Linearly coupled transverse vibrations of tensioned bolts

Abstract

In this study an analytical model for linear coupling between the two perpendicular transverse vibration modes in a tightened bolt is presented. The model shows an excellent fit with the experimental data suggesting that the coupling is dominantly linear. An experiment is performed to support the linearity claim. Two physical scenarios that could justify the theoretical model is given, one scenario involves asymmetric boundary stiffness and the other one measurements imperfection in location. This work is motivated by its application in structural health monitoring techniques, in particular those based on measuring transverse vibrations of bolts and other beam-like components.

Keywords

Transverse vibrations linear coupling bolt Bernoulli-Euler beam boundary stiffness

Introduction

Coupling between transverse modes in beam-like systems can manifest due to the presence of nonlinear or/and linear coupling terms in the system’s equation of motion. Such modal interaction may be encountered in many engineering applications, therefore investigating this coupling allows one to understand the dynamic of a structure, which can help, for example, in maintaining the healthy functioning of such a structure. Bolts are one of those mechanical systems that are widely used in engineering applications, and their failure in securing a proper tightening of mechanical components can lead to catastrophic outcomes. Thus a regular checking of proper bolt tightness is essential for safe operation. In (Brøns et al., 2021),¹ it was shown that in a tightened bolt, a coupling between the two transverse modes takes place when the bolt is subjected to a hammer impact, in particular when the bolt tension is insufficient.

In- and out-of-plane vibrations in the two perpendicular transverse directions appear in various engineering problems, for example in wind turbine, helicopter blades, airplane wings, and flexible satellites. (Haight and King, 1972)² investigated the stability of the in-plane response of a beam that is laterally excited. In their study, they included inertia nonlinearities, and showed that in-plane response can lose stability, resulting in stable out-plane motion. (Pai and Nayfeh, 1990)³ investigated the nonlinear non-planar oscillation of a laterally excited cantilever beam. While one transverse mode was directly excited, the other one was indirectly excited through the one-to-one internal (auto-parametric) resonance. The interaction between the two modes was caused by nonlinear coupling (inertia and geometric nonlinearity). It was also shown that for certain beam and load parameters, the non-planar responses can be chaotic. (Stoykov and Ribeiro, 2011)⁴ studied the interaction of vibration modes with bending, in different planes, and torsion in the free regime. They included the geometrical nonlinearity in their model, which was based on Timoshenko’s theory for bending and Saint-Venant’s for torsion. (Aghababaei et al., 2009a, 2009b)^5,6 included also linear coupling in their model, resulting from beam imperfection. They showed that even if the imperfection is small, the resulting dynamic may differ from when the beam is considered straight. However, when the beam is “perfect” (i.e; straight equilibrium axis, symmetric boundary conditions) the coupling is purely nonlinear (Nayfeh and Mook, 1979).⁷

In this work we investigate the coupling between the two transverse vibration modes of a tensioned bolt. We show that the coupling can be dominantly linear, and we give two potential scenarios that may be responsible for this linear coupling. In the following section, the experimental data are signal processed and fitted to a mathematical model. Afterwards, an explanation on the physical cause of the coupling between the two modes is given, then followed by conclusions.

Experimental data

Experimental data for hammer impact

A M12×260 mm steel bolt, mounted in a test rig, is tightened with different tensions and subjected to a hammer impact, Figure 1. The Ø100×200 mm cylindrical test rig is suspended at each end with rubber bands to mimic the free-free boundary condition. The bolt is hammered along a transverse direction “x”, and the responses are sensed using three accelerometers, that are placed along x-, y- and z-directions. Figure 2 shows an example of experimental displacement along the two transverse directions x and y. The figure shows that the amplitude of x-direction is higher than the one of y-direction since the impact is along x-direction. However as the time goes an energy exchange between the two direction modes takes place which results in a beating phenomenon reflecting a coupling between the two modes. Eventually with time, the amplitudes of the two signals decrease and die out due to the presence of damping. In the present work the experimental data given as accelerations in (Brøns et al., 2021),¹ are here displayed as displacements that are obtained by performing Inverse Fourier Transform.

Figure 1.

A transversally hammer impacted bolt with three accelerometers placed on the nut and the thread end.

Figure 2.

Experimentally obtained displacement for the two transverse directions. Blue: transverse direction parallel to the hammer impact. Black: transverse direction perpendicular to the hammer impact.

Linearity of experimental data

The equations of motion for the two transverse modes of a beam given in (Nayfeh and Mook, 1979)⁷ (ch. 7, equations. (7.1.2)-(7.1.3)) show that for a perfect beam the coupling between the modes is nonlinear. However numerical simulations suggest that mainly the linear coupling is responsible for the beating phenomenon found in our bolt experiment (Brøns et al., 2021).¹ This can be actually noted from Figure 2, where the beating period remains almost constant along the time history, independently of the decaying vibration amplitude. Considering only two modes, a mathematical model involving only linear coupling terms is:

\ddot{x} + β_{x} \dot{x} + w_{x}^{2} x + a y = 0

(1)

\ddot{y} + β_{y} \dot{y} + w_{y}^{2} y + b x = 0

(2)

where x(t) and y(t) are the modal coordinates of the two transverse beam vibration modes, and w_x and w_y the corresponding natural frequencies. Linear least square method is employed to fit the model equation (1)–(2) to the experimental data. Figures 3, 4, and 5 show a comparison between the experimental data (blue) and the mathematical model equation (1)–(2) (red) for three different bolt tensions: 11% (low), 40% (medium) and 69% (high) of the bolt’s yield stress, which 640 MPa. The good fit between the model and the experimental data is not clearly visible due to the overlapping of both signals, however in Figure 6, which is a zoom in of Figure 3, this good fit appears clearly. The initial conditions for the mathematical model are chosen equation (1)–(2) so to coincide with experimental data at a time t right after transient time. To investigate further the linearity of the coupling between the two transverse modes in the bolt, in the next section we present two physically based mechanisms that could lead to the linear mode coupling.

Figure 3.

Experimental data (blue) and theoretical results (red) for low bolt tension.

Figure 4.

As the caption in Figure 3, medium bolt tension.

Figure 5.

As the caption in Figure 3, high bolt tension.

Figure 6.

Zoom in of Figure 3.

Possible linear coupling mechanism

In this section we investigate two ways that could lead to linear coupling between the two perpendicular transverse modes. Others may exist and all of them act in combination, with different weight in different situations, our main aim is to suggest physically based mechanisms to support the experimentally observed linearity in coupling. The two coupling mechanisms investigated are unrelated to energy dissipation, so to facilitate qualitative insights damping is abstracted from in the modeling.

Boundary imperfection: Skewed boundary stiffness

The first scenario that could potentially produce a linear coupling is due to the boundary stiffness. Given that we are only interested in the first transverse natural frequency, the effect of shear deformations and rotary inertia is small as shown in (Abramovich 1992, Brøns et al.,2019).⁸¹² Therefore the bolt is modelled by a Bernoulli-Euler beam where both ends are attached to two “skewed” linear springs, each one forming an angle θ with each of the two perpendicular transverse axes, see Figure 7. That is, the principal axes of the elastic boundary do not align with the principal axes/excited direction for the bolt. Including springs in this manner proved to significantly improve, for example the prediction of natural frequencies of offshore wind turbine (Arany et al., 2015).⁹ Applying Bernoulli-Euler theory with the assumption of small slopes, we obtain the equations of motions of the beam in terms of the transverse deflections u (z, t) and v (z, t) (Inman 1996, Thomsen 2021):^10,11

\begin{array}{c} E I u^{″″} + ρ A \ddot{u} - N u^{″} & = & ((κ_{1} - κ_{2} \cos 2 θ) u^{'} + κ_{2} v^{'} \sin 2 θ) (δ (z) - δ (z - l)) \end{array}

(3)

\begin{array}{c} E I v^{″″} + ρ A \ddot{v} - N v^{″} & = & ((κ_{1} + κ_{2} \cos 2 θ) v^{'} + κ_{2} u^{'} \sin 2 θ) (δ (z) - δ (z - l)) \end{array}

(4)

where EI denotes the transverse stiffness, ρA the area density, N the tensile axial force and δ(.) is the Dirac-delta function. The dots and primes in equation (3)–(4) denote differentiation w.r.t. to t and z, respectively, while κ₁ = (k₁ + k₂)/2 denotes boundary stiffness level, and κ₂ = (k₂ − k₁)/2 boundary stiffness asymmetry, see Figure 7. The boundary spring terms (r.h.s. of equation (3)–(4)) can be calculated as ∂V/∂u and ∂V/∂v, respectively, where

V = \sum_{i = 1}^{2} k_{i} Δ l_{i}^{2} / 2

is the potential energy of the two springs, and where the relative elongation Δl_i/l at end point deformation (u, v) is (see Figure 7):

\frac{Δ l_{1}}{l} = \sqrt{{(\frac{u}{l} + \cos θ)}^{2} + {(\frac{v}{l} - \sin θ)}^{2}} - 1,

(5)

\frac{Δ l_{2}}{l} = \sqrt{{(\frac{u}{l} + \sin θ)}^{2} + {(\frac{v}{l} + \cos θ)}^{2}} - 1 .

(6)

Figure 7.

Skewed boundary stiffness - two linear springs, each one forming an angle θ with each of the two perpendicular transverse axes (a). Undeformed end springs (b). Deformed end springs (c).

Linearizing the resulting spring forces F_u = ∂V/∂u and F_v = ∂V/∂v, in u and v respectively, for small u/l and v/l, and differentiating w.r.t. the variable z gives the r.h.s. of equation (3)–(4), where the 2 θ trigonometric terms reflect the invariance, w.r.t. stiffness, of rotating the spring set 180°. The solutions u (z, t) and v (z, t) to equation (3)–(4), respectively, can be expressed as modal expansions:

u (z, t) = \sum_{j = 1}^{\infty} x_{j} (t) ϕ (z), v (z, t) = \sum_{j = 1}^{\infty} y_{j} (t) ψ (z)

(7)

Inserting equation (7) into equation (3)–(4), multiplying respectively by ϕ_i(z) and ψ_i(z), and integrating over the length of the beam give:

\begin{array}{c} A \sum_{j = 0}^{\infty} {\ddot{x}}_{j} \int_{0}^{l} ϕ_{j} ϕ_{i} d z + \sum_{j = 1}^{\infty} x_{j} & (E I \int_{0}^{l} ϕ_{j}^{''''} ϕ_{j} d z - N \int_{0}^{l} ϕ_{j}^{″} ϕ_{j} d z) = \\ \int_{0}^{l} (δ (z) - δ (z - l)) & (x_{j} (κ_{1} - κ_{2} \cos 2 θ) ϕ_{j}^{'} ϕ_{i} + y_{j} κ_{2} ψ_{j}^{'} ϕ_{i} \sin 2 θ) d z \end{array}

(8)

\begin{array}{c} ρ A \sum_{j = 0}^{\infty} {\ddot{y}}_{j} \int_{0}^{l} ψ_{j} ψ_{i} d z + \sum_{j = 1}^{\infty} y_{j} & (E I \int_{0}^{l} ψ_{j}^{''''} ψ_{j} d z - N \int_{0}^{l} ψ_{j}^{″} ψ_{j} d z) & = \\ \int_{0}^{l} (δ (z) - δ (z - l)) & (y_{j} (κ_{1} + κ_{2} \cos 2 θ) ψ_{j}^{'} ψ_{i} + x_{j} κ_{2} ϕ_{j}^{'} ψ_{i} \sin 2 θ) d z \end{array}

(9)

Carrying out the integrals in equations (8)–(9) and only considering the first transverse mode yields:

\ddot{x} + w_{x}^{2} x + a y = 0

(10)

\ddot{y} + w_{y}^{2} y + b x = 0

(11)

where w_u, w_v, a and b are given by:

w_{u}^{2} = \frac{E I}{ρ A} λ_{u}^{4} + \frac{N}{ρ A} λ_{u}^{2} + (κ_{1} - κ_{2} \cos 2 θ) (ϕ^{'} (l) ϕ (l) - ϕ^{'} (0) ϕ (0))

(12)

w_{v}^{2} = \frac{E I}{ρ A} λ_{v}^{4} + \frac{N}{ρ A} λ_{v}^{2} + (κ_{1} + κ_{2} \cos 2 θ) (ψ^{'} (l) ψ (l) - ψ^{'} (0) ψ (0))

(13)

a = \frac{κ_{2} \sin 2 θ}{ρ A L} (ψ^{'} (l) ϕ (l) - ψ^{'} (0) ϕ (0))

(14)

b = \frac{κ_{2} \sin 2 θ}{ρ A L} (ϕ^{'} (l) ψ (l) - ϕ^{'} (0) ψ (0))

(15)

Equations (12)–(15) indicates that coupling (i.e. ab ≠ 0) between the two transverse modes can occur in the case of skewed boundary stiffness if: 1) κ₂ ≠ 0, i.e. the two stiffness k₁ and k₂ are not equal, and 2) the modes shapes and slopes at both ends of the bolt (z = 0 and l) are not equal. It should be noted that similarly rotational boundary stiffness could also be included, however for simplicity only transverse boundary stiffness are included here.

Imperfection measurement directions: Misaligned accelerometers

Another situation where linear coupling between two transverse mode directions x and y can occur - or rather, seem to occur-is when the two accelerometers $\tilde{x}$ and $\tilde{y}$ intended to measure x and y are misaligned. Figure 8 shows the locations of two accelerometers $\tilde{x}$ and $\tilde{y}$ forming an angle θ_x and θ_y with x and y, respectively

Figure 8.

Schematic of the misaligned accelerometers.

The equation relating the measured accelerations of the two directions $\ddot{\tilde{x}}$ and $\ddot{\tilde{y}}$ to the ideal directions $\ddot{x}$ and $\ddot{y}$ is:

\ddot{\tilde{X}} = R \ddot{X}

(16)

where \ddot{\tilde{X}} = (\begin{array}{c} \ddot{\tilde{x}} \\ \ddot{\tilde{y}} \end{array}), \ddot{X} = (\begin{array}{c} \ddot{x} \\ \ddot{y} \end{array}), R = [\begin{array}{c} \cos θ_{x} & \sin θ_{x} + τ_{x} \\ - \sin θ_{y} + τ_{y} & \cos θ_{y} \end{array}]

(17)

and where τ_x and τ_y denote the relative transverse sensitivity (sensor crosstalk) of the two accelerometers expressing the sensitivity to vibrations in direction normal to that of each of the accelerometers main axes, for example the accelerometers used here in the experiment (B&K Type 4397) about 3%. We assume the motion in the two directions x and y are ideally uncoupled from each other such that:

\ddot{X} = - Ω^{2} X where X = (\begin{array}{c} x \\ y \end{array}), Ω^{2} = [\begin{array}{c} Ω_{x}^{2} & 0 \\ 0 & Ω_{y}^{2} \end{array}],

(18)

where Ω_x and Ω_y are the natural frequencies for the transverse modes x and y, respectively. It is also assumed that the measurement imperfections are small, i.e.

θ_{x}^{2}, θ_{y}^{2}, τ_{x}^{2}, τ_{y}^{2} ≪ 1

(the latter implies |R| ≈ I − identity matrix, so R is non-singular). Inserting (18) into (16) we obtain:

\ddot{\tilde{X}} = - R Ω^{2} R^{- 1} \tilde{X} where \tilde{X} = (\begin{array}{c} \tilde{x} \\ \tilde{y} \end{array})

(19)

with

\tilde{X}

being the displacement obtained by integrating twice the measured accelerations

\ddot{\tilde{X}}

. Writing (19) in terms of

\tilde{x}

and

\tilde{y}

and Taylor-expanding the elements of the matrix R for small θ_x and θ_y yields

\begin{array}{l} \ddot{\tilde{x}} + {\tilde{Ω}}_{x}^{2} \tilde{x} + \tilde{a} \tilde{y} = 0 \\ \ddot{\tilde{y}} + {\tilde{Ω}}_{y}^{2} \tilde{y} + \tilde{b} \tilde{x} = 0 \end{array}

(20)

where

\tilde{a} = (θ_{x} + τ_{x}) (Ω_{y}^{2} - Ω_{x}^{2})

(21)

\tilde{b} = (θ_{y} - τ_{y}) (Ω_{y}^{2} - Ω_{x}^{2})

(22)

Equations (21)–(22) indicate that misaligned accelerometers can induce apparent coupling between the two mode directions; This occurs when $\tilde{a} \tilde{b} \neq 0$ . In addition, equations (21)–(22) show that even if the two accelerometers are perfectly aligned (θ_x = θ_y = 0), the transverse sensitivity of the accelerometer (τ_x, τ_y) can induce coupling. This means that even in the case of free-free bolt with an ideal impact (along either x or y), coupling can be recorded, as long as the natural frequencies Ω_x and Ω_y for the transverse fundamental modes are not exactly identical. When Ω_x = Ω_y, i.e. with a perfectly rotational symmetric bolt and transverse symmetric boundary conditions, thus no linear coupling appears by this mechanism, even with misaligned and transversely sensitive accelerometers. However, in that case the mechanism discussed in section 3.1 would also not provide linear coupling (since κ₂ = 0), then nonlinear coupling could be the dominant mechanism (Nayfeh and Mook 1979, Sorokin et al., 2021).^7,14^13,1

Conclusion

We showed that several linear mechanisms can explain the observed coupling between two perpendicular transverse modes and derived the conditions when such linear mechanisms disappear. An analytical model with linear coupling terms gave an excellent fit with the experimental data. It was shown that for the first scenario where there is a boundary imperfection, a linear coupling manifests when the two stiffness along each of the two transverse directions, the modes shapes and slopes at both ends of the bolt (z = 0 and l) are not equal. For the second scenario, involving imperfect measurement directions, it was shown that when Ω_x = Ω_y, i.e. with a perfectly rotational symmetric bolt and transverse symmetric boundary conditions, no linear coupling appears. So, though for a theoretically “perfect” tensioned beam the coupling between transverse vibration modes may be purely nonlinear, for real structures, various sources of imperfection can contribute to linear coupling mechanisms, which dependent on circumstances (type of imperfection, excitation, and measurement set-up) may overrule the nonlinear coupling effects. Imperfection, only two of which are examined in the present work, may be e.g. geometrical (i.e. non-straight undeformed beam axis), structural (e.g. non-uniform boundary stiffness), material (e.g. non-uniform material properties), excitation-related (e.g. non-central hammer impact), or measurement-related (e.g. misaligned sensors).

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is financially supported by Independent Research Fund Denmark (Det Frie Forskningsråd), grant DFF-6111-00385.

ORCID iD

Si Mohamed Sah

References

Brøns

Thomsen

Sah

, et al. Estimating bolt tension from vibrations: transient features, nonlinearity, and signal processing. Mech Syst Signal Process 2021; 150: 107224.

Haight

King

. Stability of non-linear oscillations of an elastic rod. The J Acoust Soc America 1972; 52: 899–911.

Pai

Nayfeh

. Non-linear non-planar oscillations of a cantilever beam under lateral base excitations. Int J Non-linear Mech 1990; 25(5): 455–474.

Stoykov

Ribeiro

Nonlinear free vibrations of beams in space due to internal resonance. J Sound Vibration 2011; 330: 4574–4595.

Aghababaei

Nahvi

Ziaei-Rad

. Non-linear non-planar vibrations of geometrically imperfect inextensional beams, Part I: equations of motion and experimental validation. Int J Non-linear Mech 2009; 44: 147–160.

Aghababaei

Nahvi

Ziaei-Rad

. Non-linear non-planar vibrations of geometrically imperfect inextensional beams.Part II—Bifurcation analysis under base excitations. Int J Non-linear Mech 2009; 44: 161–179.

Nayfeh

Mook

. Nonlinear Oscillations. New York, NY: John Wiley, 1979.

Abramovich

. Natural frequencies of Timoshenko beams under compressive aaxial loads. J Sound Vibration 1992; 157(1): 183–189.

Arany

Bhattacharya

Adhikari

, et al. An analytical model to predict the natural frequency of offshore wind turbines on three-spring flexible foundations using two different beam models. Soil Dyn Earthquake Eng 2015; 74: 40–45.

10.

Inman

. Engineering Vibration. Upper Saddle River, NJ: Prentice-Hall, 1996.

11.

Thomsen

. Vibrations and Stability: Advanced Theory, Analysis, and Tools. Cham, Switzerland: Springer Nature, 2021.

12.

Brøns

Thomsen

. Experimental testing of Timoshenko predictions of supercritical natural frequencies and mode shapes for free-free beams. J Sound Vibratio 2019; 459: 114856.

13.

Sorokin

Thomsen

Brøns

. Coupled longitudinal and transverse vibrations of tensioned Euler-Bernoulli beams with general linear boundary conditions. Mech Syst Signal Process 2021; 150: 107244.