Dynamic Snap-through Research Articles

Nonlinear effect on stable state and snap-through bistability of square composite laminate

Bistable structures with the elastic instability caused by buckling are confirmed to perform significantly in micro-electromechanical systems, metamaterials, energy harvester, vibration isolation and morphing structures. The target of this paper is to explore the dynamic responses of cross-ply bistable composite laminate, focusing on the nonlinear effect of the single- and double-well vibration. Both theoretical and finite element (FE) methods are employed to simulate the nonlinear vibration and dynamic snap-through of bistable composite laminate under the foundation excitation at the center. In the theoretical model, the governing equations are established via Lagrange's equation based on the first-order shear deformation theory, von Karman nonlinear strain-displacement relation and Rayleigh-Ritz method. The fourth-order Runge-Kutta method is adopted to solve the governing equations, and the numerical results are validated by FE model. Subsequently, the details of the dynamic responses are analyzed to identify the nonlinear effects in the form of bifurcation diagram, phase portrait, time history, Poincare maps and amplitude spectrum. The dynamic responses are examined for a series of excitation parameters in both time and frequency domain. Through fixed frequency and frequency sweep tests, the nonlinear phenomenon of the single- and double-well vibrations are analyzed including superharmonic resonance, stiffness softening, hysteresis phenomenon, various periodic and chaotic vibration. The diverse responses to external inputs contribute significantly to efficiently predicting mechanical behaviors in real-world conditions, thereby offering indispensable theoretical support for structural design applications.

A rapid-response soft end effector inspired by the hummingbird beak.

Biology is a wellspring of inspiration in engineering design. This paper delves into the application of elastic instabilities-commonly used in biological systems to facilitate swift movement-as a power-amplification mechanism for soft robots. Specifically, inspired by the nonlinear mechanics of the hummingbird beak-and shedding further light on it-we design, build and test a novel, rapid-response, soft end effector. The hummingbird beak embodies the capacity for swift movement, achieving closure in less than [Formula: see text]. Previous work demonstrated that rapid movement is achieved through snap-through deformations, induced by muscular actuation of the beak's root. Using nonlinear finite element simulations coupled with continuation algorithms, we unveil a representative portion of the equilibrium manifold of the beak-inspired structure. The exploration involves the application of a sequence of rotations as exerted by the hummingbird muscles. Specific emphasis is placed on pinpointing and tailoring the position along the manifold of the saddle-node bifurcation at which the onset of elastic instability triggers dynamic snap-through. We show the critical importance of the intermediate rotation input in the sequence, as it results in the accumulation of elastic energy that is then explosively released as kinetic energy upon snap-through. Informed by our numerical studies, we conduct experimental testing on a prototype end effector fabricated using a compliant material (thermoplastic polyurethane). The experimental results support the trends observed in the numerical simulations and demonstrate the effectiveness of the bio-inspired design. Specifically, we measure the energy transferred by the soft end effector to a pendulum, varying the input levels in the sequence of prescribed rotations. Additionally, we demonstrate a potential robotic application in scenarios demanding explosive action. From a mechanics perspective, our work sheds light on how pre-stress fields can enable swift movement in soft robotic systems with the potential to facilitate high input-to-output energy efficiency.

Necessary conditions for static and dynamic bow snap-through in electrostatically actuated initially curved and latched micro-beams

A curved bistable micro-beam, subjected to electrostatic loading from an electrode facing its concave side, may produce a snap-through response with voltages as low as 54%, when compared to actuation from a convex facing electrode. Such actuation has been dubbed “bow actuation” due to the similarity of preloading an arrow onto a bow, and the resulting equilibrium shift, as “bow snap-through”. Under a certain elevation-to-thickness ratio, a bistable beam will also become latchable, allowing the beam to maintain itself in its second stable state under zero load/voltage. In the current work, necessary conditions are found for static and dynamic bow snap-through, which can be used as a tool to design and produce bow snap-through responses, promoting efficient non-volatile and low-power consumption bistable based devices. The conditions are found using an undamped dynamic single degree-of-freedom (DoF) reduced-order (RO) model, attained via Galerkin’s decomposition. Subsequent numerical calculations, conducted in the presence of ambient damping, show that the condition is necessary to attain bow snap-through responses, while also disclosing the snapping behaviour of the model.

Snap-through dynamics of buckled flexible filaments in a side-by-side configuration

The flow-induced snap-through dynamics of side-by-side buckled flexible filaments in a uniform flow were investigated using the penalty immersed boundary method. We systematically examined the effects of gap distance and bending rigidity on the filament's motion and energy harvesting performance. Four distinct modes were observed as we varied the aforementioned parameters: the dual equilibrium mode, the dual streamwise oscillation mode, the single snap-through oscillation (STO) mode, and the dual STO mode. We analyzed the corresponding wake patterns associated with each of these modes. A phase transition from out-of-phase to in-phase motion was observed at low gap distances and bending rigidities. At lower gap distances, the interaction between two filaments was pronounced, leading to a co-motion of the filaments and enhanced elastic energy. This interaction decreased as the gap distance increased. Filaments in the 2STO mode exhibited significantly higher strain energy compared to the other modes. The elastic energy was concentrated at the rear part of the filaments and increased with higher gap distances and bending rigidities. Considering both energy harvesting efficiency and space utilization rate, a gap distance equal to 1 exhibited the best performance.

Nonlinear dynamic response of a temperature-dependent FGM spherical shell under various boundary conditions and thermal shocks: Examination of dynamic snap-through

In this study, a comprehensive exploration of the nonlinear dynamic snap-through phenomenon observed in shallow spherical shells composed of functionally graded materials (FGMs) is examined. The analysis encompasses the interplay of thermomechanical characteristics within metal and ceramic phases, taking into account their temperature-dependent behaviors. To address this problem, the transient heat conduction equation in a nonlinear form, accounting for the temperature dependence factor is derived. To solve this complexity, the Generalized Differential Quadrature (GDQ) method is employed, coupled with the Crank-Nicolson time integrating scheme, leveraging an iterative approach for precision. The formulation of nonlinear dynamic motion equations is based on Hamilton's principle and incorporates the utilization of nonlinear strain-displacement equations and uncoupled thermoelasticity. In the spatial domain, GDQ proves invaluable in solving these nonlinear, coupled equations. In the temporal domain, the Newmark procedure in conjunction with the Newton-Raphson iterative method are utilized to navigate through dynamic equations, yielding insights into shell's behavior over time. In the initial stages of the investigation, we validate our spherical shell formulation and methodologies by comparing shell's response with existing, simpler works. This step ensures the reliability and accuracy of our model. The primary objective is to assess the presence and characteristics of the thermal snap-through phenomenon within the shell under these diverse conditions.

Assessment of Axisymmetric Dynamic Snap-Through and Thermally Induced Vibrations in FGM Cylindrical Shells Under Instantaneous Heating

This paper addresses the investigation of axisymmetric thermally induced vibrations in Functionally Graded Material (FGM) cylindrical shells. It considers temperature-dependent (TD) properties and geometric non-linearity (the von Karman effect). The study systematically solves a transient heat conduction equation using finite differences method and the Crank–Nicolson method. During the heating stages, the evaluation of thermal forces and moments takes place. Equations of motion are derived through the application of Hamilton’s principle. Spatial dependencies are discretized using the generalized Ritz method, while temporal dependencies are approximated using the [Formula: see text]-Newmark method with Newton–Raphson linearization. A comparative analysis validates the procedure’s efficiency and precision. Parametric studies explore the influence of parameters, including the temperature-dependency material properties, geometric nonlinearity, and shell’s power-law index, providing valuable insights into FGM shell behavior under thermal shock.

Investigations on the dynamic snap-through of MFC bonded self-resetting bistable laminates

Unsymmetric composite laminates having two stable equilibrium configurations have been studied extensively in the recent past due to their potential applications in morphing structures. Surface bonded Macro Fiber Composites (MFC) actuators have been considered as a viable solution to trigger the snap-through transition in bistable laminates. Although MFC bonded bistable laminates are widely used in morphing applications, they might require considerably high voltage inputs to achieve the required levels of actuation control during the shape transition. As a solution, other possible energy sources can be combined with active MFC patches to reduce the snap-through energy required from the single source of MFC actuation. In this work, we examine the dynamic behavior of bistable composite plates actuated using MFC actuators, where external vibration energy has been used to assist with the MFC-controlled actuation between stable states. A refined semi-analytical model based on the Rayleigh–Ritz formulation has been proposed, where the membrane energy and the bending energy are separately evaluated. Bending components are directly evaluated using the approximated transverse displacement functions, whereas the membrane components are evaluated separately by combining compatibility conditions and equilibrium equations. Results from the proposed semi-analytical framework are compared with a full geometrically nonlinear finite element framework and necessary experimental observations. The results show a significant reduction in the snap-through energy demand on MFC layers where external dynamic excitation assists the snap-through process. Additionally, a parametric study is performed using variable stiffness (VS) fiber orientation parameters, achieving bistable laminate-MFC configurations that lower snap-through requirements through the proposed morphing strategy. Thus, the study offers to aid a multi-efficient snap-through strategy for the morphing of multistable composite structures.

Snap-through dynamics of a buckled flexible filament in a channel flow

The flow-induced snap-through dynamics of a buckled flexible filament in a channel flow was explored using the penalty immersed boundary method. Two edge condition distributions were considered for comparison. One edge condition was a simply supported leading edge and a clamped trailing edge (SC); the other condition was two clamped edges. The effects of channel height and bending rigidity on the energy harvesting performance were systematically examined. The presence of the channel wall compresses the activation space of the vortex, leading to the formation of a high shear flow near the wall, which, in turn, strongly influences the wake pattern. In the snap-through oscillation mode, a wake pattern of 2S + 2P is observed in both narrow and broad channels, whereas the mechanism of vortex shedding varies between the two cases. Both cases demonstrate greater critical rigidity and greater elastic energy compared with conditions under external flow, suggesting an enhancement of the energy harvesting performance. The greater energy harvesting ability of the SC case is derived from both the larger deflection and the higher strain energy in this system. The wall effect is inversely proportional to the channel height, becoming nearly negligible when the nondimensional channel height exceeds 2. These findings provide valuable insights into the dynamics of flexible filaments in the channel flow and their potential for energy harvesting applications.

Dynamic response of piezoelectrically actuated bistable cross-ply laminates under oscillating impulse voltages

Multistable composite laminates are ideal candidates for morphing applications due to their ability to switch between multiple stable states with proper actuation mechanisms. Surface-bonded macro fiber composite (MFC) actuators to trigger snap-through between one stable shape to another by applying external voltages have attracted considerable attention recently. In this work, dynamic snap-through of bistable unsymmetric cross-ply laminate using MFC actuators under oscillating impulse voltages has been investigated. Both unimorph and bimorph laminate-MFC configurations are considered for the analysis. The stable equilibrium configurations of the MFC-actuated bistable laminate, considering the effects of MFC bondage, are discussed using the proposed finite element (FE) framework. Under the action of oscillating impulse voltages, the single-well and cross-well oscillations of the active bistable plate are analyzed in detail through time responses and phase portraits. The FE results show the advantage of actuating MFC layers through the oscillating impulse voltages as it leads to a reduction in the maximum snap-through voltage requirements.

Snap-through dynamics of a buckled flexible filament with different edge conditions

The flow-induced snap-through dynamics of a buckled flexible filament under different edge conditions were explored using the penalty immersed boundary method. Three filament edge conditions were simulated: a simply supported leading edge and a clamped trailing edge (SC), a clamped leading edge and a simply supported trailing edge, and both edges clamped. The effects of the bending rigidity and density ratio on the energy harvesting performance were systematically examined. Two different modes were observed: an equilibrium mode and a snap-through oscillation mode. The parameter range under which the modes were observed changed depending on the edge conditions. Mode transitions, induced by an increase in transverse fluid force, occurred when the bending rigidity was low. A clamped leading edge enhanced filament stability, whereas a simply supported leading edge reduced stability. Among the three configurations, the SC case showed the highest critical bending rigidity and oscillation frequency, resulting in superior energy harvesting performance. The greater energy harvesting ability of the SC case derives from the larger deflection and the higher strain energy in this system. The strain energy in the filament with SC edges tended to concentrate in two regions of the filament: the rear part and the section near the clamped end. The SC case, coupled with low density and high rigidity, offers favorable conditions for energy harvesting purposes.

A snap-through vibration-rotation modulation mechanism for high-performance low frequency energy harvester

Ambient low frequency (LF) vibrations are ubiquitous but their efficient utilization is still an open issue. To tackle this problem, this paper reports an ingenious vibration–rotation (V-R) modulation mechanism that is realized by introducing a reset spring and a pair of magnets to the conventional rack-and-pinion mechanism. By taking advantage of the magnets-enabled snap-through dynamics, the V-R mechanism can convert LF-vibrations to fast rotation. Under a pressed depth of 10 mm and trigger frequency of 2.0 Hz, a high rotation speed of 1230 rpm can be achieved by the snap-through V-R mechanism in combination with a pawl-ratchet mechanism, which is 1.4 times higher than that enabled by the linear counterpart. A rotor-based vibration energy harvester (R-VEH) designed with the snap-through V-R mechanism can generate high voltage of 4.3 V and large power of 10.8 mW as excited at 2.5 Hz. The snap-through R-VEH can also sustain the continuous work of a hygrothermograph by tapping the irregular LF-vibration of a treadmill (even with a small amplitude of roughly 2.5 mm). The snap-through R-VEH is thus a potential technology for the efficient harvesting of pervasive LF-vibrations toward self-driven sensing systems.

Snap-through dynamics of a buckled flexible filament in a uniform flow

The flow-induced snap-through dynamics of a buckled flexible filament was explored using the penalty immersed boundary method. The effects of the filament length, bending rigidity and Reynolds number on the mode transition were systematically examined. Three different modes were observed when the aforementioned parameters were varied: an equilibrium mode, a streamwise oscillation mode and a snap-through oscillation mode. Two mode transitions occurred when the bending rigidity was lowered and the length and Reynolds number were increased: a direct transition from the equilibrium mode to the snap-through oscillation mode and the successive appearance of the three modes. An increase in transverse fluid force induced the snap-through oscillation mode. A vortex-induced vibration and a self-excited vibration occurred in the streamwise oscillation mode and the snap-through oscillation mode, respectively. A wake pattern of 2S appeared in the streamwise oscillation mode, and a pattern of 2S + 2P appeared in the snap-through oscillation mode. A hysteresis was observed near the critical Reynolds number. The hysteresis loop increased in magnitude with increasing bending rigidity. The greater energy harvesting was achieved by the larger deflection and the higher strain energy. We found that most of the strain energy was concentrated in the last half of the filament.

Dynamic Snap-Through and nonlinear vibrations of bistable asymmetric Cross-Ply composite laminated cantilever shell under external excitation

The bistable asymmetric composite laminated structures have the broad applications in the design of the deformable aircrafts and energy harvester due to the unique nonlinear dynamic snap-through characteristics. This paper studies the dynamic snap-through phenomena and nonlinear vibrations of the bistable asymmetric cross-ply composite laminated cantilever shell under the external excitation. Reddy’s third-order shear deformation theory and Hamilton principle are applied to establish the differential dynamic governing equation of the bistable asymmetric composite laminated cantilever shell. Two-degree-of-freedom nonlinear dynamic system is obtained by employing Galerkin approach. The curvature-displacement relationship is given for the bistable asymmetric composite laminated cantilever shell. Chebyshev-Ritz is utilized to obtain both natural frequencies and mode shapes of the bistable asymmetric composite laminated cantilever shell. Considering the cantilever boundary condition, the free vibrations are analyzed by considering different factors of the bistable asymmetric composite laminated cantilever shell, such as the curing temperature, numbers of the layers and geometry parameters. It is observed that the dynamic snap-through behaviors and chaotic vibrations are found under varying the frequency and amplitude of the external excitation. Several validations are carried out among the obtained results with these of other references. The results of the numerical simulation and finite element software ABAQUS are in good agreement with these of the vibration experiment.

The dynamic regimes of the unsymmetric bistable laminate

A dynamic model built theoretically and simulated numerically for an unsymmetric bistable laminate is elucidated, elaborating the dynamic regimes. Peak-to-peak amplitude diagrams being amplitude-frequency and amplitude-force response curves are derived. The single-well vibrations, the limit-cycle oscillations, the multiple-period dynamic snap-through, the chaotic dynamic snap-through and the intermittency dynamic snap-through can be detected. The amplitude-frequency response curves exhibit the softening stiffness effect. The dynamic regimes are correlated with the excitation amplitude and the modal frequencies. Deploying the high-level excitation amplitude and the low-level modal frequency leads to the limit-cycle oscillations. Besides, the single-well vibrations are characterized by the period-doubling bifurcation.

Dynamic snap-through of bi-stable laminates with simply supported at four corners under impact loads

In this paper, nonlinear dynamic responses of bi-stable laminate being simply supported on four corner points, which is subjected to various impact loads, are analyzed from theoretical viewpoints. Consider a four layers asymmetric cross-ply bi-stable laminate composited by the carbon fiber reinforced resin matrix. A novel configuration function that satisfies the boundary condition is proposed to predict the equilibrium configuration. Then, employing the first order shear deformation theory, the von Kàrmàn strains and Hamilton principle, nonlinear dynamic equations of motion expressed by two curvatures are derived. To obtain the two stable configurations of the laminate, the inertia terms are ignored. With the aids of static equilibrium equation, the side length and curvature of the system are solved by using NLPSolve functions in the Maple software. The stable configurations are verified by comparing present results with those reported in available literatures. Consider the incremental load loading, the decreasing loading, the sinusoidal loading, the step loading, the triangular loading and periodic loading, the nonlinear dynamic response of the bi-stable laminate are discussed. The dynamic snap-through behavior, the intra-well oscillations and inter-wall oscillations are studied by numerical simulation in detail.

Nonlinear dynamics and snap-through regimes of a bistable buckled beam excited by an electromagnetic Laplace force

We study the nonlinear forced dynamics of a bistable buckled beam. Depending on the forcing frequency and amplitude, we observe three different regimes: (i) small intra-well oscillations in the neighborhood of one of the equilibria, (ii) transient snap-through ending into intra-well oscillations, (iii) persistent dynamic snap-through. We build experimentally and numerically phase-diagrams determining the forcing amplitude and frequency leading to each of the three regimes. The experiments leverage an original setup based on the use of the electromagnetic Laplace forces. The controlled flow of an electric current through a metallic beam immersed in an electromagnetic field is at the origin of the electromechanical coupling. This non-invasive excitation system allows us to easily tune the forcing frequency and amplitude. The results of our numerical model, based on a weakly nonlinear geometrical approximation and a three-mode Galërkin expansion for the space discretization, are in excellent agreement with the experimental findings. We show that higher-order modes, often neglected in the modal models of the literature, have a major influence on the nonlinear dynamics.

Dynamic snap-through of functionally graded shallow arches under rapid surface heating

In this paper, dynamic buckling response of a shallow arch constructed from functionally graded materials under thermal shock is investigated. It is assumed that one surface of the beam is subjected to sudden temperature elevation while the other surface is kept at reference temperature. To solve the one-dimensional Fourier heat transfer equation, Crank–Nicolson numerical method along with the central finite difference method are employed. In order to obtain the strain–deflection relations, Donnell theory of shallow arches with von Kármán simplifications are used. The governing equations are acquired based on well-known Hamilton’s principle that are converted to a set of non-linear algebraic equations using the Ritz method. Since the governing equations are non-linear equations, the Newton–Raphson method is applied and after that temporal evolution of displacements are obtained employing the Newmark time marching scheme. In the final stage, a theory to estimate the buckling load is required, which is the Bodiansky–Roth criterion in this research. To validate the formulation and solution method of this project, this work is compared with previous studies and good agreement is obtained. In addition, impacts of power law index, arch thickness, geometrical parameters, and temperature dependency on the dynamic buckling temperature are reported in this study.

Nonlinear vibrations and dynamic snap-through behaviors of four-corner simply supported bistable asymmetric laminated composite square shell

The nonlinear vibrations and dynamic snap-through behaviors are studied for a four-corner simply supported bistable asymmetric laminated composite square shell under the foundation excitation by using the theoretical, Abaqus finite element (FE) and experiment approaches. The first-order shear deformation shell theory and Hamilton principle are adopted to derive the partial differential governing equations of motion. A novel type of nonlinear strain-displacement relations is given by Sander’s strain. Galerkin approach is utilized to discretize the partial differential equations of motion into a three-degree-of-freedom system. The natural frequencies and vibration modes of the bistable shell are calculated by using Chebyshev polynomial method. The nonlinear vibrations and dynamic snap-through are investigated by using the maximum Lyapunov exponent, bifurcation diagrams, time histories, phase portraits, 3D phase portraits and Poincare maps. The theoretical results of the vibrations are well compared with those of the FE and experiment. The effects of the excitation and structural parameters on the dynamic snap-through and nonlinear vibrations are fully discussed. The influences of the initial radius of the curvature on the critical load of the dynamic snap-through behaviors and rule of the chaotic vibrations are analyzed for the bistable asymmetric laminated composite shell.

Study on snap-through actuation for the bistable composite laminated shell–MFC assembly

All available theoretical researches on the bistable composite laminated shell–MFC assembly have been fundamentally concerned with the static snap-through, while dynamic researches on the assembly have been more pronounced based on experiments for the dynamic snap-through. However, theoretical researches on both the static and dynamic snap-through for the assembly have hardly been carried out. In this article, the theoretical model for the bistable composite laminated shell–MFC assembly is elaborated, elucidating the static and dynamic snap-through by deploying the static and dynamic voltages. To initiate the static snap-through, the voltage–curvature and voltage–displacement curves are graphically presented, delivering the whole path of the process of the static actuation. To initiate the dynamic snap-through, the assembly is activated dynamically while the MFC actuator is expanded by the dynamic voltage. The dynamic responses for the assembly are characterized by the limit-cycle oscillation, the periodic, and chaotic snap-through and the secondary Hopf bifurcation. The theoretical model proofs the bistable composite laminated shell–MFC assembly to be developed and promising, which provides a validation source for the actuation strategy of the desirable snap-through and the suppression strategy of the undesirable snap-through.

Study on Dynamic Snap-Through and Nonlinear Vibrations of an Energy Harvester Based on an Asymmetric Bistable Composite Laminated Shell

Bistable energy harvesters have been extensively studied. However, theoretical research on the dynamics of bistable energy harvesters based on asymmetric bistable composite laminated plate and shell structures has not been conducted. In this paper, a theoretical model on the dynamics of an energy harvester based on an asymmetric bistable composite laminated shell is established. The dynamic snap-through, the nonlinear vibrations and the voltage output with two potential wells of the bistable energy harvester are studied. The influence of the amplitude and the frequency for the base excitation on the bistable energy harvester is studied. When the frequency for the base excitation with a suitable amplitude in the frequency sweeping is located in a specific range or the amplitude for the base excitation with a suitable frequency in the amplitude sweeping is located in a specific range, the large-amplitude dynamic snap-through, nonlinear vibrations and voltage output with two potential wells can be found to occur. The amplitude and the frequency for the base excitation interact on each other for the specific amplitude or frequency range which migrates due to the softening nonlinearity. The vibration in the process of the dynamic snap-through behaves as the chaotic vibration. The nonlinear vibrations of the bistable system behave as the periodic vibration, the quasi-periodic vibration and the chaotic vibration. This study provides a theoretical reference for the design of energy harvesters based on asymmetric bistable composite laminated plate and shell structures.