically (fig. 5) by the formulas

for volumetric stress state [15]. Tangential and von Misses stresses:

,

(26)

.

(27)

The maximum value of these stresses with static displacement

at

:

,

. Also received

.

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43

а)

b)

Fig. 5. The ratio of stress to the contact pressure (a); value of von Mises and shear stresses relative

to the distance from the contact (b).

For comparison the results of the stress-strain state analysis of half flat spring in SolidWorks [8]

with symmetrical scheme are conducted (fig. 6). A condition as no penetration between flat spring

and cylindrical support is applied in the setting of contact task. The results for the curve of

deflection and von Mises stress are an enough to near with the theoretical values (fig. 6, b and c).

Also the contact force is set as (fig. 6, a).

a)

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44

b)

c)

Fig. 6. Design model of stress-strain state (а), deflection curve (b) and von Mises stress (c)

parametric changes regarding of the half long of the flat spring.

The procedure of nonlinear analysis is done with the setting as bouded contact for the flat stress.

The maximum stress in the contact zone is . This value is almost identical to the

theoretical result.

а)

b)

Fig. 7. Diagrams of contact stresses (а – flat section, b – the value of along the parametric length of

the spring).

The pliability of intermediate supports is caused by relative displacement

of flat spring and

support at the contact zone. Determined by the following expression [15]:

,

(28)

The stiffness coefficient in the contact point

. The resulted formula has a simplified

view for contact with materials steel and steel:

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45

,

(29)

.

(30)

The last formula allows to consider the contact stiffness in intermediate support on implementation

of vibro-impact modes. In this case, parametric contact stiffness is depending from the contact force

through the instantaneous movement of local mass

. For

contact stiffness is

.

Dynamics of stress-strain state. The resulting static characteristics can be used for dynamic stress

state analysis of the vibro-impact system with kinematic’s disturbance of local mass (fig. 8).

The characteristics of local mass movement takes in the form

. Contact force

depends of the instantaneous movement of local mass according to conditions of the system:

(31)

Given a parametric dependence

contact stiffness from the contact force

, confirmed their

nonlinearity in the fig. 9.

Fig. 8. Kinematic’s disturbance scheme of the vibro-impact rod system.

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46

Fig. 9. Dependence of the stiffness coefficient of intermediate support of the magnitude of the

contact force (by the displacement

of local mass).

The equivalent bending stress is determined by the conditions of the rod system operation:

(32)

Depending in time the equivalent stresses for bending and contact presented at the fig. 10. They

have asymmetrical pulse character, caused by implementation of the vibro-impact mode scheme.

a)

b)

Fig. 10. The time dependence of bending (a) and contact stresses (b) in vibro-impact rod system.

Parametric dependents of calculated stress values from the instantaneous displacement of local mass

are shown in fig. 11. Contact stress is nonlinear from displacement in time.

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47

Fig. 11. Parametric dependence von Misses and contact stresses in the vibro-impact rod system.

Summary. The construction of resonance module capable to operate in a harmonious and vibro-

impact modes is made. Formula of the bending rigidity of flat spring to implement the given the

asymmetric elastic characteristic is obtained by finite element method. A value and character of

stress-strain state of the flat spring analytically is determined. The result is approved by modeling in

SolidWorks Simulation. Contact stiffness in intermediate cylindrical support in contact with a flat

spring is determined. Dynamics for example of the kinematic disturbance is considered and built a

parametric characteristic of equivalent stresses.

References

[1] Yoon, J. Y., & Kim, B. (2015). Vibro-Impact Energy Analysis of a Geared System with

Piecewise-Type Nonlinearities Using Various Parameter Values. Energies, 8(8), 8924-8944.

[2] Chu, S., Cao, D., Sun, S., Pan, J., & Wang, L. (2013). Impact vibration characteristics of a

shrouded blade with asymmetric gaps under wake flow excitations. Nonlinear Dynamics, 72(3),

539-554.

[3] Patent USSR № 1351696 А, B 06 B 1/14 (1987), Method to tune to resonance oscillations of the

vibratory machine with piecewise-linear characteristics of the elastic ties, in Russian

[4] Patent USSR № 1727928 А1, B 06 B 1/14 (1992), Method the settings for a given mode of

oscillation of the vibratory mathice with nonlinear elastic connections and operating mass, in

Russian.

[5] Patent USSR № 1634335 А2, B 06 B 1/14 (1991), Vibratory device, in Russian.

[6] Patent USSR № 1381282 А1, F 16 F 13/00 (1988), Eastic suspension, in Russian.

[7] Peter, Simon, Pascal Reuss, and Lothar Gaul. (2014), Identification of Sub-and Higher

Harmonic Vibrations in Vibro-Impact Systems. Nonlinear Dynamics, Volume 2. Springer

International Publishing, 131-140.

[8] Belovodskiy V. N., Bukin S. L., Sukhorukov M. Y., Babakina A. A. (2015), 2:1 Superharmonic

Resonance in Two-Masses Vibrating Machine // Journal of Vibration Engineering & Technologies,

3(2), 123-135.

[9] Ostasevicius, V., Gaidys, R., & Dauksevicius, R. (2009), Numerical analysis of dynamic effects

of a nonlinear vibro-impact process for enhancing the reliability of contact-type MEMS devices.

Sensors, 9(12), 10201-10216.

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[10] Bazrafshan, M., Ahmadian, H., & Jalali, H. (2014). Modeling the interaction between contact

mechanisms in normal and tangential directions. International Journal of Non-Linear Mechanics,

58, 111-119.

[11] Serweta, W., Okolewski, A., Blazejczyk-Okolewska, B., Czolczynski, K., & Kapitaniak, T.

(2014). Lyapunov exponents of impact oscillators with Hertz׳ s and Newton׳ s contact models.

International Journal of Mechanical Sciences, 89, 194-206.

[12] Fayyad, S. M. (2013). Analysis and Simulation of Contact Stresses of Convex Punch Analysis,

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[13] David, V. Hutton. (2004), Fundamentals of finite element analysis. Editorial McGraw− Hill,

USA.

[14] Shigley, Joseph Edward. (2011), Shigley's mechanical engineering design. Tata McGraw-Hill

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Naukova Dumka, Kiev, in Russian.

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Modelling of Station of Pumping by Variable Speed

Benretem A.

1

1 – Electromechanical Engineering Laboratory, Badji-Mokhtar Annaba University P.O. Box12, 23000, Annaba, Algeria

DOI 10.13140/RG.2.1.4707.0324

Keywords: energetic efficiency, centrifugal pump, motor, variable speed drive, inverters, modelisation, simulation.

ABSTRACT. An increased energetic efficiency will make it possible to decrease the factory operating costs and hence

to increase productivity.The centrifugal pumps are largely used because of their relatively simple operation and of their

purchase price. One analyses thorough requirements imposed by the pumping plants is decisive.It is important to keep

in mind the fact that the pumps consume approximately 20% of energy in the world. They constitute the possibility for

the most significant efficiency improvement. They can reach their maximum effectiveness only with one pressure and a

given flow. The approach suggested makes it possible to adapt with accuracy and effectiveness of system output of the

industrial process requirements. The variable speed drive is one of best and effective techniques studied to reach this

objective.

The appearance of this technique comes only after the evolution obtained in the field of power electronics systems

precisely static inverters as well as the efforts made by the researchers in the field of electric drive systems. This work

suggested is the result of an in-depth study on the effectiveness of this new technique applied for the centrifugal pumps.

1. Introduction. Due to their operating closer to the optimum, variable speed pumping plants are

reliable: energy saving, maintenance intervals saving and hence stop costs reduction. According to

the structure of the user, it is also possible, by the intermediary of a speed regulation, to reduce the

number of the pumps size. The complexity of a centrifugal pump drive makes its study and design

difficult as well as during its normal operation. To overcome this difficulty, one generally carries

out simplifications: linearization of some characteristics, addition of inertias of all the moving parts

to that of the engine, etc. in function of the objectives of modelling, these simplifications can give

satisfactory results[3]. With the aim of reducing the costs and risks of bad operations in the

evaluation phase and training of complex systems, the use of data-processing tools for digital

simulation seems a justifiable intermediate stage between the theoretical study and the tests on real

installations. Pump centrifuges P-102.b P-102.a several tools (specialized or not) for simulation are

used in the field of the machines electric and of the power electronics [4] [5]:ATOSEC5, EMTP,

SPICE, SIMNON, MATLAB, etc. [3].2. Modelisation of the System of Pumping.Figure 1

presents prone pumping plant the "of study»; unit of ammonia NH3 of the Asmidal factory -

Annaba - Algeria and more precise the centrifugal pumps P-102 intended for fed the units of nitric

acid, acid nitrate and of acid phosphates in product ammonia "to see figure 1".

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50

Fig. 1. diagram of the pumping plant of ammonia.

2.1 Mathematical modelling of the centrifugal pump.

Generally, the manufacturers do not give the characteristics of the pumps physical parameters

performances H = F (Q) is offered. Thus; knowing speed, the load height, and reference flow; it is

possible to determine the system characteristics; the following relations can be used:

4/3

60

1000

ep

sq

N

gH

Q

N

N

3

fsp

DNKQ

8.1

4

109.33.0

r

C

The modelling characteristic diagram of the centrifugal pump equations

system is presented in figure 2.[8]

Fig. 2. Structure of the centrifugal pump unit.

2.2 Modeling of the squirrel cage induction motor. The drive is designed as an integrated block

of which the goal is the optimal conversion of the electric power into mechanical energy and

hydraulics by taking into account the performance criterion defined in the specifications. The

modelling of the principal parts of the suggested pumping system (converter - engine - pump) are

gathered and presented in figure 3.

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51

Fig. 3. Modelof the pumping system.

3. Results of the system simulation.

3.1 Simulation of the system with constant speed. The figure.4 takes the stator current curve in

transient state and permanent modes as well as the magnetic flux of the centrifugal pump drive

motor with load and with no-load in the d- q plan. During the no-load, it can be noticed that flow is

inversely proportional to the stator current; the introduction of the load "pumps centrifuges" does

not present any remarkable evolution, always flow is inversely proportional to the stator current.

Fig. 4. Current stator and magnetic flux according to axes' (D, Q) in no-load and load (load

applied at the moment T = 2s).

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52

Fig. 5. Variation the motor speed: pump Flow corresponds at a speed of 2500 rev/min.

Figure.5 illustrates the variation of the pump revolutions number according to the flow. For a flow

equal to 31 m

3

/ h, the number of revolutions takes a value equal to 2500 rev/min., at the time "t = 2

s" time of introduction of the load "the centrifugal pump" generates a reduction in the number of

revolutions of the drive motor. On the other hand one records an increase in the values of the motor

torque (electromagnetic torque) for the same time "t=2 S" as it is shown in figure.6.

Fig. 6. Motor torque and angular velocity of the engine in a no-load and load (load applied at the

time T = 2s).

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53

Fig. 7. Motor stator current with no-load and load (load applied at the time T = 2s).

Figure 7 shows that the motor torque is inversely proportional to the motor revolutions number;

whereas the above mentioned torque is proportional to the motor stator current as shown in the

figure 8.d' where the two curves take the same form except that one records an increase in the

current amplitude of line beyond that time "t = 2 s".

Fig. 8. Current of line (no-load and load).

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54

Fig. 9. Voltage made up of the 3 phases (operation under load).

Both figures.9 and 10 respectively represent the voltage waveform made up and simple of the three

phases of the motor operating in load.

Fig.10. Simple voltage of the 3 phases (operation under load).

3.2 Simulation of a variable speed systemThe second part of simulation includes a new system

because of the introduction of a variable speed control system; this technique can offer substantial

cost saving in the electric power system when necessary flows or pressures variations were done in

the system. Figure 11 shows that the flow and the number of revolutions take the same form

according to time; in which the increase in flow causes the increase in the motor speed and its

reduction generates the reduction of the latter (figure 11).

It can be noticed that there is a great improvement in the parameters compared to figure 5

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55

Fig. 11. Motor speed variation and the pump flow against time.

Fig. 12. Motor torque operating with load in function of speed variation.

Figure 12 shows that the torque is always inversely proportional to the motor speed

Fig. 13. The motor Stator current operating under load with speed variation.

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56

The stator current results from this variation speed is shown in Figure 13; this current is

proportional to the motor torque evolving with the variation of the pump flow. This proves that this

technique is likely to improve the energy effectiveness of the centrifugal pump and to reduce the

energy losses, as illustrated by this load profile of Figure.14

Regular Irregular

Fig. 14. Load profile of the pump operating at variable speed.

Summary. From an economic point of view, it is often more judicious to adapt the flows desired by

the implementation of a variable speed drive.With a continuous speed variation, the consumption of

energy is adapted to the needs and there is no energy dissipation.

The wear of the pumps, the bodies of throttling and the installation decreases, since the installation

functions only in the necessary operation. The motor is definitely under loaded; and the pump has

the possibility of starting smoothly and of the unfavourable hydraulic reactions is avoided.This

intelligent method makes it possible to prolong the lifespan of the pump and the installation as a

whole and increases the availability of the installations.

In general, the results obtained confirms the reliability of modelling carried out, the possibility of

optimization of the system and minimization of the energy losses.

These satisfactory and clear results show that the use of the variable speed drive in centrifugal

pumps is an optimal solution in spite of its complexity and its relatively high price.

References

[1] Peeic (Programmer of Energy Saving in Canadian Industry), "Guide of evaluation of the output

of the energy driving systems ".Natural resources Canada 2004.

[2] Arene, "The true cost of the pumps".Hydro more, N 128.November 2002.

[3] Doumbia.M.L, "Tool of assistance to the systems design of electric drive of machine, example

of application ".Thesis of PhD of the polytechnic school of Montreal. 2000.

[4] Boubacar.N, "Design techno - economic of a system of autonomous pumping photovoltaic -

aero generator ".Memory of M.Sc.A of the polytechnic school of Montreal. February 1999.

[5] Mohan.N, Undeland.T and Robbins.W, "Simulation of Power Electronic and Motion Systems

Control - Year Overview» Proceeding of the IEEE, vol. 82, No 8, August, 1989.

[6] Etxeberria- Otadui.I, "On the system of the electronics of power dedicate has the electric

distribution - application has the quality of energy ".Thesis of doctorate of the Institut National

Polytechnique of Grenoble.September 2003.

[7] Stpanoff. A. J, "Centrifugal pump and pumps Propellers ".Dunod. Paris.1961

Mechanics, Materials Science & Engineering, July 2016 – ISSN 2412-5954

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57

[8] Betka Achour Perspectives for the sake of photovoltaic pumping development in the south thesis

for doctorate of state Biskra Science.

[9] Automatic alpha issaga diallo Sélection of the asynchronous motors in the industrial drives

thesis for obtaining Master be sciences (M.Sc) University Laval 1998.

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58

A Thermal Force Drifting Particles along a Temperature Gradient

Amelia Carolina Sparavigna

1

1 − Department of Applied Science and Technology, Politecnico di Torino, Torino, Italy

DOI 10.13140/RG.2.1.2347.7365

Keywords: Thermal gradient, Thermal transport, Thermal forces, Entropic forces.

ABSTRACT. In 1972, V. Gallina and M. Omini of the Polytechnic of Turin proposed a phenomenological model for

the thermal diffusion in liquid metals, explaining the isotope separation as provoked by a thermal force which is arising

when a temperature gradient is established in the material. Here, we discuss this thermal force and its statistical origin

from the bulk. We will see that it can be considered as a force of the form F = − S grad T, that is as a thermal/entropic

force obtained from the derivative of the Helmholtz free energy with respect to the volume.

Introduction. With the works of Ludwig Boltzmann, physics and thermodynamics started

recognizing the stochastic and probabilistic aspect of natural processes. Besides introducing a

fundamental equation widely used for thermal and charge transport [1-5], Boltzmann linked the

second law of thermodynamics to the disorder of thermodynamic systems, proposing a fundamental

relation between entropy S and probability through the formula S = k

B

ln W (k

B

is the Boltzmann's

constant and W is for Wahrscheinlichkeit, that is “probability” in German).

The Boltzmann formulation of entropy was fundamental for the “doctrine of energy and entropy”

that rules the thermodynamic world [6], a doctrine where the energy is deterministic and the entropy

is favouring randomness. In this framework of natural systems governed by energy and entropy,

forces can arise from the Helmholtz free energy A = U – T S , where U is the internal energy, T the

temperature and S the entropy. Such as the pressure, the mechanical force divided by a surface, is

coming from the derivative of internal energy U with respect to the volume, the same derivative of

the product T S is giving rise to the thermal and entropic forces. These are forces which are coming

from the statistical nature of the system, rather than from a particular microscopic interaction

existing in it [6].

Entropic forces had been proposed and used in [7,8] for the Brownian motion and for the elasticity

of polymers. For the Brownian motion, the force is in the form of a diffusional driving force or

radial force, which has a mean value <F

r

> = T<dS/dr>, where r is a radial distance [9]. We can

understand this mean value considering its dimensionality: the force is an energy - here given by the

product TS - divided by a length. However, besides the entropic force which is coming from a local

variation or gradient of entropy, we can have also a force in which it is appearing as the gradient of

temperature T multiplied by an entropy, that is F = − S grad

T. A thermal force of this kind is used

for the magnetic flux structures in superconductors [10,11]. In these references, this force is also

considered as an entropic force.

In the following discussion, we will show another example of such thermal force, which is

concerning the isotope separation in liquid metals driven by a thermal gradient. It was proposed in

1972, by V. Gallina and M. Omini of the Turin Polytechnic in a phenomenological model for the

thermal diffusion in liquid metals [12]. In their study, the authors aimed giving a formula for the

isotope separation in a liquid metal, separation which is observed when a temperature gradient is

established in the fluid [13]. In fact, the approach and the related model proposed in [12] is more

general: it embraces completely the problem of thermal diffusion in fluids.

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59

The focus of authors was in investigating the force that is moving atoms through a liquid when

there is a thermal gradient in it. In a good approximation, this force has a simple expression: F= −

k

B

grad T, where T is the temperature. In this paper we will see that it can be consider as the

thermal/entropic force F = − S grad T, [10,11]. It means that this is one of the two partners of a pair

of forces coming from the derivative of T S with respect to volume, the other member being the

entropic force F = − T grad S.

Let us start discussing the thermal force appearing in [12].

The thermal force. The central part of Reference 12 is discussing the force responsible for the

drift velocity of particles in a temperature gradient. This drift of particles, appearing when a thermal

gradient exists, is an experimental fact. It appears in observed effects and named as

thermodiffusion, thermophoresis, thermomigration or Ludwig-Soret effect [14]. This phenomenon,

which we find in mixtures of particles where the different types of particles exhibit different

responses to the temperature gradient, has a number of practical applications. In fact, due to the

different behavior, the particle types can be separated. Several recent papers on isotope separation

in silicate melts for instance, are evidencing the present interest on this subject [15-18].

In a first approximation, a particle with radius r in a solvent has a drift velocity w = F/(4πηr) [12].

η is the coefficient of shear viscosity of the solvent. F is the force drifting the particle. To give an

expression of this force, in [12], the authors evaluated first the net force acting on an atom of a solid

lattice when it is subjected to a thermal gradient. Each atom of the lattice is a scatterer of thermal

waves (phonons). In the second quantization, the displacements of waves are considered as

quasiparticles, the phonons, having energy and momentum. If we consider an atom of the lattice, we

can imagine it in a local oscillatory motion. It becomes a phonon scatterer, exhibiting a cross-

section σ(q,q’) for an elastic scattering in which a phonon having wave vector q is deviated into a

phonon of wave vector q'. In fact, this approach would be true only for an impurity scattering: in a

perfect lattice, we need at last three phonons involved in scattering processes. However, Gallina and

Omini are showing in [12], that using the general theory of phonon-phonon interactions, the same

result is obtained.

Fig. 1. An atom of the lattice as a scatterer, with transferred momentum equal to ħ(q − q').

In the diagram of Fig.1, we can see that the momentum transferred to the atom is ħ(q − q'), where ħ

is the reduced Planck constant.

Adding the contributions of all the scattering processes, which are occurring in unit time, we have

the force imparted to the atom by the thermal vibrations. The result is:

q

q

qq

qvg

N

F

1

(1)

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60

In (1), N is the total number of atoms, v

g

the group velocity, λ

q

the mean free path and g

q

the

deviation from the Bose-Einstein equilibrium function. Since g

q

is proportional to the mean free

path in the relaxation-time approximation [5], we have a result independent of the phonon mean-

free path. Therefore, in this approximation, the net force turns out to be simply F= − k

B

gradT .

This force has no real effect if the atom belongs to the solid lattice, because this force is simply

transmitted to the centre of mass of the crystal. In a liquid, where the atom is not bound to any

particular site of the system, the force becomes a driving force, moving the atom towards the cold

end of the sample [12].

Are we justified in using this force for an atom in the liquid, since it has been deduced for a solid?

[12]. In the reference, the authors are giving specific arguments for a positive answer. First, we have

a phenomenological argument: the thermal force only depends upon the atomic specific heat c =

3k

B

, and this is a quantity which is presumed to be almost unaffected by the solid-liquid transition.

Second: consider an atom O in the cage of its neighbors. Imagine we have Σ atoms surrounding O.

Since the atom can move isotropically toward any of its Σ surrounding, the probability of one of this

atoms of moving toward O is 1/Σ. Thus the atom O, at a certain instant, has the probability p = 1/Σ

of being struck by one of its neighbors. The time required for an interaction with its i-neighbor, is t

i

= a/v

i

, where a is the nearest-neighbor distance and v

i

the speed of i-atom.

After some passages, we can find that the mean force acting on O is [12]:

i

i

iB

i

i

i

O

u

a

Tk

u

a

mv

F

3

11

2

(2)

In (2),

i

u

is the unit vector from O to i-neighbor. T

i

is the temperature of the i-neighbor.

Moreover,

TuaTT

iOi

. Therefore, if we have a statistical environment which is isotropic,

we can average on the solid angle:

TkuTuakd

a

F

BiiBO

)(3

4

1

(3)

Let us consider again a particle moving in a solvent; if no other forces are present, the drift velocity

is given by:

r

Tk

wTkwr

B

B

4

04

(4)

As the authors are remarking, this force has not to be considered as due to a potential gradient, that

is, to an external action; it is a statistical force, originated from the bulk of the material, which

accounts for the possibility of an atom of making a random walk in the liquid.

To have (3), an atom which is vibrating at temperature T has an energy 3k

B

T. This is true in the

approximation of an atom considered as an Einstein harmonic oscillator. However, this energy has

to be modified when an anharmonicity exists [12].

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61

Role of surfaces. Summing the thermal force over all the atoms of the sample, we should have F=

− N k

B

grad

T . For |grad

T| = 1°C/cm and N=10

23

, we have F ≈ 100 N. Why do we not observe

this force? [12]. Let us see how the authors answered. In the case of a solid, we can have that on the

wall which is at higher temperature, there is an excess of phonon pressure with respect to the wall at

lower temperature. This extra pressure gives rise to a force directed along grad

T, which is exactly

counterbalanced by the resultant of the thermal forces. In the case of a fluid, we can see that atoms

at the surface of the material give different pressures; there is a force on surfaces which is equal and

opposite to the thermal force [12].

Fig. 2. Atoms at the surface of liquid.

For a liquid, let us take an atom of the liquid near the wall (Fig. 2). It makes a random oscillation in

a cage of the order of the atomic volume a

3

. The atom oscillates. If t is the period of the oscillation,

we can write v

x

t ≈ 2a, where v

x

is the average velocity normal to the wall. Whenever the atom

arrives at the wall, we assume an elastic reflection. The transferred momentum is 2mv

x

. If n is the

number density of atoms in the liquid, the total number of atoms facing the wall with surface Σ is

naΣ. Then the force on the wall is:

Tknvmn

t

mv

an

Bx

x

2

2

where

Tkvvvmvm

Bzyxx

2222

3

1

(5)

TkN

L

TT

kLnTknTknF

BBBBsurface

01

01

(6)

In (6), we have that an unbalanced force on surfaces exists. However, the bulk thermal force

balances this surface force. It happens when the two walls at different temperatures have the same

surface Σ. If we have different surfaces, we need to consider the role of the lateral surface, in order

to have a net force equal to zero. The conservation of momentum tells us that the net force must be

zero.

Thermal and entropic forces. Let us consider the Helmholtz free energy A = U – TS, and its

derivative with respect to the volume V. Then, let us multiply this derivative by the surface Σ. We

can consider a generalized force as F = Σ dA/dV = Σ dU/dV – Σ S dT/dV – T Σ dS/dV. The first term

is the pressure multiplied by the surface, that is, the mechanical force.

In the case that we assume the volume variation dV as equal to Σ

dx, we have a force F given by

three terms, F = Σ p – S

dT/dx – T

dS/dx. Then, besides the term containing the pressure, we have

the two thermal/entropic terms.

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62

Is the thermal force F = − N k

B

grad

T previously discussed the term – S

dT/dx ? The Boltzmann

constant has the dimensions of an entropy S; therefore, dimensionally [19], we have the thermal

force F = − N k

B

grad

T as F = − S grad

T, and a thermal/entropic force of this kind was used for

the magnetic flux structures in superconductors [11,12]. However, we can find a more convincing

reason for a positive answer in [20]. In this paper, the Debye model is used instead of the Einstein

model. In the case of a phononic assembly, the bulk force is [20]:

T

D

yy

D

Bdiffph

dyeey

T

TkNF

0

24

3

,

)1(3

(7)

If we are at low temperatures, the entropy S can be considered as S= C/3, where C is the heat

capacity [21]. In a Debye model [19]:

T

D

yy

D

B

dyeey

T

kN

C

S

0

24

3

)1(3

3

(8)

Then, the thermal force (7) given by Omini in [20], seems coincident with a thermal/entropic force,

in the abovementioned approximation. In fact, using (8) in (7), we have:

TSF

diffph

,

(9)

Also in the case of Eq.3, we can repeat the same observation given above on heat capacity and

entropy and have a force F= − S grad T.

However, besides having the force F= − S grad T, when we derive the Helmholtz energy we can

have F= − T grad S too. It means that in the cases discussed in [12] and [20], this entropic force

could exist. It is possible because entropy is depending on temperature, and temperature is a

function of the spatial coordinates. Therefore a gradient of entropy exists and consequently an

entropic force. Since this force could be written as F = − T (dS/dT) grad T, its effective role in the

system requires a further evaluation.

References

[1] Cercignani, C. (1975). Theory and application of the Boltzmann equation. Scottish Academic

Press. ISBN: 978-1-4612-6995-3.

[2] Succi, S. (2001). The lattice Boltzmann equation: for fluid dynamics and beyond. Oxford

University Press. DOI: 10.1016/S0997-7546(02)00005-5

[3] Omini, M., & Sparavigna, A. (1995). An iterative approach to the phonon Boltzmann equation

in the theory of thermal conductivity. Physica B: Condensed Matter, 212(2), 101-112. DOI:

10.1016/0921-4526(95)00016-3

[4] Omini, M., & Sparavigna, A. (1996). Beyond the isotropic-model approximation in the theory

of thermal conductivity. Physical Review B, 53(14), 9064. DOI: 10.1103/physrevb.53.9064

[5] Sparavigna, A. C. (2016). The Boltzmann equation of phonon thermal transport solved in the

relaxation time approximation – I – Theory. Mechanics, Materials Science & Engineering Journal,

3, 1-13. DOI: 10.13140/RG.2.1.1001.1923

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63

[6] Müller, I. (2007). A History of thermodynamics: The doctrine of Energy and Entropy, Springer

Science & Business Media. ISBN 978-3-540-46226-2

[7] Neumann, R. M. (1980). Entropic approach to Brownian movement. American Journal of

Physics, 48(5):354. DOI: 10.1119/1.12095.

[8] Neumann, R. M. (1977). The entropy of a single Gaussian macromolecule in a noninteracting

solvent. The Journal of Chemical Physics 66(2):870. DOI: 10.1063/1.433923

[9] Roos, N. (2014). Entropic forces in Brownian motion. American Journal of Physics, 82(12),

1161-1166.

[10] Huebner, R. P. (1979). Magnetic flux structures in superconductors, Springer Verlag. DOI:

10.1007/978-3-662-02305-1

[11] Pengcheng Li (2007). Novel transport properties of electron - doped superconductors

Pr

2-x

Ce

x

CuO

4-δ

, UMD Theses and Dissertations.

[12] Gallina, V., & Omini, M. (1972). On thermal diffusion in liquid metals. Il Nuovo Cimento,

8B(1):65-89. DOI: 10.1007/bf02743508

[13] Ott, A. (1969). Isotope separation by thermal diffusion in liquid metal. Science,

164(3877):297. DOI: 10.1126/science.164.3877.297

[14] Platten, J. K. (2006). The Soret effect: a review of recent experimental results. Journal of

applied mechanics, 73(1):5-15. DOI: 10.1115/1.1992517

[15] Huang, F., Chakraborty, P., Lundstrom, C. C., Holmden, C., Glessner, J. J. G., Kieffer, S. W.,

& Lesher, C. E. (2010). Isotope fractionation in silicate melts by thermal diffusion. Nature,

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4(8):1-7. DOI: 10.18483/ijsci.811

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270:131-139. DOI: 10.1016/s0921-4526(99)00172-6

[21] Peter Atkins, P., & De Paula, J. (2014). Atkins’ Physical Chemistry, OUP Oxford. ISBN

9780199543373

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64

Error Analysis of Method for Calculation of Non-Contact Impact on Space

Debris from Ion Thruster

Alpatov A.P.

1

, Fokov A.A.

1

, Khoroshylov S.V.

1

, Savchuk A.P.

1

1 – Institute of Technical Mechanics of National Academy of Sciences of Ukraine and State Space Agency of Ukraine,

Dnipropetrovs'k, Ukraine

DOI 10.13140/RG.2.1.3986.1361

Keywords: space debris removal, ion beam shepherd technology, spacecraft - space debris object system, contour of the

central projection, simplified calculation of the impact, error analysis, simulation of the relative motion.

ABSTRACT. A simplified approach to determine the impact on a space-debris object (a target) from the ion thruster of

a spacecraft (a shepherd), which was proposed before in the context the ion beam shepherd technology for space debris

removal, was considered. This simplified approach is based on the assumption of the validity of the self-similar model

of the plasma distribution in the thruster plume. A method for the calculation of the force impact using the information

about the contour of the central projection of the object on a plane, which is perpendicular to the ion beam axis, was

proposed within the framework of this model. The errors of this method, including the errors caused by an inaccuracy

of its realization, are analyzed. The results of the analysis justify the admissibility of the application of the specified

approach within the self-similar model of the plasma distribution. The preliminary conclusion has been made that this

simplified approach can be used to control the relative motion of the shepherd - target system as well. This conclusion is

based on the results of the simulation of the system motion, when the “real” value of the thruster impact is calculated by

the direct integration of the elementary impacts over the target surface and the value of the same impact used in the

control algorithms is determined using the information about the contour of the target. A number of factors such as the

orbital motion of the system, external perturbations, and the attitude motion of the shepherd were neglected in the

simplified model which was used for the simulation. These factors and errors in the interaction model are necessary to

consider during a more detailed analysis of this approach. The analysis of the calculation errors presented in this paper

can be used during implementation of the ion beam shepherd technology for active space debris removal.

Introduction. The technology for the removal of large debris from the low-Earth-orbit called ion

beam shepherd is presented in the papers [1], [2]. This technology provides space debris de-orbiting

due to the impact of the ion plume of the electric thruster (ET) of a spacecraft (a shepherd) located

in close proximity to space debris object (a target). A certain distance between ion beam shepherd

(IBS) and the target must be maintained to provide effectiveness of the impact of the ion beam. The

ion beam impact can be determined by integration over the surface of a target when the mechanism

of the ion interaction [3] with an elementary area of the surface and its relative position are known.

This approach can be used for modeling of the system motion but not recommended to apply in a

control loop due to its computational costs and the incomplete information about the position and

shape of the target. A simplified approach to calculate the impact from the ion beam on the target,

based on the information about the contour of its central projection on the image plane of the IBS

camera was proposed in [4, 5]. The central projection of the target on the reference plane, which is

perpendicular to the beam axis, is considered within the framework of this approach instead of its

surface. This paper justifies the application of this approach to control the relative motion of the

shepherd - target system by analyzing errors of the determination of the plasma beam impact on the

target

Calculation of the force impact by the direct integration over the target surface. The model of

the interaction of the ion beam with a space debris object, as well as the model of the ion plume

should be considered for the calculation of the impact from the ET force on the target.

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65

Neglecting the sputtering of the target material, the escaping ions from the target surface, and the

electron pressure, the elementary force

s

Fd

transmitted to the target can be calculated as follows

[6]:

smn

s

dd uvuF

, (1)

where

m

– is the ion mass;

u

– is the vector of the ion velocity;

sd

– is the elementary area of the surface, which location will be characterized by the radius

vector

s

of its midpoint;

v

– is the unit normal vector to the element of the surface;

n

– is the density of the plasma.

The force and torque transmitted from the ion beam to the target can be calculated by the

integration of the elementary forces from equation (1) over the exposed surface

S

S

s

FF d

srf

,

S

ss

FM dρ

srf

. (2)

There are ion beam models for the near and far regions of the plasma plume [7]. The far region of

the plume presents the main interest in the context of the non-contact space debris removal, because

this is where the plasma interacts with the target. Models with different degrees of complexity and

accuracy were proposed for the description of the far region of an ET beam [8]. The so-called self-

similar model of the beam is chosen for this study. Taking into account the fact that the Mach

number at the beginning of the far region of the beam is much greater than 1, the character of the

plasma distribution approaches to a cone. The plasma density can be determined for this case using

the self-similar model at an arbitrary point as follows [8]:

0

22

2

0

22

2

00

tg

3exp

tg

z

r

z

Rn

n

, (3)

where

r

,

z

– are radial (distance from the midpoint of the surface element to the axis of the beam

cone) and axial (the distance from the vertex of the cone along the axis of the beam) coordinates of

the point;

0

R

– is the radius of the beam at the beginning of the far region (at the exit of the ET nozzle,

0

2

0

tg/

Rz

);

0

n

– is the plasma density at the beginning of the far region;

0

– is the divergence angle of the beam.

Axial

z

u

and radial

r

u

velocity components of the plasma ions can be represented as follows:

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66

const

0

zz

uu

,

zruu

zr

/

0

, (4)

where

0z

u

is the axial component of the ion velocity at the beginning of the far region.

The target surface is divided into elements for determination of the integral force

srf

F

transmitted to

the target by the ion beam. After that, the coordinates of the surface elements and the unit normal

towards them are set with respect to the target reference frame (RF). Then the coordinates of the

surface elements and their normals recalculated with respect to the IBS RF, or more precisely, to

whose origin that is at the cone vertex of the ion beam. We can assume for simplicity reason that

this RF coincides with the IBS RF. The elementary force is determined for each element of the

surface using the equation (1) and taking into account its "visibility" from ET side and its position

inside or outside of the beam cone. The integral force is determined using the expression (2).

Calculation of the force impact using the contour of the target. The simplified approach [5] for

determination of the impact from the ion plume uses the information about the contour of the

central projection of the target. According to this approach the force applied to an element of the

surface is approximately equal to the force acting on the central projection of this area on a plane

that is perpendicular to the axis of the beam cone, for a example the plane of camera sensor placed

next to the ET This assumption is explained by the fact that the cone cross-section increases with

the square of the distance from the vertex of the cone and the plasma density decreases in inverse

proportion to the square of the distance from the vertex of the cone.

The equation that determines the force

Fd

transmitted through the element of the surface can be

written as follows:

dd

2

u

euF

cc

mn

,

T

0

]1//[ fyfxu

cczc

u

, (5)

0

22

22

0

22

2

00

tg

3exp

tg

f

yx

f

Rn

n

cc

c

, (6)

where

T

– is the transposition symbol;

d

– is the elementary area of the target projection on the camera image plane;

u

e

– is the unit vector of the direction of the ion velocity

u

;

c

x

,

c

y

– are coordinates of the point in the camera RF;

f

– is the focal length of the camera.

The full force

cnt

F

transmitted from the ion plume to the target is calculated by this expression:

FF d

cnt

, (7)

where

is the part of the image plane of the camera which is bounded by the contour.

Analysis of the errors caused by the simplified calculation method. Let us make a more detail

analysis of the assumption about the equality of two elementary forces from the ion plume used in

the simplified method. The first force acts on an element of the target surface and the second one is

applied to the central projection of this area on a plane, that is perpendicular to the axis of the cone

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67

beam. First, we restrict ourselves to the case when the area

sd

of the target surface is perpendicular

to the direction of the ion beam. The equations (1), (3), (4) can be rewritten for this area as follows:

smn

us

dd

2

euF

,

22

0

2

tg1

z

uu

, (8)

0

2

2

0

22

2

00

tg

tg

3exp

tg

z

Rn

mn

,

dd)cos/tg(d

2

zs

, (9)

where

– is the angle between the direction of the beam distribution and the axis of the ion beam

cone;

– is the azimuth of the direction of the ion distribution,

),(

uu

ee

.

The equations (8), (9) show that value

),(dd

ss

FF

is a function of variables

and

, and

doesn't depend from the coordinates of the element

z

of the surface. The "visible" area of the

surface element varies in the case of its arbitrary orientation depending from the inclination angle of

the element to the plane that is perpendicular to the direction of the ion beam. This variation of the

area is taken into account in the equation (1) by the factor

)( uv

, that is equal to use the

orthogonal projection of the area on the defined perpendicular plane. However, we do not deal with

the orthogonal projection during the calculation of the force impact using the target contour but

with the central one. In other words, a contour error is added during the calculations. Let's consider

the following example on Fig. 1 to evaluate this error.

Fig. 1. Analysis of error caused by the replacement of the orthogonal projection by the central

projection.

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68

We consider a part of the beam cone in the form of a small cone with the vertex angle

and name

it elementary cone. Fig. 1а shows a cross-section of the cone by a vertical plane. The elementary

cone "cuts" the element of the surface in the form of an ellipse (the segment

AB

in Fig. 1a). We

assume that the plane of the ellipse is inclined to the plane which is perpendicular to the axis

of

the elementary cone at an angle

. The area

cone

S

of the perpendicular cross-section of the

elementary cone passing through the midpoint

M

is the area of the central projection of the surface

element on plane of this section. We also define the ratio

S

k

of the area

cone

S

to the area

body

S

of

the surface element. The difference between the value of this ratio and the value of

cos

characterizes the error of the application of the equations (8) and (9). We introduce the so-called a

coincidence coefficient defined by the function:

cos/)cos(1

coin

S

kk

.

The analysis of this function is not presented here due to its cumbersome, but it still shows the

following: the independence of the coincidence coefficient of from the distance from a surface

element to the cone vertex; this kind of error appears only when the values of

are close to

90

;

the acceptable results can be achieved by reducing the cone vertex angle

even when values of

are close to

90

. Fig. 1 illustrates these conclusions by graphs of the variations of the coincidence

coefficient from the following variables: the coordinate

A

(Fig. 1b); the cone vertex angle

(Fig. 1c); the inclination angle

(Fig. 1d). One of these variables is varying while the other two

variables are fixed in these graphs.

Analysis of the errors caused by the implementation of the simplified approach. Another kind

of errors of the simplified approach can be caused by an inaccuracy of the camera placement.

According to the approach the camera must be located so as its focal point coincides with the vertex

of the imaginary cone of the ET beam. This requirement is difficult to meet from the engineering

point of view and it is worth to consider an offset of the camera from the position which stipulated

by the simplified method.

To analyze this offset we introduce the camera RF

cccc

zyxO

, whose origin

c

O

is on the camera

optical axis. The plane

cc

yx

coincides with the plane of the camera sensor. The axis

c

z

is directed

towards the target.

We consider the vector

d

, which connects the vertex of the imaginary cone of the beam and a point

P

of the target. In the ideal case of the camera placement, the coordinates of the projection point

P

on the plane of the camera sensor that included in the equation (5) are defined by following

relations:

)cam(

3

)cam(

1

d

d

fx

c

,

)cam(

3

)cam(

2

d

d

fy

c

, (10)

where

c

x

,

c

y

– are the coordinates of the projection point in the camera RF, the subscript indicates

the component number of the column

)cam(

d

corresponding to the vector

d

.

Let us consider the case where the camera is mounted with a small offset relative to the ET beam.

The target contour obtained using images from the camera with such offset is different from the one

with the ideal placement of the camera. We will illustrate this considering the example of the

contour determination.

The target is a circular cylinder. The height of the cylinder is

m2,6

. The diameter of the foundation

of the cylinder is

m2,2

. The geometric center of the target is located on the axis of the beam cone

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69

on a distance of

m7

from its vertex The line 1 in Fig. 2 shows the contour corresponding to the

case when the focal point of the camera coincidences with the vertex of the imaginary cone of the

beam and the line 2 depicts the contour for the case when the camera offset is

m2,0

.

Fig. 2. Contours of a target projection.

A method to correct coordinates of the contour points obtained using the equations (10) were

proposed in [9] for the case when the cameras focal point is placed with offset from the imaginary

vertex of the beam cone at the

x

~

and

y

~

along

cc

xO

and

cc

yO

axes, respectively. According to this

method, the coordinates can be corrected as follows:

nom

corr

~

z

x

fxx

cc

,

nom

corr

~

z

y

fyy

cc

, (11)

where

corr

c

x

,

corr

c

y

are the corrected coordinates of the point of the target projection;

nom

z

– is the nominal distance between the geometric center of the target and the vertex of

the beam cone.

Line 3 in Fig. 2 shows the contour corrected according to the equation (11) for the case when

m7

nom

z

. This figure shows that the corrected contour is almost identical to the one for the no

offset case (line 1).

The possibility to use the simplified approach to control the shepherd-target relative motion.

The preliminary assessment of the possibility to use the simplified approach to determine the beam

impact for the control of the shepherd-target relative motion is made on the basis of the simulation

results. The "real" value of the beam impact is calculated by direct integration over the target

surface and the values of the beam impact that used in the control algorithm are determined using

the information about the target contour. The confirmation of such possibility is an indirect

justification of the simplified approach for the calculation of the impact on a space debris object.

A simplified model of the motion was used for the preliminary evaluation where a number of

factors were not taken into account, such as:

- the orbital motion of the IBS and target;

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70

- external perturbations;

- the IBS attitude motion.

We start describing the reference frames that are required for a full model and then will apply the

simplifying assumptions introduced above. The following reference frames are used:

000

zyOx

is the inertial RF, the origin

O

is located in the center of the Earth, axis

0

Oy

is directed

towards the north pole of the Earth, axis

0

Oz

is the midpoint of the spring equinox;

111

zySx

,

333

zyTx

are the orbital reference frames of the IBS and target, respectively, the origin of the

RF

S

and

T

are located at their centers of mass, axis

z

is directed along the radius vector

connecting the center of the earth and the center of mass of the IBS or the target, axis

x

is in the

orbital plane and directed towards the orbital motion;

222

zySx

,

444

zyTx

are the body RF of the IBS and the target respectively, the axes of which coincide

with main central axes of inertia. For the ideal orientation of the IBS and target, these RF are

parallel and coincide with the appropriate orbital reference frame.

We will also refer to this reference frames by zero, the first ,..., the fourth according to the

subscripts which are used in their notation. The transitions matrixes from

-th RF to

-th RF are

denoted like

,

4,,1,0,

.

The IBS attitude position is determined by the following rotations of its body RF relative to its

orbital RF on the angles: pitch

2

(around the

y

axis), roll

2

(around the

x

axis) and yaw

2

(around the

z

axis). In order to avoid singularity of the kinematics relations for the case of the

uncontrolled motion of the target, the attitude position of the target is more convenient to describe

by four parameters of Rodrigues-Hamilton [10]. The parameters

0

,

1

,

2

,

3

is used to specify

the position of the body RF with respect to the orbital RF.

The dynamic equations of the attitude motion of the IBS and target is used in the Euler form [11]:

,

,

,

zyxxyzz

yzxzxyy

xzyy

z

xx

MJJJ

MJJJ

MJJJ

(12)

where

x

J

,

y

J

,

z

J

are the moments of inertia of the IBS and target with respect to their principal

central axes;

x

M

,

y

M

,

z

M

– are projections of external torques acting on the IBS (

2

) and target (

4

).

These dynamic equations are supplemented with the kinematics relations:

,sinsin

,coscossincoscos

,sincossincoscos

202222

2220222222

2220222222

z

y

x

(13)

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71

),(2

),(2

),(2

122103304

311302204

233201104

z

y

x

(14)

,tg)cossin(

,cos/)cossin(

,sincos

2222222

0222222

22222

yxz

yx

yx

(15)

,2

,2

,2

),(2

1424043

3414042

2434041

3424140

yxz

xzy

zyx

zyx

(16)

where

zyx

,,

are the projections of the angular velocities of the IBS and target to their body

RF,

4,2

;

0

– is the orbital angular velocity.

The components

1

,

4,2

of the transition matrixes from the body RF to the corresponding

orbital ones are given by:

,

222222222222

22222

222222222222

1

2

cccscssssccs

sccsc

cscssscssscc

(17)

,

)(2)(2

)(2)(2

)(2)(2

2

2

2

1

2

3

2

010322031

1032

2

1

2

3

2

2

2

03021

20313021

2

3

2

2

2

1

2

0

3

4

(18)

where

c

and

s

are

cos

and

sin

, respectively.

As mentioned above, the orbital motion (

0

0

) is neglected in this study and it is assumed that

000

zyOx

is the inertial RF and the axes of reference frames

111

zySx

and

333

zyTx

are parallel to the

corresponding axes of

000

zyOx

, and for the ideal orientation of the IBS and target their body RF

coincide with the corresponding orbital one. The RF for this case are shown in Fig. 3, where

ρ

,

d

,

R

are the radius vectors of a point

P

in the corresponding reference frames;

ST

r

is the radius

vector of center of mass (CoM) of the target in the IBS body RF;

S

r

,

T

r

are the radius vectors of

CoM the IBS and target in

000

zyOx

RF.

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72

Figure 3. Reference frames.

With these assumptions the following equations are correct:

)4(3

4

1

2

)0(1

2

)2(

)()(

T

ST

T

rd

,

)()()( 000

S

T

ST

rrr

, (19)

where the variables with superscript in parenthesis refer to the column of the vector projections to

the axes of RF with the appropriate index.

These relations are necessary to determine the impact of the ion plume on the area of the target

surface with the conditional center at a point

P

. The attitude angles and Rodrigues-Hamilton

parameters included in the matrixes

1

2

,

3

4

were determined integrating the equations (12) - (16)

during the modeling of the system motion. Due to the accepted assumptions, the values of the

components

S

r

,

T

r

, are calculated integrating the following equations:

)0()0(

SSS

Frm

,

)0()0(

TTT

Frm

, (20)

where

)0(

S

F

,

)0(

T

F

are the vectors of the resultant force

S

F

,

T

F

, applied to the IBS and target

respectively;

S

m

and

T

m

– masses of the IBS and target. In this way, according to accepted simplified

model, the system motion is described by the equations (12) - (20) under the condition

0

0

.

In the absence of external perturbations on the system, the force impact on the target is created only

by the ion plume from the shepherd ET. This impact slows the orbital velocity of the target and

allows to de-orbit it faster. The IBS concept assumes that two ETs are installed on the IBS. The

second thruster is necessary to compensate the impact on the IBS motion from the main one,

directed towards the target. However, in order to the IBS and target remain in the same orbit, the

result of the force action on CoM of the IBS has to be the same as the result of the force impact on

CoM of the target. This condition can be achieved by adjustment of the thrust of the second

compensating thruster. The algorithm to adjust the thrust

2E

F

of compensating ET was chosen in

the following simple form:

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73

1Ecnt2E

)/( FFF

TS

mm

, (21)

where

1E

F

is the thrust of the main ET.

Assuming that the IBS attitude position is unchanged, this algorithm can be simplified as follows:

xxTSx

FFmmF

_1E_cnt_2E

)/(

, (22)

where the letter

x

in the subscripts is used to denote the

x

-th component of the column, that

corresponded to the thrust vector in the RF

000

zyOx

or the body RF of the IBS.

The algorithm to determine the target contour for solving modeling tasks is described in details in

[5] and can be represented by the following steps:

– Approximate the target surface by basic elements;

– Project the central points of the surface elements on the image plane of the camera;

– Calculate the contour of the target projection by solving the problem of the polygon

construction, which covers a set of points projected on a plane.

We assume that the origin

c

O

of the camera RF coincides with the origin

S

of the body RF of the

IBS and the

c

z

axis is directed opposite to the axis

2

x

of the body RF of the IBS. Then, the

coordinates of the point

P

of the target is projected on the plane defined by the equations (10) or

calculated using the relations (11) in the case of their correction.

The components of column

(cam)

d

which are included in these equalities are given by

)2(cam

2

(cam)

dd

, (23)

where

cam

2

is the transition matrix from the body RF of the IBS to the camera RF.

Simulation results. The assumption was made for the calculations that the only forces from the

main and compensating ETs act on the IBS. The line of action of these forces passes through CoM

of the IBS.

The following values of parameters were used during the simulation.

Parameters of the ETs are the following: the radius of the beam at the beginning of the far region

m0,0805

0

R

; the plasma density at the beginning of the far region

-315

0

m104,13n

; the

divergence angle of the beam

7

0

; the axial component of the velocity of the plasma ions

m/s71580

0

z

u

; the ion mass

kg102.18

-25

m

.

The IBS parameters were chosen as follows: the matrix of inertia is

2

mkg)3,169;5,1379;4,1283(diag

; the mass is

kg500

.

The following parameters of the cylindrical target were used: the mass is

kg1000

; the height is

m6,2

; the diameter is

m2,2

. The inertia matrix of the target was calculated according to the

formulas for the moments of inertia of a hollow cylinder.

The distance of

m7

from CoM of the IBS to the target along the axis

z

of the camera RF was

chosen as nominal. The focal length of the camera is

m2,0

.

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74

The simulation results show that the difference between the nominal and the calculated distance

from the target to the IBS is less than

sm1

after

s800

of the simulation when the initial location of

CoM of the target is on the beam axis. The figures presented bellow show the results of the

simulations for the case when the initial location of CoM of the target is

m1

away from the beam

axis, the initial attitude position of the target is defined by the angle of

45

about the axis

y

of the

body RF of the target, the camera offset from its nominal position is

m2,0

along the abscissa axis

of the camera RF.

Fig. 4. The force acting on the IBS.

Fig. 4 and Fig. 5 show the graphs of the forces acting on the IBS and target, respectively. Ideally

these graphs should coincide with each other up to the scale. The variation of the target position

(Fig. 6) from the nominal one for

s800

of the simulation is no more than

sm5

(this variation is

denoted as

x

0

r

in the figure) for the ideal camera position, and does not exceed

sm8

for the case

when the camera has the position offset. The variations that obtained through the simulation (i.e. the

errors of maintaining the nominal distance to the target) are not significant, taking into account to

the real control algorithms will use the information about the distance between the target and IBS or

its estimation unlike the applied simple algorithm (21), which does not use this information.

Fig. 5. The force acting on the target. Fig. 6. The deviation of the target relative position.

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75

Fig. 7 and Fig. 8 show the graphs of the torque acting on the target and the graph the angular

velocity of the target , respectively.

Fig. 7. The torque acting on the target. Fig. 8. The angular velocity of the target.

Conclusions. The errors of the simplified approach [5] to determine the impact from the ion

thruster of a spacecraft (a shepherd) on a space debris object (a target) within the context the IBS

technology [1, 2] has been analyzed. The approach is based on the information about the contour of

the central projection of the target on a plane, which is perpendicular to the axis of the ion beam.

The plane of the camera sensor, which is installed on the shepherd, has been considered as such a

plane. The results of the error analysis justify the admissibility of the application of this simplified

approach for the determination of the force impact of the ion thruster to the space debris within self-

similar model of the plasma distribution [6]. The research results also allow to make a preliminary

conclusion about the possibility to use this simplified approach for the control of the relative motion

of the IBS- space debris object system. However, a more detailed analysis of this possibility is

needed which require to consider the orbital motion, all range of acting disturbances, errors of the

model of the interaction of the ion plume with the surface of the space debris object, and