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

Highlights

• Optimization algorithm-based approach (OABA) is proposed to predict the large deflection of cantilever beams.

• This method can predict the large deformation of uniform and non-uniform beams with high accuracy.

• This method provides a new insight into the derivation of large deflection of cantilever beam.

• This method can solve the deformation of compliant parallel-guiding mechanism.

Abstract

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.

Introduction

Beam mechanism and beam theory have been widely used in many engineering fields such as compliant mechanisms [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], flexible hinges [11], [12], [13], [14], [15], [16], [17], [18], soft robots [19], [20], [21], [22], [23], [24], piezoelectric beam-based energy harvesters [25], [26], [27], [28], and leaf springs [29], [30]. Compliant mechanisms exploit the deflection of the beams to transfer or transform the smooth and precise force and motion. Owing to utilizing fewer moveable parts, compliant mechanisms have several merits over the traditional rigid-body mechanisms such as low backlash and low wear. These advantages mentioned have attracted considerable attention from researchers and industrials, and most of them devoted to promoting the study and use of compliant mechanisms over the last decades. Leaf springs can be more compact and lightweight, compared with coil springs, for providing the same stiffness and energy storage capability, so as to enable moveable robots to provide the required elasticity at the same time meet the size and weight limitations.

To predict/characterize the deformation of beam mechanisms, vast researchers committed themselves to devise and study the beam's governing load-deformation relations. Linear beam theory has been developed and widely used to derive the small deformation of cantilever beams subject to tip loads. However, once the deformation becomes large, nonlinearity will be raised in the governing equations due to arc-length conservation and geometric nonlinearity. The linear beam theory is not valid in this case. To handle that, researchers proposed alternative methods to model the large deflection of a cantilever beam. These currently available methods can be generally categorized as elliptical integrals, finite element method, beam constraint model, energy-minimization-based solution, and pseudo-rigid body model [31]. The elliptic integrals [32], [33], [34], [35] are considered as the most accurate one, which is used as a standard solution to evaluate other methods. It is noteworthy that the axial elongation and shear of the beam are not considered in the elliptic integrals. Lyon and Howell [33] developed the elliptic integrals with no inflection point for a fixed-fixed beam, then Kimball and Tsai [34] added an inflection point to the elliptic integrals. Zhang and Chen [35] established comprehensive elliptic integrals with the consideration of the number of inflection points. With this method, the deflection of the cantilever beam can be precisely estimated regardless of the load conditions and deflection modes. Chen [36] proposed an integral approach by using the moment integral treatment, unlike the elliptical integrals, which can be applied to problems of complex force load and variable beam properties such as variable cross-section area or elasticity of materials.

Beam constraint model (BCM) [37] is a closed-form and parametric mathematical model, which is used to capture the constraint characteristics of a cantilever beam in terms of the stiffness and errors motion. Besides, in this model, the load conditions, initial and boundary conditions, and beam shapes can be taken into account. Further, the nonlinearity associated with load equilibrium is also considered here. The accuracy of this method can be guaranteed over a range of load and displacement. In particular, the maximum transverse deflection is limited to ±0.1. Then Awtar and Sen [38] developed a nonlinear strain energy formulation of the BCM based on principles of virtual work to derive the nonlinear load-displacement relations for complex flexure mechanisms. Here the internal interaction force and the load equilibrium for each constitute beam can be ignored thus significantly simplifying the calculation. To increase the range of deflection, Chen et al. [39], [40] developed a chained beam constant model (CBCM). The flexible beam was divided into a few elements, and each element is characterized by BCM.

Howell and Midha [41] proposed to approximate the beam's deflection with a pseudo-rigid body model (PRB 1R). Specifically, the cantilever beam is modeled as two rigid links connected by a pin joint along with a torsion spring. Besides, the first rigid links is fixed to the ground. The length of each rigid link and the stiffness of the torsion spring are optimized to reduce the tip locus errors between the derived results and the numerical elliptic integral data when the cantilever beam is subjected to a pure end force. To improve the accuracy and expand the load conditions, Kimball and Tsai [34] developed PRB 2R model that consists of three rigid links for modeling the deflection of a cantilever beam with an inflection point. Yu et al. [42] also developed a PRB 2R model to improve the accuracy of PRB 1R. However, the parameters in the PRB 2R model are dependent on the loading conditions. Su [43] established a PRB 3R model with high accuracy for a large range of deflection. The parameters in this model were optimized to obtain high accuracy regardless of the load conditions [44], [45]. To accurately predict the deflection of a cantilever beam with an inflection point, Yu et al. [46], [47] developed a PRB 5R model, in which six rigid links are joint at five pin joints. Four of the pin joints are accompanied by torsion springs, and the rest one is a free hinge. Unlike the aforementioned PRB 1R model [41], Verotti [48] developed a new one-DoF rigid body model, in which the stiffness coefficient was evaluated based on strain energy, and the occurrence of an inflection point was also considered here.

The aforementioned methods exploited governing equations or approximated mathematic models to derive the large deflection of cantilever beams. Note that these methods are highly dependent on the load conditions, boundary conditions, or assumptions for simplification. It is hard to guarantee high accuracy when the cantilever beam is subjected to a large range of load conditions. PRB and BCM have been proposed to predict the large deflection of cantilever beam and can solve the problems very well in some special cases. For instance, PRB models were widely used to derive the large deflection of uniform beam [41], [42], [43]. Vedant and Allison [49] proposed to use PRB to predict the large deflection of non-uniform beam. However, the parameters here need to be optimized based on the finite element analysis, and the maximum error could reach 10%. Besides, the accuracy of BCM and CBCM [37], [38], [39], [40] for modeling the large deflection of non-uniform beam still need to be further studied. Recently, from new aspects, researchers have proposed to use AI such as neural networks and optimization algorithms to solve engineering problems to which traditional approaches are ineffective or infeasible. The details of the systems such as the interaction between each component can be ignored, and the solutions can be obtained through performing sets of intelligent iterations. Shahabi and Kuo [50] devised an artificial neural network (ANN) to solve the inverse kinematics solutions of the dual-backbone continuum robot, and the forward kinematics solutions derived from PRBM were used to train the ANN. Mohamad et al. [51] employed a PSO to identify the parameters in the dynamic model of a flexible beam structure. Saffar et al. [52] developed an ANN to model the experimental data of an aluminum cantilever beam. With this trained model, the natural frequencies of the system can be predicted.

The objective of this paper is to propose a new approach to model the large deflection of uniform and non-uniform cantilever beams when subjected to a combined tip load. Unlike the existing methods, this new method, called OABA, utilized an optimization algorithm to find the locus of the beam tip through performing iteration processes. Then with that and based on the Euler-Bernoulli (E-B) beam theory, the deflection curve of the cantilever beam can be derived. The work presented here is organized in the following sequence. Section 2 introduces the working principles of the PSO-based algorithm to model the large deflection of cantilever beam. Section 3 describes the case studies including modeling of uniform cantilever beam, modeling of non-uniform cantilever beam, and analysis of compliant parallel-guiding mechanism. Section 4 discusses the extension of the presented method. Section 5 is conclusion.