On the Bolotin's reduced beam model versus various boundary conditions
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 formwhere: M = EIκ, T = EFε, 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 takeThen, from Eq. (12) and boundary conditions (28), one obtains T0 = 0 and
Function C(t) will be found later. For the additional longitudinal force, owing to the effects of inertia, one obtains
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.
References (32)
- et al.
Exact nonlinear model reduction for a von Kármán beam: slow-fast decomposition and spectral submanifolds
J. Sound Vib.
(2018) - et al.
Non-linear vibrations of a clamped beam with initial defection and initial axial displacement, part I: theory
J. Sound Vib.
(1980) - et al.
Non-linear vibrations of a clamped beam with initial defection and initial axial displacement, part II: experiment
J. Sound Vib.
(1980) - et al.
Refined models of elastic beams undergoing large in-plane motions: theory and experiment
Int. J. Solids Struct.
(2006) - et al.
Consequences of different definitions of bending curvature on nonlinear dynamics of beams
Proc. Eng.
(2017) - et al.
On the notion of curvature and its mechanical meaning in a geometrically exact plane beam theory
Int. J. Mech. Sci.
(2017) - et al.
Mathematical Models of Beams and Cables
(2013) Nonlinear structural mechanics
Theory, Dynamical Phenomena and Modeling
(2013)- et al.
Nonlinear modal interactions in clamped–clamped mechanical resonators
Phys. Rev. Lett.
(2010) - et al.
Model order reduction of nonlinear Euler–Bernoulli beam
Similitude and Approximation Theory
Nichtlineare Mechanik
On the improved Kirchhoff equation modelling nonlinear vibrations of beams
Acta Mech.
Dynamic buckling
Cited by (3)
Nonlinear oscillation of a microbeam due to an electric actuation—Comparison of approximate models
2024, ZAMM Zeitschrift fur Angewandte Mathematik und MechanikHamiltonian-based frequency-amplitude formulation for nonlinear oscillators
2021, Facta Universitatis, Series: Mechanical Engineering