Tod

delones tn o

r teloss os annguartifor

ytic95

eni

I

tv

o

Ka

t

s

s

very low value of 0.03, whereas the actual value of the frictioncoefficient for oiled steel is normally around 0.15. Haviland 3stated that loosening happens as a result of the ratcheting action oftpa

ts

m

s

Yn

ir

a

tz

iJm

angular equations of motion are established for a mathematicalmodel of threaded fasteners during the self-loosening. Eventualintegration of the angular equation of motion, with small time

ifiP

5

Dohe nut, but no mathematical model or experimental data wererovided to prove his theory. Yamamoto and Kasei 4 proposednother mechanism, which suggested that loosening occurs due tohe accumulation and release of potential energy due to the tor-ional bolt deformation. Tanaka et al. 5 conducted an axisym-etric finite element study for bolt loosening. Their model

howed that the loosening process is the same as that presented byamamoto and Kasei. Vinogradov and Haung 6 presented a dy-amic model to simulate the loosening under high-frequency load-ng 30240 kHz, which is well beyond the normal operationange for most machine vibrations. Zadoks and Yu 7,8 presenteddynamic model to simulate the loosening process of screw in a

apped hole. Their friction model assumed that friction drops toero in the circumferential direction when the friction is overcomen the transverse direction, this assumption was not a valid one.unker 9 showed that the loosening of threaded fasteners was farore severe when the joint was subjected to transverse cyclic

steps, provides information on the bolt rotation that causes theloss of its tension self-loosening. The drop in the bolt tension isdetermined from the calculated angular displacement in the loos-ening direction and the tension-rotation relationship of the bolt. AMATLAB code is developed to calculate the clamp load loss interms of the number of the transverse load cycles.

Force and Moment Analysis. A model of a typical bolted jointsystem is shown in Fig. 2; the upper plate is clamped to a tappedblock using a threaded fastener. The forces acting on the free bodydiagram Fig. 1 are detailed in Figs. 35. Figure 3 shows theunderhead bearing surface or the threads contact area divided intoa finite number n of small sectors with angle increments d. Thedifferential force vectors acting on each sector of the contact areamust be determined in order to calculate the differential bearingfriction torque component dTb that must be overcome during theloosening of the bolt.

The proposed self-loosening model takes into account somesimplifying assumptions. First, it is assumed that when the bolthead slides over the top of the block, the rotation of the bolt headwill be stopped if the shank of the bolt comes in contact with theedge of the hole. Second, it is assumed that the thread rotation isstopped when the bolt threads come in contact with the side of thetapped hole. The set of forces acting on the bolt head and on thethreads are different before and during the side contact. Third, the

Contributed by the Pressure Vessels and Piping Division of ASME for publicationn the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received June 27, 2005;nal manuscript received July 30, 2005. Review conducted by G. E. Otto Widera.aper presented at the 2005 ASME Pressure Vessels and Piping ConferencePVP2005, July 17, 2005July 21, 2005, Denver, Colorado, USA.

90 / Vol. 128, NOVEMBER 2006 Copyright 2006 by ASME Transactions of the ASMESayed A. Nassar

Basil A. Housari

Fastening and Joining Research Institute,Department of Mechanical Engineering,

Oakland University,Rochester, MI 48309

Effect ofTensionThreadeA mathematical moloosening phenomenThe study investigatthe external excitatioclamp load (fastenecause the completedifferential equationthe system propertieof the equation of awhich causes the pdeveloped and usedself-loosening. AnalDOI: 10.1115/1.234

Keywords: Self-loospitch

ntroductionThere have been several studies on the self-loosening of

hreaded fasteners; most of them are experimental. Hess 1 sur-eyed some studies on the vibration- and shock-induced looseningf threaded fasteners. Sakai 2, Haviland 3, Yamamoto andasei 4, Tanaka et al. 5, Vinogradov and Haung 6, Zadoks

nd Yu 7,8, and Junker 9 offered some insight into the effect ofransverse load on the self-loosening of threaded fasteners.

Sakai 2 presented a theoretical model of loosening based onlip at contact surfaces of the fastener. The results from his modeluggested that the friction at the contact surfaces should drop to awnloaded 12 Dec 2011 to 195.83.11.66. Redistribution subject to ASMEhread Pitch and Initialn the Self-Loosening ofFasteners

and an experimental procedure are presented to study the self-of threaded fasteners that are subjected to cyclic transverse loads.he effect of thread pitch, initial bolt tension, and the amplitude ofn the loosening of a single-bolt joint. The rate of drop in the joint

nsion) per cycle, as well as the total number of cycles that woulds of clamp load, are monitored. In the mathematical model, thef linear and angular motion of the bolt are formulated in terms ofd the external cyclic transverse excitation. Numerical integrationlar motion provides the bolt rotation in the loosening direction,al or full loss of the clamp load. An iterative MATLAB code isthe calculation of tension loss in the fastener tension due to the

al and experimental results are discussed.72

ng, vibration loosening, threaded fasteners, bolted joints, thread

loads. In his work, he concluded that the self-loosening happenswhen slippage took place between engaged threads and/or underthe bolt head/nut.

This study proposes a mathematical model and an experimentalprocedure to study the effect of thread pitch, initial bolt tension,and the amplitude of the cyclic transverse load on the loosening ofthreaded fasteners. Both the numerical and experimental resultsare presented and analyzed.

Mathematical ModelFigure 1 shows a free body diagram of the threaded fastener,

with various forces and moment/torque components. Linear and license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

ewttbatfmbttpf

a

spawd

J

Doffect of damping is not considered in this study.Figure 1 shows the free body diagrams of the bolt segments

ith no side contact. On the bolt head segment, the forces andorques acting on the bolt bearing surface are the bolt tension F,he reaction force F from the upper plate, the friction force Fbetween the bolt and the upper plate, the bolt shear force Fs cre-ted by bolt bending, the bending moment Mb, and the torque Tlransmitted from the threads to the head which is caused by allorces acting on the threads. These forces and moments deter-ine the motion of the bolt head during the loading cycle. It must

e observed that only the torque component Tl will have the po-ential to rotate the bolt in the loosening direction. This means thathe bolt has a potential to loosen if and only if the torque Tl isositive i.e., acting in the loosening direction as it overcomes theriction resistance under the bolt head.

Figures 3 and 4 show the three different coordinate systems thatre used in the modeling process. The first is a global Cartesianystem XYZ with respective unit vectors i, j, k , where X is in theositive direction of the external excitation, and Z is upward par-llel to bolt axis. The second is a local coordinate system uvw, inhich u is oriented in the radial direction, v is in the tangentialirection, and w is parallel to the global Z direction. The third

Fig. 1 Bolt-free body diagram

Fig. 2 Bolt motion under external transverse force

ournal of Pressure Vessel Technologywnloaded 12 Dec 2011 to 195.83.11.66. Redistribution subject to ASMEcoordinate system is u1v1w1, in which w1 is normal to the threadsurface, and v1 is within the thread plane in the circumferentialdirection. The relationship between the unit vectors u, v, w and i,j, k is expressed in a matrix format as follows:

Fig. 3 Forces on the bearing surface

Fig. 4 Thread contact force components

NOVEMBER 2006, Vol. 128 / 591 license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

w

wF

s

w

tm

r

watu

E

uvwT = T11 u1v1w1T 7

5

DouvwT = T ijkT 1here the superscript T indicates the transpose of a matrix, and

T is a transformation matrix given by

T = cos sin 0 sin cos 00 0 1 2here is the angle between the unit vectors i and u, as shown inig. 3.The reverse transformation from the uvw system to the XYZ

ystem is given by

ijkT = T1 uvwT 3here the superscript 1 indicates the inverse of a matrix.Similarly, the unit vectors u, v, w are given by

u1v1w1T = T1 uvwT 4he matrix elements in the first and second rows in the transfor-

ation matrix T1 are obtained using the following geometricelationships:

v1 = 0 u + cosv + sinw

w1 = tan2 + tan2 + 1 tanu + tanv + w 5here and are the thread profile angle and the thread lead

ngle, respectively, as shown in Figs. 4 and 5. The third row in theransformation matrix T1 is obtained by the cross product of thenit vectors u1, v1

w1 = u1 v1 6quation 4 may be expressed in an inverse way as follows:

Fig. 5 Thread friction force components

92 / Vol. 128, NOVEMBER 2006wnloaded 12 Dec 2011 to 195.83.11.66. Redistribution subject to ASMEShear Force. Generally, a cyclic transverse excitation may be aforce or a displacement. In this study, cyclic displacement is con-sidered. The displacement of the upper plate in Fig. 1 is given by

t = o sinti 8a

= 2f 8bwhere t is the displacement vector as function of time, 0 maxi-mum displacement amplitude, is the angular frequency of thetransverse loading, f is the frequency, and t is the time. Sliding ofthe head will take place when the bolt deflection reaches a criticalvalue cr, at which the force Fs overcomes the underhead friction.This critical displacement cr is approximated by

cr =bFiL3

3EI9

where b is the underhead friction coefficient, Fi is the bolt ten-sion, L is the grip length, E is the modulus of elasticity for the boltmaterial, and I is the moment of inertia of the cross section of thebolt. Depending on the friction coefficients under the bolt headand between the engaged threads, if motion between the threadshappens before the sliding under the bolt head, then the criticaldisplacement becomes:

crt =bFiL3

3EI xtt 10

where xtt is the displacement of the thread in the x direction attime t.

Equation 10 allows for three possible scenarios. In the firstscenario, the amplitude of the displacement cr, ocr. Thenthe bolt will be subjected to cyclic bending only. The correspond-ing shear force FsI is given by

F sIt =3EIt + xtt

L311

where xtt always opposes the direction of t. In the secondscenario, the displacement amplitude is greater than the criticaldisplacement cr and less than one half of the hole clearance .cro /2. This scenario may be divided into two phases. Inthe first phase, the bolt will bend until the displacement cr isreached; in the second phase, when tcr, the bolt head willslide until the displacement reaches its maximum amplitude o. Inthe first phase, the shear force is still given by Eq. 11; in thesecond phase, the shear force is given by

F sIIt =3EIcr

L3tt

12

where the term t / t is the unit vector in the direction of theshear force F s. In the third scenario, the displacement amplitudeo is bigger than the critical value cr and is also larger than halfof the hole clearance i.e., cr /2o. This scenario is di-vided into three phases. In the first phase, the bolt will bend untilthe displacement reaches cr and the corresponding shear force isstill given by Eq. 11. In the second phase, the bolt head willslide until the hole clearance is consumed, and the correspondingshear force is given by Eq. 12. In the third and final phase, theshank of the bolt makes contact with the side of the hole in theupper plate, and more bending is exerted on the bolt shank. Thecorresponding shear force FsIII becomes

F sIIIt =3EIt + xtt /2

L3

tt

13

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Thread-Loosening Torque. The loosening torque transmittedto the bolt head segment from the engaged threads is determinedfrom the free body diagram depicted in Fig. 1. The forces andmbtabttc

w

rplwobr

wab

f

wiu

etpe

fion

tsl

dR t = F u1,v1,w1 = T1T F + F s

n+ T1dFmt 19

J

Dooments acting on the threads include the bolt tension F, theending moment Mb, the shear force Fs, the contact force Fnnormal to the thread surface, and the thread friction force Ft. Thehread contact area is divided into a large number of small sectorss shown in Fig. 4. Although the bending moment exerted on theolt shank increases the contact pressure on one side of thehreads and reduces it on the opposite side, the pressure distribu-ion on each side of the threads is assumed to be uniform. Thehange in the axial force on each sector can be given by

Mb = 0

n

Prtd rt cos = 2

/2

/2

Prt2cosd

Mb = Prt2

/2

/2

cosd = 2Prt2

P = Mb2rt

2 P = Mb2rt

2

dFmt = Prtdk = Mb2rt

2rt2n k = Mbnrt k 14here P is the contact pressure on the thread, rt is the mean thread

adius, and dFmt is the change in bolt tension on each sector. Theositive sign corresponds to the change in the axial force on theeft half of the bolt when the moment is positive counter clock-ise and to the right half of the bolt when the moment is negative

clockwise, while the negative sign corresponds to the right halff the bolt when the moment is positive and to the left half of theolt when the moment is negative. Similarly, the change in theeaction force on each sector under the bolt head is given by

dFmb = 2Mbnrb

cosk 15

here rb is the mean bearing radius, which is determined moreccurately by Nassar et al. 10; however, an approximate meanearing radius is used in this study.

M b, rt, and rb are given by Eqs. 1618, respectively, asollows:

M b = Lw F s 16

rt =rmaj + rmin

217

rb =rmaj + + ro

218

here rmaj is the major bolt radius, rmin is the minor bolt radius, s the hole clearance, and ro is the outer radius of the contact areander the bolt head.

The bolt tension F and the shear force Fs are assumed to bevenly distributed on all sectors. The differential components ofhese two forces and the force generated by the bending are ex-ressed in the u1v1w1 coordinate system using Eqs. 26. Thequivalent force dR t on each sector has three components. Therst component is normal to the thread surface and is equal andpposite to the thread reaction force dF n. The other two compo-ents are in the thread sector plane and are equal and opposite tohe differential thread friction force dF t on the sector under con-ideration. These components are given by Eqs. 1921 as fol-ows:

ournal of Pressure Vessel Technologywnloaded 12 Dec 2011 to 195.83.11.66. Redistribution subject to ASMEdF n = dRw1 20

dF t1t = dR u1 + dR v1 21where dR u1, dR

v1

, dRw1 are the components of dR in the u1, v1,w1, respectively, and dF t1 is the thread friction force componentthat opposes the differential shear force dF s.

The maximum magnitude of the thread friction force is equal tothe normal force dF n times the coefficient of thread friction t

dFt,maxt = tdF nt 22

where t is the thread coefficient of friction. When the magnitudeof dR u1 +dR

v1

on one sector exceeds the maximum friction forcecalculated in 22, the sector becomes free to move. The magni-tude of the friction force on that sector is given by 22. On theother hand, if the magnitude dR u1 +dR

v1

is less than dFt,maxt,then part of the thread friction force dF t1 will resist the in-planeforces, as shown in Eq. 21. The sector still can offer some fric-tion resistance dFt2. The magnitude of the friction component isgiven by

dFt2 = dFt,max dFt1 23Figure 3 shows that the resultant force on sector A is greater

than the resultant force on sector B. As the displacement in-creases, the shear force on each sector increases and sector Abecomes free to move before sector B. In this case, the sector thathas the minimum resultant force will act as an instantaneous cen-ter pivot point about which the bolt can rotate within the threadand the hole clearances. This suggests that the motion in the xdirection is not pure sliding; it is more like rolling on one side ofthe bolt. For each sector, the vector rti points from the instanta-neous center of rotation to the center of the sector under consid-eration. The direction of motion for each sector is in the looseningdirection perpendicular to rti for the same sector. The instanta-neous center is found by locating the most constrained sector inthe thread area. The vector rti can be obtained by adding twovectors: the first one is the vector from the instantaneous center tothe center of the bolt, and the second is the vector from the centerof the bolt to the center of the ith sector. The resultant differentialforce dF threadt on one sector of the thread is expressed in theuvw coordinate system as follows:

dF threadt = T11dR t + dF t1t + dF t2t 24

since the w component is parallel to the bolt axis, which is parallelto the axis of rotation, then only the u and v components are usedto calculate the differential thread torque dT t of the resultantforce on each sector. This torque is given by

dT tt = rti dF thread u + dF thread v 25

The resultant force F threads on the entire thread area is calculatedby transferring the resultant force vectors on all sectors to theglobal XYZ-coordinate system, and adding them together as fol-lows:

F threadst = 1

n

T1dF threadt 26

The resultant torque on the threads is given by

NOVEMBER 2006, Vol. 128 / 593 license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Ttt = dTtt n dTtt 27emafnosdt

T

Wtfg

w

f

f

Tg

Tovccsa

Tl

Taor

Tfm

Tbt = dTbt n dTbt 36

5

Dothread 1

Underhead Friction Force and Torque Components. Differ-ntial underhead friction forces on each sector are used for deter-ining the underhead friction torque component. In-plane forces

cting on the bolt head include the shear force Fs, the bearingriction force Fb, and the torque Tt from the threads. It should beoted that the shear force acting on the bolt head is equal andpposite to the shear force acting on threads. The differentialhear force on each sector under the bolt head is obtained byividing the total shear force calculated by Eqs. 1113 by theotal number of sectors n as shown by Eq. 27

dF s = F sn

28

he magnitude of the maximum friction force is given by

dFb,maxt = bF tn

+ dFmbt 29hen the magnitude of the differential shear force dF s exceeds

he maximum value calculated by Eq. 29, the sector becomesree to slide. The magnitude of the friction force on that sector isiven by

dF bt = bF tn

+ dFmbt dF sdF s

30here the term dF s / dF s defines the unit vector in the friction

orce direction dF bt.On the other hand, if dF st dF b,maxt, then part of the

riction force dF b1 will resist the shear force; this part is given by

dF b1 = dF s 31

he sector still provides some frictional resistance dFb2, which isiven by

dFb2 = dFb,max dFs 32

he vector from the instantaneous center of rotation to the centerf the ith sector is rbi. The vector rbi is obtained by adding twoectors; namely, the vector from the instantaneous center to theenter of the bolt and the vector from the center of the bolt to theenter of the ith sector. The resultant differential force on oneector under the bolt head is given in the uvw-coordinate systems follows

dF bt = TdF st + dF b1t + dF b2t 33

he differential underhead torque dTb is determined by the fol-owing cross product:

dTbt = rbi dF bt 34he resultant force on the bolt head is equal to the integration ofll differential resultant forces on all sectors under the bolt head,r it can be approximated by the summation of all differentialesultants as follows:

F b =

underhead

dF b 1

n

dF b 35

he total underhead torque is equal to the integration of all dif-erential torques on the bearing surface, and it can be approxi-ated by the summation of all differential torques as follows:

94 / Vol. 128, NOVEMBER 2006wnloaded 12 Dec 2011 to 195.83.11.66. Redistribution subject to ASMEunderhead 1

The total force F bolt acting on the bolt is given by

F bolt = F threads + F b 37Finally, the total torque T acting on the bolt is determined by

T = T t + Tb 38Equations 838 provide the force and torque components

that are necessary for writing the linear and angular equations ofmotion of the bolt under the transverse cyclic displacement exci-tation t.

Dynamic Analysis of Self-Loosening. The linear and rotationalequations of motion of the bolt are given, respectively, by

Ft = mxt 39

Tt = Ip t 40where m is the effective mass of the bolt, Ip is the effective polarmoment of inertia of the bolt, x is the linear acceleration of thebolt head, and is the angular acceleration. The effective mass mof the bolt is equal to 33/140 of the shank mass m1 plus the fullmass of the bolt head m2 11, as expressed in

m =33

140m1 + m2 41

The polar moment of inertia Ip about the bolt axis is given by

Ip =12

m1r12 +

12

m2r22 42

where r1, r2 are the shank and bolt head radii, respectively. Thepolar moment of inertia about the instantaneous center of rotationduring the loosening process is given by

Ipc = 12m1r12 + 12m2r22 + mr12 43In this model, the friction and material damping effects are ne-glected due to the anticipated slow rate of motion. Additionally,the force and torque represented by Eqs. 39 and 40 are as-sumed to be constant for a differential interval of time dt, whichpermits the use of the following kinematic equations:

xt + dt = xdt + xt 44

xt + dt = 0.5xt + dtdt2 + xtdt + xt 45

t + dt = dt + t 46

= t + dt t = 0.5 t + dtdt2 + tdt 47As the bolt rolls, it moves in the transverse direction and themaximum rolling angle before having side contact at the threads isaffected by the thread clearance. This angle can be given by

max =t

rt48

where t is the thread clearance. The loosening angle of thebolt head reduces the bolt tension by F. The relationship be-tween F and is given by 12 as follows:

F = kbkcp2kb + kc = 49where kb is the bolt stiffness, kc is the joint stiffness, p is thethread pitch, and is the slope of the bolt tension-rotation curve.

Transactions of the ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

TbEf

weEouc

smotdttstptpaot

E

dmuimejtma

J

Dohe slope may be determined either experimentally from theolt tension-rotation curve shown in Fig. 6, or theoretically fromq. 49. The bolt and joint stiffnesses are given by Ref. 13 as

ollows:

kb =AbEb

L50

kc =AcEc

L51

here Ab is the bolt cross-sectional area, Eb is the modulus oflasticity of the bolt, Ac is the effective joint cross-sectional area,c is the modulus of elasticity of the joint, and L is the grip lengthf the bolt. The incremental reduction in the bolt tension is used topdate the force and torque values and start the next iteration ofalculations until the desired number of cycles is reached.

MATLAB Simulation Code. To obtain the reduction in bolt ten-ion, a MATLAB code is written to numerically calculate the incre-ental angle of rotation and convert it to an incremental loss

f bolt tension, due to self-loosening caused by the cyclic excita-ion. Figure 7 shows a flow chart of the MATLAB algorithm used toetermine the tension drop in the bolt. The input information tohe code include, bolt dimensions, external load information, fric-ion coefficients between contact surfaces, bolt stiffness, jointtiffness, hole clearance, initial bolt tension, time increment, andhe desired number of cycles, or until the fastener becomes com-letely loose. The MATLAB steps include iterative calculations ofhe bolt forces and torques using the equations derived in therevious sections and the calculation of linear and angular boltccelerations, velocities, and displacements. The updated valuesf the bolt tension are calculated from the displacement informa-ion.

xperimental VerificationAn experimental procedure and test setup are developed in or-

er to validate the results obtained from the new mathematicalodel that simulates the self-loosening of threaded fasteners. Fig-

re 8 shows a schematic of the self-loosening testing machine thats constructed in a similar way to the Junker machine 9. The

achine consists of a motor that drives a set of pulleys with anccentric mechanism that reciprocates the upper plate of a boltedoint in order to cause self-loosening. A load cell is used to moni-or the clamp load/bolt tension in real time. A socket cap is

ounted on the bolt head and connected to a rotation sensor usingsmall belt. The load cell and the rotation sensor are wired to a

Fig. 6 Effect of thread pitch on the self-loosening

ournal of Pressure Vessel Technologywnloaded 12 Dec 2011 to 195.83.11.66. Redistribution subject to ASMEdata-acquisition system that records the bolt tension and the smallhead rotation loosening during the test. The eccentric mecha-nism can be set to provide various levels of the transverse dis-placement amplitude to the upper plate during the self-looseningtest. Two displacement amplitudes are used in this study; namely,0.03 in. and 0.05 in. The bolts and the tapped hole are ultrasoni-cally cleaned and dried, then lubricated with SAE 5W-30 engineoil. The underhead friction coefficient b and the thread frictioncoefficient t are measured separately using FastLab ControlModule fastening system 14. The experimental values of b and

Fig. 7 Tension-rotation relationship for 1/2 in.-13 and1/2 in.-20 bolts

Fig. 8 Flow chart for the MATLAB algorithm

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o

R

slsvc

cttTtc

Twifw

5

Dot are input into the MATLAB code for the mathematical simulationf the self-loosening of threaded fasteners.

esults and AnalysisA typical self-loosening curve that shows the bolt tension ver-

us the number of transverse load cycles is depicted in Fig. 9. Theoosening curve can be approximated by a straight line, whoselope represents the rate of tension loss per cycle. The effect of thearious factors on the self-loosening of threaded fasteners is dis-ussed using the analytical and experimental data.

Effect of the Thread Pitch. Two values of thread pitch areonsidered in this study; the values correspond to fine and coursehreads: 1 /2 in.-20 and 1/2 in.-13 bolts, respectively. The pitch ofhe fine thread is 1 /20 in. and is 1 /13 in. for the coarse thread.he effect of thread pitch p may be pointed out in the model in

hree different places. First, changing the thread pitch meanshanging the threads lead angle , which is given by

= arctan pd 52

he angle is used in Eq. 5 to find the transformation matrix,hich is used in calculating the forces on the threads. This change

n has a significant effect on the magnitude of the looseningorce; this means that the pitch value affects the threshold point athich loosening begins, as shown in Fig. 10. Second, the thread

Fig. 9 Self-loosening test setup

Fig. 10 Typical self-loosening curve

96 / Vol. 128, NOVEMBER 2006wnloaded 12 Dec 2011 to 195.83.11.66. Redistribution subject to ASMEpitch p will affect the slope of the tension-rotation curve inaccordance with Eq. 47. Neglecting the effect of the fine versuscoarse thread pitch on the bolt stiffness, the slope of the tension-rotation is directly proportional to the pitch of the bolt. Hence, theslope for the 1/2 in.-13 bolt is almost 1.5 times the slope for the1/2 in.-20 bolt. This means that fine bolts need to rotate 1.5 timesthe coarse ones to loose the same amount of tension. The experi-mental data, shown in Fig. 6, is in close agreement on the slope ofthe tension-rotation curves for the fine and coarse threads in boththe tightening and loosening directions. Third, the slope of theloosening curve is also affected by the clearance between thethreads. On the one hand, coarse thread bolts normally have largerclearance between the threads; consequently, they have steeperloosening lines because the larger clearance allows motion for alonger period of time before the threads of the bolt make contactwith the side of the tapped hole. On the other hand, fine threadbolts normally have smaller thread clearances, that shortens thetime duration of the sliding motion between the threads; this slowsdown the loosening of fine thread bolts. Figure 11 shows that theexperimental and the analytical results are in good correlation forboth coarse and fine threads.

Effect of the Level of Initial Bolt Tension and ExcitationAmplitude. The correlation between the amplitude of the excita-tion and the threshold level of the initial bolt tension, below whichthe process of self-loosening begins, is inherent in the proposedmathematical model. The friction forces between the engagedthreads and under the bolt head are directly proportional to thebolt tension and to the friction coefficients. The expressions forthese forces are given by Eqs. 22 and 29, respectively. Addi-tionally the magnitude of the loosening force caused by thethreads lead angle is also proportional to both the bolt tension andthe lead angle, as shown by Eqs. 5, 19, and 21. The net effectof the forces given by Eqs. 21, 22, and 29 may be expressedin terms of two torque components, one in the loosening directionand an opposing component from the frictional resistance. Thedifference between the friction resistance and the loosening torquedetermines whether or not the bolt will start to rotate in the loos-ening direction. The friction resistance, under the bolt head andbetween the threads, is divided into two components; the firstcomponent opposes the motion in the transverse direction, whilethe second component opposes the angular motion. Prior to apply-ing the transverse displacement force, the first component iszero. Hence, the entire friction resistance would be only opposing

Fig. 11 Effect of thread pitch on the self-loosening

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ttctfcltl

fiterlpFtrps

number of cycles to complete loosening from 60 to 120 cycles.Additionally, it is observed that the 0.05 in. amplitude of the

transverse excitation is not sufficient to initiate any self-loosening

F

J

Dohe motion in the angular loosening direction. When the externalransverse displacement force is applied, the first component in-reases, while the second component decreases. At the same timehe loosening torque is increased by the increase in the normalorce on the threads. As the amplitude of the excitation is in-reased, the difference between the frictional resistance and theoosening torque becomes smaller and smaller, until the looseningorque overcomes the friction; then the bolt starts to rotate in theoosening direction.

The loosening threshold point is a function of the friction coef-cients between the threads and under the bolt head, initial bolt

ension, thread lead angle, and the amplitude of the transversexcitation. Increasing the bolt tension over the threshold point willequire larger excitation amplitude to initiate loosening. Onceoosening has started, the number of cycles required to cause com-lete loosening is directly proportional to the initial bolt tension.igures 12 and 13 show the experimental and analytical results for

he effect of initial bolt tension and the amplitude of excitation,espectively, on the self-loosening of 1/2 in.-20SAE grade 5lain bolt. Results show that the doubling of the initial bolt ten-ion from 2000 lb 8.9 kN to 4000 lb 17.8 kN, doubles the

Fig. 12 Effect of initial bolt tension on the self-loosening

ig. 13 Effect of the excitation amplitude on the self-loosening

ournal of Pressure Vessel Technologywnloaded 12 Dec 2011 to 195.83.11.66. Redistribution subject to ASMEwhen the initial bolt tension is increased to 9000 lb 40 kN,while it causes loosening when the initial bolt tension is reducedto 6000 lb 26.7 kN. Similarly, an excitation amplitude of0.03 in. is not sufficient to cause loosening when the initial ten-sion is 4000 lb 17.8 kN; it causes loosening at a lower level ofinitial bolt tension of 2000 lb 8.9 kN, however. In Fig. 13, thelower and upper horizontal lines represent bolt tensions that arehigher than the threshold points for the 0.03 and 0.05 in. ampli-tudes, respectively. They show that loosening has not occurredbecause the initial bolt tension has exceeded the threshold valuefor each of the two indicated excitation amplitudes. The resultsfrom the mathematical model correlate well with the experimentaldata.

ConclusionA new mathematical model is introduced and experimentally

verified in order to study the effect of thread pitch and bolt tensionon the self-loosening phenomenon in threaded fasteners under cy-clic loads and/or displacements. A cyclic transverse excitationmay or may not initiate the self-loosening of a tightened threadedfastener, depending on the amplitude of the excitation in relationto the level of bolt tension. For a known amplitude of the cyclicexcitation, the bolt tension will have to be below a threshold valuein order for the self-loosening to start. Once the loosening isstarted, the rate of clamp load loss in a bolted joint due to self-loosening is proportional to the thread pitch. The use of finethreads reduces the loosening rate per cycle, whereas coarsethreads will loosen at a higher rate. The threshold level of bolttension, at which the loosening begins under known excitationamplitude, is higher for fine threads as compared to coarsethreads. For both coarse and fine threads, it has been found thatonce the bolt loosening has started, the number of cycles untilcomplete loosening is reached is directly proportional to the ini-tial bolt tension. Finally, the study introduces the correlation be-tween the excitation amplitude and the required threshold value ofbolt tension above which self-loosening may be prevented. Thecorrelation concludes that bolts that are subjected to a higher levelof excitation amplitude will have to be tightened to higher pre-loads, in order to avoid self-loosening.

NomenclatureE bolt modulus of elasticityI moment of inertia of the bolt cross section

F bolt tensionL grip lengthR resultant force opposing the thread friction

forceFb bearing friction force

Fb,max maximum bearing friction forceFex external excitation forceFf friction force between the joint plates

Fhead resultant force opposing the bearing frictionforce

Mb bolt bending momentMb,max maximum bolt bending moment

Fi initial bolt tensionFl loosening force on the bolt headFn normal contact force on the threadsFs bolt shear forceFt resultant thread friction forceKb bolt stiffnessKc joint stiffnessTb bearing friction torque componentTl loosening torqueTp pitch torque component

NOVEMBER 2006, Vol. 128 / 597 license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Tt thread friction torque componentdFb differential bearing friction force

dFhead differential force opposing the bearing frictionforce

dFn differential thread contact forcedFl differential loosening forcedFs differential bolt sheardFt differential thread friction force

dFmb change in the underhead force due to bendingdFmt change in the thread force due to bending

dR subtotal differential force on one thread sectordTb differential bearing friction torquedTt differential thread friction torquedt incremental timef frequency of the external excitation

m mass of the boltm1 mass of the bolt shankm2 mass of the bolt headrb mean bearing contact radiusp thread pitchP contact pressure

rmaj major thread radiusrmin minor thread radius

ro outer bearing radiusrt mean thread radiust timex linear displacementxt threads linear displacementx linear velocityx linear acceleration

2 thread profile angle lead helix angle

b static bearing friction coefficientt static thread friction coefficient

angular position

angular velocity

angular acceleration angular frequency of the excitation

References1 Hess, D., 1998, Vibration and Shock Induced Loosening, Handbook of Bolts

and Bolted Joints, J. H. Bickford, S. Nassar, eds., Marcel Dekker, New York,pp. 757824.

2 Sakai, T., 1978, Investigations of Bolt Loosening Mechanisms, 1st Report,On Bolts of Transversely Loaded Joints, Bull. JSME, 21, pp. 13851390.

3 Haviland, G. S., 1983, Designing With Threaded Fasteners, Mech. Eng.Am. Soc. Mech. Eng., 105, pp. 1731.

4 Yamamoto, A., and Kasei, S., 1984, A Solution for the Self-LooseningMechanism of Threaded Fasteners Under Transverse Vibration, Bull. JSME,18, pp. 261266.

5 Tanaka, M., Hongo, K., and Asaba, E., 1982, Finite Element Analysis of theThreaded Connections Subjected to External Loads, Bull. JSME, 25, pp.291298.

6 Vinogradov, O., and Haung, X., 1989, On A High Frequency Mechanism ofSelf-Loosening of Fasteners, Proceedings of 12th ASME Conference on Me-chanical Vibration and Noise, Montreal, ASME, New York, pp. 131137.

7 Zadoks, R. I., and Yu, X., 1993, A Preliminary Study of the Self-Loosening inBolted Connections, ASME Nonlinear Vibrations, ASME Vol. DE-Vol. 54,7988.

8 Zadoks, R. I., and Yu, X., 1997, An Investigation of the Self-LooseningBehavior of Bolts Under Transverse Vibrations, J. Sound Vib., 208, pp. 189209.

9 Junker, G. H., 1969, New Criteria for Self-Loosening of Fasteners UnderVibration, SAE Trans., 78, pp. 314335.

10 Nassar, S. A., El-Khiamy, H., Barber, G. C., Zou, Q., and Sun, T. S., 2005,An Experimental Study of Bearing and Thread Friction in Fasteners, ASMEJ. Tribol., 127, pp. 263272.

11 Steidel, R. F., 1989, An Introduction to Mechanical Vibrations, 3rd ed., Wiley,

5

Do hole clearancet thread clearance external displacement0 external displacement amplitude sector angle in Fig. 3 slope of the bolt tension-rotation relationship

98 / Vol. 128, NOVEMBER 2006wnloaded 12 Dec 2011 to 195.83.11.66. Redistribution subject to ASMENew York.12 Bickford, J. H., 1997, An Introduction to the Design and Analysis of Bolted

Joints, 3rd ed., Marcel Dekker, New York.13 Juvinall, R. C., and Marshek, K. M., 2000, Fundamentals of Machine Compo-

nent Design, 3rd ed., Wiley, New York.14 RS Technologies, 2004, Threaded Fastener Testing System User Manual, RS

Technologies, Farmington Hills, MI.

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