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To utilize beam flexures in constraint-based flexure mechanism design, it is important to develop qualitative and quantitative understanding of their constraint characteristics in terms of stiffness and error motions. This paper provides a highly generalized yet accu-rate closed-form parametric load-displacement model for two-dimensional beam flexu...
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Citations
... A completely different treatise on geometrically nonlinear large deformation beam theories was given by Antman 33 wherein beams are modeled as constrained continuous bodies. 34 Within the framework, Simo-Reissner theory 35 was developed which is being extensively used for analyzing large deformation small strain large deflection beam problems. Due to the strong nonlinearity present in the system characteristics, updating geometry plays a crucial role for large deflection analysis of beam structures in the framework of incremental Lagrangian approach. ...
Forced vibration characteristics around large statically deformed configuration of curved beam system with rigid links and moving support are analyzed. In addition, the paper also investigates various parametric effects of height to span ratio and roller radius to height ratio coupled with support connection mechanism and profile shape, on dynamic characteristics of the curved beam system. The subject problem includes two interrelated problems: determining deformed configuration under static load and dynamic response of the loaded beam under harmonic excitation. The static problem is analyzed incrementally through the variational principle-based energy method in body-embedded curvilinear frame considering geometric nonlinearities due to combined bending-stretching and non-uniform initial curvature. In each incremental step, the nonlinear governing equation is solved iteratively, and kinematic constraints due to interactive deformation of the flexible curved beam with rigid links are imposed to obtain the global solution. Concerning the statically deformed configuration, governing equation for forced vibration is derived through Hamilton's principle in curvilinear frame. Nonlinear effects caused by combined bending-stretching modes of vibration, non-uniform curvature, and lumped mass of the dead static load are considered. The set of inhomogeneous governing equations is solved through Broyden's method. A direct substitution technique through successive relaxation is also employed to obtain an initial guess solution. Besides the forced vibration, free vibration analysis at static deformed configuration is also carried out for the completeness of the study. The theoretical model is validated through comparisons of loaded natural frequency results with finite element package. After the validation study, the effects of the mentioned geometric parameters on the dynamic response of the curved beam system are investigated and presented suitably. The physically insightful framework along with the parametric study may facilitate simulation and design of several practical structures involving moving support and rigid links subjected to continuous excitation around large statically deformed configurations.
... The CBM is a method to accurately obtain a close-form solution for flexible beams with small deflection (less than 10% of beam length), which provides the parametric and close-form expressions of load-deflection relations in the beam-by-beam constraint model. 99 The CBM divides a flexible beam into N elements, and each element is modeled by a beam constraint model. Therefore, the CBM requires fewer elements to achieve the same desired accuracy. ...
This paper presents a comprehensive review of mechanical design and synthesis methods for piezo-actuated compliant micro-positioning stages, which play an important role in areas where high precision motion is required, including bio-robotics, precision manufacturing, automation, and aerospace. Unlike conventional rigid-link mechanisms, the motion of compliant mechanisms is realized by using flexible elements, whereby deformation requires no lubrication while achieving high movement accuracy without friction. As compliant mechanisms differ significantly from traditional rigid mechanisms, recent research has focused on investigating various technologies and approaches to address challenges in the flexure-based micro-positioning stage in the aspects of synthesis, analysis, material, fabrication, and actuation. In this paper, we reviewed the main concepts and key advances in the mechanical design of compliant piezo-actuated micro-positioning stages, with a particular focus on flexure design, kineto-static modeling, actuators, material selection, and functional mechanisms including amplification and self-guiding ones. We also identified the key issues and directions for the development trends of compliant micro-positioning stages. Published under an exclusive license by AIP Publishing. https://doi.
... The BCM is based on the Euler-Bernoulli beam model and considers geometric nonlinearity and load-dependent effect associated with linearized beam curvature assumptions. It is simple, parametric, closed-form, and accurately predicts the behavior of flexure beams with intermediate deflections [45]. As aforementioned, BCM has been extended for different modeling scenarios of nonlinear deflections, including the SBCM [46,47], CBCM [27], IMBCM [29,30], TBCM [28], BCM with semi-rigid element [48]. ...
... This approach is what we commonly used in solving a typical mechanic problem which involves three components: constitutive equations, force equilibrium, and geometry compatibility [60][61][62]. The constitutive equation of each flexure beam is obtained from the BCM [45], which is explained in Appendix A. ...
Compliant mechanisms are crucial in a wide range of applications, and straight flexure beams with uniform thickness serve as their fundamental building blocks. While techniques and methods exist to estimate the sophisticated load-displacement behavior of these beams, dealing with the nonlinear large deformation and parametric design of complex compliant mechanisms using these beams remains a significant challenge. To address this challenge, an analytical model has been developed to integrate six independent geometric parameters into a general lumped-compliance beam, known as the general lumped-compliance beam model (GLBM). This approach enables the determinate synthesis of force-displacement characteristics in compliant mechanisms that feature any two flexure beams connected in series, such as the conventional lumped-compliance beam, distributed beam, inverted beam, and folded beam. The closed-form beam constraint model (BCM) is utilized to derive the GLBM that accurately captures geometric nonlinearity and load-dependent effects. To demonstrate the effectiveness of this modeling technique, we verified five specific configurations of the GLBM using nonlinear finite element analysis (FEA). In addition, we selected two representative compliant mechanisms, a revolute joint and a bistable mechanism, for nonlinear analysis and experimental validation, which further showcases the efficacy of this proposed GLBM.
... They simplified the tedious equation derivation and calculation process by using dimensionless parameters and verified the accuracy of the model by using nonlinear finite element analysis [31]. Furthermore, Awtar deduced the nonlinear strain energy expression in the form of terminal displacement for the beam bending model by using the energy method, which can be used to determine the stiffness characteristics of the bending mechanism composed of multiple beams [32]. Zhao et al have conducted extensive and indepth research on flexure structures. ...
... However, the CCSFP is unable to fully meet the requirements for space gravitational wave detection, which include a narrower rotation range, higher precision, and a higher non-linear rotation moment. This chapter uses the improved beam bending model [30] by the continuous method [31] and the energy method model [32] in light of the aforementioned particular needs. We can generalize the CCSFP model into the generalized cross-spring flexure pivot (GCSFP) model by defining the characteristic parameters. ...
... Referring to the derivation of the nonlinear strain energy equation for complex bending of flexure mechanism composed of a single spring element [32], we can obtain the expression of nonlinear strain energy of a single spring expressed by its end displacement, as shown in equation (5 ...
To control the pointing accuracy of the telescope in the space gravitational wave
detector at the nano-radian level, it is essential to design a flexure pivot with extremely
low rotational moment nonlinearity during rotation. This study derives the theoretical
equation of the nonlinear rotational moment of flexure pivot based on the beam bending
strain energy equation and completes the design of ultra-low moment noise flexure
pivot by integrating the structural form of conventional cross-spring flexure pivot
(CCSFP). The nonlinear moment noise distribution in the frequency domain of three
kinds of flexure pivots is then compared regarding TianQin’s circular high orbits. Then
the relevant finite element simulation analysis will be performed for the flexure pivot
structures, respectively, to ensure the accuracy of its rotational stiffness model and
verify the correctness of nonlinear moment theory optimization through physical
experiments. The results show that the optimized flexure pivot can effectively reduce
the nonlinearity, basically meet the nonlinear moment requirements in space
gravitational wave detection.
... In the field literature, much more attention has been given to analyses of flexure hinges in small deformations. In contrast, only a small fraction of the papers have proposed studies on flexure hinges in large deformation [36][37][38][39][40]. ...
Flexure hinges are widely used in mechanical devices, especially for micro- or even nano-scale applications, where conventional joints based on conjugate surfaces prove unsuitable. However, to achieve accurate motion of devices whose joints are flexure hinges, knowledge of stiffness models that correlate applied forces or bending moments with the resulting displacements is required. Nonlinear bending models are typically too complex and difficult to implement. Therefore, it is preferred to use linear models, which admit analytical solutions. The purpose of this paper is to show what is lost in terms of accuracy in reducing a nonlinear bending stiffness model associated with a flexure hinge when simplifications are made to obtain an analytical solution. An analysis of the simplification process leading to a linear stiffness model and its analytical solution is presented. From this study arises an increased awareness of flexure joints in terms of the accuracy obtained with their stiffness models, suggesting important information to the user in choosing the level of complexity required to model them in a specific application. A comparison between analytical and numerical results is provided in the form of maps and tables so as to make that choice as clear as possible.
... Linear analytical modeling fails to capture nonlinear stiffness characteristics and can only be applied over a relatively small range of motion. In this section, the BCM [52] was used for a generalized beam flexure, serving to accurately model the characteristics of the proposed amplifier when combing with the free-body-diagram (FBD) system modelling method. The BCM approach captures nonlinearities and leads to a closed-form model that can accurately estimate the kinetostatic behavior of a generalized beam subjected to combined loads. ...
A constant displacement amplification ratio is less investigated in compliant mechanisms. This study addresses this need by presenting an over-constraint based nearly-constant amplification ratio compliant mechanism (OCARCM) that alleviates the change in displacement amplification ratio. The free-body diagram (FBD) combined with the generic beam constraint model (BCM) method is employed to obtain the closed-form solutions that accurately and insightfully elaborate the nonlinear kinetostatic characteristics of the OCARCM. Comparative analysis is provided between the proposed OCARCM and the widely-used bridge-type compliant amplifier in terms of the ability to remain a constant amplification ratio, with and without external payloads. The closed-form models are verified by the nonlinear finite element results (FEA) with a maximum difference of 1%. In our case studies, it shows that the amplification ratio of the OCARCM changes by 1% over the range, while that of the bridge-type amplifier changes by approximately 14% under the same conditions. The results also reveal that a higher amplification ratio results in a greater variation in the ratio. An experiment based on the CNC machined aluminium alloy prototype with distributed-compliance is conducted, and experimental results show a maximum error of 3.4% for the amplification ratio compared with the analytical or FEA results.
... Elliptic integral solutions have been developed for beams with inflection points [16][17][18][19]. Awtar's beam constraint model (BCM) [20][21][22] offers a closed-form solution to accurately model geometric nonlinearities for flexure deflections within 10% of the flexure's length. Ma and Chen's bi-beam constraint model (Bi-BCM) splits a fixed-guided beam into two elements, solves each using the BCM model, and returns the closed-form load-deflection solution [14]. ...
Fixed-clamped flexures are one common component of compliant mechanisms which remain difficult to design due to their unique force- and stress-deflection profiles. In this work, an analytical stress-deflection model for fixed-clamped flexures is proposed that utilizes a modified pseudo-rigid-body model. Proof of concept mechanical testing and finite element analysis demonstrate that the model can predict forces and stresses within 3.5% for a range of steel flexure topologies. Special analysis is carried out on the characteristic radius factor, a parameter to which model accuracy is particularly sensitive. For slender flexures or large deflection scenarios, a dynamic characteristic radius factor is required to capture the resulting nonlinear axial strain. By evaluating the effects of loading, geometry, and material properties, an analytical equation that can predict an optimal value is proposed. When integrated into our model, this equation for an appropriate characteristic radius factor can predict the optimal parameter value within 0.45 +/- 0.47%, resulting in average model error of 3.45 +/- 2.09% across a large range of flexure thicknesses and deflections. The distinct combination of axial and bending stresses experienced in fixed-clamped flexures has made mechanisms which use these members challenging to design. This work provides a model that designers, engineers, and researchers can draw from to understand stress profiles present in these flexible members.
... Both the geometric nonlinearity due to the structure topology and the nonlinearity induced from the large deformation of the compliant element must be taken into consideration. Awtar [40] proposed a beam constraint model (BCM) to construct the force-displacement relation of cantilever beams, and Chen et al. ...
... The translation and rotation displacements of the free end are X i , Y i , and θ i . According to the BCM, there are [40]. ...
Bio-inspired isolators are a novel type of nonlinear isolators by mimicking the bionic structures and behaviors. Most of them are based on limb-like structures composed of rigid rods forming a geometric frame and linear springs storing elastic energy. A novel bio-inspired isolator with a compliant limb-like structure was proposed. Both the geometric constrained movement and the storage of the elastic energy are provided by the deformation of a compliant mechanism. A beam constraint model was adopted to characterize the nonlinear restoring force, and a harmonic balance method was used for the dynamic analysis. The minimum damping ratio ensuring the isolator operating within a working region was estimated, and the displacement transmissibility of the isolator was calculated. A systematic investigation was performed on the high-static-low-dynamic-stiffness characteristic and the vibration isolation performance of the proposed isolator. The investigation revealed that the compliant limb-like structure exhibits stiffness-softening, stiffness-softening-hardening, and negative-stiffness characteristics with different dimensions and deformation degrees. The dimensions can be optimized to achieve the largest high-static-low-dynamic-stiffness region, and the smallest dynamic/static stiffness ratio. The isolator exhibits a globally stable response and a weak nonlinearity if it is fully damped, while jump phenomena and unbounded response may occur with light damping. Compared with a three-spring isolator and a rigid limb-like structure, the compliant limb-like structure exhibits a strong stability, a strong load capacity, and a noticeable high-static-low-dynamic-stiffness characteristic at the same time. The combination of the bio-inspired isolation and the compliant mechanism may provide an innovative approach for the broadband vibration isolation.
... The load-deflection relationships in each element are obtained from the classic beam-constraint model [43,44]: ...
Using a small piezoelectric device to extract energy from low frequency (< 5 Hz) and ultra-low level (< 0.05 g) vibration environments is a very challenging task. The device with quasi-zero stiffness (QZS), widely used in vibration isolation, could be a good candidate for a low frequency energy harvester. However, the conventional QZS devices with complex frames are not very suitable for small-scale harvesters operating in low vibration conditions. To solve these problems, we propose a compact W-shaped compliant mechanism with the direct QZS. Both static and dynamic models have been established to investigate the performance. A compact prototype is fabricated for testing. The experimental results are in good agreement with numerical predictions and demonstrate a good energy harvesting performance under the excitation level below 0.02 g. The normalized power density can even reach 2.3 mW/cm³g² under the excitation level of 0.007 g at 3.5 Hz. This device can also serve as a compact low-frequency vibration isolator with the transmissibility of -20 dB at 10 Hz. Overall, this compact QZS device shows a great potential to scalable applications of low-frequency energy harvesting and vibration isolation.
... The improvements of Pseudorigid-body method [106,107] and the simplifications of Elliptic Integral method [77] are carried out and proved to be effective. The Beam Constraint Model is proposed in 2006 [108]. It is mainly used in small deformation with deflection less than 10% of its dimension. ...
... Chain Beam Constraint method. The Beam Constraint Model is proposed by Awtar et al. [108] to capture the close form solution of the flexible beams with small deflection (less than 10% of beam length) accurately, which can provide the parametric and close form expressions of load-deflection relations of the beam easily. The Chain Beam Constraint method divides a flexible beam into elements, and each element is modeled by Beam Constraint Model. ...
To avoid complicated force control system in precision manipulation, compliant constant force/torque mechanisms are proposed and developed continuously since the 1990s. The inherent nonlinearities due to varying stiffness raise high requirements for mechanism design, kinetostatic modeling and structural optimization. This makes the synthesis of compliant constant force/torque mechanism a multidisciplinary and multiplex task. In this work, a comprehensive survey on synthesis methods of the constant force mechanisms is conducted. Three categories of synthesis methods are summarized, i.e., the Rigid-body Replacement approach, the Building Block approach and the Structural Optimization approach. For each category, the corresponding modeling and optimization methods are introduced in details. The presented survey provides overall perspectives on current status and future challenges in this field for designers and researchers.
