Differential quadrature-based solution for non-classical Euler-Bernoulli beam theory

2021 ◽  
Vol 86 ◽  
pp. 104135
Author(s):  
Md Ishaquddin ◽  
S. Gopalakrishnan
Keyword(s):  
Beam Theory ◽  
Differential Quadrature ◽  
Bernoulli Beam ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

Analysis of bimorph piezoelectric beam energy harvesters using Timoshenko and Euler–Bernoulli beam theory

2012 ◽  
Vol 24 (2) ◽  
pp. 226-239 ◽  
Author(s):  
Gang Wang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Slenderness Ratio ◽  
Beam Theory ◽  
Bernoulli Beam ◽  
Energy Harvesters ◽  
Bernoulli Beam Theory ◽  
Spectral Finite Element ◽  
Euler Bernoulli Beam ◽  
Element Method

Single-degree-of-freedom lumped parameter model, conventional finite element method, and distributed parameter model have been developed to design, analyze, and predict the performance of piezoelectric energy harvesters with reasonable accuracy. In this article, a spectral finite element method for bimorph piezoelectric beam energy harvesters is developed based on the Timoshenko beam theory and the Euler–Bernoulli beam theory. Linear piezoelectric constitutive and linear elastic stress/strain models are assumed. Both beam theories are considered in order to examine the validation and applicability of each beam theory for a range of harvester sizes. Using spectral finite element method, a minimum number of elements is required because accurate shape functions are derived using the coupled electromechanical governing equations. Numerical simulations are conducted and validated using existing experimental data from the literature. In addition, parametric studies are carried out to predict the performance of a range of harvester sizes using each beam theory. It is concluded that the Euler–Bernoulli beam theory is sufficient enough to predict the performance of slender piezoelectric beams (slenderness ratio > 20, that is, length over thickness ratio > 20). In contrast, the Timoshenko beam theory, including the effects of shear deformation and rotary inertia, must be used for short piezoelectric beams (slenderness ratio < 5).

Case Study of the Effect of Combined Axial and Lateral Loadings on the Critical Buckling Capacity of Piping Supports

2020 ◽  
Author(s):  
Phillip Wiseman ◽  
Alex Mayes ◽  
Shreeya Karnik
Keyword(s):  
Large Deformation ◽  
Axial Loading ◽  
Beam Theory ◽  
Second Order ◽  
Lateral Acceleration ◽  
Bernoulli Beam ◽  
Deformation Method ◽  
Closed Form Solutions ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

Abstract Snubbers are used in industry to restrain piping in dynamic events which can see significant axial loading as well as lateral acceleration. Snubbers are often employed with an extension when required to bridge gaps between the piping and building structure. As a result, they are susceptible to buckling instability issues. The pipe support and restraint design by analysis buckling criteria for supports given within the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code, Section III, Division 1, Subsection NF is investigated to determine the behavior of snubber assemblies under combined axial and lateral loadings. Four types of analyses are performed on the assemblies under the action of axial loading to demonstrate finite element and closed form solutions. These include the following: linear Eigen buckling, nonlinear second order large deformation method, energy method and Euler Bernoulli beam theory. In addition, a variety of snubber assembly sizes are subjected to combined axial and lateral loading in the form of multiple magnitudes of lateral acceleration. The behavior was analyzed by the Euler Bernoulli beam theory and nonlinear second order large deformation method. The techniques of each method are compared providing explanations of the assumptions taken, relevant limitations and recommended applications.

Optimal plane beams modelling elastic linear objects

Robotica ◽  
2009 ◽  
Vol 28 (1) ◽  
pp. 135-148 ◽  
Author(s):  
Sung K. Koh ◽  
Guangjun Liu
Keyword(s):  
Motion Planning ◽  
Analytical Model ◽  
Cantilever Beam ◽  
Inverse Kinematics ◽  
Beam Theory ◽  
Analytical Models ◽  
Bernoulli Beam ◽  
Deterministic Models ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

SUMMARYThis paper discusses analytical and deterministic models for a plane curve with minimum deformation that may be utilized in planning the motion of elastic linear objects and investigating the inverse kinematics of a hyper-redundant robot. It usually requires intensive computation to determine the configuration of elastic linear objects. In addition, conventional optimization-based numerical techniques that identify the shape of elastic linear objects in equilibrium involve non-deterministic aspects. Several analytical models that produce the configuration of elastic linear objects in an efficient and deterministic manner are presented in this paper. To develop the analytical expressions for elastic linear objects, we consider a cantilever beam where the deflections are determined according to the Euler–Bernoulli beam theory. The deflections of the cantilever beam are determined for prescribed constraints imposed on the deflections at the free end to replicate various elastic linear objects. Deflections of a cantilever beam with roller supports are explored to replicate elastic linear objects in contact with rigid objects. We verify the analytical models by comparing them with exact beam deflections. The analytical model is precisely accurate for beams with small deflections as it is developed on the basis of the Euler–Bernoulli beam theory. Although it is applied to beams undergoing large deflections, it is still reasonably accurate and at least as precise as the conventional pseudo-rigid-body model. The computational demand involved in using the analytical models is negligible. Therefore, efficient motion planning for elastic linear objects can be realized when the proposed analytical models are combined with conventional motion planning algorithms. We also demonstrate that the analytical model solves the inverse kinematics problem in an efficient and robust manner through numerical simulations.

Sign in / Sign up

Export Citation Format

Share Document