Large Deflection Of Cantilever Beam Research Articles

A correction method for large deflections of cantilever beams with a modal approach

Abstract. Modal-based reduced-order models are preferred for modeling structures due to their computational efficiency in engineering problems. One of the important limitations of the classic modal approaches is that they are geometrically linear. This study proposes a fast correction method to account for geometric nonlinearities which stem from large deflections in cantilever beams. The method relies on pre-computed correction terms and thus adds negligibly small extra computational efforts during the time domain response analyses. The accuracy of the method is examined on a straight-beam model and International Energy Agency (IEA) 15 MW wind turbine blade model. The results show that the proposed method increases the accuracy of modal approaches significantly in secondary deflections due to nonlinearities such as axial and torsional motions for the two studied cases.

Being gradually softened approach for solving large deflection of cantilever beam subjected to distributed and tip loads

The solutions to small deflections of uniform cantilever beams have been obtained and widely used. However, when the deflections become large, it is quite challenging to derive the solutions due to the geometric nonlinearity, especially for non-uniform beams. To handle that, in this paper, a novel and an efficient method called being gradually softened approach (BGSA) is proposed to predict the large deflections of uniform and non-uniform cantilever beams subjected to distributed and tip loads. The cantilever beam is initially regarded as a rigid beam and then is gradually softened from the free end to the fixed end to solve the deflection of the cantilever beam. As new rigid units of the cantilever beam are softened, the direction of the force load acting on the previous softened units changes. Because of that, the deflections of the previous softened units need to be updated accordingly. Experimental results show that this method can solve the large deflections of uniform and non-uniform cantilever beams that experience distributed and tip loads with high accuracy.

Optimization algorithm-based approach for modeling large deflection of cantilever beam subject to tip load

The modeling of beam mechanisms, especially non-uniform beams, becomes complicated due to the geometric nonlinearity that is proved to be significant with large deflection. A new method, called optimization algorithm-based approach (OABA), is proposed to predict the large deflection of uniform and non-uniform cantilever beams, in which an optimization algorithm is exploited to find the locus of the beam tip. The Euler-Bernoulli beam theory is employed here. With the derived locus of the beam tip, the deflection curve of the cantilever beam can be calculated. The optimization algorithm in this paper is embodied in a particle swarm optimization (PSO) algorithm. Experimental results show that the proposed method can precisely predict the deflection of the uniform and non-uniform cantilever beams. The maximum error is limited to 4.35% when the normalized maximum transverse deflection reaches 0.75. To demonstrate the effectiveness of this method in analyzing compliant mechanisms, we also exploited this method to predict the deformation of a compliant parallel-guiding mechanism.

Large Deflection of Prismatic Cantilever Beam Exposed to Combination of End Inclined Force and Tip Moment

The large deflection of a prismatic Euler-Bernoulli cantilever beam under a combination of end-concentrated coplanar inclined force and tip-concentrated moment is investigated. The angle of inclination of the applied force with respect to the horizontal axis remains unchanged during deformation. The cantilever beam is assumed to be naturally straight, slender, inextensible and elastic. The large deflection of the cantilever beam induces geometrical nonlinearity; hence, the nonlinear theory of bending and the exact expression of curvature are used. Based on an elliptic integral formulation, an accurate numerical solution is obtained in terms of an integration constant that should satisfy the boundary conditions associated with the cantilever beam. For some special cases this integration constant is exactly found, which leads to closed form solution. The numerical solution obtained is quite simple, accurate and involves less computational time compared with other techniques available in literature. The details of elastica and its corresponding orientation curves are presented and analyzed for extremely large load combinations. A comparative study with pre-obtained results has been made to verify the accuracy of the presented solution; an excellent agreement has been obtained.

An Elliptic Integral Solution to the Multiple Inflections Large Deflection Beams in Compliant Mechanisms

In this paper, a solution based on the elliptic integrals is proposed for solving multiples inflection points large deflection. Application of the Bernoulli Euler equations of compliant mechanisms with large deflection equation of beam is obtained ,there is no inflection point and inflection points in two cases respectively. The elliptic integral solution which is the most accurate method at present for analyzing large deflections of cantilever beams in compliant mechanisms.

Non-Uniform Leaf Springs with Geometric Nonlinearity

The present study deals with the analysis of large deflection of cantilever beams for various cross sections with a transverse load at free end. The aim of this analysis is to study the vertical and horizontal displacement behaviour of leaf springs which is traditionally modeled as cantilever beams of variable cross section. Besides the free end displacement, the variation of stress, strain and the bending moment of the beam are obtained by the technique of minimization of total potential energy principle. The displacement functions are approximated by linear combination of sets of orthogonal coordinate functions, developed through Gram-Schmidt scheme and substituted in the governing equilibrium equation. The final solution of the large displacement geometric nonlinear problem is obtained iteratively with the help of matlab computaional simulation. It is observed that the free end displacements and the maximum stress at fixed end are greatly affected by the geometry of the beam cross section. The present computational method has been validated and some new results have been furnished.

Large Deflection of Cantilever Beam with Uniformly Distributed Load Using Homotopy Analysis Method

The large deflection behavior of prismatic cantilever beams subjected to uniformly distributed load is investigated. An approximate analytical solution is obtained using the homotopy analysis method (HAM). The solution is validated by the nonlinear shooting method. This reveals that the solution is accurate, efficient and convenient for cantilever beams with uniformly distributed loads.

Finding the optimal characteristic parameters for 3R pseudo-rigid-body model using an improved particle swarm optimizer

Compliant mechanisms have been used in many engineering areas where high precision and sensitivity are required. One of the major challenges of designing compliant mechanisms lies in understanding and analyzing the nonlinear deflections of flexible members. The pseudo-rigid-body model (PRBM) method, which simplifies the modeling of the nonlinear deflection by approximating it as motion of rigid links, has been accepted as one of the most important tools for synthesis and analysis of compliant mechanisms. In this paper, a review of various PRBMs is presented. The 3R PRBM whose characteristic parameters are independent of external loads is discussed in detail. For the purpose of finding the optimal set of the characteristic parameters for the 3R PRBM, a six-dimensional objective function is formulated by combining the approximation errors of both tip point and tip slope for the two extreme load cases, i.e., pure moment load and pure vertical force load. A particle swarm optimizer was employed to conduct a continuous search on the objective function. The resulting 3R PRBM with the optimized characteristic parameters shows better performance in predicting large deflections of cantilever beams over the original 3R PRBM.

Bending of Flexible Joint Driven by Linear Expandable Artificial Muscle

A flexible finger joint is proposed and applied in multi-joint manipulator. The plate-spring composes of a flexible skeleton. The linear expandable elastomeric bellows by air pressure is used as an artificial muscle. Bend angle of the joint is related to air pressure in the bellows. Static model of the joint bend has been created by assuming that the counteracting force from elastic elongation of the bellows is negligible. The problems of pure mechanics and the model have been studied in detail. The numerical solution based on the model is obtained for the case of the large deflection of cantilever beam synchronously acted on cantilever end by a couple in variable scalar along with forces changing in both scalar and direction.

Large deflection of constant curvature cantilever beam under follower load

This paper describes a method to analyze for the large deflections of curved prismatic cantilever beams with uniform curvature subjected to a follower load at the tip. The large deflection, the deflection dependent follower load and the initial curved geometry are the important features of the beam considered in this work. Shear force formulation proposed by Lee [Large deflections of cantilever beams of non-linear elastic material under a combined loading. Int J Non-Linear Mech 2002;37(3)] is used for deriving the governing equations. Using this approach, the resulting two point boundary value problem (TPBVP) can be reduced to an initial value problem (IVP). Fourth order Runge–Kutta method along with one parameter reverse shooting method is applied to the numerical solution to the problem. A novel approach presented in this paper of integrating from the free end to the fixed end of the cantilever beam simply replaces the two parameter shooting with a single parameter shooting yielding several advantages. This solution technique is demonstrated for various types of follower tip loads on curved and straight cantilever beams and is validated with existing solutions in the literature.

Semi-exact solutions for large deflections of cantilever beams of non-linear elastic behaviour

The problem of large deflections of cantilever beams made of materials obeying a Ludwick type stress–strain law under the combined action of one vertical concentrated force at the free end and a uniformly distributed load were investigated in the so-called bending moment formulation. A corrected bending moment is introduced and it is shown that the resulting governing differential equation for the slopes can be solved semi-analytically for some few selected cases of interest in which materials like acetal plastic and glass fibre enter. A discovered coupling between the material and geometrical non-linearities involved in the problem divides the general solution into two regions separated by one in which there is no coupling at all and where it is expected to exist an elastic critical behaviour.

Uniform strength for large deflections of cantilever beams under end point load

Analysis is conducted for slender beams with a varying cross-section under large non-linear elastic deformation. A thickness variation function is derived to achieve optimal - constant maximum bending stress distribution along the beam for inclined end load of arbitrary direction. Closed form solutions are derived for the large deflections that correspond to the various loading conditions. The analysis is repeated for a beam with optimally varying width (for arbitrary end force) and the width variation function is also determined.

Determining large deflections in rectangular combined loaded cantilever beams made of non-linear Ludwick type material by means of different arc length assumptions

In this study, large deflection of cantilever beams of Ludwick type material subjected to a combined loading consisting of a uniformly distributed load and one vertical concentrated load at the free end was investigated. In calculations, both material and geometrical non-linearity have been considered. Horizontal and vertical deflections magnitudes were calculated throughout Euler-Bernoulli curvature-moment relationship assuming different arc lengths. Vertical deflections were calculated by using Runge-Kutta method. More simple and easily understandable results have been obtained compared to the previous studies about the issue and compatible values have been obtained for most of the compared values.

Large deflection of cantilever beams with geometric non-linearity: Analytical and numerical approaches

Non-linear shooting and Adomian decomposition methods have been proposed to determine the large deflection of a cantilever beam under arbitrary loading conditions. Results obtained only due to end loading are validated using elliptic integral solutions. The non-linear shooting method gives accurate numerical results while the Adomian decomposition method yields polynomial expressions for the beam configuration. With high load parameters, occurrence of multiple solutions is discussed with reference to possible buckling of the beam-column. An example of concentrated intermediate loading (cantilever beam subjected to two concentrated self-balanced moments), for which no closed form solution can be obtained, is solved using these two methods. Some of the limitations and recipes to obviate these are included. The methods will be useful toward the design of compliant mechanisms driven by smart actuators.

Numerical and Experimental Analysis of Large Deflections of Cantilever Beams Under a Combined Load

We analyzed the large deflections of a uniform cantilever beam under the action of a combined load consisting of an external vertical concentrated load at the free end and a uniformly distributed load. The problem involves geometrical non-linearity and a differential equation governing the behavior of the beam is presented. The numerical solution was obtained using an algorithm based on the Runge-Kutta-Felhberg method. This algorithm was used to numerically evaluate the system, and the slopes and deflections were determined for various loads. Finally, we compared the numerical results with the experimental ones obtained in the laboratory and calculated Young's modulus for the beam material.

Large deflections of cantilever beams of non-linear elastic material under a combined loading

Large deflection of cantilever beams made of Ludwick type material subjected to a combined loading consisting of a uniformly distributed load and one vertical concentrated load at the free end was investigated. Governing equation was derived by using the shearing force formulation instead of the bending moment formulation because in the case of large deflected member, the shearing force formulation possesses some computational advantages over the bending moment formulation. Since the problem involves both geometrical and material non-linearities, the governing equation is complicated non-linear differential equation, which would in general require numerical solutions to determine the large deflection for a given loading. Numerical solution was obtained by using Butcher's fifth order Runge–Kutta method and are presented in a tabulated form.

On the Large Deflection of Cantilever Beams with End Rotational Load

On the Large Deflection of Cantilever Beams with End Rotational Load

つりざおの力学的特性と機能に関する研究 テーパ付細長い棒の大たわみ解析

This paper deals with large deflections of cantilever beams to develop fishing scientifically in the field of so called sports-leisure. The fundamental equations can be obtained in a state of equilibrium under concentrated loads at free end and calculated strictly by means of numerical solution method. From numerical results, fundamental properties for various mechanical quantities are obtained as follows: (1) The large deflections of cantilever beams can be described with dimensionless loads μeq, ratios of diameter D2/D1 and angles β respectively. The shapes of deflection curves are dependent of D2/D1 considerably. (2) Maximum curvature kmax and the displacement of free end xl in x direction are also functions of μeg and D2/D1. The location lM and lk where σMxmax and kmax occur do not coincide and the distance lM is smaller than lk. (3) The stiffness of fishing rods is proposed and discussed as the magnitude of resistance with respect to the large deflections.

Closure to “Large Deflection of Cantilever Beam”

Closure to “Large Deflection of Cantilever Beam”

Large deflections of cantilever beams of non-linear materials of the Ludwick type subjected to an end moment

This paper deals with the large deflections (finite) of thin cantilever beams of non-linear materials of the Ludwick type. The beam is subjected to an end constant moment. Large deflections of beams induce geometrical non-linearity. Therefore, in formulating the analysis, the exact expression of the curvature is used in the Euler-Bernoulli law. A closed-form solution is presented for the resulting second-order non-linear differential equation. This solution is compared to previous results assuming linear elastic materials. Deflections at the free end of beams of aluminum alloy and annealed copper are obtained.