Coupled longitudinal and transverse vibrations of tensioned Euler-Bernoulli beams with general linear boundary conditions

doi:10.1016/j.ymssp.2020.107244

Mechanical Systems and Signal Processing

Volume 150, March 2021, 107244

https://doi.org/10.1016/j.ymssp.2020.107244 Get rights and content

Highlights

•
Coupled transverse-longitudinal vibrations of tensioned beams are studied.
•
A relatively simple approach to explain the coupling phenomenon is proposed.
•
Explicit analytical expressions governing the phenomenon are obtained.
•
Effects of the tension in the beam and general linear boundary conditions are revealed.
•
Experiments are carried out for different boundary conditions and excitation types.

Abstract

The paper concerns coupling between longitudinal and transverse vibrations of tensioned Euler-Bernoulli beams with general linear boundary conditions of arbitrary elastic stiffness. The beams are assumed to be perfectly straight with symmetric boundary conditions, and excitation aligned with the longitudinal axis. Two types of excitation, harmonic and impact, are considered. A relatively simple approach is proposed to predict and qualitatively describe the coupling, allowing approximate explicit analytical expressions for the key parameters affecting the phenomenon. The beam model considered, with both ends supported by longitudinal, translational and rotational springs, can be used in particular to model tensioned bolts used in engineering. The effects of beam tension on the coupling phenomenon are revealed. A series of physical experiments is carried out to illustrate application of the obtained theoretical predictions. Numerical solution of the full nonlinear equations governing coupled transverse-longitudinal vibrations has been conducted to validate the theoretical predictions.

Introduction

The paper concerns the analysis of coupled transverse-longitudinal vibrations of straight tensioned beams under longitudinal excitation. The work was motivated by current attempts to estimate the tension in bolted joints from measured vibrations, where it was observed that even central longitudinal excitation at a bolt end could excite easily measurable transverse vibrations [15]. But the problem is of relevance more generally, for all kinds of tensioned beamlike structural members that have non-ideal end supports (i.e. of nonzero, non-infinite stiffness). The purpose of the work is to uncover and investigate the conditions under which longitudinally imparted energy can transfer into dominating transverse vibrations.

The governing equations describing coupled vibrations of tensioned beams using Euler-Bernoulli theory are well-known and given, e.g. in [3]. The solution of these equations for particular idealized boundary conditions, e.g. clamped-free and hinged-spring, and excitation types is provided in several works [1], [2], [3], [4], [5]. More recent papers [6], [7], [8], [9] can also be mentioned; for example, in [9] planar transverse-longitudinal vibrations of Timoshenko beams were studied for hinged-hinged transverse boundary conditions. In [10], [11] the dynamic response of axially loaded Euler-Bernoulli beams and their buckling due to dynamic loading applied has been studied. For perfectly straight beams with symmetric boundary conditions and excitation aligned with the longitudinal axis, the coupling between transverse-longitudinal vibrations is essentially nonlinear, and strongly depends on the boundary conditions. In the present paper, general linear boundary conditions are considered, and their effect on the coupling phenomenon discussed. Similarly to other works in the field [4], [5], [6], [7], [8], [12], [13], the boundary conditions are assumed to be linear, since taking into account nonlinearities in these conditions would make even purely numerical solution cumbersome, without providing many additional insights into the underlying physics. Two types of longitudinal excitation are considered, time-harmonic and impact. A relatively simple approach is proposed to approximately solve the corresponding governing equations, and predict the occurrence and characteristics of the coupling phenomenon. The approach implies linearizing the equation describing longitudinal vibrations. Solving this linear equation is straightforward, and allows obtaining a relatively simple equation for transverse vibrations of the beam. From this equation, conditions are determined under which relatively large transverse vibrations of the beam can be induced by longitudinal excitation. The approach provides insights into the dynamic behaviour of the beam, and allows obtaining relatively simple analytical expressions for relations between the key parameters affecting the coupling phenomenon. The approach allows for explaining the underlying physics of the phenomenon, which is important for understanding the conditions under which it can occur. In contrast to the abovementioned papers, focusing on specific boundary conditions, we here consider general linear boundary conditions. Moreover, the effects of beam tension on the coupling phenomenon are revealed and discussed.

The considered straight homogeneous Euler-Bernoulli beam with both ends supported by longitudinal, translational and rotational springs, as is shown in Fig. 1, in particular models tensioned bolts or tension members used in practical applications [14], [15]. In the present paper, results are obtained assuming straight beams with symmetric boundary conditions and excitation aligned with the longitudinal axis, which inherently precludes some other mechanisms of coupling that may be present in real situations, e.g. off-center or oblique excitation and geometrical imperfections [16]. Also, similarly to [3], [6], [7], [8], [9], plane transverse-longitudinal vibrations are considered, neglecting nonlinear transverse-transverse coupling [2], [3], [4], [5]. The consequence of these and other simplifying assumptions of the proposed approximate approach, including the use of the Euler-Bernoulli beam theory, are discussed in the paper. To validate the obtained theoretical results, numerical solution of the full nonlinear equations governing coupled transverse-longitudinal vibrations of the beam has been conducted, as has experimental analysis for different boundary conditions and excitation types.

In Section 2 equations describing coupled transverse-longitudinal vibrations of the beam and the corresponding general linear boundary conditions are presented. In Section 3, by employing the conventional modal expansion approach, these equations are reduced to a system of coupled nonlinear ODEs. Section 4 concerns the numerical solution of these equations for particular system parameters, to illustrate the emergence of relatively large transverse beam vibrations induced by the longitudinal excitation. In Section 5, the simplified approach to predict and describe the coupling phenomenon is employed, and key parameters affecting its emergence are revealed. Section 6 concerns the effects of the beam properties and boundary conditions on these key parameters. In Section 7, the theoretical results are validated by comparison with the direct numerical solution of the full nonlinear equations. Section 8 provides a discussion of the simplifying assumptions employed, and limitations of the developed approximate approach, and Section 9 describes the conducted physical experiments.

Section snippets

Governing equations

The governing equation describing longitudinal vibrations, $u (x, t)$ , of an Euler-Bernoulli beam is [12], [13], [17]: $EA \frac{\partial Λ}{\partial x} - ρ A \frac{\partial^{2} u}{\partial t^{2}} = f (x, t)$ where $f (x, t)$ is the distributed longitudinal force acting on the beam, $E$ is the Young modulus and $ρ$ density of the beam material, and $A$ is the cross-sectional area of the beam. The full axial strain $Λ (x, t)$ of the beam is governed by [3], [12], [13], [17]: $Λ (x, t) = \sqrt{{(1 + u^{'})}^{2} + {(w^{'})}^{2}} - 1 \approx u^{'} + \frac{1}{2} {(w^{'})}^{2}$ where primes denote spatial derivative, $w (x, t)$ is the transverse deflection,

Modal equations

Solutions to (3), (8) are sought in the form of modal expansions: $u (x, t) = u_{0} (x) + \sum_{n = 1}^{\infty} η_{n} (t) ψ_{n} (x)$ $w (x, t) = \sum_{j = 1}^{\infty} q_{j} (t) ϕ_{j} (x)$ where $u_{0} (x)$ is the longitudinal displacement along the beam due to longitudinal forces $N_{ext}$ applied at the ends: $u_{0} (x) = \frac{N_{ext} (2 x - l)}{2 E A}$

Inserting (11), (12) into (3), multiplying by $ψ_{r} (x)$ and integrating from 0 to l gives $- M_{r} (ω_{r}^{2} η_{r} (t) + {\ddot{η}}_{r} (t)) = - P (t) ψ_{r} (x_{0}) - \frac{1}{2} E A \int_{0}^{l} \frac{\partial}{\partial x} ({(\sum_{j = 1}^{\infty} q_{j} (t) {ϕ^{'}}_{j} (x))}^{2}) ψ_{r} (x) d x$ where $ω_{r}$ is the r’th natural frequency of the beam’s longitudinal vibrations, and $M_{r}$ is the modal mass

Numerical solution of the nonlinear modal equations

The obtained modal Eqs. (18), (19) for, respectively, the longitudinal and transverse modal coordinates $η_{r}$ and $q_{i}$ , do not allow for an explicit and exact analytical solution, even in the case of harmonic longitudinal excitation. Thus, to illustrate the coupling phenomenon, we solve them numerically for specific values of parameters. Numerical integration of the ordinary differential Eqs. (18), (19) has been performed using the NDSolve function in Wolfram Mathematica 11.1, with the convergence

A simplified approach for predicting longitudinally excited transverse oscillations

The proposed approach implies simplifying the governing equations of motions and the corresponding modal equations. For longitudinal vibrations described by (18), the nonlinear terms $\sum_{j = 1}^{\infty} \sum_{m = 1}^{\infty} L_{jmr} q_{j} (t) q_{m} (t)$ are omitted. In the absence of coupling, these terms are negligibly small, while in the presence of coupling they will affect amplitudes of the longitudinal modal coordinates, but not the coupling phenomenon itself. The same also applies to the nonlinear terms $\sum_{j = 1}^{\infty} \sum_{m = 1}^{\infty} \sum_{k = 1}^{\infty} R_{jmki} q_{j} (t) q_{m} (t) q$

Longitudinal natural frequencies

The longitudinal natural frequencies $ω_{n}$ of the beam are governed by the transcendental Eq. (24). Solutions to (24) depend on ${\tilde{k}}_{u}$ and cannot be obtained in a closed form. However, for ${\tilde{k}}_{u} < < 1$ employing the conventional perturbation procedure of expanding in a small parameter [3], [23], the solutions can be approximated by ${\tilde{ω}}_{n} = (π n + \frac{2}{π n} {\tilde{k}}_{u}), n = 1, 2, . . .$ ${\tilde{ω}}_{0} = \sqrt{2 {\tilde{k}}_{u}}$ giving $ω_{n} = \sqrt{\frac{E}{ρ}} (\frac{n π}{l} + \frac{2}{π n} \frac{k_{u}}{AE}), ω_{0} = \sqrt{2 \frac{k_{u}}{A ρ l}}$

As is seen from (41), for relatively small ${\tilde{k}}_{u}$ the longitudinal natural frequencies will increase slightly,

Harmonic excitation

As noted in Section 5, the obtained theoretical predictions qualitatively agree with the numerical results shown in Section 4. For the beam studied in Section 4, the key parameter $γ = 75$ . Taking into account that for ${\tilde{k}}_{u} > > 1$ , ${\tilde{k}}_{θ} > > 1$ , ${\tilde{k}}_{w} > > 1$ and odd values of $j + i + n$ , $P_{jni} = 0$ , implying $S_{ijn} = 0$ , $Z_{ijn} = 0$ , we obtain that for $k = 1$ , $n = 1$ condition (61) will be satisfied for $j = 5$ , $i = 2$ , giving $75 \approx \frac{π}{4} |121 - 25| = 75.40$ . Consequently, exciting at $ω_{e} \approx ω_{1}$ , the second and the fifth transverse modes should feature the most

Linearity of boundary conditions

Assuming the boundary conditions to be linear when studying nonlinear vibrations of beams, and searching solutions in the form of expansions in terms of the linear undamped free vibration modes of the beam, is a well-known and justified approach that has been used in many papers, cf. [3]. Using this approach allows us to obtain explicit Eqs. (22)–(26), governing the mode shapes and natural frequencies of the longitudinal and transverse vibrations. These expressions are then used to obtain the

Comparison with physical experiments

A series of physical experiments has been carried out to illustrate that the obtained qualitative insights into the coupling phenomenon and its dependency on beam parameters, including tension, are valid for real beams with different boundary conditions. Two bolts, M12 × 260 mm (“long”) and M12 × 140 mm (“short”), made of steel (E = 207 GPa, $ρ$ = 7850 kg/m³) with a circular cross-section with diameter $d = 12$ mm were considered. The length of the longer bolt from head to tip was 253 mm, while the

Conclusions

Coupled longitudinal-transverse vibrations of straight tensioned beams under longitudinal excitation are considered in the paper. The emergence of relatively large transverse vibrations is studied for general linear symmetric boundary conditions and two different types of excitation, harmonic, and impact. The excitation is assumed to be perfectly aligned with the longitudinal axis of the beam. A simplified approach to explain the phenomenon is proposed, and explicit conditions governing the

CRediT authorship contribution statement

Vladislav S. Sorokin: Conceptualization, Methodology, Investigation, Writing - original draft, Formal analysis, Validation, Software, Writing - review & editing. Jon Juel Thomsen: Methodology, Supervision, Writing - review & editing, Funding acquisition. Marie Brøns: Investigation, Validation, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The work is carried out with financial support from Independent Research Fund Denmark, DFF-6111-00385.

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Coupled longitudinal and transverse vibrations of tensioned Euler-Bernoulli beams with general linear boundary conditions

Highlights

Abstract

Introduction

Section snippets

Governing equations

Modal equations

Numerical solution of the nonlinear modal equations

A simplified approach for predicting longitudinally excited transverse oscillations

Longitudinal natural frequencies

Harmonic excitation

Linearity of boundary conditions

Comparison with physical experiments

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

Int. J. Non Linear Mech.

J. Sound Vib.

J. Sound Vib.

Int. J. Solids Struct.

J. Sound Vib.

Compos. Struct.

J. Sound Vib.

J. Sound Vib.

J. Sound Vib.

J. Sound Vib.

Stability of parametrically excited vibrations of an elastic rod

Dev. Theoret. Appl. Mech.

Nonlinear nonplanar dynamics of parametrically excited cantilever beams

Nonlinear Dyn.