• Smart Structures and Systems
• /
• v.4 no.1
• /
• pp.67-84
• /
• 2008

The present contribution is concerned with the static and dynamic stability of a piezo-laminated Bernoulli-Euler beam subjected to an axial compressive force. Recently, an inconsistent derivation of the equations of motions of such a smart structural system has been presented in the literature, where it has been claimed, that an axial piezoelectric actuation can be used to control its stability. The main scope of the present paper is to show that this unfortunately is impossible. We present a consistent theory for composite beams in plane bending. Using an exact description of the kinematics of the beam axis, together with the Bernoulli-Euler assumptions, we obtain a single-layer theory capable of taking into account the effects of piezoelectric actuation and buckling. The assumption of an inextensible beam axis, which is frequently used in the literature, is discussed afterwards. We show that the cited inconsistent beam model is due to inadmissible mixing of the assumptions of an inextensible beam axis and a vanishing axial displacement, leading to the erroneous result that the stability might be enhanced by an axial piezoelectric actuation. Our analytical formulations for simply supported Bernoulli-Euler type beams are verified by means of three-dimensional finite element computations performed with ABAQUS.

• Journal of applied mathematics & informatics
• /
• v.20 no.1_2
• /
• pp.17-34
• /
• 2006

Inverse coefficient identification problems associated with the fourth-order Sturm-Liouville operator in the steady state Euler-Bernoulli beam equation are investigated. Unlike previous studies in which spectral data are used as additional information, in this paper only boundary information is used, hence non-destructive tests can be employed in practical applications.

• Structural Engineering and Mechanics
• /
• v.59 no.3
• /
• pp.431-454
• /
• 2016

In this paper, the nonlinear static and free vibration analysis of Euler-Bernoulli composite beam model reinforced by functionally graded single-walled carbon nanotubes (FG-SWCNTs) with initial geometrical imperfection under uniformly distributed load using finite element method (FEM) is investigated. The governing equations of equilibrium are derived by the Hamilton's principle and von Karman type nonlinear strain-displacement relationships are employed. Also the influences of various loadings, amplitude of the waviness, UD, USFG, and SFG distributions of carbon nanotube (CNT) and different boundary conditions on the dimensionless transverse displacements and nonlinear frequency ratio are presented. It is seen that with increasing load, the displacement of USFG beam under force loads is more than for the other states. Moreover it can be seen that the nonlinear to linear natural frequency ratio decreases with increasing aspect ratio (h/L) for UD, USFG and SFG beam. Also, it is shown that at the specified value of (h/L), the natural frequency ratio increases with the increasing the values amplitude of waviness while the dimensionless nonlinear to linear maximum deflection decreases. Moreover, with considering the amplitude of waviness, the stiffness of Euler-Bernoulli beam model reinforced by FG-CNT increases. It is concluded that the R parameter increases with increasing of volume fraction while the rate of this parameter decreases. Thus one can be obtained the optimum value of FG-CNT volume fraction to prevent from resonance phenomenon.

• Magazine of the Korean Society of Agricultural Engineers
• /
• v.42 no.2
• /
• pp.108-114
• /
• 2000

This paper explores the non-linear behavior of tapered beam subjected to a floating concentration load. For applying the Bernoulli-Euler beam theory to this beam, the bending moment at any point of elastical is obtained from the final equilibrium stage. By using the bending moment equation and the Bernoulli-Euler beam theory, the differential equations governing the elastica of simple beam are derived , and solved numberically . Three kinds of tapered beam types are considered . The numerical results of the non-linear behavior obtained in this study are agreed quite well to the results obtained from the laboratory-scale experiments.

• Structural Engineering and Mechanics
• /
• v.35 no.5
• /
• pp.645-658
• /
• 2010

In this study, the Differential Transform Method (DTM) is employed in order to solve the governing differential equation of a moving Bernoulli-Euler beam with axial force effect and investigate its free flexural vibration characteristics. The free vibration analysis of a moving Bernoulli-Euler beam using DTM has not been investigated by any of the studies in open literature so far. At first, the terms are found directly from the analytical solution of the differential equation that describes the deformations of the cross-section according to Bernoulli-Euler beam theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the differential equation of the motion. The calculated natural frequencies of the moving beams with various combinations of boundary conditions using DTM are tabulated in several tables and are compared with the results of the analytical solution where a very good agreement is observed.

• Transactions of the Korean Society for Noise and Vibration Engineering
• /
• v.15 no.11 s.104
• /
• pp.1318-1323
• /
• 2005

In this paper, the vibration analysis for the Euler-Bernoulli complete and truncate wedge beams by differential Transformation method(DTM) was investigated. The governing differential equation of the Euler-Bernoulli complete and truncate wedge beams with regular singularity is derived and verified. The concepts of DTM were briefly introduced. Numerical calculations are carried out and compared with previous published results. The usefulness and the application of DTM are discussed.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
• /
• 2005.11a
• /
• pp.507-512
• /
• 2005

This paper investigated the vibration analysis fer the Euler-Bernoulli complete and truncate wedge beams by Differential Transformation Method(DTM). The governing differential equation of the Euler-Bernoulli complete and truncate wedge beams with regular singularity is derived and verified. The concepts of DTM were briefly introduced. Numerical calculations are carried out and compared with previous published results. The usefulness and the application of DTM are discussed.

• Smart Structures and Systems
• /
• v.12 no.5
• /
• pp.501-521
• /
• 2013

This paper is concerned with static and dynamic shape control of a laminated Bernoulli-Euler beam hosting a uniformly distributed array of resistively interconnected piezoelectric patches. We present an analytical one-dimensional model for a laminated piezoelectric beam with material discontinuities within the framework of Bernoulli-Euler and extent the model by a network of resistors which are connected to several piezoelectric patch actuators. The voltage of only one piezoelectric patch is prescribed: we answer the question how to design the interconnected resistive electric network in order to annihilate lateral vibrations of a cantilever. As a practical example, a cantilever with eight patch actuators under the influence of a tip-force is studied. It is found that the deflection at eight arbitrary points along the beam axis may be controlled independently, if the local action of the piezoelectric patches is equal in magnitude, but opposite in sign, to the external load. This is achieved by the proper design of the resistive network and a suitable choice of the input voltage signal. The validity of our method is exact in the static case for a Bernoulli-Euler beam, but it also gives satisfactory results at higher frequencies and for transient excitations. As long as a certain non-dimensional parameter, involving the number of the piezoelectric patches, the sum of the resistances in the electric network and the excitation frequency, is small, the proposed shape control method is approximately fulfilled for dynamic load excitations. We evaluate the feasibility of the proposed shape control method with a more refined model, by comparing the results of our one-dimensional calculations based on the extended Bernoulli-Euler equations to three-dimensional electromechanically coupled finite element results in ANSYS 12.0. The results with the simple Bernoulli-Euler model agree well with the three-dimensional finite element results.

• Advances in materials Research
• /
• v.7 no.3
• /
• pp.163-174
• /
• 2018

This article studies the free and forced vibrations of the carbon nanotubes CNTs embedded in an elastic medium including thermal and dynamic load effects based on nonlocal Euler-Bernoulli beam. A Winkler type elastic foundation is employed to model the interaction of carbon nanotube and the surrounding elastic medium. Influence of all parameters such as nonlocal small-scale effects, high temperature change, Winkler modulus parameter, vibration mode and aspect ratio of short carbon nanotubes on the vibration frequency are analyzed and discussed. The non-local Euler-Bernoulli beam model predicts lower resonance frequencies. The research work reveals the significance of the small-scale coefficient, the vibrational mode number, the elastic medium and the temperature change on the non-dimensional natural frequency.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.28 no.5
• /
• pp.477-483
• /
• 2015

In this paper, Euler-Bernoulli beam theories(EB-beam) are used, and Fast Fourier Transformation(FFT) analysis is then employed to extract their natural frequencies using both analytical approach and Co-rotational plane beam(CR-beam) EDISON program. EB-beam is used to analyze a spring-mass system with a single degree of freedom. Sinusoidal force with various frequencies and constant magnitude are applied to tip of each beam. After the oscillatory tip response is observed in EB-beam, it decreases and finally converges to the so-called 'steady-state.' The decreasing rate of the tip deflection with respect to time is reduced when the forcing frequency is increased. Although the tip deflection is found to be independent of the excitation frequency, it turns out that time to reach the steady state response is dependent on the forcing frequency.