Chained Beam Constraint Model Research Articles

Achieving high-quality and large-stroke constant torque by axial force release

Compliant constant-torque mechanisms (CCTMs) maintain constant-torque without the need for complex closed-loop feedback systems, broadening their applications in rehabilitation devices, surgical tools, and cooperative robotic arms. However, CCTMs present considerable design challenges due to the pronounced nonlinearities that arise due to large deflections and multi-axial loadings. Traditional CCTM design strategies focus on managing post-buckling phenomena, often leading to increased stresses and an imbalance in positive and negative stiffness, compromising torque consistency and stroke capacity. This study introduces a novel CCTM that effectively decouples the multi-axial loadings and releases axial forces, isolating beam bending forces. This decoupling is achieved by incorporating a parallel-guided compliant mechanism at the fixed end of the beam, which reduces stress and enhances torque stability throughout the operational range. Through the partical swarm optimization of geometric design parameters using the chained beam constraint model, this research has produced a CCTM capable of maintaining torque fluctuations below 0.39% over a rotational range of 18° to 68°. Experimental validations confirm the design’s superiority in providing an extended constant torque stroke and improved consistency, distinguishing it from conventional straight-beam CCTMs.

An adjustable stiffness vibration isolator implemented by a semicircular ring

Nonlinear vibration isolators with quasi-zero-stiffness characteristics offer broadband vibration isolations, while passive quasi-zero-stiffness isolators are unable to adapt to variable working conditions. To address the issue, a nonlinear semi-active vibration isolator based on a semicircular ring structure is proposed. As an elastic component, the semicircular ring exhibits nonlinear stiffness under a vertical compression at the centre. With a semi-active control of the horizontal distance between the two ends of the semicircular ring, a pre-deformed state is induced, and the stiffness characteristics are adjusted accordingly. A chained beam-constraint model is adopted to characterize the pre-deformed process and the vertical compression process, respectively. The adjustable nonlinear restoring force is accurately calculated and adopted for vibration analysis. A harmonic balance method is used to analyze the frequency responses of the isolator under different working conditions. The semicircular ring exhibits softening stiffness under a vertical compression, and the quasi-zero-stiffness region can be effectively adjusted by the pre-deformed process. With the vertical compression reaching a threshold, negative stiffness is presented along with a snap-through phenomenon due to the bi-stability. With the semi-active pre-deformed, the isolator demonstrates different resonant frequencies and different risks in the occurrence of the snap-through during vibration. To avoid the snap-through-induced impact, the control threshold of the pre-deformed is determined, depending on the payload mass, the excitation amplitude, and the damping coefficient. Experiments are performed to validate the adjustable stiffness and the frequency responses of the isolator, and both the jump phenomenon and the snap-through phenomenon are observed.

Design of High-Performance Triple-axis Cross Pivots

Abstract Cross-axis flexural pivots are increasingly implemented in mechanical systems due to their high precision and wide range of motion. However, designing high-performance pivots in terms of low axis drift, low rotational stiffness, and low values of maximum stress remains a challenging task. In fact, these features often behave antagonistically to each other. In this paper, the design of a novel family of high-performance cross-axis pivots is presented. The compliant joints are obtained by the composition of two crossing flexible elements with initially curved axis, and of one auxiliary flexure with straight axis. Two different arrangements, that depend on the constraints layout, are proposed. Chained beam constraint model and non-linear finite element analysis are implemented for the analysis of the kinetostatic response of the pivots. The effects of initial curvature and orientation of the flexures on axis drift, stiffness, and maximum stress, are investigated and reported in the form of design maps. A global performance index, that embodies the above-mentioned features and captures the overall kinetostatic behavior of the pivot is introduced. The design maps and the global index serve as key tools for the design procedure, giving insight into the antagonistic issue and leading to the determination of the best trade-off solutions that meet the application requirements. An experimental campaign is conducted to compare the performance improvement of triple-axis pivots with respect to a benchmark double-axis joint, and to validate the design method.

Nonlinear closed-form model for beam flexures subject to large axial loads

Closed-form models are important because they offer more design insights, require less computing time and prevent convergence problems in modeling compliant mechanisms. However, most of the available closed-form models, e.g., beam constraint models, are only appropriate for beam flexures within the intermediate deflection range. In this paper, a method for developing closed-form models based on the chained beam constraint model through the combined use of strain energy and Castigliano's first theorem is proposed. A closed-form model for beam flexures with a large prediction range is derived using this method. By modeling simple beams and compliant mechanisms using the proposed closed-form model, it is shown that the model has a significantly broader prediction range than the beam constraint model and is more efficient than the chained beam constraint model. The large prediction range and high efficiency of the proposed closed-form model make it potentially useful for optimizing compliant mechanisms.

Structural modeling and investigation of cantilever load-bearing mechanism in telescopic wing

Telescopic wing requires changing the wingspan under aerodynamic load. Structural deformation caused by forces and moments will result in increased driving forces or even stagnation. Deformation and distributed contact force of linear motion pair of the load-bearing mechanism which considers the geometric nonlinearity of guide rail is necessary to predict the driving force, the mechanical efficiency, or the stiffness matching in the optimal design process. Aiming at the cantilever load-bearing mechanism, the structural model was established in terms of the flexibility of the components, contact stiffness and other parameters on the friction of moving pairs are studied based on the modified chained-beam constraint model (CBCM). The model prediction accuracy is verified through simulation and experiment. The result shows that the friction force is related to the stiffness of guide rail, slider connector, ball bearing contact as well as the geometric parameters. Improper parameter selection will increase the friction force to more than twice the optimal solution.

Design and analysis of a contact-aided flexure hinge (CAFH) with variable stiffness

This paper presents a novel contact-aided flexure hinge (CAFH) with variable stiffness, which consists of a contact-aided segment, a flexible segment and a rigid part. The proposed CAFH can facilitate a compact design and provide an alternative for stiffness-variable designs under any loading conditions. With a mortise-tenon structure, the CAFH is trivially affected by friction. The design and deformation procedures of the CAFH are described in detail, followed by its theoretical kinetostatic modeling using the chained beam-constraint model. The deformation of all segments is considered in the kinetostatic model, which expands the space of design parameters for stiffness-variable designs. Then, the accuracy of the theoretical model and the variable stiffness design are verified by nonlinear finite element analysis (FEA) and experimental tests. In term of stiffness, the maximum relative errors of the theoretical model are 0.76% in Stage 1 and 0.70% in Stage 2, as compared with FEA, respectively. Further, the parameter sweep is carried out, followed by sensitivity analysis to identify the main test error sources. Finally, the multi-material scenarios are investigated preliminarily, and some outlooks are discussed.

Design of a Novel Large-Stroke Compliant Constant-Torque Mechanism Based on Chained Beam-Constraint Model

Abstract This study addressed the development of a novel compliant constant-torque mechanism (CCTM) that utilizes Bezier curved beams to provide a large stroke in the constant-torque operating range. Previous CCTMs are limited by their working stroke, which reduces their applicability. The proposed mechanism is based on an analytical model using the chained beam-constraint model (CBCM), which captures the kinetostatic behavior of flexible segments. A genetic algorithm based on the CBCM was used to obtain the optimal structure, which was then verified through finite element analysis and experimental results. The results show that the proposed CCTM provides good flatness with a deviation of 3.7% and a large stroke of 80 deg in the constant-torque working range, while maintaining compactness. This novel CCTM has the potential to provide a simple and effective solution for torque regulators in various applications.

Modeling and performance analysis of a curve-shaped based doubly clamped piezoelectric energy harvester (CD-PEH)

In this paper, a curve-shaped based doubly clamped piezoelectric energy harvester (CD-PEH) is explored for improving the energy harvesting performance. The harvester consists of a composed beam constructed with two arc-shaped structures and a flat beam, as well as two proof masses. A method based on chained beam constraint model theory (CBCM) is first applied to build the nonlinear restoring force model of the CD-PEH, the developed analytical model is validated by the finite element analysis (FEA). Then the electromechanically coupled model for the CD-PEH is built to investigate the effect of excitation amplitudes, geometric parameters and load resistance on the output characteristics. Due to the geometric nonlinearity caused by the arc-shaped configuration, the CD-PEH orderly exhibits quasi-linear, softening nonlinear and mixed hardening & softening nonlinearity behavior with the increasing of excitation level, which could effectively extend the frequency bandwidth of the system. For the excitation of A = 8 m/s2, the effective working bandwidth of the CD-PEH is increased by 633% compared with the effective bandwidth in the case of A = 2 m/s2. Moreover, comparison experiments demonstrate that the output voltage and the effective bandwidth are increased by 225 and 450%, respectively, compared with the typical doubly-clamped piezoelectric energy harvester (T-PEH) under the same excitation amplitude. Overall, this study provides a new way and theoretical framework for the design of high-efficiency doubly clamped piezoelectric energy harvester.

Design and testing of high-accuracy curved cross-axis flexural pivots with cable-driven actuation

The cross-axis flexural pivot proved to be a valid tool for replacing the traditional revolute joints in a variety of mechanical systems. However, the rotation axis drift limits its implementation in high-precision applications. This investigation presents a procedure for designing high-accuracy flexural pivots actuated by cable-driven systems. Kinetostatic simulations, resorting to finite element analysis and to chained beam constraint model, have been carried out to analyze the relations among the input and output parameters of the design. The effects of initial curvature on position accuracy, maximum stress, and actuation force have been investigated and several design maps have been generated to support the procedure in each step. The procedure has been applied to the design of high-accuracy pivots and an experimental campaign has been carried out to compare accuracy and kinetostatic performance of the curved configurations to the straight one.

A flexible swallowing gripper for harvesting apples and its grasping force sensing model

Securely harvesting apples without damage remains a challenge owing to their softness, vulnerability, and irregular shapes. In this study, a flexible swallowing (FS) gripper was designed and tested for harvesting apples. To analyze the adaptive grasping process of this gripper, first, a closed equation of the force-to-deformation model for the finger was proposed based on the Chained-Beam-Constraint Model (CBCM). The correctness of the model was verified by finite element analysis (FEA), and the maximum error was no more than 1.7 %. Subsequently, the grasping force sensing model related to bending angle of the finger was established based on the force-to-deformation model. Finally, taking an apple with a diameter of 80 mm as the experimental object, several groups of grasping tests were conducted with the gripper. The bending angle and force sensing error for different grasping positions of the fingers were analyzed and compared. The results demonstrated that the gripper can sufficiently perceive the grasping force, and the force sensing accuracy in the middle of the finger was the best, with an average absolute error and relative error of 0.153 N and 5.65 %, respectively; A better envelope ability and greater grasping force was obtained when grasping with the bottom of the finger; the maximum bending angle and maximum contact force were 22.6° and 5.72 N respectively. Moreover, a harvesting experiment using this gripper installed at the end of a robot arm was conducted, which further verified that this gripper has a good grasping ability for apples and can be sufficiently applied in actual robot harvesting.

Initial curvature and centroid positioning effects on cross-axis flexural pivots accuracy

Cross-axis flexural pivots have been implemented in a variety of applications and many studies have been focused on their accuracy. In this investigation, initial curvature of flexible elements and positioning of their centroids are exploited to improve the rotational performance of the cross-axis pivot. A parametric geometry based on two design variables is developed and proper accuracy criteria are introduced. Rotational performance is evaluated by means of the chained beam constraint model and by nonlinear numerical simulations. Accuracy, stiffness and stress maps are presented. • The design of high-accuracy cross-axis flexural pivots is addressed. • Initial curvature and centroids positioning are exploited to improve accuracy. • A 2-DoF parametric geometry and specific accuracy criteria are introduced. • Accuracy is evaluated by chained beam constraint model and nonlinear FEA. • Accuracy, stiffness and stress design maps are presented.

Design of mechanical metamaterial for energy absorption using a beam with a variable cross-section

In this study, the energy-dissipating characteristic of a cell of a mechanical metamaterial is developed based on a beam with a variable cross-section. An optimization model is described with the use of the chained beam constraint model (CBCM) to discretize and approximate the beam to obtain the variable cross-section beam with better energy absorption properties. The optimal configuration of the number of nodes of the CBCM and the design variables confined to the design domain, namely, the length, thickness and width of the beam, is designed and discussed to obtain the cell with high specific energy absorption. The unit cell and mechanical metamaterial are used to verify the effectiveness of the method by simulation and experiment.

Design of a Parallel Gripper Based on Topology Synthesis and Evolutionary Optimization

Abstract A compliant gripper with nearly parallel gripping motion is developed by a topology synthesis and a dimensional synthesis approach. The topology synthesis process can generate linkage-type compliant mechanisms. Suitable boundary conditions of the topology synthesis process are selected to achieve the desired functions of the device. The dimensional synthesis is based on an evolutionary optimal design process. To meet various design goals, a nondominated multi-objective genetic algorithm is selected for the optimal design process. A kinetostaic model based on the chained beam constraint model is developed for force–displacement analysis of the designs. Efficiency and accuracy of the design approach are proved by experiments. Appropriate linkage types of compliant mechanisms may be discovered by the topology optimization process before moving on to the dimensional synthesis to obtain final designs.

Kinetostatic Modeling of Planar Compliant Mechanisms With Flexible Beams, Linear Sliders, Multinary Rigid Links, and Multiple Loops

Abstract This paper presents kinetostatic models of planar compliant mechanisms with multinary rigid links, multinary joints, sliders, and multiple loops based on the chained beam constraint model. The focus is on modeling of several building blocks of the beam type compliant mechanisms to aid in their design. The modeling approaches are based on the loop-closure equations and the static equilibrium conditions. Models of the multinary rigid links, multinary joints, and sliders are presented. As a result, the kinetostatic models of the compliant mechanisms can be systematically formulated by using these building blocks. Several mechanisms constructed by the building blocks are modeled and verified by finite element analyses. A case study is provided to demonstrate the application of the developed models. These models pave the way for versatile applications of the chained beam constraint model for the design and analysis of beam type planar compliant mechanisms.

An Overview of Procedures and Tools for Designing Nonstandard Beam-Based Compliant Mechanisms

Beam-based Compliant Mechanisms (CMs) are increasingly studied and implemented in precision engineering. Straight beams with uniform cross section are the basic modules in several design concepts, which can be deemed as standard CMs. Their behavioral analysis can be addressed with a large variety of techniques, including the Euler–Bernoulli beam theory, the Pseudo-Rigid Body (PRB) method, the beam constraint model and the discretization-based methods. This variety is unquestionably reduced when considering nonstandard CMs, namely design problems involving special geometries, such as curve/spline beams, variable section beams, nontrivial shapes and contact pairs. The 3D Finite Element Analysis (FEA) provides accurate results but its high computational cost makes it inappropriate for optimization purposes. This work compares the potentialities of computationally efficient modeling techniques (1D FEA, PRB method and chained-beam constraint model), focusing on their applicability in nonstandard planar problems. The cross-axis flexural pivot is used as a benchmark in this research due to its high configurable behavior and wide range of applications. In parallel, as an attempt to provide an easy-to-use environment for CM analysis and design, a multi-purpose tool comprising Matlab and a set of modern Computer-Aided Design/Engineering packages is presented. The framework can implement different solvers depending on the adopted behavioral models. Summary tables are reported to guide the designers in the selection of the most appropriate technique and software framework. Lastly, efficient design procedures that allow to configure nonstandard beam-based CMs with prescribed behavior are examined with two design examples.

Design and modeling of a compliant gripper with parallel movement of jaws

A compliant gripper with parallel movement of jaws is developed. The gripper is composed of a bar linkage mechanism and a bistable mechanism. After a target object is grasped, the embedded bistable mechanism can maintain the gripping of the object with no energy consumption. Unstable fixation due to rotary motion of opposing gripping arms of traditional tweezer-like grippers is alleviated by the proposed gripper. Kinetostatic models of the device based on chained beam constraint model are derived to assist the design of the device. Grasp function of an object is accomplished while the gripper proceeds from its first stable equilibrium position to its second stable equilibrium position, and release of the object is achieved when the gripper returns back to its first stable equilibrium position.

Kinetostatic modeling and characterization of compliant mechanisms containing flexible beams of variable effective length

Flexible beams of variable effective length, serve both transmission and actuation functionalities in compliant mechanisms, have been employed in many devices, e.g., thermal actuators and continuum robots. This difunctional feature is favorable when utilized for operations in confined space. However, the length variation introduces modeling difficulties, which poses a new challenge to designers. To accurate model flexible beams of variable effective length and characterize the behaviors of associated compliant mechanisms form the primary objectives of this study. The chained beam constraint model is revisited and extended to model beams of variable effective length. Modeling of a chevron shape thermal in-plane microactuator and a continuum mechanism are provided to demonstrate the effectiveness of the proposed method. The predicted results have a high degree of accuracy as compared to experimental results and nonlinear finite element results.

On the modeling of a contact-aided cross-axis flexural pivot

This paper reports the study of a planar Cross-Axis Flexural Pivot (CAFP) comprising an additional contact pair. The proposed device may be useful for applications requiring a revolute joint that behaves differently when deflecting clockwise/anti-clockwise. The presence of the contact pair reduces the free length of one flexures, resulting in a considerable increment of the overall joint stiffness. The pivot behaviour is investigated, for different load cases, via the Chained-Beam-Constraint Model (CBCM), namely an accurate method to be applied in large deflection problems. A framework comprising Matlab and ANSYS is developed for testing the CAFP performances in terms of rotational stiffness, parasitic shift and maximum stress, with different combinations of geometrical aspect ratios and contact extensions. Results achieved via CBCM for a pure rotation applied to the CAFP’s output link are then verified through Finite Element Analysis. The resulting performance maps show good agreement between the numerical results. Furthermore, the CBCM shows an improved computational efficiency, which is a crucial aspect for preliminary design steps. At last, direct comparison between simulations and experiments, developed by means of two custom test rigs, confirms the efficacy of the proposed design method for the modeling of contacts in large deflection problems.

Kinetostatic Modeling and Optimization of a Novel Horizontal-Displacement Compliant Mechanism

Abstract This paper presents kinetostatic modeling of a compliant mechanism for translational motion. This mechanism arranges all compliant members in an inverted way, which enables the robustness against beam buckling due to the heavy payload. To enable quick design and analysis of the mechanism, a nonlinear analytical model is then derived based on the chained beam constraint model, which is validated by nonlinear finite element simulation. Geometric parameter optimization is further carried out for desired motion characteristics. Finally, a prototype is fabricated and tested to verify the analytical model.

A Fully Compliant Tristable Mechanism Employing Both Tensural and Compresural Segments

Abstract A multistable compliant mechanism is a device that can hold several distinct positions through the storage and release of the strain energy associated with deflections of the flexible members. This self-locking capability can benefit many applications such as threshold acceleration sensing, overload protection, and shape reconfiguration. This work presents a novel class of fully compliant tristable mechanisms called tensural–compresural tristable mechanisms (TCTMs), which forms three stable equilibrium positions through unique utilization of both tensural segments and compresural segments. To identify feasible designs, a kinetostatic model is developed using the chained beam-constraint-model (CBCM) for both tensural segments and compresural segments. Two TCTM designs accompanied with a prototype are presented to demonstrate the feasibility of this new tristable configuration and the effectiveness of the kinetostatic model.