A Mathematical Model for Vibration Induced Loosening of Preloaded Thhreaded Fasteners

1

clTfe�VsorsfiabbssvnhmpotasYdltZ

pNF

J

Do

Sayed A. Nassar

Xianjie Yang

Fastening and Joining Research Institute,Department of Mechanical Engineering,

Oakland University,Rochester, MI 48309

A Mathematical Model forVibration-Induced Loosening ofPreloaded Threaded FastenersA mathematical model is proposed for studying the vibration induced loosening ofthreaded fasteners that are subjected to harmonic transverse excitation, which oftencauses slippage between the contact surfaces between engaged threads and under thebolt head. Integral equations are derived for the cyclic shear forces as well as thebearing and thread friction torque components. They depend on the ratio of the relativerotational to translational velocities. The relationship between the dynamic thread shearforce and bending moment is developed. When the external transverse excitation is largeenough, it causes the threaded fasteners to loosen. Numerical results show that thedynamic transverse shear forces on the underhead contact surface, and between theengaged threads, decrease the bearing, and thread friction torque components. The effectof bolt preload, bearing and thread friction coefficients, the amplitude of the harmonictransverse excitation, and the bolt underhead bending on the bolt loosening are investi-gated. Experimental verification of the analytical model results of the bolt twisting torqueis provided. �DOI: 10.1115/1.2981165�

Keywords: vibration loosening, self-loosening, threaded fasteners

IntroductionThreaded fasteners have been widely used in many engineering

omponents and structures under vibratory loads. The vibratoryoading can lead to vibration-induced loosening of the fastener.here have been several studies on the self-loosening of threaded

asteners; most of them are experimental. Hess �1� surveyed sev-ral studies on the vibration and shock induced loosening. Sakai2�, Haviland �3�, Yamamoto and Kasei �4�, Tanaka et al. �5�,inogradov and Huang �6�, Zadoks and Yu �7,8�, and Junker �9�

tudied the effect of the transverse vibrations on the self-looseningf threaded fasteners. Xu et al. �10� proposed an iterative algo-ithm to identify the locations and extent of damage in boltedtructural beams using only the changes in the first few naturalrequencies. Nichols et al. �11� presented an approach for detect-ng damage-induced nonlinearities in bolted structures, and thepproach was used to detect loosening of a composite-to-metalolted joint. Sakai �2� proposed a theoretical model of looseningased on the slip between the fastener contact surfaces. The re-ults from his model suggested that the friction at the contacturfaces should drop to a very low value of 0.03; the averagealue of the friction coefficient between the oiled steel surfaces isormally around 0.15; however, Haviland �3� stated that looseningappens as a result of the ratcheting action of the nut, but noathematical model or experimental data were provided to sup-

ort his theory. Yamamoto and Kasei �4� suggested that looseningccurs due to the accumulation and release of potential energy dueo the torsional bolt deformation. Tanaka et al. �5� conducted anxisymmetric finite element study for bolt loosening. Their modelhowed that the loosening process is the same as that presented byamamoto and Kasei �4�. Vinogradov and Huang �6� presented aynamic model to simulate the loosening under high frequencyoading �30–240 kHz�, which is well beyond the normal opera-ion range for most mechanical and structural components.adoks and Yu �7,8� presented a dynamic model to simulate the

Contributed by the Technical Committee on Vibration and Sound of ASME forublication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript receivedovember 12, 2007; final manuscript received July 1, 2008; published online
ebruary 18, 2009. Assoc. Editor: Weidong Zhu.
ournal of Vibration and Acoustics Copyright © 20

wnloaded 12 Dec 2011 to 195.83.11.66. Redistribution subject to ASME

loosening process of screw in a tapped hole. Their friction modelassumed that friction dropped to zero in the circumferential direc-tion as soon as slippage began in the transverse direction; thisassumption is not valid. Junker �9� showed that the loosening ofthreaded fasteners was far more severe when the joint was sub-jected to transverse cyclic loads. In his work, he concluded thatthe self-loosening happens when slippage took place between en-gaged threads and/or under the bolt head/nut.

Nassar and Housari �12–15� introduced a mathematical modeland an experimental procedure to study the threaded fastener loos-ening phenomena under cyclic transverse loads. They investigatethe effect of thread pitch, initial bolt tension, hole clearance threadfit, bearing and thread friction, and amplitude of the cyclic trans-verse load on the loosening of threaded fasteners. They concludedthat these parameters have a significant effect on the self-loosening of threaded fasteners. However, they did not proposethe direct formulations of the variations of the bearing and threadtorques, the shear force based on the relative rotation and theslipping movement on the contact surfaces for bolted joints. Thedirect formulations are the key for exploring the mechanism of theself-loosening due to transverse cyclic loading.

In this study, a more effective mathematical model for self-loosening of bolted joint under transverse cyclic loading is pro-posed. The model is based on the relative slippage between thefriction surfaces. Integral equations are developed and are numeri-cally analyzed, and the self-loosening mechanism is explained.The effects of initial bolt tension, bearing and thread friction co-efficients, and the amplitude of the cyclic transverse displacementon the loosening of threaded fasteners are studied. The compari-son of the experimental and analytical torque variations is made.

2 Mathematical Model

2.1 Relationship Among Torque Components in the Loos-ening Direction. After the initial tightening of the bolt has beencompleted and the tightening torque is removed, there are twoscenarios for the relationship among the underhead bearing fric-tion, the thread friction, and the pitch torque components, Tb, Tt,
Tp, respectively.
APRIL 2009, Vol. 131 / 021009-109 by ASME

license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

mut

wtihFtbi

taaY

wte

wpe

cca

tctEflsr

Dttatle

wlcttcwa

0

Do

In scenario 1, if the sum of the pitch torque Tp and the maxi-um thread friction torque Ttcr is insufficient to overcome the

nderhead bearing friction torque Tb�, the correlation between thehree torques is provided by Nassar and Yang �16� as follows:

Ttwist = Tb� = Tp + Ttcr �1�

Ttcr =rt�tF cos ��sec2 � + tan2 �

1 − �t sin ��sec2 � + tan2 ��2�

here Tb�, Tp, and Ttcr are the corresponding bearing frictionorque, pitch torque, and maximum thread friction torque after thenitial tightening torque is removed, � is the lead helix angle, � isalf of the thread profile angle, �t is the thread friction coefficient,

is the clamp load, and rt is the effective thread radius. Underhis case, there is no elastic rotational springback of the bolt. Theearing friction torque reverses direction after the initial tighten-ng torque is removed. Ttcr has the same torque direction as Tp.

In scenario 2, if the maximum underhead bearing frictionorque is insufficient to overcome the sum of the pitch torque Tp

nd the thread friction torque Tt�, the corresponding relationshipmong the three torque components is provided by Nassar andang �16� as follows:

Ttwist = Tbcr = Tp + Tt� �3�

Tbcr = �brbF �4�

here �b is the underhead bearing friction coefficient, and rb ishe effective bearing radius. Under this scenario, there is a relativelastic rotational springback angle �e of the bolt as

�e =�Tp + Ttcr − Tbcr�L

GJp�5�

here G is the elastic shear modulus of bolt material, Jp is theolar moment of inertia of the bolt cross section, and L is theffective bolt length.

This relative rotational elastic springback will not result in thelamp force loss because the rotational elastic springback isaused by the relative rotation between the bolt underhead surfacend the joint member surface.

If the applied cyclic transverse shear force to joint is sufficiento cause slippage, the transverse shear force will lead to the de-rease of the critical bearing frictional torque and the thread fric-ion torque; the relationship among the three torque is the same asq. �1� or �3� if the dynamic effect can be neglected and the thread

riction torque Tt� has the same direction as Tp. When self-oosening may occur, the thread friction torque Tt� will have theame direction with Tb� and has the reverse direction with Tp; theelationship is

Tb� = Tp − Tt� �6�

uring the process of decreasing the critical bearing frictionorque Tbc� , the elastic rotation springback exists, the thread fric-ion torque Tt� may not reach the critical thread friction torque Ttc� ,nd the clamp force will stay constant. When the two frictionorques reach their critical values at the same time, the criticaloosening will occur. Therefore, the self-loosening angular accel-ration � of the bolt can be given by

J� = �Tp − Tbc� − Ttc� � �7�

here J is the moment of the inertia of the bolt, � is the self-oosening angular acceleration, Tbc� and Ttc� are the correspondingritical bearing and thread friction torques, respectively, under theransverse shear force, and � � is the Macauley sign; this meanshat if x�0, �x�=x, and if x�0, �x�=0. If there is no transverseyclic excitation, the sum of Tbc� and Ttc� is always bigger than Tphen the bearing and thread friction coefficients are not too small,

˙
s shown in Ref. �16�. Based on Eq. �7�, �=0. On the other hand,
21009-2 / Vol. 131, APRIL 2009


Tbc� and Ttc� decrease with the amplitude of the transverse cyclicexcitation. When the transverse cyclic excitation is big enough, Tp

is possible to be larger than the sum of Tbc� and Ttc� , which lead to��0 based on Eq. �7�.

The self-loosening angular velocity � is given by

� = �0 +�t

t+t

�dt �8�

where �0 is the self-loosening angular velocity at time t.The loosening angle �� is given by

�� = �0� +�0

t

�dt �9�

The clamp force loss F caused by the loosening rotation �� isgiven by

F =kbkc

kb + kc

��

2p �10�

where p is the thread pitch, and kb and kc are the correspondingstiffnesses of bolt and joint members, respectively.

2.2 Underhead Friction and Bearing Surface. Frictionforce is always opposite to the direction of the relative velocityvector on the contact surface. On the underhead surface of bolt, asshown in Fig. 1, the relative velocity va of the point on the boltunderhead referred to the joint member bearing surface consists ofthe translation velocity v1 caused by transverse shear force and thevelocity v2 caused by relative rotation. The transverse movementis assumed to be along the x-direction. Therefore, the relativetranslation velocity v1 is along the x-direction. The relative veloc-ity v2 is vertical to the position vector r of the point on boltunderhead bearing area. Figure 1 shows the kinetic analysis underthe Cartesian coordinate system x−y or the polar coordinate sys-tem r−�.

va = v1 + v2 = vb1i + ��bk� � r = �vb1 − �br sin ��i + �br cos �j

�11�

where r=xi+yj, i is the unit vector along the x-axis, j is the unitvector along the y-axis direction, �b is the relative rotation angu-lar velocity of the bolt underhead to the joint member, x and y arethe corresponding x and y coordinate values, � is the angle of thepolar coordinate system, and k is the unit vector in z-axis direc-tion and be vertical to the bearing contact surface.

Considering the bending effect of bolt underhead under trans-

Fig. 1 The relative movement on the underhead bearing sur-face to the joint members

verse loading, the pressure distribution is assumed to be

Transactions of the ASME


wafm

B

Ti

To

wt

T

Bog

Tu

J

Do

qb = qb0 + qb�x

re�12�

here qb is the underhead pressure, qb0 is the average pressure,nd qb� is the pressure variation amplitude. The direction of theriction force dFbf is opposite to the direction of relative move-ent or movement tendency.

dFbf = − dFbfva

va= −

�vb1 + �by�i − �bxj��vb1 + �by�2 + ��bx�2

dFbf �13�

ased on Coulomb friction condition, we have

dFbf = �bqdS = �bqb0 + qb�r cos �

redS �14�

he bearing friction torque dTb caused by the friction force dFbfs

dTb = r � dFbf = − �r cos �i + r sin �j�

��vb1 + �br sin ��i − �br cos �j

��vb1 + �br sin ��2 + ��br cos ��2dFbf

= dFbf

�br2 + vb1r sin �

�vb12 + 2�bvb1 sin � + �b

2r2k �15�

he bearing friction torque Tb can be presented by integrating dTbn the contact bearing area �bearing:

Tb = ��bearing

dTb

= ��bearing

�bqb0 + qb�r cos �

re

��br2 + vb1r sin �

�vb12 + 2�bvb1 sin � + �b

2r2dS

= �bqb0�ri

re

r2dr�0

2 � b sin � + r�d�

� b2 + 2 br sin � + r2

+�bqb�

re�

ri

re

r3dr�0

2 � b sin � + r�cos �d�

� b2 + 2 br sin � + r2

�16�

here b=vb1 /�b called the bearing translation-rotational ratio inhis paper. As we know,

�ri

re

r3dr�0

2 � b sin � + r�cos �d�

� b2 + 2 br sin � + r2

t = sin �=

��ri

re

r3dr�0

0 � bt + r�dt

� b2 + 2 brt + r2

= 0 �17�

herefore,

Tb = �bqb0�ri

re

r2dr�0

2 � b sin � + r�d�

� b2 + 2 br sin � + r2

�18�

earing friction shear force increment dFbs along the x-directionn the small area dS of the underhead bearing contact surface isiven by

dFbs = dFbf · i = − dFbf

va · i

va= −

dFbf�vb1 + �br sin ��vb1

2 + �b2r2 + 2�bvb1r sin �

�19�

herefore, the transverse bearing friction shear force Fbs on the
nderhead bearing contact surface is given by
ournal of Vibration and Acoustics


Fbs = ��bearing

dFbs = − �bqb0�ri

re

rdr�0

2 � b + r sin ��d�

� b2 + r2 + 2 br sin �

�20�

In summary, the bearing frictional torque Tb and transverse bear-ing friction shear force Fbs can be presented as the functions ofthe bearing translation-rotational ratio b as follows:

RTb = � Tb

�bqb0� =�

ri

re

r2dr�0

2 � b sin � + r�d�

� b2 + 2 br sin � + r2

�21�

RFb = � Fbs

�bqb0� =�

ri

re

rdr�0

2 � b + r sin ��d�

� b2 + r2 + 2 br sin �

�22�

where qb0 is the average pressure on the contact bearing surface.The integral equations �21� and �22� are numerically solved forRTb and RFb in terms of the bearing translation-rotational ratio b.Figure 2 shows the numerical results for a bearing friction coef-ficient �b��bcr. Obviously, when the underhead bearing frictionforce is smaller than the critical value, there will be no relativemovement at the underhead interface. However, there exists thetendency for relative movement when the shear force is less thanthe critical one. This means that if there exists a shear force Fbs,the direction of the friction force will change with the shear forceeven though the friction force does not reach the critical one.From Eqs. �21� and �22�, when b→� �this means no relativerotation or rotation tendency between the bolt head and the jointmember�, the bearing friction torque Tb=0 and Fbs=�bF. On theother hand, when b=0 �this means no relative translation or rela-tive translation tendency between the bolt head and the joint mem-bers�, Eqs. �21� and �22� will lead to Tb=�brbF and Fbs=0. Equa-tions �21� and �22� characterize the friction properties of theinterface between the bolt underhead and joint member. Based onthe analysis, Eqs. �21� and �22� can be used either with the relativemovement or without the relative movement. On the other hand, ifthere is no relative movement on the friction surfaces, �b in Eqs.�21� and �22� is smaller than the critical static bearing frictioncoefficient as the practical friction force should be smaller thanthe critical friction force.

From Fig. 2, it is found that the bearing friction torque Tb willdecrease and the shear force will increase with increasing bearingtranslation-rotational ratio b. When the ratio b=vb1 /�b is largeenough �e.g., b�100�, Tb will approach zero while the shearforce Fbs approaches to its saturation. Figure 3 shows that theratio Fbs /Tb significantly increases with increasing ratio b. If the

Fig. 2 The RTb−�b and RFb−�b relationships under the trans-verse shear force Fbs, where �b=vb1 /�b

shear force is larger enough to cause relative movement, the bear-

APRIL 2009, Vol. 131 / 021009-3


iRrf

ita

wc

mw

T4

Ft

0

Do

ng friction torque is significantly decreased with increasing b.elative movement leads to the elastic rotational springback or

educes initial bolt elongation, which reduces the joint clamporce.

2.3 Thread Friction Analysis. As shown in Fig. 4, consider-ng the joint member moves along the x direction, the relativeranslation velocity vt1 of bolt thread to the joint member can bessumed as

vt1 = vtxi + vtzk �23�

here vtx is the x-axis component of vt1 and vtz is the z-axisomponent of vtz.

If the relative angular velocity of the bolt thread to the jointember is �t, the total relative velocity vector v of the bolt threadith respect to the joint member will be

v = vt + vr = �vtxi + vtzk� + ��tk� � r = �vtx − �tr sin ��i

+ �tr cos �j + vtzk �24�

he unit normal vector w1 of the thread surface as shown in Fig.can be presented as

ig. 3 „Fbs /Tb…−�b and „Fts /Tt…−�t relationships under theransverse shear force Fts, where �b=vb1 /�b and �t=vt1 /�t

Fig. 4 Schematic relative move

21009-4 / Vol. 131, APRIL 2009


w1 =1

�sec2 � + tan2 ��tan � cos � + tan � sin �, tan � sin �

− tan � cos �, 1 �25�If the contacting surface between the bolt threads and the joint

threads keeps contact under the loading, the relative velocity vec-tor v of the contacting bolt thread should keep tangent on thecontact thread surface; therefore

w1 · v =1

�sec2 � + tan2 ��vtx − �tr sin ��tan � cos �

+ tan � sin �� − �tr cos ��tan � cos � − tan � sin �� + vtz�

= 0 �26�

The velocity component vtz can be given by

vtz = − vtx�tan � cos � + tan � sin �� + �tr tan � � − vtx tan � cos �

+ �tr tan � �27�

Substituting Eq. �27� into Eq. �24�, we have

v = �vtx − �tr sin ��i + �tr cos �j + ��tr tan �� − vtx tan � cos ��k�28�

Considering the bending effect of bolt, the thread pressure vectorqt is assumed to be

qt = − qt0 + qt�x

rmajw1 �29�

where qt0 is the average value of the thread pressure, qt� is theincrement amplitude caused by the bending effect, rmaj is the boltthread major radius, x is the x-coordinate value, and w1 is the unitoutward normal vector of the thread surface, as shown in Fig. 4.

As the friction force direction is always opposite to the relativemovement or the tendency of the friction surfaces, the thread fric-tion force increment vector dFtf on the area dS� of the threadcontact surface, as shown in Fig. 4, can be given by

ment on the thread surface



C

Tb

T

T�g

Isb

R

J

Do

dFtf = − dFtfv

v= − dFtf

�vtx − �tr sin ��i + �tr cos �j + �tan ��tr − vtx tan � cos ��k�vtx

2 �1 + tan2 � cos2 �� + sec2 ��t2r2 − 2vtx�tr�sin � + tan �tan � cos ��

�30�

onsidering tan �� 1, we have the thread friction force increment dFtx along x-direction

dFtx = dFtf · i = − dFtf

�vtx − �tr sin ��vtx

2 �1 + tan2 � cos2 �� + sec2 ��t2r2 − 2vtx�tr�sin � + tan �tan � cos ��

=

− dFtf

�vtx − �tr sin ��vtx

2 �1 + tan2 � cos2 �� + �t2r2 − 2vtx�tr sin �

�31�

aking the surface integral of the dFtx on the contact thread surface �thread, the thread friction shear force Fts along the x-direction cane given by

Fts =��thread

−

�tqt0 + qt�r cos �

rmaj�vtx − �tr sin ��dS�

�vtx2 �1 + tan2 � cos2 �� + �t

2r2 − 2vtx�tr sin �

� − �tqt0�sec2 � + tan2 ��

rmin

rmaj

rdr�0

2 � t − r sin ��d�

� t2�1 + tan2 � cos2 �� + r2 − 2 tr sin �

�32�

he thread friction torque increment dTt caused by the friction force increment dFtf can be given by

dTt = �r � dFtf� · k = −dFtfr�cos �i + sin �j� � ��vtx − �tr sin ��i + �tr cos �j − vtx tan � cos �k�

�vtx2 �1 + tan2 � cos2 �� + �t

2r2 − 2vtx�tr sin �· k

=− dFtfr��tr − vtx sin ��

�vtx2 �1 + tan2 � cos2 �� + �t

2r2 − 2vtx�tr sin ��33�

aking the surface integral of dTt on the contact thread surfacethread, the thread friction torque Tt caused by friction force is

iven by

Tt = −��thread

�bqt0 + qt�r cos �

rmajr��tr − vtx sin ��dS�

�vtx2 �1 + tan2 � cos2 �� + �t

2r2 − 2vtx�tr sin �

= − �tqt0�sec2 � + tan2 �

��rmin

rmaj

r2dr�0

2 �r − t sin ��d�

� t2�1 + tan2 � cos2 �� + r2 − 2 tr sin �

�34�

n summary, the thread frictional torque Tt and the thread frictionhear force Fts along the x-direction under transverse loading cane given by

RTt = � Tt

�tqt0� = �sec2 � + tan2 �

��rmin

rmaj

r2dr�0

2 �r − t sin ��d�

� t2�1 + tan2 � cos2 �� + r2 − 2 tr sin �

�35�

RFt = � Fts

�tqt0� = �sec2 � + tan2 �

��rmin

rmaj

rdr�0

2 � t − r sin ��d�

� t2�1 + tan2 � cos2 �� + r2 − 2 tr sin �

�36�The integral equations �35� and �36� are numerically solved for

Tt and RFt in terms of the thread translation-rotational ratio t.



Figure 5 shows the numerical results for a thread friction coeffi-cient �t��ts. When the thread friction force does not reach thecritical one, there is no any relative movement between the twocontact thread surfaces. However, there exists the tendency ofrelative movement when the shear force is less than the criticalone. This means that if there exists the shear force Fts, the direc-tion of the friction force will change with the shear force. Basedon the analysis, Eqs. �35� and �36� can be used either with therelative movement or without the relative movement. On the otherhand, under no relative movement, �t in Eqs. �35� and �36� issmaller than the critical static thread friction coefficient.

From Fig. 5, it is found that the thread friction torque Tt willdecrease and the shear force will increase with increasing threadtranslation-rotational ratio t. When the ratio t is large enough�e.g., t�100�, Tt will approach zero, and the shear force ap-proaches to its saturation. Figure 3 shows that the ratio Fts /Tt

Fig. 5 RTt−�t and RFt−�t relationships under the transverse
shear force Fts, where �t=vtx /�t
APRIL 2009, Vol. 131 / 021009-5


sffmeat�w

3v

wvama

tttu

wEl

adcfTi

wbi

0

Do

ignificantly increases with increasing the ratio t. If the shearorce is larger enough to make the relative movement, the threadriction torque will decrease very significantly. The relative move-ent will lead to the elastic rotation springback or the bolt tension

longation decrease. If the sum of the bearing friction torquend the thread friction torque is smaller than the pitch torque,he tension elongation decrease will appear based on Eqs.7�–�10�, which leads to the clamp force loss. The self-looseningill happen.

Shear Force and Bending Moment Due to Trans-erse Cyclic LoadingWhen bolt is subjected to transverse cyclic loading, the bolt

ill have bending deformation. The correlation among the trans-erse displacement at bolt underhead, the transverse shear forcest the underhead bearing or thread surfaces, and the bending mo-ent at bolt underhead play an important role for self-loosening

nalysis.

3.1 Effect of Bending Angle. The bolt is subjected to theransverse shear force Fbs and the reaction bending moment M� athe bolt underhead, as shown in Fig. 6. Based on elastic beamheory, the transverse displacement � and bending angle � at thenderhead are given by

� =FbsL

3

3EI−

M�L2

2EI�37�

� =FbsL

2

2EI−

M�L

EI�38�

here I is the axial moment of inertia for the cross sectional area,is Young’s elastic modulus, and L is the effective bolt bending

ength.The underhead pressure distribution is affected by the bending

ngle �. When the bending angle � is bigger, the compressiveeformation of the joint member bearing surface will be signifi-antly unevenly distributed, which caused by the angle �. There-ore, the angle � leads to the nonuniform pressure distribution.he correlation between the underhead pressure qb and the bend-

ng angle � is assumed to be

qb = qb0 + ��x

re�39�

here � is a constant, which is dependent on the bolt underheadending stiffness. Therefore, the bending moment M� on the bear-

Fig. 6 The force diagram of the bolt

ng surface caused by the pressure distribution qb is given by

21009-6 / Vol. 131, APRIL 2009


M� = ��bearing

qb0 + ��x

rexdS = ��

1

re�

ri

re

r3dr�0

2

cos2 �d�

=��

4re�re

4 − ri4� �40�

Considering Eqs. �38� and �40�, we have

M� =��re

4 − ri4�

2

FbsL2

4EIre + �L�re4 − ri

4�=

�FbsL2

2�L +8EIre

�re4 − ri

4��41�

Substituting Eq. �41� into Eq. �37�, we have

Fbs =EI�

L3

3−

�L4

4�L +16EIre

�re4 − ri

4�

�42�

From Fig. 6, based on the static equilibrium condition, the bend-ing movement M on the thread surface is given by

M = FbsL −��re

4 − ri4�

2

FbsL2

4EIre + �L�re4 − ri

4�

= FbsL

8EIre

�re4 − ri

4�+ �L

8EIre

�re4 − ri

4�+ 2�L

�43�

The bending moment M on the bolt thread surfaces leads to thenonuniform pressure distribution on the thread surfaces.

3.2 Shear Force on Bolt Thread Surface. Considering thenonuniform pressure distribution on thread surfaces caused by thebending moment M, the bolt thread pressure vector qt on thethread surface is assumed to be

qt = − qt0 + qt�x

rmajw1 = − qt0 + qt�

r cos �

rmajw1 �44�

where qt0 is the average thread pressure, qt� is the thread pressureamplitude caused by bending moment, x is the x-axis coordinatevalue, rmaj is the bolt thread major radius, and w1 is the unitoutward normal vector of the thread surface. The bolt thread shearforce component FxM along the x-direction caused by the non-uniform pressure distribution on thread surfaces is

FxM =��thread

�qt · i�ds

=�rmin

rmaj

rdr�0

2 qt0 + qt�r cos �

rmaj

�tan � cos � + tan � sin �

�sec2 � + tan2 ��sec2 � + tan2 �d�

=�rmin

rmaj

rdr�0

2 qt0 + qt�r cos �

rmaj�tan � cos � + tan � sin ��d�

=qt�

rmaj�

rmin

rmaj

r2dr�0

2

cos ��tan � cos � + tan � sin ��d�

=qt� tan �

�rmaj3 − rmin

3 � �45�
3rmaj


Bb

Cf

w

fb

C

Ef

4

tmhafi

swfibs

J

Do

ased on the correlation between the thread pressure and theending moment M, we have

M = −��thread

��qt · k�x + �qt · i�z�dS =��xy

qt0 +qt�r cos �

rmaj

��r cos � + ��rmaj − r�tan � +�

2p��tan � cos �

+ tan � sin ��rdrd� =qt�

rmaj�

6�rmaj

4 − rmin4 � + �rmaj tan2 �

+ tan �

2− tan �p�1

3�rmaj

3 − rmin3 ��

qt�

rmaj�

6�rmaj

4 − rmin4 �

+ �1

9rmaj + 0.2854p��rmaj

3 − rmin3 �� 46�

onsidering Eqs. �45� and �46�, the correlation between the shearorce and the bending moment on the thread surfaces is given by

FxM = tan �M�rmaj

3 − rmin3 �

2�rmaj

4 − rmin4 � + 1

3rmaj + 0.8562p�rmaj

3 − rmin3 �

= tan �M

rmin��4 − 1�2��3 − 1�

+ 1

3rmaj + 0.8562p �47�

here �=rmaj /rmin, and p is the bolt thread pitch length.If the bolt does not contact with the joint hole’s wall, the thread

riction shear force component Fts along the x-direction is giveny

Fts = Fbs + FxM = Fbs +tan �M

rmin��4 − 1�2��3 − 1�

+ 1

3rmaj + 0.2725p

�48�onsidering Eqs. �43� and �48�, we have

Fts = Fbs�1 +tan �L

rmin��4 − 1�2��3 − 1�

+ 1

3rmaj + 0.2725p

�

8EIre

�re4 − ri

4�+ �L

8EIre

�re4 − ri

4�+ 2�L� �49�

quation �49� gives the correlation between the underhead shearorce Fbs and the thread friction shear force Fts.

Contact Surface Slippage AnalysisAs shown in Figs. 7 and 8, ��t� represents the transverse exci-

ation of the upper joint plate, xh�t� denotes the absolute move-ent of the bolt head, and xt�t� is the bolt thread movement. The

ole clearance for bolt shank is assumed to be and the clearancet the bolt-nut thread interface is assumed to be 1. The followingve scenarios are presented and analyzed.Scenario I. If the transverse displacement of upper joint plate is

mall enough, the bolt head will move with the moving joint plateithout relative movement as the resulting shear force is not suf-cient to overcome the underhead friction force. Likewise, theolt thread surfaces have no relative movement when the resulting
hear force will not be sufficient to overcome the thread friction


force. Considering Eq. �42�, the bolt deflection at the bolt head isexpressed as follows:

�xh�t� − xt�t�� = �xh�t�� = ��t�� =kFbsL

3

3EI�50�

where the bending factor k can be defined as k=1−3�L / �4�L+16EIre /�re

4−ri4�� based on Eq. �42�.

Under this scenario, the bolt deflection is equal to the transverseexcitation displacement of the joint member �J, which is given by

�J = �0 sin��t� �51�

where �0 is the transverse displacement amplitude, and �� is theangular velocity for the excitation. Therefore, the shear force ap-plied on the underhead surface is given by

Fbs =3EI

kL3 �0 sin��t� �52�

The thread friction shear force Fts of bolt along the x-direction isgiven by Eq. �49�, so

Fts =3EI

kL3 �0 sin��t��1 +tan �L

rmin��4 − 1�2��3 − 1�

+ 1

3rmaj + 0.2725p

�

8EIre

�re4 − ri

4�+ �L

8EIre

�re4 − ri

4�+ 2�L� �53�

Scenario II. When there is a relative movement at the threadinterface for 0� �xt�t��1 /2 and no relative movement at theunderhead contact interface, the shear force applied to the under-head surface is given by

Fbs =3EI

kL3 �xh�t� − xt�t�� =3EI

kL3 ��t� − xt�t�� 54�

Scenario III. As shown in Fig. 8, when there is a relative move-ment at the thread interface for 0� �xt�t��1 /2 and there is arelative movement at the underhead contact interface with ��t�−xh�t�� /2, the shear force applied to the underhead surface isgiven by

Fbs =3EI

kL3 �xh�t� − xt�t�� 55�

Scenario IV. When �xt�t��1 /2, and there is a relative move-ment at the underhead contact interface with ��t�−xh�t�� /2,the shear force applied to the underhead surface is given by

Fbs =3EI

3 ��xh�t�� − 1/2� �56�

Fig. 7 The schematic diagram for bolt joint under transversecyclic loading

kL

APRIL 2009, Vol. 131 / 021009-7


−mjTtcu

wt

0

Do

Scenario V. As shown in Fig. 9, when �xt�t��1 /2 and ��t�xh�t�� /2, this means that the bolt shank contact with the jointembers and relative slippage between the thread surfaces and

oint member thread surface is caused just by elastic deformation.he contact between the bolt shank and the joint members leads to

he resistance force FJs. The sum of the resistance force FJsaused by the upper joint plate and the shear force applied to thenderhead surface is given by

FJs + Fbs =3EI

kL3 ��t�� − /2 − 1/2 − ubx − utx� �57�

here ubx is the elastic contact deformation displacement betweenhe bolt shank and the joint hole along the x-axis direction caused

Fig. 8 Schematic diagram under scenaforce diagram of the bolt

Fig. 9 The schematic for scenario V

21009-8 / Vol. 131, APRIL 2009


by FJs, and utx is the contact deformation displacement betweenthe bolt threads and the joint member threads along the x-axisdirection, as shown in Fig. 9. In practical condition, the deforma-tion displacements ubx and utx are often much smaller than /2and 1 /2, so Eq. �57� can be simplified as

FJs + Fbs =3EI

kL3 ��t�� − /2 − 1/2� �58�

5 Results and DiscussionIn this section, the thread clearance 1 is assumed to be zero,

the transverse displacement amplitude �0 is assumed to be much

III and IV: „a… The kinetic diagram „b…
rios
under transverse cyclic loading



strri

sc=dbrlfmb

DtvstttFspdfs

tfv1tWtmttzt

Ud

Frm

J

Do

maller than half of the hole clearance /2 for simplification ofhe numerical simulation. Elastic Young’s modulus and Poissonatio for bolt and joint plates are assumed to be 200 GPa and 0.3,espectively. The effective bolt length is 30 mm, and the fasteners M12�1.75.

If the pressure distribution on bearing contact surface is as-umed to be presented as Eq. �39�, the thread friction shear forceomponent along the x-direction is presented by Eq. �49�. Let A8EIre /�re

4−ri4� and B=�L. The ratio �=B /A represents the un-

erhead bending stiffness. The simulation results can be obtainedy using Eqs. �6�, �21�, �22�, �35�, �36�, �52�, and �53�. When theatio is larger, the bending angle at the underhead of the bolt willead to larger resistant bending moment caused by bearing sur-ace. In the Secs. 5.1–5.3, the ratio � is assumed to be zero. Thiseans that the resistant bending moment at underhead caused by

earing surface is assumed to be ignored there.

5.1 Effect of Amplitude of Cyclic Transverseisplacement. If the bolt preload is 30,000 N, the bearing and

hread friction coefficients are 0.15 and 0.15, respectively, Theariation of the critical bearing friction torque resistant to thelippage under the different transverse cyclic displacement excita-ions of 0.02–0.2 mm is shown in Fig. 10. Figure 10 shows thathe torque changes significantly with the different amplitudes ofhe transverse displacement excitations. When �0=0.2 mm, fromig. 10, the critical bearing friction torque will approach to zero inome time period of a cycle. This means that the transverse dis-lacement will lead to the increase of the shear force and theecrease of the critical bearing friction torque. When the bearingriction torque approaches to zero, the fastener may lead to theelf-loosening.

If the bolt preload is 30,000 N and the bearing and thread fric-ion coefficients are all 0.15, the variation of the critical threadriction torque resistant to the slippage under the different trans-erse cyclic displacement excitations is shown in Fig. 11. Figure1 shows that the thread friction torque changes significantly withhe increase of the amplitude of the transverse displacement.

hen �0�0.06 mm, from Fig. 11, the critical thread frictionorque will approach to zero in some time period of a cycle. This

eans that the transverse displacement will lead to the increase ofhe thread shear force and the decrease of the thread frictionorque. When the bearing and thread friction torque approach toero, the transverse excitation may lead to the self-loosening ofhe threaded fastener.

5.2 Effect of Bearing and Thread Friction Coefficients.nder the specific bolt preload of 30,000 N and the transverse

ig. 10 The variations of the critical bearing friction torque Tbesistant to slippage under different transverse cyclic displace-ent excitations

isplacement amplitude of 0.1 mm with �t=0.15, Fig. 12 displays



the variation of the critical bearing friction torque with time underthe transverse cyclic excitation. From Fig. 12, the variation of thecritical bearing friction torque depends on the bearing frictioncoefficient, the larger bearing friction coefficient can increase thecritical bearing friction torque level, and the critical bearing fric-tion torque under �b=0.10 can approach to zero in some period ofa cycle. This means that the larger bearing friction coefficient canimprove the resistance to the self-loosening of the threaded boltjoints. If the bearing friction coefficient keeps constant and thethread friction coefficient changes in some range, the similar re-sults of the thread friction torque variation with the ones of bear-ing friction torque variation can be obtained.

5.3 Effect of Bolt Preload. If the bearing and thread frictioncoefficients are all 0.15 and the amplitude of transverse displace-ment �0 is 0.15 mm, the variation of the bearing friction torqueunder three preloads of 10,000 N, 20,000 N, and 30,000 N isshown in Fig. 13. From Fig. 13, it is indicated that the level of thecritical bearing friction torque is strongly dependent on the clampload level, and the significant decrease of the critical bearing fric-tion torque occurs when the preload decreases. Especially, whenthe preload is just 10,000 N, and 20,000 N, the torques can ap-proach to zero in some period of a cycle. As we know, when thecritical bearing friction torque becomes smaller, it is easier tomake the threaded fastener self-loosening. If the thread frictioncoefficient �t and the amplitude of the transverse displacementkeep constant, the same results for the effect of the preloads on thecritical thread friction torque variation can be obtained. In sum-

Fig. 11 The variations of the critical thread friction torque Ttunder different transverse cyclic displacement excitations

Fig. 12 The effect of bearing friction coefficient �b on thevariation of the critical bearing friction torque Tb under trans-
verse cyclic excitation
APRIL 2009, Vol. 131 / 021009-9


mlp

slhfh=aaitrbnj

6

tttsttw

Fb

Ft

0

Do

ary, the larger preload can improve the resistance to the self-oosening of the threaded fastener under a specific transverse dis-lacement excitation.

5.4 Effect of Underhead Bending Stiffness. In fact, the re-istant bending moment at underhead of bolt will affect the self-oosening behavior of bolt joint. The underhead bending stiffnessas affected the pressure distributions on bearing surface. There-ore, the underhead bending stiffness affects the resisting under-ead bending moment of the bolt. When �t=0.25, �b=0.25, �00.05 mm, and F=30,000 N, the variations of the critical threadnd bearing friction torques under the ratios � of 0,1,10, and 100re shown in Figs. 14 and 15, respectively. Figures 14 and 15ndicate that the variations of the critical thread and bearing fric-ion torques in a cycle are dependent on the ratio �. Decreasing theatio � will improves the resistance to the self-loosening of theolted joint. This means that the weaker underhead bending stiff-ess improves the resistance to the self-loosening of the boltedoint.

Experimental and Theoretical ComparisonIn this section, an experimental procedure is employed in order

o compare with the results of the analytical model proposed byhis study. In Sec. 5, the variations of the critical bearing frictionorque and thread friction torque are studied. Only when relativelippage occurs can the critical values of the bearing frictionorque Tb and the thread friction torque Tt be obtained. Generally,he friction force Fs is always in the range −�cN�Fs��cN,here N is the contact normal force and �c is the critical friction

ig. 13 The effect of preload F on the variation of the criticalearing friction torque Tb under transverse cyclic excitation

ig. 14 The bending effect on the variation of the critical
hread friction torque Tt under transverse cyclic excitation
21009-10 / Vol. 131, APRIL 2009


coefficient, in which the value of Fs will be determined by usingstatic equilibrium or dynamic condition. Therefore, the actualbearing friction torque Tb� is in the range −�Tbc� ��Tb�� Tbc� �, whilethe actual thread friction torque Tt� is in the range −�Ttc� ��Tt�� Ttc� �.

To experimentally verify the mathematical model, the actualtwisting torque in shank of the bolt is measured by using straingauges. After initial tightening, the initial value Tb0� of the bearingfriction torque is equal to Tt0� +Tp, which is the twisting torque inthe bolt after the removal of the tightening torque. The twistingtorque causes an elastic twisting angle �=Tb0� L /GJp. The elastictwisting energy creates the tendency for the bolt to springbackunder subsequent cyclic transverse loading. The values of the fric-tion torques Tb� and Tt� should be determined based on static equi-librium conditions if the critical friction condition cannot be met.After tightening, the twisting torque will decrease from its initialvalue Tb0� to Tp due to springback effect. The initial reduction ofthe twisting torque Ttwist can be determined using Eqs. �21�, �22�,�35�, �36�, �53�, and �55�.

In order to proceed with the experimental verification, the fol-lowing variables are introduced:

�b =Tb�L

GJp�59�

vb1 = xh�t� − xt�t� �60�

b = vb1/�b �61�

After the twisting torque decreases to Tp, most of �b is rigidbody rotation of the bolt caused by self-loosening under cyclictransverse excitation. The twisting torque Ttwist is equal to Tb�=Tp−Tt�. On the one hand, when the bearing friction coefficient�b and preload Fi are high enough, Ttwist will be equal to Tpbecause there is no relative slippage between the bolt underheadand the joint surface. On the other hand, a relatively low preloadmay lead to the self-loosening of the bolt. In such case, the sum ofthe critical thread friction torque and the critical bearing frictiontorque �which resist the self-loosening� will be intermittentlysmaller than Tp during a part of each half cycle of the transversevibration. Vibration-induced loosening of the bolt is caused byincremental rotation motion of the bolt that occurs in an intermit-tent fashion over each cycle. This means that the dynamic rota-tional acceleration � in Eq. �7� is very small, which causes theright hand side �Tp−Tbc� −Ttc� � to approach zero even though the

Fig. 15 The bending effect on the variation of the critical bear-ing torque Tb under transverse cyclic excitation

self-loosening occurs.



F„modified junker machine…

preload of 9780 N

Journal of Vibration and Acoustics

Downloaded 12 Dec 2011 to 195.83.11.66. Redistribution subject to ASME

To experimentally measure the twisting torque Ttwist, straingauged bolts are used in the vibration loosening apparatus �modi-fied Junker machine�, as shown in Fig. 16. Prior to the vibrationloosening test, a separate test is conducted for determining thebearing friction coefficient �b=0.165 and the thread friction co-efficient �t=0.16. The two friction coefficients are assumed to beconstant in our proposed analytical model.

Figures 17 and 18 show the comparison of the experimental andanalytical results for the twisting torque Ttwist under the variouscyclic transverse excitations and preload levels. Figure 17 showsthe experimental and analytical results for the bearing frictiontorque Tb� and the thread friction torque Tt� versus time for a cyclictransverse excitation �0=0.254 mm and a bolt preload of13,928 N; no vibration self-loosening was observed. The resultsshow a rapid decrease in the first cycle, which is showed by Tb�approaching Tp, while Tt� approaches zero. Data show a goodcorrelation between the experimental and analytical results eventhough the experimental data show some small fluctuation at thenoise level.

Similarly, Fig. 18 shows the experimental and analytical val-ues of the twisting torque Ttwist variation under the excitation

ta on the twisting and thread friction3,928 N

ta on the twisting torque for a bolt

ig. 16 A schematic for the vibration loosening test machine

Fig. 17 Experimental and analytical datorques variation under the preload of 1

Fig. 18 Experimental and analytical da

APRIL 2009, Vol. 131 / 021009-11


�tcc

7

abatnftwbfmfvtehclwastttada

A

Ar

N

0

Do

0=0.508 mm for a bolt preload of 9780 N. The initial twistingorque variation is shown during the first few cycles. The analyti-al and experimental data for the twisting torque Ttwist are in goodorrelation.

ConclusionsBased on the relative slip movement at the contact interfaces

nd some assumptions of the bearing and thread pressure distri-utions, a new mathematical model for the contact friction torquesnd the shear forces under transverse excitation is derived to studyhe effect of different thread parameters on the self-loosening phe-omenon in threaded fasteners. In the model, the new integralormulation for the bearing friction torque, thread friction torque,he bearing shear force, and the thread shear force are developed,hich are dependent not only on the frictional coefficients, theolt preload, but also on the relative slippage of the contact sur-aces and the bolt bending effect. The effect of the bending mo-ent on the bearing shear forces and the thread surface shear

orce is taken into account. The model shows that a cyclic trans-erse excitation may or may not initiate the self-loosening of aightened threaded fastener, depending on the amplitude of thexcitation in relation to the level of bolt tension, the bolt under-ead bending stiffness, and the bearing and thread friction coeffi-ients. In order to minimize the potential of vibration-inducedoosening, bolts that are subjected to a higher vibration amplitudeill have to be tightened to higher preloads, have higher bearing

nd thread friction coefficients, and have a lower ratio nondimen-ional value that represents the bolt material, size, grip length, andhe underhead contact pressure distribution. The comparison ofhe theoretical and experimental data on the twisting torque showshat the proposed vibration loosening model can accurately char-cterize the mechanism of the self-loosening of the bolted jointsue to the vibration loads in the transverse direction to the boltxis.

cknowledgmentThe authors acknowledge the support from U.S. Army Tank

utomotive Command during the course of this study as part ofesearch Contract No. DAAE07-03-L110.

omenclatureE � Young’s modulus of boltF � bolt tension

Fbs � transverse bearing friction shear forceFbs max � maximum transverse bearing friction shear

forceFi � initial bolt tension �preload�

FJs � resistant contact force between the bolt shankand the hole surface in the upper plate.

Fts � thread friction shear force component along thex-direction

FxM � the bolt thread shear force component alongx-direction, caused by the nonuniform threadpressure

G � Shear elastic modulus of boltI � axial moment of inertia for the bolt cross sec-

tion areaJ � moment of inertia of the bolt

Jp � polar moment of inertia for the bolt cross sec-tion area

L � effective bolt lengthM � bending moment on the thread surfaces

M� � the reaction bending moment of bolt at the bolt
underhead
21009-12 / Vol. 131, APRIL 2009


Tb � bearing friction torqueTb� � bearing friction torqueTb� � actual bearing friction torque

Tb0� � initial actual bearing friction torque after tight-ening and removing the tightening torque

Tbc� � critical bearing friction torque under transversecyclic excitation

Tl � loosening torqueTp � pitch torque componentTt � thread friction torque componentTt� � thread friction torqueTt� � actual thread friction torqueTt0� � initial actual thread friction torque after tight-

ening and removing the tightening torqueTtc� � critical thread friction torque under transverse

cyclic excitationTtcr � maximum thread friction torque

Ttwist � the twisting torque of boltdFbf � bearing friction force increment vector on the

contact bearing surface area dSdFbf � bearing friction force increment on the contact

bearing surface area dSdFtf � thread friction force increment vector on the

contact thread surface area dS�dFtf � thread friction force increment on the contact

thread surface area dS�dFtx � thread friction force increment along

x-direction on the contact thread surface areadS�

dS � the contact bearing surface area incrementdS� � the contact thread surface area incrementdTb � bearing friction torque increment caused by

dFbfdTt � thread friction torque increment caused by dFtfdt � time increment

i � unit vector of x-axisj � unit vector of y-axisk � unit vector of z-axisk � bending factor

kb � bolt stiffnesskc � joint stiffnessp � thread pitch

qt � bolt thread pressure vectorqt � thread contact pressure

qt0 � average thread contact pressureqt� � thread pressure variation amplitudeqb � underhead contact pressure

qb0 � average underhead contact pressureqb� � underhead pressure variation amplituder � the position vector of a pointr � the radius in the polar coordinate system

rb � effective bearing contact radiusrmaj � major thread radiusrmin � minor thread radius

re � outer contact radius under bolt headri � internal contact radius under bolt headrt � effective thread radiust � time

ubx � elastic deformation displacement of bolt-upperjoint plate contact along the x-direction

utx � elastic deformation displacement of the boltthread-joint member thread contact along thex-direction

v � total relative velocity vector of the bolt threadwith respect to the joint thread

v1 � relative translation velocity vector of the boltunderhead



J

Do

v2 � relative velocity vector of the bolt underheadcaused by the relative rotation with respect tothe contact joint bearing surface

va � relative velocity vector of the bolt underheadwith respect to contact joint bearing surface

va � absolute value of the relative velocity vavb1 � relative translation velocity of bolt head along

the x-directionvtx � relative bolt thread translation velocity along

the x-directionvtz � relative bolt thread translation velocity along

the z-directionw1 � unit outward normal vector of the contact bolt

thread surfacex � x-axis coordinate

xh�t� � bolt head displacement along the x-directionxt�t� � bolt thread displacement along the x-direction

y � y-axis coordinate�bearing � contact bearing surface area�thread � contact bolt thread surface area

�xy � projected area of the contact bolt thread sur-face area �thread in x-y plane

� joint hole clearancet � thread clearance

�� loosening angle at time t+t�0� � loosening angle at time t

� � half of thread profile angle� � lead helix angle� � external transverse displacement

�0 � amplitude of the transverse cyclic displacement� � bending angle at the bolt underhead� � ratio of the major radius to the minor radius of

the bolt thread: �=rmaj /rmin� � constant dependent on the bending stiffness of

bolt head b � bearing translation-rotational ratio: vb1 /�b t � thread translation-rotational ratio: vtx /�t�b � bearing friction coefficient under bolt head�t � thread friction coefficient

�ts � static thread friction coefficient� � angle in the polar coordinate system

�e � relative elastic springback angle�� bolt rotation angle�� angular velocity of the transverse displacement

excitation



�0 � angular velocity at time t for self-loosening�b � relative angular velocity of the bolt underhead

with respect to the joint bearing surface�t � relative angular velocity of the thread surfaces

with respect to joint thread surface� � self-loosening angular acceleration of bolt� � self-loosening angular velocity at time t+t� � ratio �L / �8EIre /�re

4−ri4��

References�1� Hess, D., 1998, “Vibration and Shock Induced Loosening,” Handbook of Bolts

and Bolted Joints, J. H. Bickford and S. Nassar, eds., Dekker, New York, pp.757–824.

�2� Sakai, T., 1978, “Investigations of Bolt Loosening Mechanisms, 1st Report:On Bolts of Transversely Loaded Joints,” Bull. JSME, 21, pp. 1385–1390.

�3� Haviland, G. S., 1983, “Designing With Threaded Fasteners,” Mech. Eng.�Am. Soc. Mech. Eng.�, 105, pp. 17–31.

�4� Yamamoto, A., and Kasei., S., 1984, “A Solution for the Self-LooseningMechanism of Threaded Fasteners Under Transverse Vibration,” Bull. Jpn.Soc. Precis. Eng., 18, pp. 261–266.

�5� Tanaka, M., Hongo, K., and Asaba, E., 1982, “Finite Element Analysis of theThreaded Connections Subjected to External Loads,” Bull. JSME, 25, pp.291–298.

�6� Vinogradov, O., and Huang, X., 1989, “On A High Frequency Mechanism ofSelf-Loosening of Fasteners,” Proceedings of 12th ASME Conference on Me-chanical Vibration and Noise, Montreal, Quebec, Canada, DE Vol. 18, n pt4,1989, pp. 131–137.

�7� Zadoks, R. I., and Yu, X., 1993, “A Preliminary Study of the Self-Loosening inBolted Connections,” ASME Proceedings of the 14th Conference of Mechani-cal Vibration and Noise, New York, DE Vol. 54, 1993, pp. 79–88.

�8� Zadoks, R. I., and Yu, X., 1997, “An Investigation of the Self-LooseningBehavior of Bolts Under Transverse Vibrations,” J. Sound Vib., 208, pp. 189–209.

�9� Junker, G. H., 1969, “New Criteria for Self-Loosening of Fasteners UnderVibration,” SAE Trans., 78, pp. 314–335.

�10� Xu, G. Y., Zhu, W. D., and Emory, B. H., 2007, “Experimental and NumericalInvestigation of Structural Damage Detection Using Changes in Natural Fre-quencies,” ASME J. Vibr. Acoust., 129, pp. 686–700.

�11� Nichols, J. M., Trickey, S. T., Seaver, M., Motley, S. R., and Eisner, E. D.,2007, “Using Ambient Vibrations to Detect Loosening of a Composite-to-Metal Bolted Joint in the Presence of Strong Temperature Fluctuations,”ASME J. Vibr. Acoust., 129, pp. 710–717.

�12� 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, pp. 590–598.

�13� Nassar, S. A., and Housari, B. A., 2006, “Study of the Effect of Hole Clear-ance and Thread Fit on the Self-Loosening of Threaded Fasteners Due toCyclic Transverse Loads,” ASME J. Mech. Des., 128, pp. 586–594.

�14� Housari, B. A., and Nassar, S. A., 2007, “Effect of Thread and Bearing Fric-tion Coefficients on the Vibration-Induced Loosening of Threaded FastenersUnder Cyclic Transverse Loads,” ASME J. Vibr. Acoust., 129, pp. 1–9.

�15� Housari, B. A., 2007, “Vibration-Induced Loosening of Threaded Fasteners,”Ph.D. thesis, 2007, Oakland University, Rochester.

�16� Nassar, S. A., and Yang, X., 2007, “Novel Formulation on the Break-AwayTorque Components and the Torque-Tension Relationship in Threaded Fasten-ers,” ASME J. Pressure Vessel Technol., 129, pp. 653–663.

APRIL 2009, Vol. 131 / 021009-13


A Mathematical Model for Vibration Induced Loosening of Preloaded Thhreaded Fasteners

Documents

Transcript of A Mathematical Model for Vibration Induced Loosening of Preloaded Thhreaded Fasteners