Application of consistent geometric decomposition theorem to dynamic finite element of 3D composite beam based on experimental and numerical analyses

doi:10.1007/s11709-020-0625-4

Front. Struct. Civ. Eng.

2020, Vol. 14

Issue (3) : 675-689 https://doi.org/10.1007/s11709-020-0625-4

RESEARCH ARTICLE

Application of consistent geometric decomposition theorem to dynamic finite element of 3D composite beam based on experimental and numerical analyses

Iman FATTAHI, Hamid Reza MIRDAMADI(

), Hamid ABDOLLAHI

Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 8415683111, Iran

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Abstract

Analyzing static and dynamic problems including composite structures has been of high significance in research efforts and industrial applications. In this article, equivalent single layer approach is utilized for dynamic finite element procedures of 3D composite beam as the building block of numerous composite structures. In this model, both displacement and strain fields are decomposed into cross-sectional and longitudinal components, called consistent geometric decomposition theorem. Then, the model is discretized using finite element procedures. Two local coordinate systems and a global one are defined to decouple mechanical degrees of freedom. Furthermore, from the viewpoint of consistent geometric decomposition theorem, the transformation and element mass matrices for those systems are introduced here for the first time. The same decomposition idea can be used for developing element stiffness matrix. Finally, comprehensive validations are conducted for the theory against experimental and numerical results in two case studies and for various conditions.

Keywords composite beam dynamic finite element degrees of freedom coupling experimental validation numerical validation

Corresponding Author(s): Hamid Reza MIRDAMADI

Just Accepted Date: 22 April 2020 Online First Date: 17 June 2020 Issue Date: 13 July 2020

Cite this article:

Iman FATTAHI,Hamid Reza MIRDAMADI,Hamid ABDOLLAHI. Application of consistent geometric decomposition theorem to dynamic finite element of 3D composite beam based on experimental and numerical analyses[J]. Front. Struct. Civ. Eng., 2020, 14(3): 675-689.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-020-0625-4
https://academic.hep.com.cn/fsce/EN/Y2020/V14/I3/675

Fig.1 The cross section of the 3-D beam with a piezoelectric layer.

Fig.2 The composite cantilevered beam used for numerical validation.

parameter (SI unit)	value
density of substructure ( $k g / m 3$ )	2700
density of piezoelectric layer ( $k g / m 3$ )	7750
density of end mass ( $k g / m 3$ )	7810
Young’s modulus of substructure ( $G P a$ )	70
Young’s modulus of piezoelectric layer ( $G P a$ )	61
Young’s modulus of end mass ( $G P a$ )	210
width of the cross section, a (mm)	1.5
thickness of substructure, b (mm)	0.4
thickness of piezoelectric layer, c (mm)	0.2
active length of the beam, L (mm)	33
$l x$ mm	3.6
$l y$ mm	1.5
$l z$ mm	3

Tab.1 Specifications of the composite cantilevered beam used for numerical validation

Fig.3 Natural frequencies of the composite cantilevered beam in different modes: (a) natural frequency; (b) error between theoretical model (MATLAB code) and ABAQUS results.

Fig.4 The first mode shape of the structure: (a) ABAQUS and (b) theoretical model.

Fig.5 The second mode shape of the structure: (a) ABAQUS and (b) theoretical model.

Fig.6 The third mode shape of the cantilevered harvester: (a) ABAQUS and (b) theoretical model.

Fig.7 The results of (

a z rel / a z base

) over a frequency range for the composite cantilevered beam for (a) c = 0.1 mm, (b) c = 0.2 mm, and (c) c = 0.4 mm.

Fig.8 Parametric study and numerical validation for different lengths of the beam: L = 15, 24, and 33 mm.

Fig.9 Numerical validation of the beam with two different substructures (aluminum and steel).

Fig.10 The composite cantilevered beam prototype used for experimental validations.

parameter (SI unit)	value
density of substructure ( $k g / m 3$ )	2700
density of steel patches and end mass ( $k g / m 3$ )	7810
Young’s modulus of substructure ( $G P a$ )	70
Young’s modulus of steel patches and end mass ( $G P a$ )	210
thickness of substructure, b (cm)	0.7
thickness of steel patches, c (cm)	0.2
active length of the beam, L (cm)	33.5
$l x$ (cm)	5
$l y$ (cm)	2.7
$l z$ (cm)	3.5
$l 1$ , $l 2$ (cm)	5
$l 3$ (cm)	4

Tab.2 Specifications of the composite cantilevered beam used for experimental validation

Fig.11 The whole test setup of the composite cantilevered beam used for experimental validation.

Fig.12 The results of tip displacement for experimental validation for excitation amplitudes of (a) 0.07 mm, (b) 0.1 mm, and (c) 0.13 mm.

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