On the Bolotin's reduced beam model versus various boundary conditions

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Highlights

  • Simplified beam equations with inertial terms and nonlinear curvatures are derived.

  • They can be treated as the first approximation to the Kirchhoff nonlinear equations.

  • Simplified equations versus boundary conditions are illustrated and discussed.

  • Nonlinear modes of beam vibrations are constructed for various boundary conditions.

Abstract

This paper is devoted to the construction of asymptotically correct simplified models of nonlinear beam equations for various boundary conditions. V.V. Bolotin mentioned that in some cases (e.g., if compressed load is near the buckling value), the so-called “nonlinear inertia” must be taken into account. The effect of nonlinear inertia on the oscillations of the clamped-free beam is investigated in many papers. Bolotin used some physical assumption and did not compare the order of nonlinear terms in original equations. Below we propose our method for deriving those, which we will named “Bolotin's equations”. This approach is based on fractional analysis of original boundary value problems.

Introduction

Beams are commonly used as structural elements in macro-, micro- or nanoscales [1,2]. Consequently, models for their analysis are currently met in any field of civil and industrial engineering. Beams are frequently used in many practical applications, for example in buildings, bridges, mining supports, railroads, biomechanics etc.

Following the trend to downscale electronic devices, mechanical devices are also entering the micro- and even nanometer regime [3]. To read out the motion of a beam, it has to be coupled to an electronic circuit. These systems are the so-called micro-electromechanical systems (MEMS) and they find commercial applications in accelerometers, gyroscopes, mass sensing, pressure sensing, band-pass filters and scanning probe microscopy. The increasing demand for realistic simulations leads to a higher level of detail during the modeling phase, especially in nanomechanics and biology (for example, for describing mechanical behavior of DNA). The resulting complicated non-linear PDEs can be solved by discrete methods (finite elements, finite differences, etc.). But the time required to solve high-dimensional discretized models remains a bottleneck towards efficient and optimal design of structures. To simplify the original equations, model order reduction methods are widely used [4,5]. This approach is based on partial discretization followed by an analysis of the high-dimensional system. An alternating approach is fractional analysis [6], based on the detection of small parameters using non-dimensionalization and normalization with following asymptotic splitting.

Our paper is devoted to the construction of asymptotically correct simplified models of non-linear beam equations for various boundary conditions. The paper is organized as follows. First, we employ the traditional Kirchhoff's approximation. In Section 3, we obtain Bolotin's equations for clamped–clamped beam. Section 4 deals with generalization for different boundary conditions. Section 5 presents an example of non-linear normal mode construction. Section 6 is devoted to study correct dynamical equations of a buckled beam. Finally, Section 7 presents concluding remarks.

Section snippets

Kirchhoff's approximation

Kirchhoff [7] proposed simple approximate equations of non-linear beam vibration, which became very popular [8,9]. Let us briefly discuss this approximation. Consider the governing equations of non-linear beam vibration in the following formρF2Wt2+2Mx2x(TWx)=0,ρF2Ut2Tx=0,where: M = EIκ, T = EFε, κ=2Wx2,ε=Ux+0.5(Wx)2; E is the Young's modulus; F,  I are the area and the static moment of transversal beam cross section, respectively; κis the curvature; U, W are the longitudinal

Bolotin's approximation for a clamped–clamped beam

Bolotin [13] mentioned that in some cases (e.g., if compressed load is near the buckling value), the so-called “non-linear inertia” must be taken into account. The effect of non-linear inertia on the vibrations of the clamped-free beam (with a non-linear curvature expression different from ours) is investigated in [14,15].

Bolotin used some physical assumption and did not compare order of linear and non-linear terms in original equations. Below we propose our way for obtaining the mentioned

Bolotin's approximation for various boundary conditions

Consider the construction of Bolotin's approximation, limiting ourselves to a linear approximation for the curvature. First, consider the beam with the free axis conditions in the axial direction, i.e. we takeT=0atx=0,L.Then, from Eq. (12) and boundary conditions (28), one obtains T0 = 0 andu0=120x(w0x)2dx+C(t).

Function C(t) will be found later. For the additional longitudinal force, owing to the effects of inertia, one obtainsT1x=ρF2u0t2=ρF22t2[0x(w0x)2dx+C(t)x].

Integrating

Non-linear normal modes (NNMs) for a clamped–clamped beam

Concept of NNMs for discrete systems plays important role in nonlinear dynamics of lumped mass mechanical systems (see for details [21,22]). Kirchhoff model allows for an exact separation of spatial and time variables for some type of boundary conditions. Wah [23] was the first who used this possibility and constructed NNMs for continuous system. Though Bolotin's equation does not allow for exact separation of spatial and time variables, but NNMs can be constructed approximately.

Let us

Correct nonlinear dynamic equation of buckled beam

Consider a construction of the correct equations for nonlinear beam vibrations under boundary conditions, when employed axial loads are close to the buckling value. It should be emphasized that in a general case, in order to fit appropriately the experimental results, the boundary conditions can have more complex form [27]. If this circumstance is not taken into account, a comparison of theoretical and experimental results raises questions.

Starting from papers [28,29], for analysis of the

Concluding remarks

1D non-linear thin-walled structures (rods, beams, arches, rings, etc.) are commonly used as structural elements in macro-, micro- or nanoscales. Consequently, models for their analysis are currently met in any field of civil and industrial engineering. The resulting complicated non-linear PDEs can be solved by discrete methods (FEM, FD, etc.). However, the time required to solve high-dimensional discretized models remains a bottleneck towards the efficient and optimal design of structures. In

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.

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