Snap-through Behavior Research Articles

Experimental and numerical studies on nonlinear vibrations and dynamic snap-through phenomena of bistable asymmetric composite laminated shallow shell under center foundation excitation

This paper experimentally and numerically investigates the nonlinear vibrations and dynamic snap-through behaviors of the bistable asymmetric carbon fiber-reinforced [90n/0n] composite laminated shallow shell. In the experimental research, in order to produce the dynamic snap-through phenomena, the center of a bistable asymmetric composite laminated shallow shell is clamped on an electromechanical shaker. The shaker provides the controlled frequency and amplitude of the foundation excitation. A laser displacement testing system collects the data of the vibration signals for the bistable asymmetric composite laminated shallow shell. A high-speed camera captures the steady-state vibrations of the bistable asymmetric composite laminated shallow shell. The experimental results demonstrate the influence of the structural parameters on the dynamic snap-through phenomena and nonlinear vibrations of the bistable asymmetric composite laminated shallow shell. The amplitude-frequency response curves are obtained by the experimental results. The bifurcation diagrams, phase portraits, time histories, power spectrums, and Poincaré maps are obtained to experimentally and numerically illustrate the single-well periodic, chaotic and double-well chaotic vibrations of the bistable asymmetric composite laminated shallow shell under the center foundation excitation. In numerical research, ABAQUS/CAE is used to simulate the dynamic responses of the bistable asymmetric composite laminated shallow shell. The subspace iteration method is utilized to give the vibration modal analysis. The vibration modes in the experimental research are qualitatively consistent with the results of the ABAQUS numerical simulation. The phenomenon of the energy transfer from the high-order to low-order primary resonance is discovered. The results of the experiment and numerical simulation are well verified with each other.

A geometrically exact model for predicting statics and snap-through behaviors of bi-stable composite laminates

The present study aimed to analyze the stable states and the snap-through forces of square and rectangular bi-stable composite laminates with [03/903] stacking sequence and large to very large displacements and medium to large rotations. Also, the effect of lateral length and aspect ratio of these laminates on their stable states and snap-through forces were analyzed. For this purpose, a realistic geometric exact model of plate (GEMP) was developed based on continuum mechanics. In addition, the analysis was also performed with a commonly used model based on classical laminated‐plate theory (CLPT) with Von-Karman strains. FEM simulations were used to validate the results of these theoretical models. The static equations of the system were derived using the principle of virtual work and Rayleigh-Ritz method. Good consistency was observed between the results of the proposed GEMP and the FEM models, especially for laminates with very large displacements and large rotations.

LOW-VELOCITY IMPACT RESPONSES OF FG-GRC SANDWICH BEAMS WITH VISCOELASTIC CORES

The dynamic snap-through and oscillation behaviors are studied of a post-buckled functionally graded graphene reinforced composite (FG-GRC) beam with a viscoelastic core under low-velocity impact. The modified Halpin Tsai meso-mechanical model is used to predict the material properties of FG-GRC. The Hertz point contact model is used to calculate the contact force between the impactor and the post-buckled beam. Considering the axial prestress, the constitutive relations of the composite layer and the Kelvin type viscoelastic constitutive model of the damping layer are proposed. A generalized higher-order shear deformation zig-zag beam theory is utilized to model the nonlinear displacement fields. Based on the Hamilton energy variational principle, the governing equations of dynamics are derived. Through two-step analysis, the post-buckling equilibrium paths are obtained as the initial state condition for the impact problem analysis. Further, combined with the fourth-order Runge Kutta method, the two-step perturbation-Galerkin method is extended to simulate the time history curves of the contact force and the dynamic response curves of the post-buckled beams. Compared with the results based on other beam models, the correctness of the material model, theoretical model and computation method is validated. The dynamic characteristics of bi-stable large amplitude vibration of the post-buckled beams under single and two collisions are studied. The effects of axial load, impact velocity, viscoelastic damping characteristics and impactor materials on the contact force and the deflection time history curves of the post-buckled beams are discussed. The results suggest that the contact force is only sensitive to the impact velocity. A certain structural collision parameter design can change the response of the post-buckled beam from single potential energy trap motion to double trap large oscillation, and the contact force is approximately unchanged. The results of this work can be of reference significance for the design of bi-stable energy capture system with collision.

An analysis for snap-through behavior of bi-stable hybrid symmetric laminate with cantilever boundary

This paper focuses on a study of the snap-through behavior of a family of bi-stable plates constructed by hybrid symmetric laminates with a cantilever boundary. Involving shape change and actuation demand, a deep understanding of the snap-through behavior is of particular importance since this bi-stable laminate is often served as a host structure of energy harvester or a morphing structure. The snap-through behavior is obtained using two displacement-controlled finite element procedures provided in the commercial software ABAQUS and the results are verified by experiments. The entire equilibrium path of reaction force-displacement response with a couple of zero load crossings is captured by the ‘Static, Riks’ procedure to reveal the potential nonlinear structural behavior. The practical situation of the reaction force-displacement curve is obtained by the ‘Static, General’ procedure which exhibits consistent results with experiments. Several characteristics, such as snap-down, hysteresis, negative stiffness, and even zero stiffness are captured. A parametric study including lay-up design and length is also carried out using the finite element model and verified by experiments. The results show that the lay-up design not only can fundamentally change the snap-through response but also influences other characteristics of this bi-stable laminate including stable shape, and the length is a relatively reliable parameter to alter the snap-through features, e.g. peak force and hysteresis performance.

Rotational snap-through behavior of multi-stable beam-type metastructures

Metastructures consisting of planar arrangements of bi-stable snap-through beams are able to exhibit multiple stable configurations. Apart from the expected translational state transition, when all beam elements snap through, rotational states may exist as well. In this paper we explore the rotational properties of multi-stable metastructures on the basis of both experimental and theoretical investigations, and define the conditions for achieving rotational stable states. Results show that the metastructure is able to realize both translational and rotational states, while the rotational transitions require less energy as compared to their translational counterparts. The influence of geometric parameters on rotational stability is investigated via parametric studies. Furthermore, to determine the design criteria for rotational stability, a theoretical investigation based on mode superposition principle is performed to predict the nonlinear-deformation of a unit cell. The theoretical analysis predicts well the rotational snap-through transitions that are observed in finite element simulations. It is found that the rotational stability is determined by setting proper values for h/L and t/L (h, t, L represent apex height, thickness and span of the bi-stable beam structure, respectively). Finally, we experimentally demonstrate that the proposed metastructure with multiple layers is able to achieve large rotations and translations.

Snap-through of a pinned-clamped elastica with arbitrarily movable support at the clamped end

This paper studies the snap-through of a pinned-clamped elastica when the support of the clamped end can be moved arbitrarily in plane. The universal snap curve, which describes the critical boundary conditions of the pinned-clamped elasticas, is firstly obtained by determining the saddle-node bifurcation points of the moment-rotation response curves. Based on the universal snap curve, the stability of the pinned-clamped elastica can be determined when the support at the clamped end is moved. The critical boundary can also be directly obtained, where the elastica loses stability and the snap-through occurs between the non-inverted shape and the inverted shape. This study can be useful to reveal the snap-through behavior for some other complex systems where movable supports exist.

Geometric design of triangulated bistable scissor structures taking into account finite hub size

Pre-assembled scissor structures can be transformed from a compact bundle of elements to a fully deployed configuration, offering a considerable volume expansion. Intended geometrical incompatibilities during transformation can be introduced as a design strategy to obtain bistability, which allows instantaneously achieving some structural stability in the deployed state. Because of these incompatibilities, some specific members bend during transformation, resulting in a controlled potentially tunable snap-through behaviour. Geometric design methodologies were proposed in the literature to obtain a compatible geometry (i.e. with all of the beams straight) in the folded and the deployed configurations. However, most of these approaches do not consider finite hub sizes or introduce extra incompatibilities in the geometry by adding hub legs. In this contribution, deployability conditions are derived taking the finite hub size, i.e. the spacing between the connections of the different beams to the hub, into account to make triangulated bistable scissor modules fully geometrically compatible in the folded and the deployed configuration.

On static and dynamic snap-throughs of the imperfect post-buckled FG-GRC sandwich beams

Post-buckled beams have been the basic elements of many mechanical devices used for energy harvesting, low-frequency vibration isolation, etc. This paper focuses on the theoretical analysis of nonlinear bending and forced vibration of the post-buckled sandwich beams with graphene reinforced composites (GRC) face sheets. It is assumed that the graphene nanoplatelets are directional in the beam's length and piecewise functionally graded (FG) along with the thickness. Also, the initial imperfection is supposed to possess the same shape as the first-order vibration mode and the material properties are temperature dependent. Based on a modified higher-order shear deformation zig-zag theory, the governing equations considering the thermal effects are derived. To obtain the closed form solutions of the strongly nonlinear system, a combination of the two-step perturbation with the modified Lindstedt-Poicaré method is proposed. The accuracy of the results based on the current models and computation methods is validated. Subsequently, the imperfection sensitivity of bending and principal resonance characteristics under different axial compressions is also discussed. Numerical results reveal the static and dynamic snap-through behaviors and provide the optimal graphene reinforcement type of sandwich beams under different core-to-face sheet thickness ratios.

Thermal buckling of sisal and glass hybrid woven composites: Experimental investigation

The influence of hybridization of sisal woven fabric with glass fabric on thermal buckling behaviour of composite beam has been analysed experimentally. Initially, non-uniform heating (upward-downward, downward and downward-upward) has been achieved by positioning infrared (IR) heaters at the chosen locations along the longitudinal direction of the composite beam and deflection of the composite beam in the lateral direction measured through LabVIEW program and linear variable displacement transducer. Experiment results indicate that deflection behaviour of the beams under thermal load is entirely different from the deflection under mechanical load. The beams exhibited snap-through deflection behaviour with multiple inflection points due to the thermal load. Results revealed that the sisal woven fabric reinforced composite has high deflection under thermal load than hybrid composites. High elastic modulus and sandwiching effect as a result of hybridization provides more resistance against thermal deflection of the hybrid composites. Furthermore, the layering sequence and type of heating also influences the performance of the composites under thermal load. Results revealed that compared to cross ply sisal composite, an angle ply composite enhanced the resistance against deflection. Synergy effect of sandwiching and ply lay-up enhances resistance against deflection of the composite beams under non-uniform thermal load.

Dynamic buckling of classical/non-classical curved beams by nonlocal nonlinear finite element accounting for size dependent effect and using higher-order shear flexible model

This paper investigates the dynamic snap-through buckling of classical and non-classical curved beams subjected to a suddenly applied step load. The small scale effect prevalent in non-classical beams, viz, micro and nanobeam, is modeled using the nonlocal elasticity approach. The formulation accounts for moderately large deflection and rotation. The governing equilibrium equations are derived using the dynamic version of the principle of virtual work and are subsequently simplified in terms of the generalized displacements for the development of a nonlocal nonlinear finite element model. The spatial domain comprises of 3-noded higher-order curved beam elements based on shear flexible theory associated with sine function. The nonlinear governing equations are solved using the incremental stiffness matrices and by adopting direct time integration method. The critical dynamic buckling load is identified by the smallest load at which there is a sudden rise in the amplitude of vibration. The efficacy of model here is compared against the available analytical studies for the local and nonlocal beams. A detailed study is made to highlight the effects of the geometric parameter, initial condition, nonlocal parameter, load duration, and boundary conditions on the dynamic stability of both classical and non-classical curved beams. The nature and degree of participation of various eigen modes accountable for the dynamic snap-through behavior are examined a posteriori using the modal expansion approach. Some interesting observations made here are valuable for the optimal design of such structural members against fatigue and instability.

Experimental study of multi-stable morphing structures actuated by pneumatic actuation

In this paper, we proposed a new pneumatic actuation method inspired from a soft robot, which could drive the multi-stable carbon fiber-reinforced polymer (CFRP) structure transition among five stable configurations under air pressure. Multi-stable structure was manufactured by the carbon fiber prepreg laid in asymmetrical and anti-symmetric layup methods and stacked at the joint during the preparation process according to the designed scheme. In experiments, multi-stable structures with two different fibers layups were prepared to study the snap-through behaviors. The snap-through process of two kinds of multi-stable structures was investigated by experiment. The parameters used to characterize efficiency of the pneumatic actuator actuating the multi-stable structures, including air pressure and snap time, were measured. The influence of the pneumatic actuator on the curvature of multi-stable structures was discussed by comparing the shape characteristic of multi-stable structures with or without pneumatic actuation, and it showed pneumatic actuator has little effect to the structure. The results demonstrated that the proposed pneumatic actuation could be used to drive multi-stable morphing structure flexibly. The combination of multi-stable laminates develops the function of multiple point bending and shape retention of pneumatic actuation.

Design of metastructures with quasi-zero dynamic stiffness for vibration isolation

A metastructure consisting of numerous unit cells with quasi-zero stiffness property has been designed and researched in this paper. The quasi-zero stiffness property is realized in each unit cell by combining snap-through behavior of a sinusoidal beam and bending dominated support of two semicircular arches. Mechanical behavior of the metastructure is derived theoretically based on the property of all the unit cell components. The unit cell and metastructure samples are designed and fabricated by the FDM additive manufacturing technology. Experiments and simulations are carried out on the samples to investigate static characteristics of the metastructure and verify the theoretical solution. Based on that, dynamic analysis of the metastructure under displacement control vibration is carried out to study the vibration isolation performance of the metastructure by theoretical and simulation methods. Results show that the presented metastructure can achieve good quasi-zero stiffness property by designing the unit cells properly. It exhibits excellent vibration isolation performance and has great potential to be used as continuousstructures or materials in the vibration isolation for small-scale equipment which is difficult to be realized by installation of the spring mechanism.

A symmetrical bistable mechanism from combination of pre-shaped microbeams

A bistable mechanism is proposed with symmetrical snap-through behavior between the two equilibrium positions. The mechanism is monolithic and does not require external axial load, as buckled beams. It consists of two pre-shaped beams of opposite curvature with similar properties and dimensions, which are connected after fabrication. The combination of the opposite pre-shaped beams, each having non-symmetrical bistability, results in total symmetric bistability. This symmetrical bistable mechanism provides the advantage of having the same switching force in the forward and backward directions. The concept of the new mechanism is presented, analyzed based on the snapping force curves, and tested experimentally on microfabricated prototypes.

Nonlinear thermal behavior of shear deformable FG porous nanobeams with geometrical imperfection: Snap-through and postbuckling analysis

Present paper deals with the nonlinear thermal bending response, thermal postbuckling behavior, and snap-through phenomenon due to lateral mechanical load in a thermally preloaded functionally graded (FG) porous perfect/imperfect nanobeam subjected to two types of thermal loading, including heat conduction across the thickness and uniform temperature rise. Heterogenous material properties of the porous nanobeams are assumed to be position/temperature-dependent, where dependency is obtained according to the modified rule of mixture and Touloukian formulation. Two types of porosity distribution and also geometrical imperfection of the nanobeam are considered. Assuming the Timoshenko beam model and von-Karman nonlinearity, the nonlinear equilibrium equations are extracted on the basis of the nonlocal theory of elasticity. With the establishment of the principle of virtual displacement and Chebyshev polynomial of the first kind as the basic functions, the Ritz method is utilized to obtain the matrix form of the nonlocal governing equations. Two different strategies, including the Newton-Raphson technique and direct displacement control scheme, are first proposed to extract the nonlinear bending and postbuckling trajectories of the graded porous nanobeam. Afterwards, to trace the snapping behavior beyond limit loads of the graded porous nanobeam with thermal preloading, the cylindrical arch-length scheme as a path-following method is adopted. Numerical parametric investigations are given to discuss the influences of the power-law index, two types of porosity distribution and thermal loads, size-dependent nonlocal parameter, imperfection amplitude, and boundary conditions on the snap-through behavior and the nonlinear thermal stability of the FG nanobeam.

Analytical Study of the Snap-Through and Bistability of Beams With Arbitrarily Initial Shape

Abstract We derive the snap-through solution and the governing snapping force equations for an arbitrarily preshaped beam deflected under a mid-length lateral point force. The exact solution is obtained based on the classical theory of elastic beams as a superposition of the initial shape and the modes of buckling. Two kinds of solutions are identified depending on the axial force level. The two solutions, bifurcation conditions, bistability conditions, and the snapping force equations are derived and discussed. The snap-through and snapping force solutions are then calculated for two common beam initial shapes, the curved (first buckling shape) and the inclined one (V-shape). In both cases, explicit expressions are obtained describing the snap-through behavior. The analytical modeling results show excellent agreement with finite element simulations. The comparison between the two cases shows a similar snap-through behavior qualitatively, while several differences and similarities are noticed quantitatively.

Velocity-amplified monostable dual-charged electret dome energy harvester using low-speed finger tapping

Power generation from linear finger-tapping-based electrostatic energy harvesting (FTEEH) devices is hindered by the slow capacitance variation under low-speed finger-tapping (FT) motion. Herein, a velocity amplification mechanism is proposed, which exploits the snap-through behavior of a dual-charged electret monostable dome structure and thus greatly enhances the power generation of FTEEH devices from slow FT motion. The kinetic energy and velocity amplification during the buckling event were effectively predicted for various specimens using the modified Föppl–von Kármán equations and Hamilton's principle. A high degree of dynamic velocity amplification was demonstrated both theoretically and experimentally and quantified with respect to the velocity gain and power gain. Specifically, the velocity of the capacitance variation of the designed FTEEH device, driven by a slow FT motion at 2.7 cm/s, was substantially increased to 18.5 cm/s, affording a high velocity gain of 6.9 and a correspondingly large power gain of 6.8. The proposed velocity-amplified nonlinear FTEEH device was compared with recently developed linear FTEEH devices that do not utilize this velocity amplification mechanism and found to yield a large pulse width of 90.0 ms (full width) and a high volumetric power density of 1015.7 μW/cm3.

Modeling and design of a class of hybrid bistable symmetric laminates with cantilever boundary configuration

Multistable laminates have been widely analyzed in the recent past for their potential in morphing applications. However, all the analytical models developed up until now have taken into account only the free-free boundary condition. In this work two objectives are met: (a) an analytical model is developed, which extends the previously available models in literature to account for the cantilever boundary condition for a special class of hybrid bistable symmetric laminates (HBSL); (b) the previously proposed HBSL is modified by replacing the aluminum layers with bi-direction glass-epoxy prepregs in the layup. It is observed that the modified layup has a curvature similar to the previously proposed HBSL while maintaining bistability. The analytical model developed here successfully captures the equilibrium shapes and the snap-through behavior for this special class of laminates which is validated against the results obtained using ABAQUS® and experiments. The developed model is then subsequently used to study the design space and bistability characteristics of the HBSL and the proposed modified layup (m-HBSL) in the cantilever boundary condition.

Snap-Through of Shallow Circular Arches with Variable Horizontal Supports under Unilateral Displacement Control

AbstractThis study proposes a numerical method to solve the snap-through problem of elastic shallow inextensible circular arches. Specifically, the snap-through behavior of arches subjected to a do...

Hill-top branching: Its asymptotically expanded and visually solved bifurcation equations

Hill-top branching (HTB) is an exceptional stability problem in which the bifurcation point (BP) coincides exactly with a limit point (LP). In such cases, snap-through and branching instability occur simultaneously. Then, HTB is often obscured in snap-through behavior and is often overlooked in stability design. The compound instability is in fact not well realized in the community of computational mechanics, when the rank deficiency and number of negative eigenvalues of the stiffness matrix are carelessly examined. No report of the relevant literature has established diagnosis of path branching or proposed a branch-switching procedure. Furthermore, few HTB models are available in the literature.This report proposes a set of asymptotic bifurcation equations to address this academically important and exciting issue. There might be a single or two critical eigenvectors of the singular stiffness matrix. Higher-derivatives of the stiffness matrix with respect to nodal displacements can be computed exactly using hyper-dual numbers (HDNs). The resulting simultaneous equations can be solved visually using graphical tools available in popular mathematical software packages such as MATLAB®. From the locations of existing real roots displayed on a graphic monitor, the crossover situation of path branching is visually clear above and below the hill-top. For the present study, three HTB examples are computed to verify the proposed algorithm strategy using HDNs and graphical tools. The obtained numerical results show excellent agreement with analytical and finite element predictions. The proposed asymptotically expanded and visually solved bifurcation equations reliably diagnose asymmetric and unstable symmetric HTB in structural stability problems and function well in computational engineering problems.

Novel multidirectional negative stiffness mechanical metamaterials

This paper presents a novel tri-directional negative stiffness (NS) mechanical metamaterial, consisting of disk structure elements arranged in three-dimensions. The NS behavior was achieved through the elastic instability of the disk structure. Through the combination of numerical simulations and experiments, the NS property, the deformation mechanisms, and the basic mechanical performance of the disk structure and the novel NS mechanical metamaterial were systematically investigated. The influence of the geometric parameters on the snap-through behaviors of the NS metamaterial was investigated through the experimentally verified numerical method, and further designs were conducted on the disk structure to tailor its mechanical responses. Also based on the instability of the disk structure, several novel multidirectional NS structures, which can achieve NS behavior along five and seven loading directions, were firstly proposed. This innovation improves the homogeneity of the NS metamaterials, and is expected to expand the application scope of such metamaterial, considering that the previous reported NS metamaterial can exhibit NS behavior along three-dimensions at most. Moreover, the mechanical performance of the proposed NS metamaterial is tunable, benefited from the special structure form of itself. To our knowledge, this work is first to design NS metamaterial using the disk structures, and report the multidirectional (more than tri-directions) NS metamaterials.