An Experimental Approach to the Design of PVDF Modal Sensors

2001 ◽  
Author(s):  
Daniel Cuhat ◽  
Patricia Davies
Keyword(s):  
Boundary Conditions ◽  
Experimental Approach ◽  
Experimental Methodology ◽  
Laser Doppler Velocimeter ◽  
Piezoelectric Film ◽  
Bernoulli Beam ◽  
Beam Structure ◽  
Arbitrary Boundary ◽  
Classical Boundary ◽  
Euler Bernoulli Beam

Abstract The principle of modal sensing is based on the use of a shaped PVDF piezoelectric film measuring strains on the surface of a bending beam and acting as a modal filter. So far, the use of this type of sensors has remained confined to studies involving uniform structures with classical boundary conditions. The goal of this paper is to present an experimental methodology for the design of a shaped modal sensor applicable to an non-uniform Euler-Bernoulli beam with arbitrary boundary conditions. This approach is illustrated with test data collected on a cantilever beam structure with a laser Doppler velocimeter.

Dynamics of a Pipe Conveying Fluid Flexibly Supported at the Ends

2014 ◽  
Author(s):  
Mojtaba Kheiri ◽  
Michael P. Païdoussis ◽  
Giorgio Costa del Pozo
Keyword(s):  
Boundary Conditions ◽  
Open Systems ◽  
Equation Of Motion ◽  
Bernoulli Beam ◽  
Flexible Supports ◽  
Classical Boundary ◽  
The Subject ◽  
The Galerkin Method ◽  
Euler Bernoulli Beam ◽  
Conveying Fluid

The subject of this paper is the study of dynamics and stability of a pipe flexibly supported at its ends and conveying fluid. First, the equation of motion of the system is derived via the extended form of Hamilton’s principle for open systems. In the derivation, the effect of flexible supports, modelled as linear translational and rotational springs, is appropriately considered in the equation of motion rather than in the boundary conditions. The resulting equation of motion is then discretized via the Galerkin method in which the eigenfunctions of a free-free Euler-Bernoulli beam are utilized. Thus, a general set of second-order ordinary differential equations emerge, in which, by setting the stiffness of the end-springs suitably, one can readily investigate the dynamics of various systems with either classical or non-classical boundary conditions.

Free Vibrations of Beams with Arbitrary Boundary Conditions Based on Wave Propagation Method

2011 ◽  
Vol 66-68 ◽  
pp. 753-757
Author(s):  
Wan You Li ◽  
Hai Jun Zhou ◽  
Jun Dai ◽  
Bing Lin Lv ◽  
Dong Hua Wang ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Boundary Conditions ◽  
Beam Theory ◽  
Free Vibrations ◽  
Bernoulli Beam ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions ◽  
Wave Propagation Method ◽  
Wave Numbers ◽  
Euler Bernoulli Beam

Under the Euler-Bernoulli beam theory, the wave propagation method is used for the vibration analysis of beams with arbitrary boundary conditions. The boundary conditions end the beam could be arbitrary that all the conventional homogeneous beam boundary conditions can be included by setting the stiffnesses of the springs be infinity or zero. In this paper, the flexural displacement of the beam is expressed in the wave propagation form including wave numbers. The wavenumber could be obtained in a known form for conventional boundary conditions. So the results are obtained through the boundary conditions and the known wavenumbers and compared with the numerical results. In order to validate the correctness, results with different stiffness are compared with those obtained by previous published papers.

Calculation of the natural frequencies and mode shapes of a Euler–Bernoulli beam which has any combination of linear boundary conditions

2019 ◽  
Vol 25 (18) ◽  
pp. 2473-2479 ◽  
Author(s):  
Paulo J. Paupitz Gonçalves ◽  
Michael J. Brennan ◽  
Andrew Peplow ◽  
Bin Tang
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
New Method ◽  
Mode Shapes ◽  
Linear Boundary ◽  
Bernoulli Beam ◽  
Classical Boundary ◽  
Euler Bernoulli Beam

There are well-known expressions for natural frequencies and mode shapes of a Euler-Bernoulli beam which has classical boundary conditions, such as free, fixed, and pinned. There are also expressions for particular boundary conditions, such as attached springs and masses. Surprisingly, however, there is not a method to calculate the natural frequencies and mode shapes for a Euler–Bernoulli beam which has any combination of linear boundary conditions. This paper describes a new method to achieve this, by writing the boundary conditions in terms of dynamic stiffness of attached elements. The method is valid for any boundaries provided they are linear, including dissipative boundaries. Ways to overcome numerical issues that can occur when computing higher natural frequencies and mode shapes are also discussed. Some examples are given to illustrate the applicability of the proposed method.

Medium Frequency Vibration Analysis of Beam Structures Modeled by the Timoshenko Beam Theory

2020 ◽  
Author(s):  
Yichi Zhang ◽  
Bingen Yang
Keyword(s):  
Shear Deformation ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Beam Structure ◽  
Medium Frequency ◽  
Beam Structures ◽  
Euler Bernoulli Beam

Abstract Vibration analysis of complex structures at medium frequencies plays an important role in automotive engineering. Flexible beam structures modeled by the classical Euler-Bernoulli beam theory have been widely used in many engineering problems. A kinematic hypothesis in the Euler-Bernoulli beam theory is that plane sections of a beam normal to its neutral axis remain normal when the beam experiences bending deformation, which neglects the shear deformation of the beam. However, as observed by researchers, the shear deformation of a beam component becomes noticeable in high-frequency vibrations. In this sense, the Timoshenko beam theory, which describes both bending deformation and shear deformation, may be more suitable for medium-frequency vibration analysis of beam structures. This paper presents an analytical method for medium-frequency vibration analysis of beam structures, with components modeled by the Timoshenko beam theory. The proposed method is developed based on the augmented Distributed Transfer Function Method (DTFM), which has been shown to be useful in various vibration problems. The proposed method models a Timoshenko beam structure by a spatial state-space formulation in the s-domain, without any discretization. With the state-space formulation, the frequency response of a beam structure, in any frequency region (from low to very high frequencies), can be obtained in an exact and analytical form. One advantage of the proposed method is that the local information of a beam structure, such as displacements, bending moment and shear force at any location, can be directly obtained from the space-state formulation, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated with the FEA in numerical examples, where the efficiency and accuracy of the proposed method is present. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are illustrated through comparison of the Timoshenko beam theory and the Euler-Bernoulli beam theory.

Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model

2018 ◽  
Vol 24 (3) ◽  
pp. 559-572 ◽  
Author(s):  
Yuanbin Wang ◽  
Kai Huang ◽  
Xiaowu Zhu ◽  
Zhimei Lou
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Nonlocal Parameter ◽  
Beam Theory ◽  
Nonlocal Model ◽  
Bernoulli Beam ◽  
Two Phase ◽  
Differential Model ◽  
Euler Bernoulli Beam

Eringen’s nonlocal differential model has been widely used in the literature to predict the size effect in nanostructures. However, this model often gives rise to paradoxes, such as the cantilever beam under end-point loading. Recent studies of the nonlocal integral models based on Euler–Bernoulli beam theory overcome the aforementioned inconsistency. In this paper, we carry out an analytical study of the bending problem based on Eringen’s two-phase nonlocal model and Timoshenko beam theory, which accounts for a better representation of the bending behavior of short, stubby nanobeams where the nonlocal effect and transverse shear deformation are significant. The governing equations are established by the principal of virtual work, which turns out to be a system of integro-differential equations. With the help of a reduction method, the complicated system is reduced to a system of differential equations with mixed boundary conditions. After some detailed calculations, exact analytical solutions are obtained explicitly for four types of boundary conditions. Asymptotic analysis of the exact solutions reveals clearly that the nonlocal parameter has the effect of increasing the deflections. In addition, as compared with nonlocal Euler–Bernoulli beam, the shear effect is evident, and an additional scale effect is captured, indicating the importance of applying higher-order beam theories in the analysis of nanostructures.

Influence of Axial Force on the Vibration of Euler-Bernoulli Beam Structures Solved by DQEM Using EDQ

2006 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Boundary Conditions ◽  
Axial Force ◽  
Differential Quadrature ◽  
Analysis Model ◽  
Bernoulli Beam ◽  
Beam Structures ◽  
Differential Quadrature Element Method ◽  
Quadrature Element Method ◽  
Euler Bernoulli Beam ◽  
Element Boundary

The influence of axial force on the vibration of Euler-Bernoulli beam structures is analyzed by differential quadrature element method (DQEM) using extended differential quadrature (EDQ). The DQEM uses the differential quadrature to discretize the governing differential eigenvalue equation defined on each element, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of the beam. Numerical results solved by the developed numerical algorithm are presented. The convergence of the developed DQEM analysis model is efficient.

Influence of Axially Distributed Force on the Vibration of Euler-Bernoulli Beam Structures Solved by DQEM Using EDQ

2007 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Boundary Conditions ◽  
Numerical Algorithm ◽  
Differential Quadrature ◽  
Analysis Model ◽  
Bernoulli Beam ◽  
Beam Structures ◽  
Differential Quadrature Element Method ◽  
Quadrature Element Method ◽  
Euler Bernoulli Beam ◽  
Element Boundary

The influence of axially distributed force on the vibration of Euler-Bernoulli beam structures is analyzed by differential quadrature element method (DQEM) using extended differential quadrature (EDQ). The DQEM uses the differential quadrature to discretize the governing differential eigenvalue equation defined on each element, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of the beam. Numerical results solved by the developed numerical algorithm are presented. The convergence of the developed DQEM analysis model is efficient.

scholarly journals The Vibration of a Beam under a Traversing Load

2021 ◽  
Author(s):  
Kan-Chen Jane Wu
Keyword(s):  
Boundary Conditions ◽  
Constant Velocity ◽  
Equation Of Motion ◽  
Linear Strain ◽  
Bernoulli Beam ◽  
Simply Supported ◽  
Runge Kutta Method ◽  
Extended Hamilton’S Principle ◽  
Gyroscopic Forces ◽  
Euler Bernoulli Beam

The objective of this study is to investigate the response of an Euler-Bernoulli beam under a force or mass traversing with constant velocity. Simply-supported and clamped-clamped boundary conditions are considered. The linear strain-displacement scenario is applied to both boundary conditions, while the von Kármán nonlinear scenario is applied only to the former boundary condition. The governing equation of motion is derived via the extended Hamilton's principle. Simulations are performed with the fourth-order Runge-Kutta method via Matlab software. The equation of motion is first validated and then used to investigate the effects of the beam second moment of area, the magnitude of the traversing velocity, and centrifugal and gyroscopic forces.

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