Non-linear normal modes of a simply supported beam: continuous system and finite-element models

Mazzilli, Carlos Eduardo Nigro; Soares, Mário Eduardo Senatore; Neto, O.G.P. Baracho

doi:10.1016/j.compstruc.2004.07.007

Cited by 17 publications

(9 citation statements)

References 12 publications

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“…First, variations of equations (3) to (5) are substituted into equation (2) and simplified. Next, the variations along the longitudinal and transverse directions are put to zero respectively to obtain the following equations of motion [8] m€ v v{ EAe y À Á '~0 ð6Þ…”

Section: Fig 1 Non-linear Euler-bernoulli Beam Modelmentioning

confidence: 99%

“…The strain displacement and curvature displacement relation of any point on a beam undergoing a large displacement and a small rotation are given as ( [1][2][3][4][5][6][7][8])…”

Section: Non-linear Beam Modelmentioning

confidence: 99%

“…In such an approach the partial differential equation (PDE) describing the system dynamics is mapped to a finite space resulting in a finite number of ordinary differential equations (ODEs), which is done using various methods of discretization including the lumped mass technique [2], the finite element methods [3,8], the finite difference technique [9,10], or using Galerkin projection-based problem-oriented orthogonal basis functions (known as proper orthogonal decomposition) [11,12]. This set of ODEs is then used to design the control action.…”

Section: Introductionmentioning

confidence: 99%

“…In the ATD approach the idea is to first come up with a low-dimensional reduced (truncated) model, which retains the dominant modes of the system and then using this truncated model (which is often a finite-dimensional lumped parameter approximate model) to design the controller. In such an approach the partial differential equation (PDE) describing the system dynamics is mapped to a finite space resulting in a finite number of ordinary differential equations (ODEs), which is done using various methods of discretization including the lumped mass technique [2], the finite element methods [3,8], the finite difference technique [9,10], or using *Corresponding author: Department of Civil Engineering, Indian Institute of Science, Bangalore, Karnataka 560012, India. email: sk.faruque.ali@gmail.com; skfali@civil.iisc.ernet.in Galerkin projection-based problem-oriented orthogonal basis functions (known as proper orthogonal decomposition) [11,12].…”

Section: Introductionmentioning

confidence: 99%

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Active vibration suppression of non-linear beams using optimal dynamic inversion

Ali

Padhi

2009

Proceedings of the Institution of Mechanical Engineers, Part I:

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Euler-Bernoulli beams are distributed parameter systems that are governed by a non-linear partial differential equation (PDE) of motion. This paper presents a vibration control approach for such beams that directly utilizes the non-linear PDE of motion, and hence, it is free from approximation errors (such as model reduction, linearization etc.). Two state feedback controllers are presented based on a newly developed optimal dynamic inversion technique which leads to closed-form solutions for the control variable. In one formulation a continuous controller structure is assumed in the spatial domain, whereas in the other approach it is assumed that the control force is applied through a finite number of discrete actuators located at predefined discrete locations in the spatial domain. An implicit finite difference technique with unconditional stability has been used to solve the PDE with control actions. Numerical simulation studies show that the beam vibration can effectively be decreased using either of the two formulations.

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Section: Fig 1 Non-linear Euler-bernoulli Beam Modelmentioning

confidence: 99%

“…The strain displacement and curvature displacement relation of any point on a beam undergoing a large displacement and a small rotation are given as ( [1][2][3][4][5][6][7][8])…”

Section: Non-linear Beam Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Active vibration suppression of non-linear beams using optimal dynamic inversion

Ali

Padhi

2009

Proceedings of the Institution of Mechanical Engineers, Part I:

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“…In this context they have been investigated in depth and developed along various research paths, with reference to various mechanical and structural systems, which include shallow arches and cables [20][21], single beams [17], frames [22], rotor blades [23] and MEMS [24]. In this paper, they are used to detect homoclinic orbits.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

Dimension reduction of homoclinic orbits of buckled beams via the non-linear normal modes technique

Lenci

Rega

2007

International Journal of Non-Linear Mechanics

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Abstract. The problem of detecting the homoclinic orbits of an initially straight buckled beam is addressed. Two families of b.c. are identified and investigated in detail. For the first family, the homoclinic orbits belong to a planar invariant manifold, and are easily computed in closed form. For the second family, the manifold is no longer planar, and is detected via the nonlinear normal modes technique by obtaining approximate expressions which are sufficient to highlight the effects of the non-flatness. A hierarchy of reduced order, single degree of freedom, models is determined. These are obtained by taking into account increasing degrees of nonlinearity in the potential energy, which allow for a more and more refined computation of the homoclinic solution. The various models are compared with each other and discussed in detail, and the nonplanarity of the manifold is illustrated through examples.

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Modal Analysis of Nonlinear Mechanical Systems

Kerschen,

Vakakis

2022

Handbook of Experimental Structural Dynamics

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Non-linear normal modes of a simply supported beam: continuous system and finite-element models

Cited by 17 publications

References 12 publications

Active vibration suppression of non-linear beams using optimal dynamic inversion

Active vibration suppression of non-linear beams using optimal dynamic inversion

Dimension reduction of homoclinic orbits of buckled beams via the non-linear normal modes technique

Modal Analysis of Nonlinear Mechanical Systems

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