A Study of Self Loosening of Bolted Joints Due to Repetition of Small Amount of Slippage at Bearing...

8/12/2019 A Study of Self Loosening of Bolted Joints Due to Repetition of Small Amount of Slippage at Bearing Surface

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Journal of Advanced Mechanical Design,Systems, andManufacturing

Vol. 1, No. 3, 2007

358

A Study of Self-Loosening of Bolted Joints Due

to Repetition of Small Amount of Slippage at

Bearing Surface*

Shinji KASEI**

**Shinshu University,

4-17-1, Wakasato, Nagano, 380-8553 JAPAN

E-mail: [email protected]

Abstract

This paper deals with the mechanism and relating matters on self-loosening rotation

of bolted joints in the cases when very small slippages occur repeatedly at bearing

surfaces under transverse loads. It can be supposed that accumulation of this kind

of loosening finally produces serious states for bolted joints. Based on an

assumption that a restoring force caused by an elastic torsion of a bolt shank which

arises from a relative displacement between bolt and nut threads drives loosening

rotation, a theoretical explanation is presented about how loosening rotation occurs

and grows larger. Experiments in quasi-static states show results which agree with

the theory of this paper. Additionally, consumption of transverse work and

anti-loosening performance being taken up as relating subjects, more information

about self-loosening is shown through examinations on some kinds of test samples.

Key words: Bolted Joint, Transverse Load, Loosening Rotation, Minute Slippage,

Bolt-Shank Torsion, Loosening Mechanism, Consumed Work,

Anti-Loosening Performance

1. Introduction

Self-loosening of screw threads is one of major subjects relating directly to reliability of

bolted joints. Accordingly, many efforts have been made to understand this phenomenon

and prevent its occurrence. It is well known that there exist two kinds of self-loosening, the

one due to rotation of a bolt or a nut and the other not due to it. Their respective features

and mechanisms have been known gradually(1)~(4)

. Especially, many researchers have

exerted themselves for self-loosening due to rotation since it has a dangerous inclination to

bring greater loss of a fastening force than the case not due to rotation. G.H.Junker

(5)

pointedout clearly that loosening rotation occurs easily under repetition of transverse loads and thus

we should take notice of this case. A.Yamamoto and S.Kasei(6),(7)

presented a hypothesis of

the loosening mechanism, in which behavior of an elastic torsion of a bolt shank performs

an important role, and verified it experimentally. As recent achievements, many papers have

been published by R.I.Zadoks & X.Yu(8)

, P.Wolfsteiner & F.Pfeiffer(9)

, N.G.Pai &

Hess(10),(11)

, Y.Jiang, et al.(12),(13)

, S.Izumi, et al.(14),(15)

and so on. Each paper has respectively

contributed to explication of the self-loosening problems. It is noticeable that three

dimensional FEM was used as a tool of analyses in some papers(10)~(12),(14),(15)

. This fact

makes us feel that an era in which problems like self-loosening can be numerically analyzed

has come. Among them, S.Izumi, et al.(14)

showed the analytical results supporting the

results by the author, et al.

(6),(7)

.The author, et al.

(16),(17)pointed out that loosening rotation can be induced even when*Received 24 Jan., 2007 (No. 07-0054)

[DOI: 10.1299/jamdsm.1.358]


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Vol. 1, No. 3, 2007

359

slippages at bearing surfaces are very small and its situation should be considered as a sign

capable to connect to a serious state. And a theoretical consideration about the loosening

mechanism was developed by basing on the hypothesis(6),(7)

mentioned above. Subsequently,

several researches intending to this case were performed(10)~(13),(15)

. In particular, S.Izumi, et

al. classified contacting situations of a mating surface on threads and a bearing surface with

nine kinds of combinations and explained loosening behavior in each combination.

This study was planned to follow up the previous report (16)

, especially about

experimental considerations which have been remained not enough. So, it is the main

purpose to prove the theory experimentally by using an improved experiment apparatus and

consequently contribute explication of the self-loosening mechanism. In addition, to present

more information about self-loosening of bolted joints, consumptions of transverse work

and anti-loosening performances are examined about some test bolt-nut samples.

2. Theory

Along the hypothesis of the previous report(16)

, a theoretical approach will be promoted.

Basically, the author considers that loosening rotation is caused by a restoring action of an

elastic torsion of a bolt shank which is due to a relative motion at a mating surface on

threads. Figure 1 shows an explanatory illustration about this relation in the case of

right-hand threads. Speaking relatively, it can be said that whichever side is fixed between a

bolt and a nut, it does not make an essential difference.

2.1 Behavior of elastic torsion of bolt shank

Figure 2 illustrates slide loci(6)

at the mating surface on threads under the transverse

load. The side of bolt-threads is replaced with a concentrated point and the slide loci which

the point draws on a thread surface of the nut are expressed schematically in processes of

sliding-up and -down (the nut side is fixed). Where, : a half angle of thread, : a lead angle

of a helix, r: a radial position from the bolt axis, u: a circumferential displacement

(corresponding to the torsion), Rr: a force which moves the bolt-threads up or down along

the thread surface of the nut,Fa: a force which moves the bolt-threads down along the helix,

andFt: a restoring force due to the torsion along the helix. The direction of a resultant force

ofRr,FaandFtis considered to coincide with the tangential direction of the loci and to be

opposite to the direction of a frictional force on the thread surface. In the mating surface on

threads, parts of sliding-up and sliding-down coexist and the slide brings the torsion angle

T. By taking a pitch diameter of thread dpinto account, an equation pT du2= is obtained.

If the nut side is fixed throughout the processes, ucan reach to a critical value ucr(or Tcr, a

critical value of T) in a critical condition ta FF = . However, once the nut slides at the

bearing surface, the torsion is released by a rotation of a unified body composed of the boltand the nut

(6),(16). Since the bolt-threads descend along the helix of the nut-threads with

FtFa

Rr

A

B

aFriction force

Screw surface

of nut

Circumfer-

ential dir.

Axial dir.

Radial dir.

u

r

In sliding-up

Slide locus

a u

r

A'

B'

Slide locus

In sliding- down

Fig.2 Slide loci on mating surface on threads

Fixed plate

Movable plate

Bolt shank

TorsionRestoring

force

Fig.1 Transverse displacementand torsion of bolt-shank


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Vol. 1, No. 3, 2007

360

increasing of u and thus an axial tension Ff reduces by a quantity corresponding to a

deformation of tanu , the above rotation becomes a loosening one in the result.

The slide locus is formulated by the following equation(6),(7)

.

coscos

cos

d

d

ta

r

FF

R

u

r

= (1)

Denoting a part ofFfasFf', a coefficient of friction assand a part of a spring constant

of the boltKuasKu', the following relations are induced for Eq.(1)(6)

.

( ) ( )2''

'

,

'

)cos4(,sin

costan1costandown)(slidingup)(sliding

putfa

s

fs

r

s

fs

r

duKFFF

FR

FR

==

+

==

Consequently, the critical torsion angle Tcr is

known as

tan2

tan2 '

'

u

fp

u

fp

TcrK

Fd

K

Fd== (2)

whereKuandKu'are quantities with the unit of

[Nm/rad].

2.2 Progress of loosening rotation

Typically, the case where an initial torsion

u0due to a bolt clamping is greater than ucr is

discussed. Figure 3 conceptually illustrates two

typical relations between the number ofloading cycle n and a behavior of u. The one

denoted as (a) is in a state where the nut does

not slide and conversely the other (b) is in a

state where the slide of the nut exists at the

bearing surface little by little. In the case (a), u

approaches gradually to ucrfrom an upper side

because of a gradual release of uat the mating

surface on threads, but no loosening rotation

occurs. In the case (b), at an initial stage of the

transversely repeated loading u goes down

beneath ucrdue to a slide at the mating surface on threads and/or the bearing surface of thenut. Afterward, u increases toward ucrwhen no slippage occurs at the bearing surface and

conversely decreases when the slippage occurs. In Fig.3, a zigzag line expresses a process

of progress of the loosening conceptually and means that the loosening rotation occurs

cyclically in sections from uun to uln. It is considered that uun and uln depend on the axial

tension, the transverse load, frictional conditions, etc. Moreover, the difference between uun

and ulnis considered to have a close relation with the magnitude of loosening rotation.

2.3 Relation between transverse load and transverse displacement

Figure 4 shows two basic models expressing the relation between the transverse load W

and the transverse displacement x. The models correspond to cases with some losses of

energy at the mating surface on threads and/or the bearing surface. S 0is a starting point ofthe loading. In the case (a), the slippage at the mating surface on threads only exists (S4'S5'

Number of cycle n

Tors

ion

ofbo

lt-s

han

k

u

looseningOccurrence oflooseningOccurrence of

uln

uun

ucr

(a)

u0

(b)

Fig.3 Explanatory figure of relation between

torsion and number of loading cycle

W

S3

S5

S0

S1

S2

S4

S5

S6

S7 xS7

S2

S4

W

xS0

S1

(a) (b)

Fig.4 Two basic models between

transverse load and displacement


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Vol. 1, No. 3, 2007

361

and S7'S2'), but there is no

slippage at the bearing

surface. The case (b) has, in

addition to the part of the

slippage at the mating surface

on threads (S4S5 and S7S2),the part of the slippage at the

mating surface on threads

plus at the bearing surface

(S2S3 and S5S6). The case (a)

corresponds to the one of

Fig.3(a) and the case (b) to

the one of Fig.3(b).

Incidentally, sections of S0S1, S2'S4' and S5'S7' in the former and sections of S0S1, S3S4and

S6S7 in the latter express processes in which the bolt and the nut displace together like a

unified body.

3. Experiment apparatus and experimentalconditions

Fig.5 shows an outline of the experiment

apparatus. Roughly speaking, it consists of a main

body, a driving part and a system for measurement

and control. In the main part, many steel balls being

inserted in-between, a movable plate is clamped to a

fixed plate by a test bolt and a test nut under a

required force. In the driving part, through a

differential screw mechanism a servomotor gives

the movable plate a transverse alternatingdisplacement. The servomotor has a reduction gear

drive and a rotational encoder, and its rated output

performances are 111W and 30rpm. In addition, the

differential screw mechanism in which a

combination of pitch of 1.5 and 1.25mm is

employed is used to convert a rotational motion to a

translational one and make a small and smooth

quasi-static transverse displacement. A motion

control is performed for the amplitude of the

transverse displacement of the movable plate.

Figure 6(a) shows a detailed drawing of the sideview of the main body. Parts numbered are: the

test bolt, the test nut, a attachment for

measuring the bolt shank's torsion angle or the nut's

rotation angle (additionally, the nut's slippage at the

bearing surface), the movable plate, the fixed plate, a load-cell for measuring

the axial tension of the bolt, a fixture for fixing firmly the test bolt to a inner side of .

A bearing plate with which the test nut keeps contact was heat-treated and its both end

surfaces were ground.

Quantities to be measured are the axial tension Ff, the transverse displacement of the

movable plate x, the transverse load W, the torsion angle of the test bolt shank T, the

rotation angle of the test nut and the slippage at the bearing surface of the test nut s. As

shown in Figure 6(b), fixing the attachment to the test bolt shank,Tis obtained basing

on subtraction of outputs of two gap sensors A and B, and instead fixing to the test nut,

Pulse Board

Strain Amp

A/D Board

Strain Gage

Control Unit

Fixed plateBolt/nut

Servo motor

Torsion,

Slippage,

Rotation

Differential

screw mech.

Axial forceExt.

force

Displacement

Movable plate

Electronic

micrometer

Non-contact

sensors

Strain gages

Strain amp.

A/D board

Control unit

Pulse board

Fig.5 Outline of the experiment apparatus

(a) Side view of the main body

(b) Attachment for measurement

of torsion, etc.

Fig.6 Detailed explanation on the

apparatus


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Vol. 1, No. 3, 2007

362

and s are obtained basing on subtraction

and sum of outputs from A and B,

respectively.

Table 1 shows the summary of

conditions for the experiments. Test thread

sizes are M10 and M101.25, and a griplength is 28mm. An initial axial tension of

the bolt is always set to be 10kN and

machine oil added MoS2 powder is applied

to the thread surfaces and the bearing

surfaces as a lubricant.

4. Experimental results and

considerations

4.1 State of progress of loosening

Figure 7 shows changes of therotation angle of the nut under

transversely repeated loads.

Amplitudes of the transverse

displacement x are 15m to 40m.

The negative side of means the

loosening direction. Changes of in

these cases are small and especially

in ~ the changes are smaller

and can be recognized to stop on the

way. In the cases of ~, it is

supposed that the changes are not dueto loosening rotation, but due to

releases of initial torsions of the bolt

shanks and initial deflections of the

bolt shanks which are forced to

generate in clamping operations.

These kinds of rotations only connect

to initial tiny losses of Ff. On the

other hand, in and , real

loosening rotations probably add to

some extent and so their values

steadily grow greater little by little.According to Fig.7, it can be

supposed that loosening rotation will

go forward when the amplitude of x

is greater than around 30m in the

present experiments.

4.2 Torsion of bolt shank and

loosening rotation

Figure 8 shows experimental

curves of the torsion angle T and

corresponding curves of Ffwhen the

amplitudes of x are 15 and 60m.

About T, the direction indicated with

-30

-25

-20

-15

-10

-5

0

0 100 200 300 400 500 600 700 800

Numbe r of cycle n

Rotation

ofnut

(

)

B:15m A: 15m

A:30m B: 30mA: 35m A: 40m

Fig.7 Progresses of the nut rotation (A: M10 hexagon nut,

B:M101.25 hexagon nut with flange)

0

2

4

6

8

Tors

ion

T

(

) 15m

60mTcr

2

4

6

8

10

0 50 100 150 200 250 300Number of cycle n

Axialtension

F

f

(kN) 15m

60m

Fig.8 Changes of torsion angle and axial tension (M10hexagon nut)


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Vol. 1, No. 3, 2007

363

the arrow mark in Fig.1 is employed

as the positive side. It is known that

these curves of T show a similar

tendency to Fig.3. In the case of15m, T approaches gradually to a

value regarded as Tcr from an initial

value forced by the clamping

operation and only the tiny loss ofFf

can be recognized in the early stage

as mentioned in 4.1. On the other

hand, in the case of 60m, T drops

down rapidly just after a start of the

transverse loading and afterward

repeats up and down in the area between zero and the value regarded as Tcr. Moreover, it

can be seen that progress of loosening corresponds to the behavior of T. By substitutingFf=10kN, =3.03 , dp=9.03mm and Ku=1.5910

3Nm/rad to Eq.(2), Tcr=1.5210

3rad=5.22is found. Here, the value of Ku is from a calculation about the test bolt. This

value agrees fairly well with the experimental value indicated in Fig.8.

To examine in detail, a behavior of Twas detected in a process which consists of a

stage of a manual clamping by a spanner, five cycles of loading (the amplitude of x is

100m) and a manual loosening by the spanner. Figure 9 shows its result. In the figure, A1

is equivalent to a point at which the fastening torque reaches a maximum, A2 a removal

point of the fastening torque, A3a point of a maximum loosening torque and A4an ending

point of the loosening. According to Fig.9, in the process of transverse loading except an

initial part, T behaves up and down similarly and periodically every half cycle. Such

feature is because the test bolt-nut is almost under the same kinetic conditions every halfcycle in whichever direction the transverse load acts. However, it is supposed that if there

are some reasons like geometrical deviations (e.g. perpendicularity between a bolt's axis

-10

-5

0

5

10

15

20

25

0 100 200 300 400

T

ors

ion

alang

le

T

(

)

A1

A2

A3

A4

100m5cycles

Time (s)

Fig.9 Behavior of torsion angle of bolt-shank inmanual-fastening plus transverse-loading plusmanual-loosening (M10 hexagon nut)

-0.8

-0.4

0

0.4

0.8

Tra

nsverseloadW

kN

0

2

4

6

Torsion

T

(

)

-35

-30

-25

-20

-15

-100 -50 0 50 100

Nutrota

tion

Transverse displacement x(m)

S3

S6

S4

S7

S7S4

S3

S6

S2

S2

S2

S5

S5

S3

S6 S7

S5

S4

Fig.10 Mutually related diagrams oftransverse load, torsion angle ofbolt-shank and rotation of nut(M10 hexagon nut)

0

2

4

6

-120 -60 0 60 120

x (m)

0

2

4

6

-120 -60 0 60 120

x (m)

T

Fig.11 Additive diagrams of torsion angle ofbolt-shank (M10 hexagon nut)

Table 2 Slip widths and rotation angles of nut in

a half cycle (n=5, M10,hexagon nut)


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Vol. 1, No. 3, 2007

364

and a bearing surface), relating conditions will change and consequently T will show

some different behavior. And, during transverse loading T exhibits a bias with a positive

value. This is considered to meet the hypothesis that an elastic torsion of a bolt shank plays

a role to drive loosening rotation.

Figure 10 shows diagrams of W, Tand relating to xby arranging these in parallel.

These diagrams were obtained from a simultaneous detection in the same experiment (theamplitude of x is 100m). Since the case is supposed to correspond to the state shown by

Fig.4(b), the symbols used there are attached at the corresponding points supposed. During

sections of S3S4 and S6S7 in which the bolt and the nut are regarded to be unified, T

changes just only a little, but increases during sections of S4S5and S7S2in which the mating

threads are regarded to be forced to slip. And it is known that the nut rotates in the manner

synchronizing with sections of S2S3and S5S6which are regarded as sections involving the

slippage at the bearing surface. Figure 11 shows the diagrams of T versus x when the

amplitudes ofxare 60 and 80m. Similarity is clearly recognized between the diagrams of

Figs.10 and 11. It is considered that such behaviors and relationships show a validity of the

theory explained in 2. It is supposed that geometrical and tribological conditions at the

mating threads and the bearing surface are influence factors which bring fluctuations tothese relations.

In the experiments, progresses of loosening rotation are clearly recognized when the

amplitude ofxis greater than 60m. Table 2 shows experimental values of the width of the

slippagesat the bearing surface and the increment of during a half cycle at the fifth cycle

(n=5). We can confirm thatsis very small on the whole, loosening rotation occurs even in

the case wheresis less than 10m and its value grows larger with increase of the magnitude

of slippage.

M10, hex. nut M101.25, hex. nut M10, hex. nut with locknut

Fig.12 Changes of axial tension and consumed work per a cycle (amplitude ofxis 80m)

n = 2

n=100

n=300

- 0 . 8

- 0 . 4

0

0. 4

0. 8

- 100 - 50 0 50 10 0

x (m)

W

(kN

50 100-100

0.8

0.4

-0.4

-0.8

W

(kN)

0

-0.8

-0.4

0

0.4

0.8

-100 -50 0 50 100x (m)

W

(kN)

n = 2

100

n = 2

100

300

x (m)

-1

-0.6

-0.2

0.2

0.6

1

-100 -50 0 50 100x (m)

W

(kN)

n = 2

100

300

M10, hex. nut M101.25, hex.nut

M10, hex. nut with locknut

Fig.13 Diagrams between transverse load and transverse displacement corresponding to the cases of Fig.12


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365

4.3 Consumed work

Since occurrence of loosening rotation must be closely related with the slippages at the

mating surface on threads and the bearing surface, a work (energy) consumed by them is

considered to involve information about loosening. For anti-loosening performances, G. H.

Junker, et al.(19),(5)

employed a quantity relating to the external work of vibration as an

evaluation factor. Here, the work consumed per one loading cycle designated as Lwill beexamined. L is considered to equal an area surrounded by diagrams of W versus x like in

Fig.4. So, measurements of L are tried every cycle through a numerical treatment with

regard to Wandx.

Figure 12 shows experimental examples aboutFfandLversus n. The nut samples are as

follows, (a) a M10 hexagon nut, (b) a M101.25 hexagon nut and (c) a M10 hexagon nut

with a locknut (so-called a "double-nut"). Figure 13 shows diagrams of W versus x

corresponding to the cases of Fig.12. According to these figures, the following can be

stated. In the case of (a), bothFf andLfall most rapidly. It seems to be caused by a reason

that the slippages are easy to occur at the mating surface on threads and the bearing surface.

In the case of (b), both of Ff andLshows a tendency of slow falling. It is supposed that the

slippage at the mating surface on threads is hard to occur to some extent. Lastly, in the caseof (c),Ffshows no falling tendency, butLshows a quite different feature. Namely, although

the value ofLis initially in a higher level and keep the level for a while, afterward it falls

down steadily. It can be pointed out that the slippage is forced to occur only at the bearing

surface, protecting strongly the slippage at the mating surface on threads by a complete

locking by means of a locknut. In the above example, it is known that the slippage at the

bearing surface is getting harder to occur at n=300 than at n=2 and 100. Perhaps, such a

situation comes from tribological changing of the bearing surface.

4.4 Tests for anti-loosening

performance

Anti-loosening perform-ances of several kinds of nut

samples are compared one

another by evaluating a loss

rate of the axial tension .

Table 3 presents M10 or

M10 1.25 test samples in

detail and the marks assigned

for them. In the case of a M10

hexagon nut with a locknut (a

"double-nut"), a locking

operation is completely doneto assure locking effect.

Figure 14 shows the

values of which are

obtained from the axial

tension losses between n=0 to

300 under five different

amplitudes of xup to 100m.

Each point doted expresses

the average of three test

results. Speaking from the

results shown in Table 2, theslippage at the bearing

0

20

40

60

80

100

Amp. of displacement (m)

Lossrateofaxialtension

(%)

15 30 60 80 100

Fig.14 Results of anti-loosening tests

With washer

Coarse(1.5)

Fine(1.25)

Hex. Nut

Thread

pitch (mm)

Without washer

Hex. nutHex. nut

with flange

Hex. nut

with locknut

(double-

nut)

"Washer" means a special plain washer which was heat-treated and

ground

Table 3 Samples for anti-loosening tests


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Vol. 1, No. 3, 2007

366

surface is estimated to be very small.

According to Fig.14, the "hexagon nut", which is regarded as the basic sample, has the

lowest performance. Other samples, although degrees of the effect are not alike, exhibit

some respective effects. However, an excellent performance is only recognized in the case

of the "hexagon nut with a locknut ("double-nut")". Other cases seem to have insufficient or

limited effects. Progress of loosening rotation is affected by the conditions of the bearingsurface and the mating surface on threads. The performance of the "hexagon nut with a

locknut ("double-nut")" comes from a hard obstruction against the slippage at the mating

surface on threads. Basically, situations in which slippages may occur at both parts of the

mating surface on threads and the bearing surface should not be permitted for preventing

loosening rotation. Fig.14 suggests such a point well.

5. Conclusions

(1) With regard to the cases under repetition of small slippages at the bearing surface, a

theoretical explanation about the mechanism for occurrence and progress of loosening

rotation was presented. It is based on the hypothesis that the basic driving force of

loosening rotation is the restoring force due to the elastic torsion of the bolt shank.(2) Experiments under repetition of the transverse displacement were performed about M10

or M101.25 samples in quasi-static states. It can be said that experimental results show

well agreements with the theory of this paper and thus the basic mechanism of loosening

rotation can be explained well by this paper.

(3) The consumed work in the mating surface on threads and the bearing surface was taken

up as a factor suggesting the feature of loosening. With some examples detected in the

experiments, relations between its value and tendency and loosening behaviors were

explained.

(4) Anti-loosening performances about some kinds of test samples were inspected under

repetition of small slippages at the bearing surface. Among the present samples, only the

case employing the "hexagon nut with a locknut", which was subjected to a stronglocking operation, showed an excellent performance. It is emphasized that when a high

anti-loosening performance is required a firm obstruction of the slippages at the mating

surface on threads and/or the bearing surface must be ensured.

Acknowledgment

The author expresses sincere thanks to Prof. Emeritus, Akira Yamamoto, Tokyo

Institute of Technology for his guidance and suggestions over long years. The author is

grateful to Mr. Satoshi Yoshida, Mr. Hisanori Ishibashi and Mr. Takato Komura for their

co-operations and contributions to this study as graduate students, and also grateful to other

students who took part in this study. Additionally, the author wishes to thank the Ministry of

Education, Culture, Sports, Science and Technology, and Japan Society for the Promotion ofScience for their financial support (the Grant-in Aids for scientific research) to some parts

of this study.

References

(1)Yamamoto,A., Principles and Design of Threaded Joints, (1995), Chap. 4, Yokendo. (in

Japanese)

(2)The Japan Research Institute for Screw Threads and Fasteners (JFRI) ed., Guide to Fastening

Technology of Bolted Joints, (2004), Chap. 6, JFRI. (in Japanese)

(3)Yoshimoto,I. ed.,Points of Design of Threaded Joints (revised ver.), (2002), Chap. 6, Japanese

Standards Association. (in Japanese)(4)Wiegand,H. et al., Schraubenverbindungen, (1988), Kapitel 9, Springer.


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Vol. 1, No. 3, 2007

367

(5)Junker,G.H., New Criteria for Self-Loosening of Fasteners under Vibration, SAE Transactions,

No.690055, pp.314-335.

(6)Yamamoto,A. and Kasei,S., Investigations on the Self-loosening of Threaded Fasteners under

Transverse Vibration A Solution for Self-loosening Mechanism, Bulletin of the Japan

Society of Precision Engineering, Vol.18, No.3, (1984), pp.261-266. (originally in Japanese)

(7)Yamamoto,A. et al., Investigations on the Self-loosening of Threaded Fasteners underTransverse Vibration Theorization of Locking Performance Curve,Journal of the Japan

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A Study of Self Loosening of Bolted Joints Due to Repetition of Small Amount of Slippage at Bearing...

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