Beams In Compliant Mechanisms Research Articles

Modeling Method for Static Large Deflection Problem of Curved Planar Beams in Compliant Mechanisms Based on a Novel Governing Equation

Abstract Slender beams serve as typical flexible components to transfer motion, force, and energy in compliant mechanisms. Therefore, the accurate and efficient kinetostatic modeling for the slender beams is highly needed in the synthesis and analysis of compliant mechanisms. Several impressive and great modeling methods have been completed by the pioneering researchers based on the governing equation of slender beams, which can solve the deflection of the beam while the beam-tip loads are known. However, parts of the beam-tip loads are unknown in some scenarios and the traditional governing equation becomes inefficient in these scenarios. The aim of this paper is to propose a novel modeling method for the compliant mechanism, which can solve the large deflection problem without iterative algorithm. In this research, a novel governing equation of slender beams has been established based on Castigliano’s principle and simplified by the differential geometry. In the proposed governing equation, the statically force equilibrium equations and geometric constraint equations between different slender beams can be established in the boundary conditions, therefore the model of compliant mechanisms can be solved directly even if parts of the beam’s tip loads and displacements are unknown. The proposed governing equation provides the deformed shapes and cross-sectional loads of the slender beams, which help the designer to check the mechanical interference and strength in the design process. The numerical examples are presented to demonstrate the feasibility of proposed method.

Modeling Large Spatial Deflections of Slender Beams of Rectangular Cross Sections in Compliant Mechanisms

Abstract Modeling large spatial deflections of flexible beams has been one of the most challenging problems in the research of compliant mechanism. This study presents an approach called chained power series model for modeling large spatial defections of flexible beams with uniform rectangular cross section. This approach is based on the power series model developed in our previous work for modeling spatial deflections of rectangular beams in the intermediate deflection range. The chained power series model splits a rectangular beam into several elements and models each element by the power series model, and then, the deflections of all elements are assembled to form the deflection of the beam through transformations using quaternions. The effectiveness of the approach is demonstrated by comparing with the nonlinear finite element model preformed in ansys and the chained 3D pseudo-rigid-body model. Several examples are demonstrated to show the capability of the chained power series model for solving the deflections of rectangular beams in compliant mechanisms.

A generalized pseudo-rigid-body PPRR model for both straight and circular beams in compliant mechanisms

The pseudo-rigid-body (PRB) models have been widely applied for modeling of straight beams in compliant mechanisms. In this paper, a generalized PRB PPRR model with two mutually perpendicular linear springs is established for both straight and circular beams in compliant mechanisms. In the developed model, the static equations of the deflected beams are derived from Bernoulli’s equation. The characteristic parameters of the developed model are identified by the reasonably organized particle swarm optimization (PSO) algorithm. A simulation experiment of tip locus of a circular beam is conducted to evaluate the developed model. The simulation results show the developed model has higher accuracy than the PRB 2R, 3R, and RRHR model. In addition, two numerical examples are further implemented to demonstrate the effectiveness of the developed model.

Modeling Large Deflections of Initially Curved Beams in Compliant Mechanisms Using Chained Beam Constraint Model

Understanding and analyzing large and nonlinear deflections are the major challenges of designing compliant mechanisms. Initially, curved beams can offer potential advantages to designers of compliant mechanisms and provide useful alternatives to initially straight beams. However, the literature on analysis and design using such beams is rather limited. This paper presents a general and accurate method for modeling large planar deflections of initially curved beams of uniform cross section, which can be easily adapted to curved beams of various shapes. This method discretizes a curved beam into a few elements and models each element as a circular-arc beam using the beam constraint model (BCM), which is termed as the chained BCM (CBCM). Two different discretization schemes are provided for the method, among which the equal discretization is suitable for circular-arc beams and the unequal discretization is for curved beams of other shapes. Compliant mechanisms utilizing initially curved beams of circular-arc, cosine and parabola shapes are modeled to demonstrate the effectiveness of CBCM for initially curved beams of various shapes. The method is also accurate enough to capture the relevant nonlinear load-deflection characteristics.

Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms

There are several design equations available for calculating the torsional compliance and the maximum torsion stress of a rectangular cross-section beam, but most depend on the relative magnitude of the two dimensions of the cross-section (i.e., the thickness and the width). After reviewing the available equations, two thickness-to-width ratio independent equations that are symmetric with respect to the two dimensions are obtained for evaluating the maximum torsion stress of rectangular cross-section beams. Based on the resulting equations, outside lamina emergent torsional joints are analyzed and some useful design insights are obtained. These equations, together with the previous work on symmetric equations for calculating torsional compliance, provide a convenient and effective way for designing and optimizing torsional beams in compliant mechanisms.

Kinematic and dynamic analysis of compliant mechanisms considering both lateral and axial deformations of flexural beams

The kinematic and dynamic analysis of compliant mechanisms is investigated comprehensively in this work. Based on the pseudo-rigid-body model, a new PR model is proposed to simulate both the lateral and axial deformations of flexural beams in compliant mechanisms. An optimization for the characteristic factors and a linear regression for the stiffness coefficients of PR pseudo-rigid-body model are presented. Compared with the 1R and 2R pseudo-rigid-body model, the advantage of the PR model is well illustrated. The dynamic modeling of flexible beams in compliant mechanisms is then developed based on the PR pseudo-rigid-body model. The dynamic equation of a PR pseudo-rigid-body dynamic model is derived and the dynamic responses are then presented. The kinematic and dynamic analysis of a compliant slider-crank mechanism is presented by the 1R, 2R and PR model, respectively. The effectiveness of pseudo-rigid-body models and the superiorities of the PR pseudo-rigid-body model and PR pseudo-rigid-body dynamic model are shown clearly in the numerical example.

5R pseudo-rigid-body model for inflection beams in compliant mechanisms

A 5R pseudo-rigid-body model (PRBM) is proposed for inflection beams in compliant mechanisms. This new model consists of six rigid links connected by five joints. Four joints with torsional spring are added at the joints to simulate the deflection and a free hinge without spring is used to present the inflection point of the flexural beam. An objective function is established according to relative angular displacement between the two rigid links jointed by the free hinge for finding the optimal characteristic parameters of the 5R PRBM. The spring stiffness coefficients of the 5R PRBM are obtained using a linear regression technique. Numerical examples are presented and the results are shown by comparing with elliptical integral solutions. This new pseudo-rigid-body model behaves as load independent and can be used to predict both the tip locus and inflection position of flexural beams.

Bi-BCM: A Closed-Form Solution for Fixed-Guided Beams in Compliant Mechanisms

A fixed-guided beam, with one end is fixed while the other is guided in that the angle of that end does not change, is one of the most commonly used flexible segments in compliant mechanisms such as bistable mechanisms, compliant parallelogram mechanisms, compound compliant parallelogram mechanisms, and thermomechanical in-plane microactuators. In this paper, we split a fixed-guided beam into two elements, formulate each element using the beam constraint model (BCM) equations, and then assemble the two elements' equations to obtain the final solution for the load–deflection relations. Interestingly, the resulting load–deflection solution (referred to as Bi-BCM) is closed-form, in which the tip loads are expressed as functions of the tip deflections. The maximum allowable axial force of Bi-BCM is the quadruple of that of BCM. Bi-BCM also extends the capability of BCM for predicting the second mode bending of fixed-guided beams. Besides, the boundary line between the first and the second modes bending of fixed-guided beams can be easily obtained using a closed-form equation. Bi-BCM can be immediately used for quick design calculations of compliant mechanisms utilizing fixed-guided beams as their flexible segments (generally no iteration is required). Different examples are analyzed to illustrate the usage of Bi-BCM, and the results show the effectiveness of the closed-form solution.

Modeling Large Spatial Deflections of Slender Bisymmetric Beams in Compliant Mechanisms Using Chained Spatial-Beam Constraint Model

Modeling large spatial deflections of flexible beams has been one of the most challenging problems in the research community of compliant mechanisms. This work presents a method called chained spatial-beam constraint model (CSBCM) for modeling large spatial deflections of flexible bisymmetric beams in compliant mechanisms. CSBCM is based on the spatial-beam constraint model (SBCM), which was developed for the purpose of accurately predicting the nonlinear constraint characteristics of bisymmetric spatial beams in their intermediate deflection range. CSBCM deals with large spatial deflections by dividing a spatial beam into several elements, modeling each element with SBCM, and then assembling the deflected elements using the transformation defined by Tait–Bryan angles to form the whole deflection. It is demonstrated that CSBCM is capable of solving various large spatial deflection problems either the tip loads are known or the tip deflections are known. The examples show that CSBCM can accurately predict large spatial deflections of flexible beams, as compared to the available nonlinear finite element analysis (FEA) results obtained by ansys. The results also demonstrated the unique capabilities of CSBCM to solve large spatial deflection problems that are outside the range of ansys.

Modeling Large Planar Deflections of Flexible Beams in Compliant Mechanisms Using Chained Beam-Constraint-Model1

Modeling large deflections has been one of the most fundamental problems in the research community of compliant mechanisms. Although many methods are available, there still exists a need for a method that is simple, accurate, and can be applied to a vast variety of large deflection problems. Based on the beam-constraint model (BCM), we propose a new method for modeling large deflections called chained BCM (CBCM), which divides a flexible beam into a few elements and models each element by BCM. The approaches for determining the strain energy stored in a deflected beam and the stress distributed on it are also presented within the framework of CBCM. Several typical examples were analyzed and the results show CBCMs capabilities of modeling various large deflections of flexible beams in compliant mechanisms. Generally, CBCM can serve as an efficient and versatile tool for solving large deflection problems in a variety of compliant mechanisms.

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.

A Comprehensive Elliptic Integral Solution to the Large Deflection Problems of Thin Beams in Compliant Mechanisms

The elliptic integral solution is often considered to be the most accurate method for analyzing large deflections of thin beams in compliant mechanisms. In this paper, a comprehensive solution based on the elliptic integrals is proposed for solving large deflection problems. By explicitly incorporating the number of inflection points and the sign of the end-moment load in the derivation, the comprehensive solution is capable of solving large deflections of thin beams with multiple inflection points and subject to any kinds of load cases. The comprehensive solution also extends the elliptic integral solutions to be suitable for any beam end angle. Deflected configurations of complex modes solved by the comprehensive solution are presented and discussed. The use of the comprehensive solution in analyzing compliant mechanisms is also demonstrated by examples.

Modeling and Experiments of Buckling Modes and Deflection of Fixed-Guided Beams in Compliant Mechanisms

This paper explores the deflection and buckling of fixed-guided beams used in compliant mechanisms. The paper’s main contributions include the addition of an axial deflection model to existing beam bending models, the exploration of the deflection domain of a fixed-guided beam, and the demonstration that nonlinear finite element models typically incorrectly predict a beam’s buckling mode unless unrealistic constraints are placed on the beam. It uses an analytical model for predicting the reaction forces, moments, and buckling modes of a fixed-guided beam undergoing large deflections. The model for the bending behavior of the beam is found using elliptic integrals. A model for the axial deflection of the buckling beam is also developed. These two models are combined to predict the performance of a beam undergoing large deflections including higher order buckling modes. The force versus displacement predictions of the model are compared to the experimental force versus deflection data of a bistable mechanism and a thermomechanical in-plane microactuator (TIM). The combined models show good agreement with the force versus deflection data for each device.