basic experimental considerations on bending...

Proc. Schl. Eng. Tokai Univ., Ser. E （2013）-

Vol. XXXVIII, 2013

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*1 Graduate Student, Course of Science and Technology *2 Graduate Student, Course of Mechanical Engineering *3 Junior Associate Professor, Department of Prime

Mover Engineering *4 Professor, Department of Prime Mover Engineering

Basic Experimental Considerations on Bending Levitation Control by Electromagnetic Force for Flexible Steel Plate

by

Takayoshi NARITA*1, Hiroki MARUMORI*2, Shinya HASEGAWA*3 and Yasuo OSHINOYA*4

(Received on Mar. 30, 2013 and accepted on Jul. 11, 2013)

Abstract

In the transport system of a thin-steel-plate production line, there is a problem that the quality of the plate surface deteriorates over time because the plate is usually in contact with rollers. To solve this problem, electromagnetic levitation technologies have been studied. However, when a flexible thin steel plate with a thickness of less than 0.3 mm is to be levitated, levitation control becomes difficult because the thin plate undergoes increased flexure. We propose the levitation of an ultrathin steel plate that is bent to an extent that does not induce plastic deformation. It has been confirmed that vibrations with mainly low frequencies are generated when a steel plate is bent and levitated. In this study, bending levitation experiments were carried out on the basis of the optimal control theory using thin steel plates with a thickness of 0.18, 0.24 or 0.30 mm to examine the levitation stability of thin steel plates of different thicknesses and to compare the levitation performance. The results show it was found that the thin steel plates with thicknesses less than 0.30 mm are stably levitated by bending levitation, and that an electromagnet tilt angle that is approximately 80% of the natural deflection angle is most effective for stably levitating thin steel plates.

Keywords: Flexible Steel Plate, Electromagnetic Levitation System, Optimal Control, Bending Levitation Control

1. Introduction Thin steel plates are widely used in automobiles, electrical appliances, can manufacturing and in other products. In recent years, it has become possible to manufacture ultrathin steel plates, which are required in various fields with ever-increasing demands for surface quality. However, in the transport system in a thin-steel-plate production line, there is the problem that the quality of the plate surface deteriorates over time because the plate is usually in contact with rollers.

To solve this problem, studies of electromagnetic levitation technology have been carried out1-3). However, as the steel plate becomes thinner, the vibration caused by minute unpredictable factors, including the nonlinearity of the attractive force of the electromagnet and the change in the

resistance due to heat generation by the electromagnet, makes it difficult to maintain the levitation state.

Furthermore, when an ultrathin steel plate with a thickness of less than 0.3 mm is targeted for levitation, levitation control becomes difficult because the thin plate undergoes increased flexure. We propose the levitation of an ultrathin steel plate that is bent to an extent that does not induce plastic deformation4,5). It has been confirmed that vibrations with mainly low frequencies are generated when a steel plate is bent and levitated6).

In this study, bending levitation experiments were carried out on the basis of the optimal control theory using thin steel plates with a thickness of 0.18, 0.24 or 0.30 mm to examine the levitation stability of thin steel plates of different thicknesses and to compare the levitation performance.

2. System for Control Experiment

Figure 1 and Figure 2 shows an outline of the control

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Proc. Schl. Eng. Tokai Univ., Ser. E38 (2013) 47-52

Takayoshi NARITA, Hiroki MARUMORI, Shinya HASEGAWA and Yasuo OSHINOYA

Proceedings of the School of Engineering Tokai University, Series E

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system and experimental apparatus. Figure 3 shows a photograph of the experimental apparatus. The object of electromagnetic levitation is a rectangular zinc-coated steel plate (SS400) with length a = 800 mm, width b = 600 mm. The properties of SS400 are as follows. Young's modulus is 206 GPa, and density is 7.9×10 3kg/m3. To accomplish noncontact support of a rectangular thin steel plate using 5 pairs of electromagnets (Nos. 1-5) as if the plate was hoisted by strings, the displacement of the steel plate is measured using five eddy-current gap sensors. Here, the electric circuits of paired electromagnets are connected in series, while an eddy-current gap sensor is positioned between the two magnets of each pair. The detected displacement is converted to velocity using digital differentiation. In addition, the current in the coil of the electromagnets is calculated from the measured external resistance, and a total of 15

DSP

40MHz

x

y

z

Electromagnet and gap sensor

No.1

No.3

D/Aconverter

i1 i3i2 i4 i5 z1 z3 z4 z5

A/D converter

No.5No.4

No.2

Steel plate

AMP5

AMP1AMP2AMP3AMP4

z2

(TMS320C31)

(800mm×600mm×0.18mm) Fig. 1 Electromagnetic levitation control system

Front view

Electromagnet

Gap sensor

Electromagnet

Gap sensor

A

A'Side view(A-A' section)

Steel plate ( 800mm×600mm×0.18 mm, 0.24mm or 0.30mm)

Fig. 2 Schematic illustration of experimental apparatus

measured values are input into the digital signal processor (DSP) via an A/D converter to calculate the control law. A control voltage is output from the D/A converter into a current-supply amplifier to control the attractive force of the 5 pairs of electromagnets in order that the steel plate is levitated below the surface of the electromagnets by 5 mm. The electromagnets are arranged such that the attractive force of each electromagnet on the flatly levitated thin steel plate is uniform and the flexure of the bent thin steel plate is small. Figure 4 shows the arrangement of the electromagnets. Among the 5 pairs of electromagnets, the 4 pairs at the corners (Nos. 1-4) can be inclined, as shown in the front view in Fig. 2. Note that the positions of the electromagnets are controlled so that the attractive force is applied to the fixed positions on the thin steel plate even when the thin steel plate is bent (Fig. 5).

Electromagnet

Steel plate

Fig. 3 Photograph of experimental apparatus

155 80

800

600

85

0

y

x

Fig. 4 Arrangement of electromagnets

(800mm×600mm)



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system and experimental apparatus. Figure 3 shows a photograph of the experimental apparatus. The object of electromagnetic levitation is a rectangular zinc-coated steel plate (SS400) with length a = 800 mm, width b = 600 mm. The properties of SS400 are as follows. Young's modulus is 206 GPa, and density is 7.9×103kg/m3. To accomplish noncontact support of a rectangular thin steel plate using 5 pairs of electromagnets (Nos. 1-5) as if the plate was hoisted by strings, the displacement of the steel plate is measured using five eddy-current gap sensors. Here, the electric circuits of paired electromagnets are connected in series, while an eddy-current gap sensor is positioned between the two magnets of each pair. The detected displacement is converted to velocity using digital differentiation. In addition, the current in the coil of the electromagnets is calculated from the measured external resistance, and a total of 15

DSP

40MHz

x

y

z

Electromagnet and gap sensor

No.1

No.3

D/Aconverter

i1 i3i2 i4 i5 z1 z3 z4 z5

A/D converter

No.5No.4

No.2

Steel plate

AMP5

AMP1AMP2AMP3AMP4

z2

(TMS320C31)

(800mm×600mm×0.18mm) Fig. 1 Electromagnetic levitation control system

Front view

Electromagnet

Gap sensor

Electromagnet

Gap sensor

A

A'Side view(A-A' section)

Steel plate ( 800mm×600mm×0.18 mm, 0.24mm or 0.30mm)

Fig. 2 Schematic illustration of experimental apparatus

measured values are input into the digital signal processor (DSP) via an A/D converter to calculate the control law. A control voltage is output from the D/A converter into a current-supply amplifier to control the attractive force of the 5 pairs of electromagnets in order that the steel plate is levitated below the surface of the electromagnets by 5 mm. The electromagnets are arranged such that the attractive force of each electromagnet on the flatly levitated thin steel plate is uniform and the flexure of the bent thin steel plate is small. Figure 4 shows the arrangement of the electromagnets. Among the 5 pairs of electromagnets, the 4 pairs at the corners (Nos. 1-4) can be inclined, as shown in the front view in Fig. 2. Note that the positions of the electromagnets are controlled so that the attractive force is applied to the fixed positions on the thin steel plate even when the thin steel plate is bent (Fig. 5).

Electromagnet

Steel plate

Fig. 3 Photograph of experimental apparatus

155 80

800

600

85

0

y

x

Fig. 4 Arrangement of electromagnets

(800mm×600mm)

Proceedings of the School of Engineering,Tokai University, Series E－ 48－



Vol. XXXVIII, 2013

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(a)θ = 0°

Steel plate Electromagnet175mm

(g)θ =30°

165mm

90mm

°30

(b)θ= 5°

210mm 5°

15mm

(c)θ =10°

°205mm 10

25mm

°

(d )θ =15°

200mm15

50mm

°

(e)θ =20°

195mm

65mm

20

( f )θ=25°

°

185mm

75mm

25

Fig. 5 Relationship between tilt angle of electromagnet and shape of steel plate

3. Modeling

In this model, independent control is carried out, in

which information on detected values of displacement, velocity and coil current of the electromagnet under study at one position are fed back only to the same electromagnet. Therefore, as shown in Fig. 6, the steel plate is divided into 5 hypothetical masses and each part is modeled as a lumped constant system (local model).

In an equilibrium levitation state, magnetic forces are determined so as to balance with gravity. The equation of small vertical motion around the equilibrium state of the steel plate subjected to magnetic forces is expressed as

zz fzm 2 (1)

where mz =m/5 [kg], z: vertical displacement [m], and fz: dynamic magnetic force [N]. Figure 7 shows a schematic illustration of the electro-magnet. The number of turns of the electromagnet coil is 1005 (wire diameter is 0.5 mm), and the sectional area passing the magnetic flux of the E-type core, which was made from ferrite, is 225mm2. The characteristics of the electromagnet are estimated on the basis of the following assumptions:

The permeability of the core is infinite, the eddy current inside the core is negligible, and the inductance of the electromagnetic coil is expressed as the sum of the component inversely proportional to the gap between the steel plate and magnet and the component of leakage inductance.

If deviation around the static equilibrium state is very small, the characteristic equations of the electromagnet are linearized as

zz

zzz i

IFz

ZFf 22

0

(2)

zz

zz

zz

z

effz v

Li

LRz

ZI

LL

idtd

21

220

(3)

lea0

effz L

ZL

L (4)

Using the state vector, the equations (1) ～ (4) are written as the following state equations:

zzz vBzAz (5)

zizz z

mz

z

Iz+iz

Steel plate(rigid body)

fzfz

Electromagnet forlevitation control

Eddy current typegap sensor

Fig. 6 Theoretical model of levitation control of the steel plate (Local model)

88

14 1660 15

28

836

Coil

Ferrite

(1005 turns)

Fig. 7 Configuration of electromagnet

T


Vol. XXXVIII, 2013

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(a)θ = 0°


(g)θ =30°

165mm

90mm

°30

(b)θ= 5°

210mm 5°

15mm

(c)θ=10°

°205mm 10

25mm

°

(d )θ =15°

200mm15

50mm

°

(e)θ =20°

195mm

65mm

20

( f )θ=25°

°

185mm

75mm

25


3. Modeling




zz fzm 2 (1)




zz

zzz i

IFz

ZFf 22

0

(2)

zz

zz

zz

z

effz v

Li

LRz

ZI

LL

idtd

21

220

(3)

lea0

effz L

ZL

L (4)


zzz vBzAz (5)

zizz z

mz

z

Iz+iz


fzfz




88

14 1660 15

28

836

Coil

Ferrite

(1005 turns)


T


Vol. XXXVIII, 2013

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(a)θ = 0°


(g)θ =30°

165mm

90mm

°30

(b)θ= 5°

210mm 5°

15mm

(c)θ=10°

°205mm 10

25mm

°

(d )θ =15°

200mm15

50mm

°

(e)θ =20°

195mm

65mm

20

( f )θ=25°

°

185mm

75mm

25


3. Modeling




zz fzm 2 (1)




zz

zzz i

IFz

ZFf 22

0

(2)

zz

zz

zz

z

effz v

Li

LRz

ZI

LL

idtd

21

220

(3)

lea0

effz L

ZL

L (4)


zzz vBzAz (5)

zizz z

mz

z

Iz+iz


fzfz




88

14 1660 15

28

836

Coil

Ferrite

(1005 turns)


T

Vol. ⅩⅩⅩⅧ, 2013 － 49－




―4―

z

zz

z

eff

zz

z

z

z

LR

ZI

LL

ImF

ZmF

20

202

010

20

0zA

zL2100zB

where Fz: magnetic force of the coupled magnets in the equilibrium state [N], Z0: gap between the steel plate and electromagnet in the equilibrium state [m], Iz : current of the coupled magnets in the equilibrium state [A], iz: dynamic current of the coupled magnets [A], Lz: inductance of one magnet coil in the equilibrium state [H], Rz: resistance of the coupled magnet coils [Ω], vz: dynamic voltage of the coupled magnets [V], and Llea: leakage inductance of the one magnet coil [H].

4. Control Theory

In this study, a control system is constructed using a discrete time system; therefore, the evaluation function of a continuous system is digitized, and the optimal control law is obtained based on the optimal control theory of the discrete time system. Here, the following discrete time system is considered.

iii ddd ΓvΦzz 1 (6)

sTAΦ exp , BAΓ dsT

0exp

Here, the evaluation function of the discrete time system

is expressed as follows.

0i

ddT

dddT

dd iiii vrvzQzJ (7)

MΦΓMΓΓrMΓΦQMΦΦM TTd

Td

T 1 (8)

ddod zFv (9)

MΦΓMΓΓrF TTdd

1 (10)

Where Qd and rd are weighting coefficients, M is the solution of the algebraic matrix, the Riccati equation, and Ts is a sampling interval. MATLAB command “ lqrd ” was used to solve eq. (8) and the digital controller was designed by using SIMLINK in the DSP.

5. Experiment of Bending Levitation 5.1 Condition of experiment

The optimal control theory is applied for levitation control of the thin steel plates to compare the results under different electromagnet tilt angles. In the bending levitation experiment using thin steel plates, the electromagnet tilt angle is increased from 0o. The position of each electromagnet is determined on the basis of an assumed shape of a thin steel plate hoisted by strings, because the distance between the surface of the electromagnets and the thin steel plate is controlled to be constant at 5 mm under all conditions.

The natural deflection angle is the deflection angle at the two supporting points when the distributed load due to gravitational force is applied to the thin steel plate, under the assumption that a thin steel plate not supported at the center can be regarded as a both-ends-free beam. The natural deflection angle θ is obtained as eq. (11),

222 6

2al

Ehgl

(11)

where E is the Young’s modulus of the thin steel plate [N/m2], h is the plate thickness [m], ρ is the plate density [kg/m3], g is the acceleration due to gravity [m/s2], l is the plate length [m] and a is distance from edge of steel plate to support point [m]. Table 1 shows the natural deflection angles of steel plates of different thicknesses.

In this study, the standard deviation of displacement and levitation probability were measured. The standard deviation of displacement was measured 10 times for each electromagnet tilt angle for every thin steel plate sample, and the mean was used as the experimental result. To avoid the effects of the transient state of the thin steel plates, the measurement was started approximately 10 s after the start of

Table 1 Natural deflection angle

Plate thickness h [mm]

Natural deflection angle [degree]

0.18 19.2

0.24 10.8

0.30 6.9

T



―4―

z

zz

z

eff

zz

z

z

z

LR

ZI

LL

ImF

ZmF

20

202

010

20

0zA

zL2100zB


4. Control Theory




0exp



0i

ddT

dddT

dd iiii vrvzQzJ (7)


Td

T 1 (8)

ddod zFv (9)

MΦΓMΓΓrF TTdd

1 (10)





222

62

alEhgl

(11)






0.18 19.2

0.24 10.8

0.30 6.9

T



―4―

z

zz

z

eff

zz

z

z

z

LR

ZI

LL

ImF

ZmF

20

202

010

20

0zA

zL2100zB


4. Control Theory




0exp



0i

ddT

dddT

dd iiii vrvzQzJ (7)


Td

T 1 (8)

ddod zFv (9)

MΦΓMΓΓrF TTdd

1 (10)





222

62

alEhgl

(11)






0.18 19.2

0.24 10.8

0.30 6.9

T




Vol. XXXVIII, 2013

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levitation. Levitation is considered successful when it continues for at least 30 s and the levitation performance is calculated as a percentage of successful levitations among 50 trials.

5.2 Levitation experiment The bending levitation experiments were carried out on

the basis of the optimal control theory using thin steel plates with a thickness of 0.18, 0.24 or 0.30 mm. A weighting factor is determined so that the standard deviation of the displacement of the levitated steel plate at a tilt angle of 0o for repeated experiments becomes constant. Figures 8 and 9 show the standard deviation of displacement and levitation probability for the tilt angle under optimal control, respectively.

For thin steel plates with a thickness of 0.18 mm, the standard deviation of displacement decreases and the levitation probability increases with increasing electromagnet tilt angle from 0o. The standard deviation of displacement is the smallest and the levitation probability is the highest (98%) at a tilt angle of 15o. At tilt angles higher than 15o, the standard deviation of displacement increases and the levitation probability decreases. The levitation probability

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

0 5 10 15 20 25

Stan

dard

dev

iatio

n of

dis

plac

emen

t [m

m]

Tilt angle of electromagnet [ °]

Plate thickness =0.18mm


Plate thickness =0.30mmh

h

h

Fig. 8 Relationship between tilt angle of electromagnet and

standard deviation of displacement at sensor No.1

0

10

20

30

40

50

60

70

80

90

100

0 5 10 15 20 25

Levi

tatio

n pe

rfor

man

ce [

%]





h

h

h

Fig. 9 Relationship between tilt angle of electromagnet

and levitation probability

becomes 0% at a tilt angle of 25o. The above results indicate that stable levitation of thin steel plates with a thickness of 0.18 mm can be realized by bending the thin steel plate under optimal control. The reason behind the levitation probability of 0% at a tilt angle of 25o is that a tilt angle of 25o markedly exceeds the natural deflection angle (18.5o) of the thin steel plate with a thickness of 0.18 mm, leading to difficulty in levitation because of a large restoring force applied to the thin steel plate. As mentioned above, the natural deflection angle of the thin steel plate with a thickness of 0.18 mm is 18.5o and the optimal tilt angle for stable levitation is 15o, indicating that the optimal tilt angle of electromagnets is approximately 80% of the natural deflection angle.

For thin steel plates with a thickness of 0.24 mm, the standard deviation of displacement decreases and the levitation probability increases with increasing electromagnet tilt angle, as in the case of thin steel plates with a thickness of 0.18 mm. When the tilt angle is 80% of the natural deflection angle shown in Table 1 in the levitation experiment, the best levitation performance is obtained. When the thin steel plate is levitated at a tilt angle that exceeds the natural deflection angle, the levitation performance deteriorates; the levitation becomes impossible at a tilt angle of 13o. From these results, the tilt angle of electromagnets equivalent to approximately 80% of the natural deflection angle is the most effective for levitating thin steel plates with a thickness of 0.24 mm, as in the case of thin steel plates with a thickness of 0.18 mm.

For thin steel plates with a thickness of 0.30 mm, the standard deviation of displacement increases and levitation probability decreases with increasing electromagnet tilt angle. Therefore, bending is not necessary during levitation for the thin steel plates with a thickness of 0.30 mm; in other words, the levitation at a tilt angle of 0o is the most stable. From the above results, it was confirmed that bending levitation is not necessary for thin steel plates with thicknesses of 0.30 mm or greater, whereas an electromagnet tilt angle that is approximately 80% of the natural deflection angle is the most effective for stably levitating thin steel plates with thicknesses of 0.18 mm and 0.24mm.

6. Conclusion

From the results, in the case of experiment conditions

and steel plates, it was found that thin steel plates with thicknesses of 0.18 mm and 0.24 mm were stably levitated by bending levitation. On the other hand, bending levitation was not necessary for thin steel plates with thicknesses of 0.30mm or greater. In the future, we will examine the levitation performance of a continuous steel model instead of a


Vol. XXXVIII, 2013

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levitation. Levitation is considered successful when it continues for at least 30 s and the levitation performance is calculated as a percentage of successful levitations among 50 trials.

5.2 Levitation experiment The bending levitation experiments were carried out on

the basis of the optimal control theory using thin steel plates with a thickness of 0.18, 0.24 or 0.30 mm. A weighting factor is determined so that the standard deviation of the displacement of the levitated steel plate at a tilt angle of 0o for repeated experiments becomes constant. Figures 8 and 9 show the standard deviation of displacement and levitation probability for the tilt angle under optimal control, respectively.

For thin steel plates with a thickness of 0.18 mm, the standard deviation of displacement decreases and the levitation probability increases with increasing electromagnet tilt angle from 0o. The standard deviation of displacement is the smallest and the levitation probability is the highest (98%) at a tilt angle of 15o. At tilt angles higher than 15o, the standard deviation of displacement increases and the levitation probability decreases. The levitation probability

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

0 5 10 15 20 25

Stan

dard

dev

iatio

n of

dis

plac

emen

t [m

m]




Plate thickness =0.30mmh

h

h

Fig. 8 Relationship between tilt angle of electromagnet and

standard deviation of displacement at sensor No.1

0

10

20

30

40

50

60

70

80

90

100

0 5 10 15 20 25

Levi

tatio

n pe

rfor

man

ce [

%]





h

h

h

Fig. 9 Relationship between tilt angle of electromagnet

and levitation probability

becomes 0% at a tilt angle of 25o. The above results indicate that stable levitation of thin steel plates with a thickness of 0.18 mm can be realized by bending the thin steel plate under optimal control. The reason behind the levitation probability of 0% at a tilt angle of 25o is that a tilt angle of 25o markedly exceeds the natural deflection angle (18.5o) of the thin steel plate with a thickness of 0.18 mm, leading to difficulty in levitation because of a large restoring force applied to the thin steel plate. As mentioned above, the natural deflection angle of the thin steel plate with a thickness of 0.18 mm is 18.5o and the optimal tilt angle for stable levitation is 15o, indicating that the optimal tilt angle of electromagnets is approximately 80% of the natural deflection angle.

For thin steel plates with a thickness of 0.24 mm, the standard deviation of displacement decreases and the levitation probability increases with increasing electromagnet tilt angle, as in the case of thin steel plates with a thickness of 0.18 mm. When the tilt angle is 80% of the natural deflection angle shown in Table 1 in the levitation experiment, the best levitation performance is obtained. When the thin steel plate is levitated at a tilt angle that exceeds the natural deflection angle, the levitation performance deteriorates; the levitation becomes impossible at a tilt angle of 13o. From these results, the tilt angle of electromagnets equivalent to approximately 80% of the natural deflection angle is the most effective for levitating thin steel plates with a thickness of 0.24 mm, as in the case of thin steel plates with a thickness of 0.18 mm.

For thin steel plates with a thickness of 0.30 mm, the standard deviation of displacement increases and levitation probability decreases with increasing electromagnet tilt angle. Therefore, bending is not necessary during levitation for the thin steel plates with a thickness of 0.30 mm; in other words, the levitation at a tilt angle of 0o is the most stable. From the above results, it was confirmed that bending levitation is not necessary for thin steel plates with thicknesses of 0.30 mm or greater, whereas an electromagnet tilt angle that is approximately 80% of the natural deflection angle is the most effective for stably levitating thin steel plates with thicknesses of 0.18 mm and 0.24mm.

6. Conclusion

From the results, in the case of experiment conditions

and steel plates, it was found that thin steel plates with thicknesses of 0.18 mm and 0.24 mm were stably levitated by bending levitation. On the other hand, bending levitation was not necessary for thin steel plates with thicknesses of 0.30mm or greater. In the future, we will examine the levitation performance of a continuous steel model instead of a

Vol. ⅩⅩⅩⅧ, 2013 － 51－




―6―

single-degree-of-freedom model and the levitation performance when other control theories are adopted.

Acknowledgments

We are pleased to acknowledge the considerable

assistance of Mr. Kei Miura.

References

1) T. Namarikawa, D. Mizutani, H. Kuroki: The Institude of Electrical Engineers of Japan 126-10, D, (2006), 1319-1324.

2) M.Sase，S.Torii : International Journal of Applied Electromagnetics and Mechanics，13，1-4，（2001/2002）， 129-136．

3) Y. Oshinoya, N. Nakamura, S. Hasegawa, K. Ishibashi, H. Kasuya, The 15th MAGDA Conference, (2006), 304-305.

4) Y. Oshinoya, N. Nakamura, S. Hasegawa, K. Ishibashi, H. Kasuya: The 23rd Symposium on Electromagnetics and Dynamics, (2007)，37-38．

5) Y. Oshinoya, S. Hasegawa, K. Ishibashi:Transactions of the Japan Society of Mechanical Engineers, Series C,74-740， (2008)，823-832．

6) T. Masaki, K. Urakawa, T. Narita, Y. Oshinoya, S. Hasegawa, K. Ishibashi, H. Kasuya: The 16th Conference on Kanto Branch of The Japan Society of Mechanical Engineers，(2010)， 177-178.



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