Notes on nonlinear beam theory

1987 ◽  
Author(s):  
DONALD DANIELSON ◽  
DEWEY HODGES
Keyword(s):  
Beam Theory ◽  
Nonlinear Beam

Investigation on Free Vibration of Buckled Beams

2012 ◽  
Vol 433-440 ◽  
pp. 41-44 ◽  
Author(s):  
Ming Hsu Tsai ◽  
Wen Yi Lin ◽  
Kuo Mo Hsiao ◽  
Fu Mio Fujii
Keyword(s):  
Free Vibration ◽  
Virtual Work ◽  
Beam Theory ◽  
Taylor Series Expansion ◽  
Element Formulation ◽  
Governing Equations ◽  
Nonlinear Beam ◽  
Buckled Beams ◽  
Buckled Beam ◽  
D’Alembert Principle

The objective of this study is to investigate the deformed configuration and free vibration around the deformed configuration of clamped buckled beams by co-rotational finite element formulation. The principle of virtual work, d'Alembert principle and the consistent second order linearization of the nonlinear beam theory are used to derive the element equations in current element coordinates. The governing equations for linear vibration are obtained by the first order Taylor series expansion of the equation of motion at the static equilibrium position of the buckled beam. Numerical examples are studied to investigate the natural frequencies of clamped buckled beams with different slenderness ratios under different axial compression.

scholarly journals A direct approach to evaluate interaction forces between self-adhesive polymeric films subjected to T-peeling

2020 ◽  
Author(s):  
Konstantin Naumenko ◽  
Behnaz Bagheri
Keyword(s):  
Beam Theory ◽  
Peel Test ◽  
Exponential Sum ◽  
Polymeric Films ◽  
Direct Approach ◽  
Adhesive Interaction ◽  
Opening Displacement ◽  
Equilibrium Conditions ◽  
Interaction Forces ◽  
Nonlinear Beam

AbstractIn this paper, a direct procedure to identify interaction forces between self-adhesive flexible polymeric films is developed. High-resolution photographs of the deformed shape within and outside the zone of adhesive interaction are taken at different instances of the T-peel test. To describe the deformed centerline, an approximate analytical solution to the equations of the nonlinear beam theory is derived. The obtained function is the exponential sum satisfying both kinematic and static boundary conditions for the T-peel configuration. The interaction forces computed with the developed function satisfy equilibrium conditions. The procedure provides characteristics of the adhesive interaction such as the energy of adhesion, maximum force and critical opening displacement. Furthermore, the developed direct approach is applicable to generate the whole traction–separation curve.

scholarly journals Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150790 ◽  
Author(s):  
F. dell’Isola ◽  
I. Giorgio ◽  
M. Pawlikowski ◽  
N. L. Rizzi
Keyword(s):  
Large Deformations ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Beam Theory ◽  
Beam Model ◽  
Computationally Efficient ◽  
Nonlinear Beam ◽  
Pantographic Structures ◽  
Diffusion Of Technology ◽  
Second Gradient

The aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call ‘pantographic structures’. The interest in these materials was increased by the possibilities opened by the diffusion of technology of three-dimensional printing. They can be regarded, once choosing a suitable length scale, as families of beams (also called fibres) interconnected to each other by pivots and undergoing large displacements and large deformations. There are, however, relatively few ‘ready-to-use’ results in the literature of nonlinear beam theory. In this paper, we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by Piola to formulate a continuum fully nonlinear beam model. The homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems. Furthermore, we consider pantographic structures, find the corresponding homogenized second gradient deformation energies and study some planar problems. Numerical solutions for these two-dimensional problems are obtained via minimization of energy and are compared with some experimental measurements, in which elongation phenomena cannot be neglected.

Modeling and Experimentation of a Ribbed Caudal Fin for Aquatic Robots

2017 ◽  
Author(s):  
Oscar Rios ◽  
Ardavan Amini ◽  
Hidenori Murakami
Keyword(s):  
Large Deformation ◽  
Beam Theory ◽  
Fe Model ◽  
Beam Model ◽  
Caudal Fin ◽  
Nonlinear Beam ◽  
Beam Elements ◽  
Side Walls ◽  
Thin Beams ◽  
Shear Deformable

Presented in this study is a mathematical model and preliminary experimental results of a ribbed caudal fin to be used in an aquatic robot. The ribbed caudal fin is comprised of two thin beams separated by ribbed sectionals as it tapers towards the fin. By oscillating the ribbed caudal fin, the aquatic robot can achieve forward propulsion and maneuver around its environment. The fully enclosed system allows for the aquatic robot to have very little effect on marine life and fully blend into its respective environment. Because of these advantages, there are many applications including surveillance, sensing, and detection. Because the caudal fin actuator has very thin side walls, Kirchhoff-Love’s large deformation beam theory is applicable for the large deformation of the fish-fin actuator. In the model, it is critical to accurately model the curvature of beams. To this end, C1 beam elements for thin beams are developed by specializing the shear-deformable beam elements, developed by the authors, based upon Reissner’s shear-deformable nonlinear beam model. Furthermore, preliminary experiments on the ribbed fin are presented to supplement the FE model.

Free Vibrations of a Rotating Inclined Beam

2006 ◽  
Vol 74 (3) ◽  
pp. 406-414 ◽  
Author(s):  
Sen Yung Lee ◽  
Jer Jia Sheu
Keyword(s):  
Natural Frequencies ◽  
Beam Theory ◽  
Free Vibrations ◽  
Physical Parameters ◽  
Series Solutions ◽  
Vibration System ◽  
Nonlinear Beam ◽  
Consistent Linearization ◽  
Inclined Beam ◽  
Dynamic Subsystem

By utilizing the Hamilton principle and the consistent linearization of the fully nonlinear beam theory, two coupled governing differential equations for a rotating inclined beam are derived. Both the extensional deformation and the Coriolis force effect are considered. It is shown that the vibration system can be considered as the superposition of a static subsystem and a dynamic subsystem. The method of Frobenius is used to establish the exact series solutions of the system. Several frequency relations that provide general qualitative relations between the natural frequencies and the physical parameters are revealed without numerical analysis. Finally, numerical results are given to illustrate the general qualitative relations and the influence of the physical parameters on the natural frequencies of the dynamic system.

Limit-Cycle Oscillation of High-Aspect-Ratio Wings

2014 ◽  
Vol 556-562 ◽  
pp. 4329-4332
Author(s):  
Yan Ping Xiao ◽  
Yi Ren Yang ◽  
Peng Li
Keyword(s):  
Aspect Ratio ◽  
Limit Cycle ◽  
High Aspect Ratio ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Structural Equations ◽  
Limit Cycle Oscillation ◽  
Nonlinear Beam ◽  
Structural Nonlinearity ◽  
Cycle Oscillation

In this paper structural equations of motion based on nonlinear beam theory and the unsteady aerodynamic forces are gained to study the effects of geometric nonlinearity on the aerodynamic response of high-aspect-ratio wings. Then the Galerkin’s method is used to discretize the equations of motion. The results of HALE wing show good agreement with references. And other results investigate the effects of geometric structural nonlinearity on the response of a wing. Also the complex changes of the limit-cycle oscillation with speed increasing is carefully studied.

An experiment in nonlinear beam theory

2004 ◽  
Vol 72 (8) ◽  
pp. 1008-1012 ◽  
Author(s):  
A. Valiente
Keyword(s):  
Beam Theory ◽  
Nonlinear Beam

scholarly journals Effects of geometric nonlinearity in an adhered microbeam for measuring the work of adhesion

2018 ◽  
Vol 474 (2211) ◽  
pp. 20170594
Author(s):  
Wenqiang Fang ◽  
Joyce Mok ◽  
Haneesh Kesari
Keyword(s):  
Three Dimensional ◽  
Beam Theory ◽  
Work Of Adhesion ◽  
Element Analysis ◽  
Nonlinear Beam ◽  
Force Microscopy ◽  
Atomic Force ◽  
Microelectromechanical Devices ◽  
The One ◽  
Good Agreement

Design against adhesion in microelectromechanical devices is predicated on the ability to quantify this phenomenon in microsystems. Previous research related the work of adhesion for an adhered microbeam to the beam's unadhered length, and as such, interferometric techniques were developed to measure that length. We propose a new vibration-based technique that can be easily implemented with existing atomic force microscopy tools or similar metrology systems. To make such a technique feasible, we analysed a model of the adhered microbeam using the nonlinear beam theory put forth by Woinowsky–Krieger. We found a new relation between the work of adhesion and the unadhered length; this relation is more accurate than the one by Mastrangelo & Hsu (Mastrangelo & Hsu 1993 J. Microelectromech. S. , 2 , 44–55. ( doi:10.1109/84.232594 )) which is commonly used. Then, we derived a closed-form approximate relationship between the microbeam's natural frequency and its unadhered length. Results obtained from this analytical formulation are in good agreement with numerical results from three-dimensional nonlinear finite-element analysis.

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