Keywords

1 Introduction

Vibration-induced loosening of preloaded threaded fastener systems can lead to joint leakage at pressure boundaries, fatigue failure, or joint separation leading to significant warranty costs, catastrophic failure, safety risks, or loss of human life. Although there are some studies on the self-loosening of threaded fastener due to vibrations [1,2,3,4,5,6,7,8,9,10,11,12,13], none of the existing models has incorporated the thread profile angle as an independent variable that may have a significant effect on the vibration-induced loosening performance of preloaded threaded fastener systems. For example, Hess [7] provided a survey of some experimental and theoretical studies [4,5,6,7,8] on the vibration- and shock-induced loosening of threaded fasteners. The mechanism of vibration-induced loosening of threaded fasteners is primarily attributed to a reduction in the friction torque resistance, which is caused by the relative slippage between contact surfaces such as the interface between engaged threads and/or the interface between the bolt underhead and contacting joint surface [1,2,3,4,5,6,7,8]. Shoji and Sawa [13] proposed a three-dimensional finite element model to simulate the self-loosening of the threaded fastener under transverse cyclic loading. Their FEA simulation has been helpful to the understanding of the mechanics of the self-loosening process.

Nassar and Housari [14,15,16] developed an experimentally validated linear mathematical model for studying the loosening phenomenon in a preloaded threaded fastener system under a cyclic transverse load. The main variables in their model included the thread pitch, bolt preload, hole clearance, thread fit, bearing and thread friction coefficients, and the excitation amplitude of the cyclic transverse load. It was concluded that these parameters had a significant effect on the self-loosening of threaded fasteners. Their linear model has resulted in a linear correlation between the number of cycles-to-complete loss of bolt tension and the initial bolt preload. Subsequently, Nassar and Yang [17, 18] developed a more accurate nonlinear vibration-induced loosening model that was based on process, based on the relative slippage between two sets of contact surfaces, namely the contacting surfaces of the engaged bolt and nut threads (that had standard profile angles of 60°) and between the bearing contact between the bolt underhead and joint surfaces.

In this study, Nassar and Yang’s model [17] is extended to incorporate the effect of the nut thread profile angle (as an independent design variable) on the loosening performance of a preloaded threaded joint model in which the engaged bolt threads have a standard 60° profile angle. Other variables investigated in this study include thread fit clearance, bolt hole clearance, frictional coefficients, and bolt preload level. An experimental procedure and test setup are developed for validating the proposed model.

2 Formulation of Frictional Torque and Shear Forces

This section is primarily dedicated to incorporating the engaged nut thread profile angle (as an independent variable) into the thread friction analysis of engaged bolt threads in a preloaded bolt–nut system that is subjected to a cyclic transverse load. However, frictional analysis of the bolt underhead contact surface with that of the joint surface, shown in Fig. 1, would be the same as that provided in Nassar and Yang model [17] as summarized in the following section.

Fig. 1
figure 1

Schematic of a bolt–nut model under harmonic transverse excitation

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2.1 Underhead Bearing Friction Analysis

Slippage and friction force illustrations on the underhead bearing contact surface are shown in Fig. 2. A transverse bearing friction shear force \(\overline{F}_{\text{bs}}\) and the corresponding bearing friction torque \(\overline{T}_{\text{b}}\) are given by Nassar and Yang [17] in terms of a newly defined speed ratio \(\eta_{\text{b}}\) as follows:

Fig. 2
figure 2

Schematic diagram of underhead bearing sliding speeds bearing friction shear and torque

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$$\overline{{F}}_{{{bs}}} = {f}_{{{{3}}{b}}} \left( {{\eta}_{{b}} } \right) = \left| {\frac{{{F}_{{{bs}}} }}{{{\mu}_{{b}} {q}_{{{b}{{0}}}} }}} \right| = \int\limits_{{{r}_{{i}} }}^{{{r}_{{o}} }} {{rdr}} \int\limits_{{{0}}}^{{{{2}}{\pi}}} {\frac{{\left( {{\eta}_{{b}} + {r}\sin{\theta}} \right){d\theta }}}{{\sqrt {{\eta}_{{b}}^{{{2}}} + {r}^{{{2}}} + {{2}}{\eta}_{{b}} {r}\sin{\theta}} }}}$$
(1)
$$\overline{T}_{\text{b}} = f_{{1{\text{b}}}} \left( {\eta_{\text{b}} } \right) = \left| {\frac{{T_{\text{b}} }}{{\mu_{\text{b}} q_{\text{b0}} }}} \right| = \int\limits_{{r_{\text{i}} }}^{{r_{\text{o}} }} {r^{2} {\text{d}}r} \int\limits_{0}^{2\pi } {\frac{{\left( {\eta_{\text{b}} \sin \theta + r} \right){d}\theta }}{{\sqrt {\eta_{\text{b}}^{2} + r^{2} + 2\eta_{\text{b}} r\sin \theta } }}}$$
(2)
$$\eta_{\text{b}} = v_{\text{bx}} /\omega_{\text{t}}$$
(3)

where \(\eta_{\text{b}}\) is simply the ratio of the bolt underhead translational (sliding) speed \(v_{{1{\text{b}}}}\) in the x-direction to the bolt head angular (loosening) speed \(\omega_{\text{t}}\), \(T_{\text{b}}\) is the sliding underhead bearing friction torque, \(F_{\text{bs}}\) is the underhead bearing friction shear force, \(\mu_{\text{b}}\) is the underhead bearing friction coefficient, \(q_{\text{b0}}\) is the average underhead contact pressure, \(r_{\text{i}}\) and \(r_{\text{o}}\) are, respectively, the minimum and maximum underhead contact radii; r and \(\theta\) are the polar coordinates within the underhead contact area.

Equations (1) and (2) are numerically solved for ratios \(\overline{F}_{\text{bs}}\) and \(\overline{T}_{\text{b}}\) in terms of the speed ratio \(\eta_{\text{b}}\) as a variable. Figure 3 shows that the bearing friction torque ratio \(\overline{T}_{\text{b}}\) (solid line) rapidly decreases with increasing \(\eta_{\text{b}}\) in the range \(0 \le \eta_{\text{b}} \le 75\), and it asymptotes to a nearzero value for \(\eta_{\text{b}} \ge 300\). However, the bearing shear force ratio \(\overline{F}_{\text{bs}}\) (dotted line) rises sharply from zero (at \(\eta_{\text{b}} = 0\)) to a saturation value of \(A_{\text{b}} = \pi \left( {r_{\text{o}}^{2} - r_{\text{i}}^{2} } \right)\) when \(\eta_{\text{b}} \ge 50\), approximately. The condition \(\eta_{\text{b}} = 0\) represents the case where the relative slippage is only due to bolt head rotation, as the force \(F_{\text{bs}}\) becomes zero in Eq. (1); the corresponding bearing friction torque in Eq. (2) is \(T_{\text{b}} = \mu_{\text{b}} F_{\text{b}} r_{\text{b}}\), where \(F_{\text{b}}\) is the bolt tension and \(r_{\text{b}}\) is equivalent bearing friction radius. As \(\eta_{\text{b}} \to \infty\), however, the relative slippage becomes pure linear translational movement against the bearing friction force that peaks at \(F_{\text{bs}} = \mu_{\text{b}} F_{\text{b}}\) (with \(T_{\text{b}} = 0\)).

Fig. 3
figure 3

Bearing friction ratios \(\overline{T}_{\text{b}}\) and \(\overline{F}_{\text{bs}}\) versus bearing speed ratio \(\eta_{\text{b}}\) for M10 × 1.5 bolt–nut model

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2.2 Incorporation of Nut Thread Profile Angle into Frictional Analysis of Engaged Thread Surfaces

With reference to Fig. 4, since the (lead) helix angle is fairly small for standard threaded fastener products, its effect would be insignificant to the relationship between the x- and z-displacements at Point A and the bending angle of the thread flank. The kinetic relationship is hence established between the bending angle of the thread flank and the slippage of bolt threads with respect to the nut threads. At point A, the relative movements between the bolt and the nut are bending rotation angle \({\Delta} \theta_{\text{A}}\), and linear x- and z-displacements \({\Delta} x_{\text{A}}\) and \({\Delta} z_{\text{A}}\), respectively. The key parameter \({\Delta} \theta_{\text{A}}\) for the effect of geometry clearance on the loosening performance is investigated in this study. The \(x - z\) coordinate system is fixed on bolt thread while the \(x^{\prime} - z^{\prime}\) system is fixed on nut thread. Before any relative bending rotation occurs between bolt and nut, the original two coordinate systems coincide and have the same original point A. When a bending rotation occurs, the \(x^{\prime} - z^{\prime}\) coordinate system has a relative translational displacement and a relative bending rotation with respect to the \(x - z\) system. The relationship between the two coordinate systems is given by

Fig. 4
figure 4

Simplified 2-D schematic of local bolt and nut axes and relative slippage between engaged threads

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$$\left\{ {\begin{array}{*{20}c} x \\ z \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {{\Delta} x_{\text{A}} } \\ {{\Delta} z_{\text{A}} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} {\cos {\Delta} \theta_{\text{A}} } & { - \sin {\Delta} \theta_{\text{A}} } \\ {\sin {\Delta} \theta_{\text{A}} } & {\cos {\Delta} \theta_{\text{A}} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {x^{\prime } } \\ {z^{\prime } } \\ \end{array} } \right\}$$
(4)

For the thread surface shown in Fig. 4, we have

$$z = \pm x\tan \alpha$$
(5)
$$z^{\prime} = \pm x^{\prime}\tan \alpha$$
(6)

where \(\alpha\) is half of the thread profile angle.

Equation (4) gives the following values for x and z as follows:

$$x = {\Delta} x_{\text{A}} + x^{\prime } - {\Delta} \theta_{\text{A}} z^{\prime }$$
(7)
$$z = {\Delta} z_{\text{A}} + {\Delta} \theta_{\text{A}} x^{\prime } + z^{\prime }$$
(8)

For point C on the nut thread where \(x = r_{ {\min} }\), \(z = r\tan \alpha_{ {\min} }\), \(z^{\prime} = x^{\prime}\tan \alpha\), we have

$${\Delta} x_{\text{A}} + x^{\prime } \left( {1 - {\Delta} \theta_{\text{A}} \tan \alpha } \right) = r_{ {\min} }$$
(9)
$$r_{{\min} } \tan \alpha = {\Delta} {z}_{\text{A}}+x^{{\prime }} \left( {\Delta} {\theta}_{\text{A}}+ \tan \alpha \right)$$
(10)

where \(r_{ {\min} }\) is the minor thread radius of the nut.

Solving Eqs. (9) and (10) and ignoring higher order terms gives the kinematic equation for point A as follows:

$${\Delta} z_{\text{A}} - {\Delta} x_{\text{A}} \tan \alpha = - {\Delta} \theta_{\text{A}} \left[ {r_{ {\min} }\left( 1+\tan^{2} \alpha \right) } \right]$$
(11)

Similarly, for point D on the bolt thread (Fig. 4), using the geometric relations \(x^{\prime} = - r_{\text{maj}} ,z^{\prime} = r_{\text{maj}} \tan \alpha\), and \(z = - x\tan \alpha\), where \(r_{\text{maj}}\) is the major thread radius and combing with Eq. (11) gives the x- and z-translational displacements \({\Delta} x_{\text{A}}\) and \({\Delta} z_{\text{A}}\) of point A and the corresponding translational velocity components \(v_{\text{Ax}}\) and \(v_{\text{Az}}\) as follows:

$${\Delta} x_{\text{A}} = \frac{{(r_{\text{maj}} + r_{ {\min} }) }}{{\sin 2\alpha_{\text{A}} }}{\Delta \theta_{\text{A}}}$$
(12)
$${\Delta} z_{\text{A}} = \left( {\frac{{r_{\text{maj}} - r_{ {\min} } }}{2}\sec^{2} \alpha_{\text{A}} }{\Delta}{\theta}_{\text{A}} \right)$$
(13)
$$v_{\text{Ax}} = \frac{{r_{\text{maj}} + r_{{\min} } }}{{\sin 2\alpha }} {\Delta} {\theta}_{\text{A}}$$
(14)
$$v_{\text{Az}} = \left( {\frac{{r_{\text{maj}} - r_{{\min} } }}{2}} \right) \sec^{2} \alpha \cdot {\theta}_{\text{A}}$$
(15)

where \(\omega_{\text{A}}\) is the time derivative \({\text{d}}\theta_{\text{A}} /{\text{d}}t\) of the bending angle of the bolt thread.

Equations (14) and (15) characterize the relationship between the translational velocity components of point A and the bending angular speed of thread contact in x-z plane. However, the relative tangential velocity vector \(\varvec{v}_{\text{t}}\) on the engaged threads would also include a relative rotational component about the bolt axis z. Thus, at any point on the bolt thread contact surface, the relative velocity \(\varvec{v}_{\text{t}}\) with respect to the engaged nut threads is given by

$$\begin{aligned} \varvec{v}_{\text{t}} & = \varvec{v}_{\text{A}} + \left( { - \omega_{\text{A}} \varvec{j}} \right) \times \varvec{r} + \left( {\omega_{\text{t}} \varvec{k}} \right) \times \varvec{r} = \left[ {v_{\text{Ax}} - \omega_{\text{A}} \left( {\frac{p}{2\pi }\theta + r\tan \alpha } \right) - \omega_{\text{t}} r\sin \theta } \right]\varvec{i} \\ & \quad + \omega_{\text{t}} r\cos \theta \varvec{j} + \left( {v_{\text{Az}} + \omega_{\text{A}} r\cos \theta } \right)\varvec{k} \\ \end{aligned}$$
(16)

where p is the thread pitch, \(\omega_{\text{t}}\) is the relative angular velocity of the thread surface about the z-axis with respect to nut thread surface.

The differential thread friction force vector \({\text{d}}\varvec{F}_{\text{t}}\) on the representative area element of the engaged thread contact area is given by

$${\text{d}}\varvec{F}_{\text{t}} = \mu_{\text{t}} q_{\text{t}} {\text{d}}S\frac{{\varvec{v}_{\text{t}} }}{{\left| {\varvec{v}_{\text{t}} } \right|}}$$
(17)

where \(\mu_{\text{t}}\) is the thread friction coefficient, \(q_{\text{t}}\) is the thread contact pressure, and \(\frac{{\varvec{v}_{\text{t}} }}{{\left| {\varvec{v}_{\text{t}} } \right|}}\) is a unit vector in the tangential direction. The corresponding transverse friction force component in the x-direction would be obtained from the dot product of an x-unit vector i with vector Eq. (17); the total transverse friction force \(F_{\text{ts}}\) on the thread surface would then be given by

$$\begin{aligned} F_{\text{ts}} & = \iint\limits_{{\varOmega_{\text{thread}} }} {\mu_{\text{t}} q_{\text{t}} \frac{{\varvec{v}_{\text{t}} \cdot \varvec{i}}}{{\left| {\varvec{v}_{\text{t}} } \right|}}{\text{d}}S} \\ & = \int\limits_{{r_{{\min} } }}^{{r_{\text{maj}} }} {r{\text{d}}r} \int\limits_{0}^{2n\pi } {\frac{{\mu_{\text{t}} q_{\text{t0}} \left\{ {\frac{{v_{\text{Ax}} }}{{\omega_{\text{t}} }} - \frac{{\omega_{\text{A}} }}{{\omega_{\text{t}} }}\left( {\frac{p}{2\pi }\theta + r\tan \alpha } \right) - r\sin \theta } \right\}\sqrt {\sec^{2} \alpha + \tan^{2} \beta } {\text{d}}\theta }}{{\sqrt {\left[ {\frac{{v_{\text{Ax}} }}{{\omega_{\text{t}} }} - \frac{{\omega_{\text{A}} }}{{\omega_{\text{t}} }}\left( {\frac{p}{2\pi }\theta + r\tan \alpha } \right) - r\sin \theta } \right]^{2} + r^{2} \cos^{2} \theta + \left[ {\frac{{v_{\text{Az}} }}{{\omega_{\text{t}} }} + \frac{{\omega_{\text{A}} }}{{\omega_{\text{t}} }}r\cos \theta } \right]^{2} } }}} \end{aligned}$$
(18)

where \(\varOmega_{\text{thread}}\) is the overall thread contact surface area between all engaged n threads, \(q_{\text{t0}}\) is average thread contact pressure, and \(\beta\) is the lead helix angle of the bolt and nut threads.

Additionally, using Eq. (17), the thread friction torque increment \({\text{d}}T_{\text{t}}\) would be given by

$$\begin{aligned} dT_{\text{t}} & = \left( {\varvec{r} \times {\text{d}}\varvec{F}_{\text{t}} } \right) \cdot \varvec{k} \\ & = \left| {\begin{array}{*{20}c} \varvec{i} & \varvec{j} & \varvec{k} \\ {r\cos \theta } & {r\sin \theta } & {r\tan \alpha + \frac{p\theta }{2\pi }} \\ {v_{\text{tx}} } & {v_{\text{ty}} } & {v_{\text{tz}} } \\ \end{array} } \right| \cdot \varvec{k}\frac{{\mu_{\text{t}} q_{\text{t0}} }}{{v_{\text{t}} }}{\text{d}}S = \left( {r\cos \theta v_{\text{ty}} - r\sin \theta v_{\text{tx}} } \right)\frac{{\mu_{\text{t}} q_{\text{t0}} }}{{v_{\text{t}} }}{\text{d}}S \\ \end{aligned}$$
(19)

where \(v_{\text{tx}}\), \(v_{\text{ty}}\), and \(v_{\text{tz}}\) are the corresponding components of the sliding velocity vector \(\varvec{v}_{t}\) along x- , y- and z-directions, respectively. Taking the surface integral on the engaged thread surface \(\varOmega_{\text{thread}}\) gives the total thread friction torque \(T_{\text{t}}\) as follows:

$$\begin{aligned} T_{\text{t}} & = \iint\limits_{{\varOmega_{\text{thread}} }} {\mu_{\text{t}} q_{\text{t0}} \frac{{\left( {r\cos \theta v_{\text{ty}} - r\sin \theta v_{\text{tx}} } \right)}}{{v_{\text{t}} }}{\text{d}}S} \\ & = \int\limits_{{r_{{\min} } }}^{{r_{\text{maj}} }} {r{\text{d}}r} \int\limits_{0}^{2n\pi } {\frac{{\mu_{\text{t}} q_{\text{t0}} \left\{ {r^{2} - r\sin \theta \left[ {\frac{{v_{\text{Ax}} }}{{\omega_{\text{t}} }} - \frac{{\omega_{\text{A}} }}{{\omega_{\text{t}} }}\left( {\frac{p}{2\pi }\theta + r\tan \alpha } \right)} \right]} \right\}\sqrt {\sec^{2} \alpha + \tan^{2} \beta } {\text{d}}\theta }}{{\sqrt {\left[ {\frac{{v_{\text{Ax}} }}{{\omega_{\text{t}} }} - \frac{{\omega_{\text{A}} }}{{\omega_{\text{t}} }}\left( {\frac{p}{2\pi }\theta + r\tan \alpha } \right) - r\sin \theta } \right]^{2} + r^{2} \cos^{2} \theta + \left[ {\frac{{v_{\text{Az}} }}{{\omega_{\text{t}} }} + \frac{{\omega_{\text{A}} }}{{\omega_{\text{t}} }}r\cos \theta } \right]^{2} } }}} \\ \end{aligned}$$
(20)

Using Eqs. (18) and (20) and following a procedure similar to that employed in Sect. 2.1, the transverse thread force \(\overline{F}_{\text{ts}}\) and thread torque \(\overline{T}_{\text{t}}\) are obtained as follows:

$$\overline{F}_{\text{ts}} = \frac{{F_{\text{ts}} }}{{n\mu_{\text{t}} q_{\text{t0}} }} = \int\limits_{{r_{ {\min} } }}^{{r_{\text{maj}} }} {r{\text{d}}r\int\limits_{0}^{2\pi } {\frac{\begin{aligned} & \left\{ {\left[ {\frac{{r_{\text{maj}} + r_{{\min} } }}{\sin 2\alpha } - \left( {\frac{p}{2\pi }\theta + r\tan \alpha } \right)} \right]\eta_{\text{t}} - r\sin \theta } \right\} \\ & \quad \sqrt {\sec^{2} \alpha + \tan^{2} \beta } {\text{d}}\theta \\ \end{aligned} }{\begin{aligned} & \sqrt {\left\{ {\left[ {\frac{{r_{\text{maj}} + r_{{\min} } }}{\sin 2\alpha } - \left( {\frac{p}{2\pi }\theta + r\tan \alpha } \right)} \right]^{2} + \left[ {\frac{{r_{{{\text{ma}}j}} - r_{{\min} } }}{2}\sec^{2} \alpha + r\cos \theta } \right]^{2} } \right\}\eta_{\text{t}}^{2} } \\ & \quad \sqrt { + r^{2} - 2\left[ {\frac{{r_{\text{maj}} + r_{{\min} } }}{\sin 2\alpha } - \left( {\frac{p}{2\pi }\theta + r\tan \alpha } \right)} \right]\eta_{\text{t}} r\sin \theta } \\ \end{aligned} }} }$$
(21)
$$\overline{T}_{\text{t}} = \frac{{T_{\text{t}} }}{{n\mu_{\text{t}} q_{\text{t0}} }} = \int\limits_{{r_{{\min} } }}^{{r_{\text{maj}} }} {r{\text{d}}r} \int\limits_{0}^{2\pi } {\frac{{\left\{ {r^{2} + \eta_{\text{t}} \left( {\frac{p\theta }{2\pi } + r\tan \alpha - \frac{{r_{\text{maj}} + r_{{\min} } }}{\sin 2\alpha }} \right)r\sin \theta } \right\}\sqrt {\sec^{2} \alpha + \tan^{2} \beta } {\text{d}}\theta }}{\begin{aligned} & \sqrt {\left\{ {\left[ {\frac{{r_{\text{maj}} + r_{{\min} } }}{\sin 2\alpha } - \left( {\frac{p}{2\pi }\theta + r\tan \alpha } \right)} \right]^{2} + \left[ {\frac{{r_{\text{maj}} - r_{{\min} } }}{2}\sec^{2} \alpha + r\cos \theta } \right]^{2} } \right\}\eta_{\text{t}}^{2} } \\ & \quad \sqrt { + r^{2} - 2\left[ {\frac{{r_{\text{maj}} + r_{{\min} } }}{\sin 2\alpha } - \left( {\frac{p}{2\pi }\theta + r\tan \alpha } \right)} \right]\eta_{\text{t}} r\sin \theta } \\ \end{aligned} }}$$
(22)

where the bending-to-torsional angular (sliding) speed ratio \(\eta_{\text{t}}\) on the threads is defined by

$$\eta_{\text{t}} = \frac{{\omega_{\text{A}} }}{{\omega_{\text{t}} }}$$
(23)

Integral Eqs. (21) and (22) are numerically solved for \(\overline{F}_{\text{ts}} \left( {f_{{3{\text{t}}}} \left( {\eta_{\text{t}} } \right)} \right)\) and \(\overline{T}_{\text{t}} \left( {f_{{ 1 {\text{t}}}} \left( {\eta_{\text{t}} } \right)} \right)\) in terms of the bending-to-torsional rotational sliding speed ratio \(\eta_{\text{t}}\). Figure 5 shows that the (sliding) thread friction torque \(\overline{T}_{\text{t}}\) (solid line) decreases rapidly in the range \(0 \le \eta_{\text{t}} \le 20\); its value approaches zero for \(\eta_{\text{t}} > 30\), although the limit of the right-hand side of Eq. (22) goes to zero as \(\eta_{\text{t}} \to \infty\). The thread friction transverse shear force ratio \(\overline{F}_{\text{ts}}\) (dotted line) rises rapidly from zero at \(\eta_{\text{t}} = 0\) to its saturation value.

Fig. 5
figure 5

Thread friction ratios \(\overline{T}_{\text{t}}\) and \(\overline{F}_{\text{ts}}\) versus thread speed ratio \(\eta_{\text{t}}\) for M10 × 1.5 bolt–nut model

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Under the transverse cyclic load (Fig. 1), intermittent vibration-induced loosening of the bolt would only occur when the value of system’s pitch torque component \(T_{\text{p}} = \frac{p}{2\pi }F_{\text{b}}\) from bolt–nut preloading [19] exceeds the sum of the sliding bearing friction torque component \(T_{\text{b}}\) and the sliding thread friction torque component \(T_{\text{t}}\). From Eqs. (2) and (22), the condition for vibration-induced bolt loosening is given by

$$\left[ {T_{\text{p}} = \frac{p}{2\pi }F_{\text{b}} } \right] > \left[ {\left( {T_{\text{b}} = \mu_{\text{b}} nq_{\text{b0}} \overline{T}_{\text{b}} } \right) + \left( {T_{\text{t}} = \mu_{\text{t}} nq_{\text{t0}} \overline{T}_{\text{t}} } \right)} \right]$$
(24)

2.3 Transverse Shear Force on Bolt Thread Surface Due to Bending

The contact pressure distribution on preloaded engaged thread surfaces varies significantly along the bolt axis, with the highest contact pressure being on the first fully engaged thread near the face of the nut [20, 21]. Thread contact pressure rapidly decreases with the distance from the first fully engaged thread. Bolt bending due to the externally applied transverse excitation would cause non-uniform distribution on each thread surface as illustrated by Fig. 6. A nonlinearly varying thread contact pressure vector \(\varvec{q}_{\text{t}}\) is assumed as follows:

Fig. 6
figure 6

Schematic of thread contact pressure distribution on threads

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$$\varvec{q}_{\text{t}} = - \left[ {1 - \frac{p\theta }{{3\pi h_{0} }} - \frac{1}{3}\left( {\frac{p\theta }{{2\pi h_{0} }}} \right)^{2} } \right]\left( {q_{{{\text{t}}0}}^{{\prime }} + \lambda_{\text{t}} \varphi_{\text{Nt}} \frac{r\cos \theta }{{r_{\text{maj}} }}} \right)\varvec{w}_{1}$$
(25)

where \(q_{\text{t0}}^{{\prime }}\) is the average thread contact pressure on the first engaged thread surface, z is the axial location, x is the transverse location, \(\lambda_{\text{t}}\) is a constant, \(\varphi_{\text{Nt}}\) is the bending angle at bolt threads, \(h_{0}\) is the thread engagement length, \(\varvec{w}_{1}\) is the unit vector normal to the thread surface, which is given by

$$\varvec{w}_{1} = \frac{{ - \left( {\tan \alpha \cos \theta + \tan \beta \sin \theta } \right)\varvec{i} - \left( {\tan \alpha \sin \theta - \tan \beta \cos \theta } \right)\varvec{j} + \varvec{k}}}{{\sqrt {\sec^{2} \alpha + \tan^{2} \beta } }}$$
(26)

The bolt preload \(F_{\text{b}}\) is obtained by integrating the contact pressure overall engaged thread surfaces as follows:

$$\begin{aligned} {F_b} & = \iint\limits_{{\Omega _{{\text{thread}}}}} {\left| {{{\text{q}}_t} \cdot {\text{k}}} \right|}ds \approx \int\limits_{{r_{{\text{maj}}}}}^{{r_{{\text{min}}}}} r dr\int\limits_0^{2n\pi } {q_{t0}^\prime } \left[ {1 - \frac{{p\theta }}{{3\pi {h_0}}} - \frac{1}{3}{{\left( {\frac{{p\theta }}{{2\pi {h_0}}}} \right)}^2}} \right]d\theta \\ \quad \approx q_t^\prime 0n\pi \left( {r_{{\text{maj}}}^2 - r_{{\text{min}}}^2} \right)\left\{ {1 - \frac{1}{3} - \frac{1}{9}} \right\} = \frac{5}{9}q_t^\prime 0n\pi \left( {r_{{\text{maj}}}^2 - r_{{\text{min}}}^2} \right) \\ \end{aligned} $$
(27)

Thus, the average thread contact pressure \(q_{\text{t0}}^{\prime }\) on the surface of the first engaged thread would be given by

$$q_{\text{t0}}^{\prime } = \frac{{9F_{\text{b}} }}{{5n\pi \left( {r_{\text{maj}}^{2} - r_{ {\min} }^{2} } \right)}}$$
(28)

The total transverse thread shear force component \(F_{\text{xM}}\) (along x-direction) caused by the non-uniform pressure distribution on all thread surfaces would be through dot product by the unit vector i as follows:

$$\begin{aligned} F_{\text{xM}} & = \iint\limits_{{\varOmega_{\text{thread}} }} {\left| {\varvec{q}_{\text{t}} \cdot \varvec{i}} \right|{\text{d}}s} \\ & \approx \int\limits_{{r_{ {\min} } }}^{{r_{\text{maj}} }} {r{\text{d}}r} \int\limits_{0}^{2n\pi } {\left( {\tan \alpha \cos \theta } \right.} \\ & + \left. {\tan \beta \sin \theta } \right)\left[ {1 - \frac{p\theta }{{3\pi h_{0} }} - \frac{1}{3}\left( {\frac{p\theta }{{2\pi h_{0} }}} \right)^{2} } \right]\lambda_{\text{t}} \varphi_{\text{Nt}} \left( {\frac{r\cos \theta }{{r_{\text{maj}} }}} \right) d\theta\\ & \approx \frac{{5n\pi \tan \alpha \lambda_{\text{t}} \varphi_{\text{nt}} }}{{27r_{\text{maj}} }}\left( {r_{\text{maj}}^{3} - r_{{\min} }^{3} } \right) \\ \end{aligned}$$
(29)

The corresponding bending moment \(M_{\text{Nt}}\) caused by the non-uniform thread contact pressure \(\varvec{q}_{\text{t}}\) about y-axis at the nut axis center is formulated as follows:

$$\begin{array}{c} {M_{{\rm{Nt}}}} = \iint\limits_{{\Omega _{{\rm{xy}}}}} \left\{ {r\cos \theta + \left[ {r\tan \alpha + \frac{\theta }{{2\pi }}p} \right]\left( {\tan \alpha \cos \theta + \tan \beta \sin \theta } \right)} \right\}\left[ {1 - \frac{{p\theta }}{{3\pi {h_0}}}} \right. \\ \quad \left. { - \frac{1}{3}{{\left( {\frac{{p\theta }}{{2\pi {h_0}}}} \right)}^2}} \right]{\lambda _{\rm{t}}}{\varphi _{{\rm{Nt}}}}\frac{{r\cos \theta }}{{{r_{{\rm{maj}}}}}}r{\rm{d}}r{\rm{d}}\theta = \frac{{{\lambda _{\rm{t}}}{\varphi _{{\rm{nt}}}}}}{{{r_{{\rm{mx}}}}}} \\ \int_{{r_{\min }}}^{{r_{{\rm{maj}}}}} {{r^3}} {\rm{d}}r\int\limits_0^{2n\pi } {\left[ {\cos \theta - \left( {\theta \tan \beta + \tan \alpha } \right)\left( {\tan \alpha \cos \theta + \tan \beta \sin \theta } \right)} \right]\cos \theta d\theta } \\ \quad \approx \frac{{n\pi {\lambda _{\rm{t}}}{\varphi _{{\rm{Nt}}}}}}{{{r_{{\rm{maj}}}}}}\left( {\left. {r_{{\rm{maj}}}^4 - r_{\min }^4} \right)\left[ {\frac{5}{{36}}{{\sec }^2}\alpha + \frac{{7n\pi }}{{72}}\tan \alpha \tan \beta } \right]} \right. = {\beta _{{\rm{Nt}}}}{\varphi _{{\rm{N}}t}} \\ \end{array}$$
(30)

where \(\varOmega_{\text{xy}}\) is the projected area of the thread contact area on x-y plane, and the bending stiffness \(\beta_{\text{Nt}}\) of the engaged threads is given by

$$\beta_{\text{Nt}} \approx \frac{{n\pi \lambda_{\text{t}} }}{{r_{\text{maj}} }}\left( {r_{\text{maj}}^{4} - r_{ {\min} }^{{4}} } \right) \left[ {\frac{5}{36}\sec^{2} \alpha + \frac{7n\pi }{72}\tan \alpha \tan \beta } \right]$$
(31)

From Eqs. (29) and (30), the relationship between the thread shear force \(F_{\text{xM}}\) and the corresponding bending moment \(M_{\text{Nt}}\) on the thread surfaces may be expressed as follows:

$$F_{\text{xM}} = \frac{{M_{\text{Nt}} }}{{\rho_{\text{M}} }}\text{ , }\rho_{\text{M}} = \frac{{r_{min}\left(\gamma^4-1 \right) \left( 30\sec^{2} \alpha +21 n \pi\tan \alpha \tan \beta \right) }}{{40\tan \alpha \left( {\gamma^{3} - 1} \right)}}$$
(32)

where the effective bending moment arm \(\rho_{\text{M}}\) and the number of fully engaged threads in Fig. 6 is \(n = h_{0} /p\).

As long as bolt bending (due to \(M_{\text{Nt}}\)) does not consume the hole clearance (to causes direct contact between the bolt shank and the internal surface of the hole), the thread friction shear force component \(F_{\text{ts}}\) in the x-direction would be given by

$$F_{\text{ts}} = F_{\text{bs}} + M_{\text{Nt}} /\rho_{\text{M}}$$
(33)

2.4 Rate of Bolt Loosening Due to Cyclic Transverse Load

In this section, various scenarios are used to formulate the rate of vibration-induced loosening per cycle in a preloaded bolt–nut model that incorporates the nut thread profile angle as an independent variable. Classical elastic beam theory is used in connection with the preceding formulation of the bearing shear force \(F_{\text{bs}}\), bolt bending moment \(M_{\text{b}}\), and twisting torque components \(T_{\text{b}}\) and \(T_{\text{tp}}\).

Scenario I: No slippage on any contact surfaces (bolt underhead, nut face, or thread surfaces)

The bolt deflection \(\delta_{\text{b}}\) and the bending angle \(\varphi_{\text{b}}\) at the junction with the bolt head (Fig. 6) are, respectively, given by

$$\delta_{\text{b}} = \frac{{F_{\text{bs}} L^{3} }}{3EI} - \frac{{M_{\text{b}} L^{2} }}{2EI} + \varphi_{\text{N}}^{\prime } L$$
(34)
$$\varphi_{\text{b}} = \frac{{F_{\text{bs}} L^{2} }}{2EI} - \frac{{M_{\text{b}} L}}{EI} + \varphi_{\text{N}}^{\prime }$$
(35)

where L is bolt grip length, E is Young’s modulus of bolt material, I is moment of inertia of the bolt cross-sectional area, \(M_{\text{b}}\) is bending moment acting at the bolt underhead surface, the angle \(\varphi_{\text{N}}^{{\prime }}\) is the sum of the nut rotational angle \(\varphi_{\text{Ns}}\) and the relative bending angle \(\varphi_{\text{Nt}}\) between the bolt threads and nut threads as follows:

$$\varphi_{\text{N}}^{\prime } = \varphi_{\text{Ns}} + \varphi_{\text{Nt}}$$
(36)

As the bolt and nut have small masses, when the transverse vibration frequency and excitation amplitude are not very high, the dynamic inertia force and moment can be ignored. The equilibrium conditions of the bolt–nut connection can make the following relationships

$$F_{\text{bs}} L = M_{\text{b}} + M_{\text{Ns}}, \quad F_{\text{bs}} = F_{{{\text{N}}s}},\quad M_{\text{Ns}} = M_{\text{Nt}}$$
(37)

where \(F_{\text{Ns}}\) is the shear force on the nut bearing contact surface along x-direction, and \(M_{\text{Ns}}\) is the bending moment on nut bearing contact surface.

The relationships between the bending moments and the corresponding rotations of the bolt underhead, nut bearing surface and the engaged threads are assumed to be expressed as follows:

$$M_{\text{b}} = \beta_{\text{b}} \varphi_{\text{b}},\quad M_{\text{Ns}} = \beta_{\text{Ns}} \varphi_{{{\text{N}}s}},\quad M_{\text{Nt}} = \beta_{\text{Nt}} \varphi_{\text{Nt}}$$
(38)

where \(\beta_{\text{b}}\), \(\beta_{\text{Ns}}\), and \(\beta_{\text{Nt}}\) are, respectively, the bending stiffness of bolt underhead to joint, nut bearing-to-joint, and bolt threads-to-nut. The combined bending stiffness \(\beta_{\text{N}}^{'}\) of the nut bearing-to-joint and the engaged threads to nut is expressed as follows:

$$\frac{1}{{\beta_{\text{N}}^{'} }} = \frac{1}{{\beta_{\text{Ns}} }} + \frac{1}{{\beta_{\text{Nt}} }}$$
(39)

From Eqs. (35) to (38), the following equations are also obtained

$$\begin{aligned} \delta_{\text{b}} & = \frac{{F_{\text{bs}} L^{3} }}{3EI}\left\{ {1 + \frac{{\frac{3EI}{{L\beta_{\text{N}}^{\prime } }} - \frac{3L}{4EI}\beta_{\text{b}} }}{{1 + \frac{{\beta_{\text{b}} L}}{EI} + \frac{{\beta_{\text{b}} }}{{\beta_{\text{N}}^{'} }}}}} \right\},\quad\varphi_{\text{b}} = \frac{{3EI\delta_{\text{b}} }}{{L^{2} }}\frac{{\frac{L}{2EI} + \frac{1}{{\beta_{\text{N}}^{\prime } }}}}{{1 + \frac{3EI}{{\beta_{\text{N}}^{\prime } L}} + \frac{{\beta_{\text{b}} L}}{4EI} + \frac{{\beta_{\text{b}} }}{{\beta_{\text{N}}^{'} }}}},\quad \\ \varphi_{\text{Nt}} & = \frac{{3EI\delta_{\text{b}} }}{{\beta_{\text{Nt}} L^{2} }}\frac{{1 + \frac{{\beta_{\text{b}} L}}{2EI}}}{{1 + \frac{3EI}{{\beta_{\text{N}}^{\prime } L}} + \frac{{\beta_{\text{b}} L}}{4EI} + \frac{{\beta_{\text{b}} }}{{\beta_{\text{N}}^{\prime } }}}} \end{aligned}$$
(40)

If the bolt underhead and nut bearing contact pressures are assumed to vary linearly along the x-direction (Fig. 6), the bending moments \(M_{\text{b}}\) and \(M_{\text{Ns}}\) are given by

$$M_{\text{b}} = \frac{{\lambda_{\text{b}} \pi }}{{4r_{\text{e}} }}\left( {r_{\text{e}}^{4} - r_{\text{i}}^{4} } \right)\varphi_{\text{b}} = \beta_{\text{b}} \varphi_{\text{b}},\quad M_{\text{Ns}} = \frac{{\lambda_{\text{Ns}} \pi }}{{4r_{\text{Ne}} }}\left( {r_{\text{Ne}}^{4} - r_{\text{Ni}}^{4} } \right)\varphi_{\text{Ns}} = \beta_{{{\text{N}}s}} \varphi_{{{\text{N}}s}}$$
(41)

where \(\lambda_{\text{b}}\) and \(\lambda_{\text{Ns}}\) are constants which affect \(\beta_{\text{b}}\) and \(\beta_{\text{Ns}}\), respectively, and they are presented as follows:

$$\beta_{\text{b}} = \frac{{\lambda_{\text{b}} \pi }}{{4r_{\text{e}} }}\left( {r_{\text{e}}^{4} - r_{\text{i}}^{4} } \right),\quad \beta_{{{\text{N}}s}} = \frac{{\lambda_{\text{Ns}} \pi }}{{4r_{\text{Ne}} }}\left( {r_{\text{Ne}}^{4} - r_{\text{Ni}}^{4} } \right)$$
(42)

Equation (40) shows the linear relationship between the bolt deflection \(\delta_{\text{b}}\) and the underhead friction shear force \(F_{\text{bs}}\) given by Eq. (1), which would also determine the value of the sliding underhead friction torque \(T_{\text{b}}\). Similarly, Eqs. (34), (38), and (40) give the transverse shear force \(F_{\text{ts}}\) on the engaged threads in terms of \(F_{\text{bs}}\) as follows:

$$\begin{aligned} F_{\text{ts}} & = F_{\text{bs}} + M_{\text{Nt}} /\rho_{\text{M}} = F_{\text{bs}} \left[ {1 + \left( {\frac{L}{{\rho_{\text{M}} }}} \right)\frac{{1 + \frac{{\beta_{\text{b}} L}}{2EI}}}{{1 + \frac{{\beta_{\text{b}} L}}{EI} + \frac{{\beta_{\text{b}} }}{{\beta_{\text{N}}^{\prime } }}}}} \right] = \chi^{\prime}F_{\text{bs}} , \\ \quad \chi^{\prime} & = \left[ {1 + \left( {\frac{L}{{\rho_{\text{M}} }}} \right)\frac{{1 + \frac{{\beta_{\text{b}} L}}{2EI}}}{{1 + \frac{{\beta_{\text{b}} L}}{EI} + \frac{{\beta_{\text{b}} }}{{\beta_{\text{N}}^{\prime } }}}}} \right] \end{aligned}$$
(43)

The expression for \(\chi^{\prime}\) in Eq. (43) represents the ratio of the thread friction force \(F_{\text{ts}}\) to the bearing friction shear force \(F_{\text{bs}}\), which is obviously larger than 1.

Scenario II: Slippage only on the engaged thread surfaces (without slippage under the bolt head or the nut face)

When slippage occurs on the thread contact surface (as the contact friction force has reached its critical value), the increment \({\Delta} \delta_{\text{b}}\) in bolt deflection and the corresponding bending angle increment \({\Delta} \varphi_{\text{b}}\) of the bolt axis at the bolt head is, respectively, given by

$${\Delta} \delta_{\text{b}} = \frac{{{\Delta} F_{\text{bs}} L^{3} }}{3EI} - \frac{{{\Delta} M_{\text{b}} L^{2} }}{2EI} + {\Delta} \varphi_{\text{N}}^{\prime } L,\quad {\Delta} \varphi_{\text{b}} = \frac{{{\Delta} F_{\text{bs}} L^{2} }}{2EI} - \frac{{{\Delta} M_{\text{b}} L}}{EI} + {\Delta} \varphi_{\text{N}}^{\prime }$$
(44)

Following the same procedure in scenario I, expressions for \({\Delta} \delta_{\text{b}}\), \({\Delta} \varphi_{\text{b}}\), \(\varphi_{\text{Nt}}\), and \(F_{\text{ts}}\) are obtained as follows:

$$\begin{array}{c} {\Delta} \delta_{\rm{b}} = \frac{{{\Delta} F_{\rm{bs}} L^{3} }}{3EI}\left\{ {1 + \frac{{\frac{3EI}{{L\beta_{\rm{N}}^{\prime \prime } }} - \frac{3L}{4EI}\beta_{\rm{b}} }}{{1 + \frac{{\beta_{\rm{b}} L}}{EI} + \frac{{\beta_{\rm{b}} }}{{\beta_{\rm{N}}^{\prime \prime } }}}}} \right\},\\ \quad {\Delta} \varphi_{\rm{b}} = \left( {\frac{{3EI{\Delta} \delta_{\rm{b}} }}{{L^{2} }}} \right)\frac{{\frac{L}{2EI} + \frac{1}{{\beta_{\rm{N}}^{\prime \prime } }}}}{{1 + \frac{3EI}{{\beta_{\rm{N}}^{\prime \prime } L}} + \frac{{\beta_{\rm{b}} L}}{4EI} + \frac{{\beta_{\rm{b}} }}{{\beta_{\rm{N}}^{\prime \prime } }}}},\\ \quad {\Delta} \varphi_{\rm{Nt}} = \left( {\frac{{3EI{\Delta} \delta_{\rm{b}} }}{{\beta_{\rm{Nt}}^{\prime} L^{2} }}} \right)\frac{{1 + \frac{{\beta_{\rm{b}} L}}{2EI}}}{{1 + \frac{3EI}{{\beta_{\rm{N}}^{\prime \prime } L}} + \frac{{\beta_{\rm{b}} L}}{4EI} + \frac{{\beta_{\rm{b}} }}{{\beta_{\rm{N}}^{\prime \prime } }}}} = {\Delta} \delta_{\rm{b}} /\sigma^{\prime\prime},\\ \quad {\Delta} F_{\rm{ts}} = {\Delta} F_{\rm{bs}} + \frac{{{\Delta} M_{\rm{Nt}} }}{{\rho_{\rm{M}} }} = {\Delta} F_{\rm{bs}} \left[ {1 + \left( {\frac{L}{{\rho_{\rm{M}} }}} \right)\frac{{1 + \frac{{\beta_{\rm{b}} L}}{2EI}}}{{1 + \frac{{\beta_{\rm{b}} L}}{EI} + \frac{{\beta_{\rm{b}} }}{{\beta_{\rm{N}}^{\prime \prime } }}}}} \right]\\ = \chi^{\prime\prime}{\Delta} F_{\rm{bs}} \end{array}$$
(45)

where \(\beta_{\text{Nt}}^{\prime }\) (which much less than \(\beta_{\text{Nt}}\) in Eq. (39)) is the contact bending stiffness of the bolt threads with respect to the nut threads as relative slippage between engaged threads occurs, and \(\beta_{\text{N}}^{\prime \prime }\) is the combined bending stiffness (bolt threads to nut and nut to clamped joint) during the slippage of the threads given by

$$\frac{1}{{\beta_{\text{N}}^{\prime \prime } }} = \frac{1}{{\beta_{\text{Ns}} }} + \frac{1}{{\beta_{\text{Nt}}^{\prime } }},$$
(46)

and \(\sigma^{\prime\prime}\) is given by

$$\sigma^{\prime\prime} = \frac{{1 + \frac{3EI}{{\beta_{\text{N}}^{\prime \prime } L}} + \frac{{\beta_{\text{b}} L}}{4EI} + \frac{{\beta_{\text{b}} }}{{\beta_{\text{N}}^{\prime \prime } }}}}{{1 + \frac{{\beta_{\text{b}} L}}{2EI}}}\left( {\frac{{\beta_{\text{Nt}}^{\prime } L^{2} }}{3EI}} \right)$$
(47)

A linear relationship exists in Eq. (45) between the incremental bolt deflection \({\Delta} \delta_{\text{b}}\) and the incremental underhead shear force \({\Delta} F_{\text{bs}}\) . The total value of the bearing transverse friction shear force \(F_{bs}^{\prime }\) would be

$$F_{\text{bs}}^{\prime} = F_{\text{bs}} + {\Delta} F_{\text{bs}}$$
(48)

Similar to Scenario 1, the value of the underhead friction shear force \(F_{\text{bs}}^{'}\) will determine the value of the sliding underhead friction torque \(T_{\text{b}}\) by using Eqs. (1) and (2). Equation (45) also gives the relative bending angle increment \({\Delta} \varphi_{\text{Nt}}\) between the bolt threads and nut threads in terms of bolt deflection increment \({\Delta} \delta_{\text{b}}\) . If the thread angle increment \({\Delta} \varphi_{\text{Nt}}\) reaches its critical value \({\Delta} \varphi_{\text{Ntcr}}\), which is determined by the thread geometry and the class of thread fit, slippage between the engaged threads would stop (i.e., loosening rotation would stop under this condition). Further increase of the transverse displacement of the joint would only cause relative transverse slippage between the nut face and the joint surface, which does not cause any loosening.

Scenario III: Relative slippage occurs simultaneously between all of the three sets of contact surfaces (bolt underhead–joint, nut face–joint, and engaged bolt–nut threads)

As outlined in Scenarios I and II, bolt loosening will only take place when each of the two surfaces is sliding against its respective contact. That requires the bolt underhead surface to be sliding against the contacting joint surface, which must also be simultaneous with relative sliding occurring between the contact surfaces of engaged threads. This is further explained in the following two sections:

  1. (1)

    Simultaneous slippage on the bolt underhead–joint and the engaged bolt–nut thread surfaces (without nut face slippage)

The relative displacement \({\Delta} \delta_{\text{BJ}}\) between bolt underhead and the fixed joint is given by

$${\Delta} \delta_{\text{BJ}} = \delta_{\text{J}} - {\Delta} \delta_{\text{b}} - {\Delta} x_{\text{A}} = \delta_{\text{J}} - \sigma^{\prime\prime}{\Delta} \varphi_{\text{Nt}} - {\Delta} x_{\text{A}}$$
(49)
$$\delta_{\text{J}} = \delta_{0} \sin \omega^{\prime } t$$
(50)

where \(\delta_{\text{b}}\) is the bolt deflection that corresponds to the relative displacement between the bolt underhead and its threaded end, \(\delta_{\text{J}}\) is the cyclic joint transverse excitation, \({\Delta} x_{\text{A}}\) is the relative displacement of the bolt threads, \({\Delta} \varphi_{\text{Nt}}\) is the relative bolt thread bending angular increment with respect to nut, \(\delta_{0}\) is the amplitude of cyclic transverse excitation, and \(\omega^{\prime}\) is the angular frequency of the transverse excitation.

Differentiating Eq. (49) with respect to time, and the translational-to-angular speed ratios \(\eta_{\text{b}}\) and \(\eta_{\text{t}}\) as defined earlier (Sects. 2.1 and 2.2) gives bolt rotational speed \(\omega_{\text{t}}\) of bolt threads relative to engaged nut threads as follows:

$$\omega_{\text{t}} = \frac{{\dot{\delta }_{\text{J}} }}{{\eta_{\text{b}} + \eta_{\text{t}} \left( {\frac{{r_{\text{maj}} + r_{{\min} } }}{{\sin 2\alpha}}} +\sigma^{\prime\prime} \right)}}$$
(51)

The relative bending rotation increment \({\Delta} \varphi_{\text{Nt}}\) with respect to the nut is obtained by integration as follows:

$${\Delta} \varphi_{\text{Nt}} = \int\limits_{{t_{1} }}^{t} \omega_{\text{A}} {\text{d}}t = \int\limits_{{t_{1} }}^{t} {\eta_{\text{t}} \omega_{\text{t}} {\text{d}}t}$$
(52)

where \(t_{1}\) indicated the start of loosening and t is current time (i.e., loosening duration = t−t1).

The critical value \({\Delta} \varphi_{\text{Ntcr}}\), which is the maximum bending angle between the bolt and the nut, is approximately determined by

$${\Delta} \varphi_{\text{Ntcr}} = \frac{{{\Delta} d_{\text{th}} }}{{\sqrt {d^{2} + H^{2} } }}$$
(53)

where \({\Delta} d_{\text{th}}\) is the thread fit clearance between the bolt threads and nut threads (Fig. 7: based on pitch diameter), d is the bolt nominal diameter, H is the thread engagement length (Fig. 6). If \({\Delta} \varphi_{\text{Nt}}\) reaches its critical value \({\Delta} \varphi_{\text{Ntcr}}\), the bolt threads cannot have further relative bending angle increment with nut threads. The critical value \({\Delta} \varphi_{\text{Ntcr}}\) is dependent on the thread clearance \({\Delta} d_{\text{th}}\) (fit class), the thread engagement length H, and the thread major–minor diameters. As a result, \({\Delta} \varphi_{\text{Ntcr}}\) would be proportional to the thread fit clearance, and it would decrease with the increasing the thread engagement length H and thread diameter. The horizontal resisting force would balance the external transverse shear force. The thread friction torque will increase significantly by the transverse resistant force. Under this scenario, the loosening condition is not satisfied and the loosening rotation of the bolt will stop.

Fig. 7
figure 7

Schematic of the thread clearance between engaged bolt and nut threads

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On the other hand, if the absolute value \(\left| {{\Delta} \delta_{\text{BJ}} } \right|\) of the relative displacement reaches its critical value, which is equal to the joint hole diameter clearance \({\Delta} d_{\text{h}}\), the bolt shank will contact with the cylindrical surface of the hole. The contact pressure on the hole surface will produce additional friction torque resistance to the bolt loosening rotation. The additional friction torque can prevent the self-loosening of the threaded fastener.

If \({\Delta} \varphi_{\text{Nt}} < {\Delta} \varphi_{\text{Ntcr}}\) and \(\left| {{\Delta} \delta_{\text{BJ}} } \right| < {\Delta} d_{\text{h}}\), the incremental clamp load loss \({\text{d}}F_{\text{b}}\) due to loosening caused by the transverse cyclic loading is given by

$${\text{d}}F_{\text{b}} = \frac{{k_{\text{b}} k_{\text{c}} }}{{k_{\text{b}} + k_{\text{c}} }}\frac{p}{2\pi }\omega_{\text{t}} {\text{d}}t = \frac{{k_{\text{b}} k_{\text{c}} }}{{k_{\text{b}} + k_{\text{c}} }}\frac{p}{2\pi }\frac{{\left| {\dot{\delta }_{\text{J}} } \right|{\text{d}}t}}{{\eta_{\text{b}} + \eta_{\text{t}} \left( {\sigma^{\prime\prime} + \frac{{r_{\text{maj}} + r_{{\min} } }}{\sin 2\alpha }} \right)}}$$
(54)
  1. (2)

    Simultaneous slippage on the joint–nut face and bolt–nut engaged thread surfaces (without slippage on the bolt underhead surface)

The relative displacement \({\Delta} \delta_{\text{NJ}}\) for the nut and the joint is given by Eq. (49). If \({\Delta} \varphi_{\text{Nt}} < {\Delta} \varphi_{\text{Ntcr}}\) and \(\left| {{\Delta} \delta_{\text{NJ}} } \right| < {\Delta} d_{\text{h}}\), the incremental clamp load loss \(dF_{\text{b}}\) due to the loosening caused by the transverse cyclic loading would be given by

$$\begin{aligned} {\text{d}}F_{\text{b}} & = \frac{{k_{\text{b}} k_{\text{c}} }}{{k_{\text{b}} + k_{\text{c}} }}\left( {\frac{p}{2\pi }} \right)\omega_{\text{t}} {\text{d}}t \\ & = \frac{{k_{\text{b}} k_{\text{c}} }}{{k_{\text{b}} + k_{\text{c}} }}\left( {\frac{p}{2\pi }} \right)\frac{{\left| {\dot{\delta }_{\text{J}} } \right|{\text{d}}t}}{{\eta_{\text{N}} + \eta_{\text{t}} \left( {\sigma^{\prime\prime} + \frac{{r_{\text{maj}} + r_{{\min} } }}{\sin 2\alpha }} \right)}}, \quad {\text{with}}\;\;\eta_{\text{N}} = v_{\text{Nx}} /\omega_{\text{t}} \end{aligned}$$
(55)

3 Results and Discussion

Figures 8, 9, 10, 11, 12, 13, and 14 show the proposed model prediction of loosening performance represented by Eq. (54), and the corresponding experimental validation, of a steel M10 × 1.5 Class 8 bolt–nut system (Fig. 1) that is subjected to harmonic transverse displacement with amplitude \(\delta_{0} = 0.71\;{\text{mm}}\). Typical values of 200 MPa and 0.3 are, respectively, used for Young’s modulus and Poisson’s ratio. While the bolt thread profile angle is fixed in this study (at its standard 60° value), two different values are investigated for the engaged nut thread profile angle, namely the nonstandard profile with \(2\alpha = 120^\circ\) and the standard nut thread are profile angle with \(2\alpha = 60^\circ\). Commercially available nuts with nonstandard profile angle \(2\alpha = 120^\circ\) are referred to as Spiralock nuts [7]. Coefficients of underhead bearing friction \(\mu_{\text{b}}\), thread friction \(\mu_{\text{t}}\) were determined from separate precision torque-tension tests; their experimental values ranged from 0.065 to 0.13, for the various lubricity conditions between the underhead bearing and thread contact surfaces of test part. Bolt preload in this study is varied between 5 and 20.5 kN, which is 14–57% of the bolt proof load of 34.8 kN. Experimental validation of proposed model results on loosening performance was generated using a [modified] Junker vibration loosening test system [18] with a cyclic transverse displacement control.

Fig. 8
figure 8

Thread friction torque ratio \(\overline{T}_{\text{t}} = \frac{{T_{\text{t}} }}{{\mu_{\text{t}} F_{\text{b}} }}\) versus thread friction transverse shear ratio \(\overline{F}_{\text{ts}} = \frac{{F_{\text{ts}} }}{{\mu_{\text{t}} F_{\text{b}} }}\) (for nonstandard nut profile angle 2α = 120° and standard nut profile angle 2α = 60°)

Full size image
Fig. 9
figure 9

Experimental validation of model prediction of nut loosening performance for two thread profile angles (standard nut with a thread profile angle 2α = 60° versus nonstandard nut with 2α = 120°)

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Fig. 10
figure 10

Model prediction of the effect of thread clearance on the loosening performance of standard nut-bolt connection

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Fig. 11
figure 11

Model prediction of the effect of hole diameter clearance on loosening performance of standard nuts

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Fig. 12
figure 12

Model validation of the loosening performance of nonstandard nut with thread profile angle 2α = 120° (for bolt preload = 5600 N)

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Fig. 13
figure 13

Model validation of the loosening performance of nonstandard nut with thread profile angle 2α = 120° (for bolt preload = 8,700 N)

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Fig. 14
figure 14

Model validation of the effect of bolt preload on the loosening performance for standard nuts (2α = 60°)

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3.1 Effect of Nut Thread Profile Angle \(2\alpha\) on the Loosening Performance

For the purpose of this study, two values of the nut thread profile angle \(2\alpha = 120^\circ\) and \(2\alpha = 60^\circ\) are used (Fig. 4) for engaging a standard bolt thread profile angle \(2\alpha = 60^\circ\). A commercially available specialty nut profile angle \(2\alpha = 120^\circ\) [7] is analytically and experimentally compared with standard female thread profile angle 600 [23] in terms of their respective loosening performance. For simplicity, it is assumed that the thread friction coefficient \(\mu_{\text{t}}\) would not be significantly affected by the thread profile angle, when test nuts have been cleaned.

Figure 8 displays the relationship between \(\frac{{T_{\text{t}} }}{{\mu_{\text{t}} F_{\text{b}} }}\) and \(\frac{{F_{\text{ts}} }}{{\mu_{\text{t}} F_{\text{b}} }}\) for the standard nut thread profile angle \(2\alpha = 60^\circ\) and nonstandard \(2\alpha = 120^\circ\) nuts when separately engaged with standard bolt threads. It is obvious that the ratio \(\frac{{F_{\text{ts}} }}{{\mu_{\text{t}} F_{\text{b}} }}\) for critical transverse friction force of the larger thread profile angle nut \(\left( {2\alpha = 120^\circ} \right)\) is much higher than that for the standard nut threads with \(2\alpha = 60^\circ\). If the transverse thread friction force \(F_{\text{ts}}\) is the same, the critical slipping thread friction torque \(T_{\text{t}}\) for \(2\alpha = 120^\circ\) would be much higher than that for standard nut. This means that the nut with larger thread profile angle is more resistant loosening as compared to nuts with standard thread profile angle \(2\alpha = 60^\circ\).

Figure 9 also shows better loosening performance of the nonstandard (larger) thread profile angle \(2\alpha = 120^\circ\), for various combinations of thread and bearing friction coefficients at two levels of bolt preload, namely 5000 and 8700 N. For an initial bolt preload of 8700 N the proposed model shows that the using a nut with a larger thread profile angle has completely prevented any loosening, while the standard nut \(2\alpha = 60^\circ\) has become completely loose (zero bolt tension) after only 150 vibration cycles. For a lower bolt preload of 5,000 N, the larger nut thread profile angle of \(2\alpha = 120^\circ\) has still out-performed the standard nut (with \(2\alpha = 60^\circ\)) in terms of providing more resistance to loosening. The former withstood more than 400 vibration cycles before it became completely loose (with no residual bolt tension), as compared to less than 80 cycles for the latter.

3.2 Effect of Thread Fit

Figure 10 shows model prediction of the effect of thread fit clearance on the vibration loosening performance under harmonic transverse excitation. Thread fit class (tolerance) is often described as tight, medium, or loose fit [22]. An ISO designation of 6H/6g is assigned to a standard tolerance (fit) for commercially available metric nuts (6H)/bolts (6g). Since the nut thread clearance has already been directly incorporated as an independent variable in the proposed model, four arbitrary values of thread clearance are used, namely 0.10, 0.14, 0.20, and 0.22 mm. Results in Fig. 10 show that tighter thread fit would lower the rate of bolt loosing per cycle (i.e., increases the number of cycles to complete loosening). The number of the cycles until complete loosening is 240, 171, 121, and 113, for thread fit tolerance of 0.1 mm, 0.14 mm, 0.20 mm, and 0.22 mm, respectively.

3.3 Model Prediction of the Effect of Hole Clearance

Figure 11 shows loosening prediction of a of preloaded M10 × 1.5 bolt for hole diameter clearances of 0.3, 0.5, and 1.0 mm. A smaller hole clearance would reduce the vibration loosening rate, and the opposite is also true. For an initial bolt preload of 8700 N, the respective number of vibration cycles until complete loosening has been reached (i.e., zero bolt tension) for those three values of hole clearance are 285, 171, and 134.

3.4 Effect of Bearing Friction Coefficient on the Loosening Performance of Nuts with Nonstandard Thread Profile Angle (\(2\alpha = 120^\circ\))

Figures 12 and 13 show experimentally validated model prediction of the effect of bearing friction coefficient \(\mu_{\text{b}}\) on the vibration loosening performance of a preloaded M10 × 1.5 bolt that is engaged with a nonstandard nut with a thread profile angle \(2\alpha = 120^\circ\) for two levels of bolt preload, namely 5600 and 8700 N. For both preload levels, the model prediction (solid lines) and experimental data (dotted lines) are in reasonable agreement in showing that increasing the bearing friction coefficient would reduce the bolt loosening rate per cycle. The difference between model prediction and the experimental data is lower for the higher preload level (Fig. 13).

3.5 Model Validation of the Effect of Bolt Preload

Finally, Fig. 14 shows experimental validation of the proposed model in terms of the effect of preload level on the loosening performance of a standard M10 × 1.5 bolt that is engaged with a standard nut \(\left( {2\alpha = 60^\circ} \right)\). Three preload levels are used, namely 20, 18, and 16 kN; solid lines represent model prediction, while the dotted lines represent the corresponding experimental validation. Obviously, increasing the initial bolt tension would increase the number of vibration cycles-to-complete-loosening, in a nonlinear fashion. A reasonable agreement exists between the model prediction and test data.

4 Conclusion

This paper proposes an experimentally validated analytical model for investigating the effect of nut thread profile angle on the loosening performance of a preloaded bolt–nut system under a cyclic transverse excitation. Additional variables investigated include thread fit clearance, hole clearance, initial preload, and friction coefficients. Experimentally validated model prediction of the loosening performance shows that thread profile angle of the nut has significant effect on the loosening performance of a preloaded standard thread bolt. As compared to the standard nuts with 60° thread profile angle, nonstandard nuts with a larger thread profile angle of 120° (engaged with a preloaded standard M10 × 1.5 bolts) showed a significantly higher resistance to loosening under a harmonic transverse excitation. Results show also that reducing either of thread fit clearance (i.e., higher thread fit class) and/or reducing the clearance of bolt hole would increase the resistance to vibration-induced loosening of preloaded bolt–nut system. The opposite is also true; increasing either thread clearance (loose fit) and/or increasing the bolt hole clearance would reduce resistance to vibration-induced loosening of a preloaded bolt–nut system.