zaki_effect of conical angle and thread pitch on the self-loosening performance of preloaded...

Amro M. Zaki

Sayed A. Nassar

Xianjie Yang

Department of Mechanical Engineering,

Fastening and Joining Research Institute,

Oakland University,

Rochester, MI 48309

Effect of Conical Angle andThread Pitch on theSelf-Loosening Performance ofPreloaded Countersunk-HeadBoltsThis paper investigates the effect of the countersunk conical angle and thread pitch onthe loosening performance of preloaded countersunk-head bolts that are subjected toharmonic transverse loading. A nonlinear mathematical model is used to predict the loos-ening performance. Cumulative differential loosening rotation of the bolt head is con-verted to a gradual loss in the bolt tension and joint clamp load. Model prediction of theself-loosening behavior is experimentally validated. [DOI: 10.1115/1.4005058]

Keywords: self-loosening, countersunk fasteners, conical angle, thread pitch

1 Introduction and Literature Survey

In many applications, it is critically important that the initiallevel of fastener preload (clamp force) be sufficient for adequateclamping of the joint; equally important is controlling the clampload decay over the life of the joint. In many cases, the clamp loaddecay is caused by gradual self-loosening under dynamic serviceloads.Junker [1] investigated the self-loosening behavior of threaded

fasteners subjected to cyclic transverse excitation and concludedthat the loosening is more severe when the joint is subjected todynamic loads perpendicular to the thread axis (shear loading).Tanaka et al. [2] later stated that in modeling the self-looseningbehavior, the eccentric load distribution on the threads when thejoint is subjected to shear loads should be taken into account.Their simulation results showed that the loosening starts with aslip between the threads of the bolt and the nut, followed by a slipat the bearing surface. Pai and Hess [3] did an experimental inves-tigation in which they concluded that it is not necessary to have acomplete slip between the contact surfaces in order to start theself-loosening process. The slip starts at some regions and buildsup until a complete slip takes place. The model presented byZhang et al. [4] suggests that the micro-slip between the contact-ing threads is initiated due to the variation in contact pressurebetween the threads caused by the uneven bending moment.Nassar and Housari [57] presented a mathematical model for

predicting the self-loosening behavior of hex head fasteners. Themodel is based on the fact that when the bolt is subjected to abending moment, the bolt pivots on one side, and the frictionalforces on the opposite side is reduced. More recently, Nassar andYang [8] and Yang and Nassar [9] developed a nonlinear mathe-matical model that took into account the 3D geometry of thethreads and various pressure distribution scenarios on the contactsurfaces under the bolt head and between engaged threads. Thiswork extends the work done by Yang and Nassar [9] and Zaki etal. [10,11] to investigate the effect of the conical bolt head angleand thread pitch on the loosening performance of fasteners with acountersunk head configuration.

2 Formulation

Figure 1 shows a schematic of the bolted joint model developedin Refs. [10,11], where the countersunk bolt threads are engagedinto a tapped hole in the bottom block. The upper block movestransversely subjecting the bolt to cyclic shear loading while thebottom plate is constrained. The model predicts the loosening rateduring the stage of self-loosening characterized by significantbacking-off of the nut/bolt and rapid loss of clamping force. Therolling friction between the clamped plates is negligible as theproposed model describes the post-state of friction loss where thefriction between the clamped parts deteriorated to a negligiblevalue. The transverse load is carried entirely by the bolt and thebolt shank does not contact the hole wall. The model is based onthe force and moment as well as the kinematic relationships dur-ing the transverse motion of the upper plate. The angular equationof motion is numerically integrated with a sufficiently small timestep to provide the incremental loosening rotation of the bolt,which is in turn correlated to the incremental drop in the bolttension using a MATLAB code.The self-loosening model takes into account some simplifying

assumptions. First, it is assumed that the contact area under thebolt head is small compared to the conical surface of the counter-sunk head of the bolt. This is justified by the existence of manu-facturing tolerances that permits 62 deg variation in the bolt headcone angle [12]. Second, it is assumed that the bolt bending due tothe transverse excitation will cause the contact pressures betweenthe conical surfaces to be concentrated on two small areas,namely, one small axisymetric angular area at the upper and oneat the bottom portions of the conical bolt head. This assumption isbased on the fact that those two contact areas had been visuallynoticed during lab tests and appeared to have been subjected tohigh contact stresses when sliding occurs between the conicalcontact surfaces. This assumption was also validated using finiteelement analysis (FEA) of the model taking into account the man-ufacturing tolerances.

2.1 Condition for Self-Loosening. At the start of looseningprocess, the torque components acting on the fastener are brokendown into two components. One component trying to loosen thebolt; namely, the pitch torque component Tp, while the threadfriction torque component Tt and the bearing friction torque

Contributed by the Pressure Vessel and Piping Division of ASME for publicationin the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received May 13, 2011;final manuscript received August 15, 2011; published online January 25, 2012.Assoc. Editor: Maher Y. A. Younan.

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component Tb are maintaining the bolt from loosening. Therelationship among the three torque components at the start ofloosening is given by

Jp _x Tp Tb Tt

(1)

where Jp is the polar moment of inertia of the bolt cross section,and _x is the loosening angular acceleration of the bolt. A suffi-ciently large transverse shear force will lead to a decrease in thebearing and thread friction torque components Tb and Tt,respectively.

2.2 Countersunk-Head Bearing Friction Analysis. Figure 2shows a schematic of the local Cartesian coordinate system usedin the analysis of the relative velocity of any point Q on the frus-tum surface of the bolt under head with respect to the referencepoint O. The velocity vector~v of point Q is given by

~v ~vOQ xbk^ ~a x1 j^ a* (2)

where ~vOQ is the relative velocity of point Q with respect to thereference point O, x1 is the angular speed of the bolt head (aboutthe y-axis) due to bolt head bending, xb is the angular speed ofthe bolt head (about the z-axis) due to bolt rotation in the loosen-ing direction, and ~a is the vector connecting the reference point Oto point Q.Figure 3 shows the kinematic analysis of the sliding motion.

When the countersunk bolt is subjected to transverse excitation,the bolt head might slide upward against the joint conical holecausing an increase in bolt tension due to the shift in position ofpoint O to point O0.Using the instantaneous center I.C. approach, the velocity vec-

tor~v of any point Q on the conical bolt head surface is given by

~v x1LB cosu xbrx sin h x1 rx ri tanu i^ xbrx cos h j^ x1 LB sinu ri rx cos h k^

(3)

where u is the complementary cone angle (Fig. 2), rx is a variablethat can stand for ri or ro depending on the analysis location, andLB is the distances from the instantaneous center I.C. to point B.Using the coordinates of points A, A0, B, and B0 (Fig. 3), the dis-placements of points A and B due to bolt head sliding are given byDxA and DxB, as follows:

DxA 1m1 cos w/

ri re 2h2

q re ri

DxB

m(4)

where / is the bending angle of the bolt head about y-axis (Fig.3), h is the height of the bolt head, m is a constant defined as theratio of the change DxB=DxA, and w is a constant angle from thebolt head geometry. The displacement of point O in x-direction isgiven by DxO as follows:

DxO DxA m riri re m 1

(5)

Figure 4 shows a free body diagram of the bolt head. The normalforces on the conical bearing surface are simplified by two normalforces F1 and F2 acting at points B and A, respectively. The pres-sure distribution along the peripheral of the bolt head is assumedto vary sinusoidally with the angular location h on small area at Aand B reaching a maximum pressure at the line of action of thetransverse excitation.The normal force vector ~F1 acting at B is obtained by integrat-

ing the normal pressure q1 on the surface S shown with the shadedarea in Fig. 4 as follows:

Fig. 2 Velocity vector for an arbitrary point Q on the bolt underhead

Fig. 3 Schematic of bolt head sliding and instantaneouscenter analysis

Fig. 4 Schematic of bolt head free body diagram and contactpressure distribution

Fig. 1 Model schematic

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~F1 riDhcosu

3p2

p2

q1 cos hj jn^dh q1Dhri p2tanui^ 2 cosuk^

h i(6)

where n^ is the unit normal vector acting perpendicular to the bolthead surface (Fig. 4), and Dh is the vertical height of the smallarea dS upon which the pressure q is acting.Following the same procedure, the normal force vector ~F2 act-

ing at A is given by

~F2 reDhcosu

p2

p2

q2 cos hj jn^dh q2Dhre p2

tanui^ 2 cosuk^h i

(7)

Since the frictional forces are always opposing the sliding tendency,the direction of the frictional forces ~Fbf1 and ~Fbf2 would be oppositeto the velocity vectors of points B and A, respectively. Substitutingthe corresponding radii in Eq. (3) and defining the ratio gb as thetranslational-to-rotational speed ratio of the bolt head (gb vO=xb),the velocity vectors~vbf1 and~vbf2 are given as follows:

~vbf1 gbLBLO

cosu ri sin h

i^ ri cos h j^

gbLO

LB sinu ri ri cos h

k^ (8)

~vbf2 gbLBLO

cosu re sin h gbLO

re ri tanu

i^

re cos h j^ gbLO

re cos h LB sinu ri

k^ (9)

The frictional forces ~Fbf1 and ~Fbf2 are obtained by integrating thefrictional force vectors on their corresponding contact surface(Fig. 4) as follows:

~Fbf1 lb3p=2p=2

qh1~vbf1

~vbf1 dS

lbq13p=2p=2

Fbf1xi^ Fbf1y j^ Fbf1zk^

dh (10a)

~Fbf2 lbp=2p=2

qh2~vbf2

~vbf2 dS

lbq2p=2p=2

Fbf2xi^ Fbf2y j^ Fbf2zk^

dh (10b)

where Fbf1x, Fbf1y, and Fbf1z are, respectively, the x, y, and z com-ponents of the force ~Fbf1 divided by q1lb. Similarly Fbf2x, Fbf2y,and Fbf2z are, respectively, the x, y, and z components of the force~Fbf2 divided by q2lb.From the free body diagram of the fastener in Fig. 4, the fric-

tional shear force Fbs under the bolt head can be obtained fromthe equilibrium condition as follows:

Fbs q1 Dhri p2tanu

lb

3p=2p=2

Fbf1xdh

" #

q2 Dhre p2

tanu

lbp=2p=2

Fbf2xdh

" #(11)

The bearing friction torque Tb is obtained by integrating the crossproduct of the forces ~Fbf1, ~F1, ~Fbf2, and ~F2 with their respectivemoment arm vector as follows:

Tb ~ae q2 lbp=2p=2

~Fbf2dhp=2p=2

~Fdh2

!" # k^

~ai q1 lb3p=2p=2

~Fbf1dh3p=2p=2

~F1dh

!" # k^ (12)

where ~ae and ~ai are the radial vectors from the reference point Oto the point of action of the forces ~F2 and ~F1, respectively.Numerical integration of Eqs. (11) and (12) gives the bearing

frictional shear force Fbs and the bearing frictional torque Tb,respectively. Figure 5 shows Tb and Fbs verses the speed ratio gb.It is observed that when there is no relative rotational movementbetween the bolt head and the joint g!1 , the bearing fric-tional torque Tb would be equal to zero, while the shear frictionalforce Fbs would quickly asymptote to a constant (critical) value.On the other hand, when there is no relative translational speedgb 0 , the bearing frictional torque Tb will be maximum(resisting the relative movement), while the shear frictional forceFbs would be equal to the under head normal forces projected inx-direction.

2.3 Thread Friction Analysis. Nassar and Yang [8] pro-posed a 3D model of the threads relative motion under transverseexcitation. The thread friction shear force Fts was given asfollows:

Fts ltqt0sec2 a tan2 b

p rmajrmin

rdr

2p0

gt r sin h g2t 1 tan2 a cos2 h r2 2gtr sin h

p dh (13)

where lt is the thread friction coefficient, qt0 is the average thread contact pressure, a is half the thread profile angle, b is the helix angle,rmin is the minimum thread contact radius, rmaj is the maximum thread contact radius, and gt is the translational-to-rotational speed ratioof the thread surface (gt vtx=xt).The thread friction torque Tt was given as follows:

Tt Xt

ltqtosec2 a tan2 b

p rmajrmin

r2dr

2p0

r gt sin h g2t 1 tan2 a cos2 h r2 2gtr sin h

p dh (14)

Figure 6 shows the thread friction shear force Fts and the threadfriction torque Tt as a function of the speed ratio gt, obtained

through the numerical integration of Eqs. (13) and (14),

respectively.

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2.4 Shear Force and Bending Moment due to TransverseExcitation. The external transverse cyclic excitation subjects thebolt to elastic deformation due to bending and shear loading. Thebending deformation is simplified assuming that the bolt behavesas a cantilever beam subjected to a transverse shear force Fex anda bending moment M0b, as shown in Fig. 7. Since the displacementof point O in x-direction DxO is opposite to the direction of exter-nal excitation, the total deflection dT of the bolt head is given by

dT FbsL3

3EIMbL

2

2EI DxO (15)

where Fbs is the bearing frictional shear force of the conical bolthead opposing the external shear force Fex, Mb is the reactionbending moment of the bolt opposing the external bendingmoment M0b due to the joint upper plate sliding, and L is the boltgrip length as shown in Fig. 7.The bending moment M0b of the bolt head depends on the bend-

ing stiffness of the bolt head, the joint bending stiffness, and con-tact area. A linear relationship is assumed between the bendingmoment M0b and the head rotation angle / as follows:

M0b k/ (16)

where k is the bending stiffness of the bolt head. In this study, thebending stiffness k is obtained using finite element analysis, usingthe tolerance outlined in Ref. [12]. Substituting Eqs. (5) and (16)into Eq. (15), the bearing frictional shear force Fbs is given asfollows:

Fbs 3EIL3

dT k/L2

2EI DxA m ri

ri re m 1

(17)

From static equilibrium, the shear force acting on the thread sur-face Fts is related to the bearing shear force under the bolt headFbs as follows:

Fts Fbs Mt tan armin c4t 1 2 c3t 1

13rmaj 0:2725p

(18)

where ct is the ratio of the major-to-minor thread radii, and Mt isthe bending moment acting on the threads given by Ref. [8] asfollows:

Mt q0t

rmaj

p6

r4maj r4min

19prmaj 0:2854p

r3maj r3min

(19)

where q0t is the amplitude increase in thread contact pressure dueto bolt bending.The speed ratio gb can be obtained through equating the fric-

tional shear force Fbs obtained from Eqs. (11) and (17). The corre-sponding bearing frictional torque Tb for the same speed ratio gbis then calculated using Eq. (12). A similar procedure is followedfor calculating the thread frictional torque Tt component. Figure 8shows the various torque components during one full excitationcycle. When the pitch torque Tp is larger than the sum of the bear-ing and thread friction torques, there will be a net loosening tor-que component according to Eq. (1).

Fig. 6 Thread friction torque Tt and thread shear force Fts ver-sus the translational-to-rotational speed ratio gt

Fig. 7 Bolt bending due to external transverse excitation

Fig. 8 Illustration of the cyclic torque components and shearforce

Fig. 5 Bearing friction torque Tb and bearing shear force Fbsversus the translational-to-rotational speed ratio gb

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3 Experimental Procedure and Test Setup

In this section, an experimental procedure and test setup areestablished in order to validate the analytical prediction of theeffect of thread pitch and conical countersunk angle on the loosen-ing performance. Fasteners used in this study are 1=2 in.20 2.5in. and 1=2 in.13 2.5 in. SAE Grade 8 countersunk fastenerswith two conical head angles 82 deg and 100 deg. Test boltsare uncoated cold rolled fasteners having an average Rockwellhardness (RHC) of 42.6, purchased from the same commercialsupplier. The washers used are conical washers that are heattreated and hardened to a RHC of 65 to make it harder than themating fastener. The average conical angle of the washers was82:4 deg6 0:6 deg and 100:3 deg6 0:8 deg, corresponding tonominal conical angles 82 deg and 100 deg, respectively. Sincefasteners were obtained from the same commercial supplier, itwas presumed that they conformed to applicable standards for tol-erances [12].A self-loosening testing machine that is similar to the Junker

machine [1] is used for model verification. The machine consistsof a motor that drives a set of pulleys with an eccentric mecha-nism that reciprocates the upper plate shown in the bolted jointschematic (Fig. 1) in order to cause self-loosening. The tests aredisplacement controlled to apply a predetermined transverse exci-tation of amplitude 0.7mm and frequency of 10Hz. An embeddedload cell is used to monitor the loss in joint clamp load/bolt ten-sion (due to loosening) in real time during the test. Tightening iscontrolled by achieving a predetermined bolt preload of 20 kN,corresponding to 26.4% and 23.4% of the fastener proof load forcoarse and fine threaded fasteners, respectively. Seven tests wereconducted for each set of variables. The average self-looseningcurve for each variable combination is compared with the dataobtained from the analytical model. Figure 9 shows a sample of

the self-loosening testing conducted for 1=2 in.13 2.5 in. Flat82 deg fasteners.Thread and bearing friction coefficients are experimentally

obtained from separate torque-tension tests using a fastener testingsystem following the test procedure outlined in Ref. [11]. Themean values of the thread and bearing friction coefficients of fivetest samples for each variable combination are shown in Table 1.The friction coefficients are then input into the MATLAB code thatprovides the model prediction of self-loosening.Ultrasonic cleaning is performed on all parts prior to testing.

All test bolts, nuts, and washers were ultrasonically cleaned priorto testing. Cleaning continued for 5min before the parts wereremoved and allowed to dry at room temperature. New fastenersand washers are tested each time for both the fastener testing sys-tem and the self-loosening testing system. After drying, the fasten-ers were dipped in oil to allow for a consistent frictional behavior.Lower plate tapped hole fixture was cleaned every five tests to getrid of excess oil.

4 Results and Discussion

Analytical and experimental results are presented in this sec-tion. Two variables are investigated for their effect on the self-loosening rate, namely, the thread pitch p and the conical counter-sunk angle u. Model prediction of the fastener loosening behavioris experimentally validated.Table 2 shows the average loosening rate per cycle for 1=2 in.-13

Flat 82 deg, 1=2 in.-20 Flat 82 deg, and 1=2 in.-13 Flat 100 deg coun-tersunk fasteners tightened to initial preload of 20 kN. The loosen-ing rate is calculated assuming a linear loosening rate in the range20kN F 5kN, as the self-loosening curve becomes nonlinearbelow 5 kN.Figures 10 and 11 show the average experimental loosening

rate for coarse 1=2-13 Flat 82 deg and fine 1=2-20 Flat 82 deg fasten-ers, respectively, versus the analytical results obtained from theself-loosening model. The drop in bolt preload for fine threadedfasteners at the first few loading cycles is attributed to embedmentas well as alignment errors and clearance stack in the fine threadedfixture that engage with the test bolts. The model predicts theloosening rate to be 230.77 N/cycle and 43.73 N/cycle for coarse

Fig. 9 Sample self-loosening tests for coarse threads fasten-ers (82deg conical angle)

Table 1 Experimental friction coefficient data for countersunkbolts

Fastener typeThread frictioncoefficient lt

Bearing frictioncoefficient lb

1=2 in.-13 Countersunkplain Flat 82 deg

0.166 0.242

1-r Scatter 9% 6%1=2 in.-20 Countersunk

plain Flat 82 deg0.16 0.296

1-r Scatter 2.8% 4.2%1=2 in.-13 Countersunkplain Flat 100 deg

0.27 0.35

1-r Scatter 40% 18.8%

Table 2 Experimental self-loosening results of investigatedfasteners

Fastener

1=2 in.-13 2.5 in.Flat 82 deg

1=2 in.-20 2.5 in.Flat 82 deg

1=2 in.-13 2.5 in.Flat 100 deg

Self-looseningrate (N/cycle)

267.87 38.18 159.57

1-r Scatter 5.7% 19.8% 19.6%

Fig. 10 Self-loosening results for coarse threads fasteners(82deg conical angle)

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and fine fasteners, respectively. The percent difference in self-loosening rate between the analytical and experimental modelresults are 16% and 12.7% for fine and coarse fasteners,respectively.Figure 12 shows the average experimental loosening rate for

1=2-13 Flat 100 deg fasteners verses the analytical results. Theaverage experimental loosening rate is 159.57 N/cycle, model pre-dicts the loosening rate to be 163.04 N/cycle. The percent differ-ence, in self-loosening rate, between the analytical andexperimental model results is 2.12%.

4.1 Effect of Fastener Thread Pitch. Figure 13 shows theaverage loosening rates obtained from experimental testing for

fine 1=2 in.-20 and coarse 1=2 in.-13 Flat 82 deg countersunk fasten-ers. It is observed that the loosening rate for fine threaded fasten-ers is significantly lower than that for coarse threaded fasteners.The decrease in thread pitch p from 1.95mm to 1.27mm causedthe self-loosening rate to decrease from 267.87 N/cycle to 38.18N/cycle (85.7% reduction). It is concluded that the thread pitchhas a very significant effect on the self-loosening behavior asreported by Housari and Nassar [7].

4.2 Effect of the Countersunk Conical Angle u. The coni-cal complementary angle u of countersunk fasteners under head,shown in Fig. 1, is an important variable affecting the fastenerself-loosening behavior. Commercially available countersunk fas-teners come in two different conical angles, namely, 82 deg and100 deg, equivalent to u 49 deg and u 40 deg, respectively.Figure 14 shows the model prediction of the effect of the coni-

cal angle u on the self-loosening behavior of 1=2 in.-13 counter-sunk fasteners tightened to initial preload of 17.5 kN. It isobserved that the angle u increase causes an increase in the fas-tener self-loosening resistance. The increase of the conical angleu from 40 deg to 49 deg caused the self-loosening rate to decreasefrom 236.4 N/cycle to 187.6 N/cycle (20.6% reduction).Figure 15 shows the effect of the conical angle u on the analyti-

cal bearing torque component Tb given by Eq. (12), which resiststhe bolt rotation in the loosening direction. It is observed thatincreasing the conical angle u caused an exponential increase inthe frictional bearing torque Tb, which is interpreted as moreresistance to fastener loosening. The increase in fastener

Fig. 11 Self-loosening results for fine threads fasteners(82deg conical angle)

Fig. 12 Self-loosening results for coarse threads fasteners(100deg conical angle)

Fig. 13 Effect of thread pitch p on the self-loosening behavior(82deg conical angle)

Fig. 14 Effect of under head conical angle u on the self-looseningbehavior for fasteners tightened to initial preload of 17:5 kN

Fig. 15 Effect of under head conical angle u on the frictionalbearing torque component Tb

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loosening resistance is also attributed to the increase in the boltshear resistance component created by the projected contact pres-sure along x-axis. It is concluded that the fastener conical angle uhas a significant effect on the fastener loosening behavior.

5 Conclusion

The effect of conical angle u and the thread pitch p on thevibration-induced loosening of preloaded countersunk-head boltsis investigated using a nonlinear mathematical model. The pro-posed model correctly predicts a higher resistance to loosening forfine threads as compared to coarse threads. Increasing the conicalangle u enhances the loosening resistance significantly, as theresisting bearing friction torque component Tb increases exponen-tially with the conical angle u. Model results are in good correla-tion with the experimental data on the loosening behavior ofcountersunk-head fasteners.

Acknowledgment

The authors would like to acknowledge the US Army TARDECfor their support as part of Contract No. DAAE07-03-C-L110.Authors would also like to thank Mr. Zhijun Wu for helping withthe finite element simulations.

Nomenclature

E Youngs modulus of the boltF clamp force/bolt preloadF1 normal force acting at B, under the bolt headF2 normal force acting at A, under the bolt head

Fbf1 frictional force acting at B, under the bolt headFbf1x x-direction component of the vector ~Fbf1=lbq1Fbf1y y-direction component of the vector ~Fbf1=lbq1Fbf1z z-direction component of the vector ~Fbf1=lbq1Fbf2 frictional force acting at A, under the bolt headFbf2x x-direction component of the vector ~Fbf2=lbq2Fbf2y y-direction component of the vector ~Fbf2=lbq2Fbf2z z-direction component of the vector ~Fbf2=lbq2Fbs reaction frictional shear force under the bolt headFts thread frictional shear force acting along x-directionI moment of inertia about y-axis for the bolt cross section

Jp polar moment of inertia of the bolt cross section due totorsion

Kb bolt stiffnessLB distance from the instantaneous center I.C. to point BLO distance from the instantaneous center I.C. to point OLg effective bolt grip lengthM0b external bending moment acting on the bolt head due to

transverse excitationMt bending moment acting on the thread surfaceMb reaction bending moment of the bolt headTb bearing friction torque componentTp pitch torque componentTt thread friction torque component~a vector connecting any point Q to the reference point O

can stand for~ae or~ai~ae vector connecting any point Q on the upper part of the

bolt head to the reference point O shown in Fig. 2~ai vector connecting any point Q on the lower part of the

bolt head to the reference point O shown in Fig. 2h height of the countersunk fastener headm ratio of the change Dri=Dre due to bolt head slidingn^ unit vector perpendicular to the contact surface under the

bolt headp thread pitchq1 contact pressure acting on the lower portion of the bolt

head surfaceq2 contact pressure acting on the upper portion of the bolt

head surface

qto average thread pressureq0t increment pressure amplitude caused by the bending

effectqh contact pressure variation as a function of the radial loca-

tion hre outer contact radius under the bolt head (Fig. 1)ri inner contact radius under the bolt head (Fig. 1)

rmaj maximum thread contact radiusrmin minimum thread contact radiusrx variable radius of the bolt head can stand for ri or revtx velocity of any point on the thread surface along

x-direction with respect to the joint surface~v general velocity vector of any point under the bolt head

with respect to the joint surface~vOQ relative velocity of point Q with respect to reference point

ODh the height of a small area dS upon which the under head

pressure actsDxA displacement of point A in x-direction due to bolt head

slidingDxB displacement of point B in x-direction due to bolt head

slidingDxO displacement of point O in x-direction due to bolt head

slidinga half the thread profile angleb lead helix anglect ratio of major-to-minor thread radiidT transverse displacement of the joint upper plategb ratio of the translational-to-rotational velocity of the bolt

headgt ratio of the translational-to-rotational velocity of the

thread surfaceh angular locationk bending stiffness of the bolt headlb coefficient of bearing frictionlt coefficient of thread friction/ bending angle of the bolt head due to external excitationu half the bolt under head complementary cone angle,

shown in Fig. 1w a constant angle that depend on the bolt head geometryx1 relative angular velocity of the bolt under head surface

with respect to the joint bearing surface about y-axisxb relative angular velocity of the bolt under head surface

with respect to the joint bearing surface about z-axisxt relative angular velocity of the thread surface with respect

to the joint thread surface_x self-loosening angular acceleration of the bolt

References[1] Junker, G. H., 1969, New Criteria for Self-Loosening of Fasteners Under

Vibration, SAE Trans., 78, pp. 314335.[2] Tanaka, M., Hongo, K., and Asaba, E., 1982, Finite Element Analysis of the

Threaded Connections Subjected to External Loads, Bull. JSME, 25, pp.291298.

[3] Pai, N., and Hess, D., 2002, Experimental Study of Loosening of ThreadedFasteners Due to Dynamic Shear Loads, J. Sound Vib., 253(3), pp.585602.

[4] Zhang, M., Jiang, Y., and Lee, C. H., 2004, Finite Element Modeling of Self-Loosening of Bolted Joints, ASME Publications PVP, PVP2004-2618, Vol.478, pp. 1927.

[5] Nassar, S. A., and Housari, B. A., 2006, Effect of Thread Pitch on the Self-Loosening of Threaded Fasteners Due to Cyclic Transverse Loads, ASME J.Pressure Vessel Technol., 128(4), pp. 590598.

[6] Nassar, S. A., and Housari, B. A., 2007, Study of the Effect of Hole Clearanceand Thread Fit on the Self-Loosening of Threaded Fasteners, ASME J. Mech.Des., 129(6), pp. 586594.

[7] Housari, B. A., and Nassar, S. A., 2007, Effect of Thread and Bearing FrictionCoefficients on the Vibration-Induced loosening of Threaded Fasteners,ASME J. Vib. Acoust., 129(4), pp. 484494.

[8] Nassar, S. A., and Yang, X., 2009, A Mathematical Model for Vibration-Induced Loosening of Preloaded Threaded Fasteners, Trans ASME, J. Vib.Acoust., 131, 021009.

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[9] Yang, X., and Nassar, S. A., 2011, Analytical and Experimental Investigationof Self-Loosening of Preloaded Cap Screw Fasteners, Trans. ASME J. Vib.Acoust., 133, 031007.

[10] Zaki, A. M., Nassar, S. A., and Yang, X., 2010, Vibration Loosening Modelfor Preloaded Countersunk-Head Bolts, 2010 ASME Pressure Vessels and Pip-ing Division Conference, PVP2010-25069.

[11] Zaki, A. M., Nassar, S. A., and Yang, X., 2010, Effect of Thread andBearing Friction Coefficients on the Self-Loosening of PreloadedCountersunk-Head Under Periodic Transverse Excitation, ASME J. Tribol.,132, 031601.

[12] Industrial Fasteners Institute, 1988, Fastener Standards, 6th ed., IndustrialFasteners Institute, Cleveland, OH.

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