Bistable Beam Research Articles

On the Nonlinear Snap-Through of Arch-Shaped Clamped–Clamped Bistable Beams

Abstract Bistable compliant elements offer excellent advantages in many applications ranging from high precision sensing to energy harvesting. The essential nonlinear mechanics of such elements are strongly coupled with their buckling mode, geometric parameters, and loading conditions. The force–displacement plot of bistable curved beams could contain a displacement limit point, which cannot be well modeled by the commonly used smooth cubic function and would cause operational problems due to incorrect predictions of the bistability. In this technical brief, the nonlinear bistable mechanics of a compliant curved beam with both ends fixed is analyzed based on the large deflection finite element theory. By using the multistep displacement loading method, the deformation behaviors and their transition from symmetric to asymmetric modes are numerically studied, which provides insights into the force–displacement curve and the multiple snapping pathways. Furthermore, the influences of the structure parameters on bistable mechanics are analyzed, and a quality factor for identifying the occurrence of displacement limit points is introduced for different loading conditions. Finally, a method for achieving a single smooth snapping pathway is proposed, providing a theoretical basis to the design and control of the bistable compliant structures.

Bi-stable buckled beam nonlinear energy sink applied to rotor system

In this paper, a bi-stable nonlinear energy sink (BNES) with buckled beam is developed to suppress the vibration of unbalanced rotor system. The structure of the BNES is developed. The specific structures and working principles of the buckled beam are introduced. Based on these, the dynamic equations of the rotor-BNES system are established. Then, the transient and steady state responses of the rotor-BNES system are numerically studied and compared with the linear visco-elastic damper. In addition, the vibration suppression ability and frequency suppression range of the BNES are analyzed. Finally, experiments are carried out and compared with the simulations to confirm the vibration suppression effectiveness of the BNES. The numerical and experimental results show that the designed BNES has a strong vibration suppression effect on the rotor system and can withstand a wide range of energy.

Snap-through of elastic bistable beam under contactless magnetic actuation

We propose the modeling of contactless switching of a bistable slender beam using Laplace force actuation. The model beam is based on the elastica approach which allows large amplitudes of the cross-section rotation and large elastic transformations. Furthermore, the extensibility of the elastic beam is accounted for and it plays a crucial role in the switching process. The study is devoted particularly to the actuation mechanism of the bistable beam uniformly loaded by a density of lineic force which is permanently perpendicular to the beam deformation. Such actuation can be produced by the Laplace force due to an electric current traveling along the beam placed in a magnetic induction. The mechanism of the bistable switching is analyzed in detail using a variational formulation and stable and unstable equilibria are described. We investigate numerically the post-buckling behavior to capture the exact buckling modes that follow the path in the unstable region. The numerical simulations are also used to obtain the bistable response in terms of actuating force (or electric current amplitude), beam end-shortening and mid-point displacement of the beam. We also perform a finite element computation and provide a validation of the results obtained by solving the set of equations of the boundary value problem.The second part of the work is focussed on the experimental validation of the switching mechanism of the bistable beam presented in the analytical part. We design an experimental set-up for the fine measurement of the mid-point displacement of the bistable beam as a function of the electric current traveling the beam. The region of bistable instability is revealed by experimental adjustment of the bifurcation point associated with the actuating force. All the results extracted from experimental tests are compared to those coming from the modeling investigations, which ascertains with good accuracy the approach of the proposed model for bistable beam.

Design of a flexure-based mechanism possessing low stiffness and constant force

This paper presents a novel design of a flexure-based constant force mechanism with a long travel stroke. Unlike the conventional force control method using a force sensor and feedback controller to obtain constant force output, the proposed compliant mechanism provides a constant force utilizing the unique mechanical property of the mechanical structure. The constant force is generated by using the combination of a negative-stiffness and a positive-stiffness mechanism. In order to achieve a low driving force, the negative stiffness is realized by a special bistable beam, which is a step beam with structural holes. Meanwhile, the positive stiffness is generated by the leaf flexure hinges with structural holes. The regular structural holes can reduce the mass and stiffness of the whole mechanism. Furthermore, the elliptic integral method and the pseudo-rigid-body approach are utilized to establish the model of the constant force mechanism. Based on the established model, the performance of the constant force mechanism is evaluated computationally. Additionally, the parametric model of the proposed mechanism is investigated using the nonlinear finite element analysis. Finally, a prototype is fabricated using 3D printing technique. The open-loop and the closed-loop experimental tests are implemented to investigate the performance of the developed constant force mechanism. It is noted that the constant force mechanism can be robustly controlled by a proportional-integral-derivative control method. Experimental results demonstrate that the developed constant force mechanism has a constant force with slight fluctuation with a range of 500 μm.

ON THE DISCRETIZATION OF A BISTABLE CANTILEVER BEAM WITH APPLICATION TO ENERGY HARVESTING

A typical setup for energy harvesting is that of a cantilever beam with piezoceramics excited by ambient base vibrations. In order to get higher energy output for a wide range of excitation frequencies, often a nonlinearity is introduced by intention in that way, that two magnets are fixed close to the free tip of the beam. Depending on strength and position of the magnets, this can either result in a mono-, bi- or tristable system. In our study, we focus on a bistable system. Such systems have been investigated thoroughly in literature while in almost all cases the beam has been discretized by a single shape function, in general the first eigenshape of the linear beam with undeflected stable equilibrium position.There can be some doubts about the suitability of a discretization by a single shape function mainly due to two reasons. First: In case of stochastic broadband excitations a discretization, taking into consideration just the first vibration shape seems not to be reasonable. Second: as the undeflected position of the considered system is unstable and the system significantly nonlinear, the question arises, if using just one eigenshape of the linear beam is a suitable approximation of the operation shapes during excited oscillations even in the case of harmonic excitation. Are there other, e.g. amplitude dependent, possibilities and/or should multiple ansatz functions be considered instead?In this paper, we focus mainly on the second point. Therefore, a bistable cantilever beam with harmonic base excitation is considered and experimental investigations of operation shapes are performed using a high-speed camera. The observed operation shapes are expanded in a similar way as it is done in a theoretical analysis by a corresponding mixed Ritz ansatz. The results show the existence of distinct superharmonics (as one can expect for a nonlinear system) but additionally the necessity to use more than one shape function in the discretization, covering also the amplitude dependence of the observed operation shapes.

Analytical modeling for rapid design of bistable buckled beams

Double-clamped bistable buckled beams demonstrate great versatility in various fields such as robotics, energy harvesting, and microelectromechanical system (MEMS). However, their design often requires time-consuming and expensive computations. In this work, we present a method to easily and rapidly design bistable buckled beams subjected to a transverse point force. Based on the Euler–Bernoulli beam theory, we establish a theoretical model of bistable buckled beams to characterize their snapthrough properties. This model is verified against the results from a finite element analysis (FEA) model, with maximum discrepancy less than 7%. By analyzing and simplifying our theoretical model, we derive explicit analytical expressions for critical behavioral values on the force-displacement curve of the beam. These behavioral values include critical force, critical displacement, and travel, which are generally sufficient for characterizing the snap-through properties of a bistable buckled beam. Based on these analytical formulas, we investigate the influence of a bistable buckled beam's key design parameters, including its actuation position and precompression, on its critical behavioral values, with our results validated by FEA simulations. Our analytical method enables fast and computationally inexpensive design of bistable buckled beams and can guide the design of complicated systems that incorporate bistable mechanisms.

Nonlinear dynamics of shape memory alloys actuated bistable beams

The phenomenon of bi-stable behaviour has been widely used in the structural design, as it can provide large deformation by switching between two stable equilibrium positions. This paper aims to investigate the intrinsic nonlinear dynamic characteristics of an actively controlled bistable beam using a simplified spring-mass model. The dynamic model for an active (heated) SMA wire driven bistable beam is established based on a polynomial constitutive equation to describe the thermomechanical behaviour of the shape memory alloy. The actively controlled bistable beams are designed, fabricated and experimentally tested to achieve the morphing behaviour snapping-through form one position to another. The results obtained from the experimental testing and the theoretical simulation are compared to validate the proposed model. Dynamic behaviour of the proposed SMA wires actuated bistable beam under varying external excitation is investigated to show the influence of the thermomechanical loadings. Analysis of the experimental data and simulation results shows that the SMA wires actuated bistable structure can be well-performed for the bistable switching. It also approved that the different behaviours of the system, including periodic responses, complex responses and chaos can be accurately predicted using the proposed simplified model.

Dramatic bandwidth enhancement in nonlinear metastructures via bistable attachments

We report amplitude-dependent substantial enhancement of the frequency bandwidth in locally resonant metamaterial-based finite structures (metastructures) via bistable attachments. The bistable magnetoelastic beam attachments of the unit cells exhibit linear intrawell, nonlinear intrawell, and nonlinear interwell oscillations for low, moderate, and sufficiently high intensity excitations, respectively. As a result, the overall metastructure leverages linear locally resonant bandgaps under low amplitudes and nonlinear attenuation due to wideband chaotic vibrations of the bistable attachments under large amplitudes. The concept was first demonstrated through a linear mass-spring chain with bistable attachments in a numerical case study. Experimental results and validations are then presented for a base-excited cantilever beam hosting seven bistable unit cells. Transition from linear locally resonant bandgaps to nonlinear attenuation is observed, and the amplitude-dependent bandwidth enhancement is shown. The bandwidth offered by nonlinear interwell oscillations is substantially wider than the linear locally resonant bandgap that is limited by the added mass.

Dynamic release condition in latched curved micro beams

Curved bistable beams subjected to transverse loading may exhibit latching, namely remain in their buckled state under zero force. Under such circumstances, an opposite force is required for snapping-back (release) of the beam to its initial configuration. For an electrostatically actuated beam, two electrodes located at either side of the beam may therefore be required for bidirectional actuation. In this study, a release procedure conducted by the same electrode, which is used for the snapping, is considered. The approach involves preloading of the latched beam using electrostatic force, followed by a sudden release of the voltage and snapping back motion. A reduced order (RO) model, resulting from the Galerkin decomposition with buckling modes of a straight beam as base functions is used for the formulation of a necessary release condition, which is then verified numerically.

Stochastic P-bifurcation in a nonlinear viscoelastic beam model with fractional constitutive relation under colored noise excitation

In this paper, we study the stochastic P-bifurcation problem for an axially moving bistable viscoelastic beam with fractional derivatives of high-order nonlinear terms under colored noise excitatio...

Buckled MEMS Beams for Energy Harvesting from Low Frequency Vibrations.

Vibration energy harvesters based on the resonance of the beam structure work effectively only when the operating frequency window of the beam resonance matches with the available vibration source. None of the resonating MEMS structures can operate with low frequency, low amplitude, and unpredictable ambient vibrations since the resonant frequency goes up very high as the structure gets smaller. Bistable buckled beam energy harvester is therefore developed for lowering the operating frequency window below 100Hz for the first time at the MEMS scale. This design does not rely on the resonance of the MEMS structure but operates with the large snapping motion of the beam at very low frequencies when input energy overcomes an energy threshold. A fully functional piezoelectric MEMS energy harvester is designed, monolithically fabricated, and tested. An electromechanical lumped parameter model is developed to analyze the nonlinear dynamics and to guide the design of the nonlinear oscillator based energy harvester. Multilayer beam structure with residual stress induced buckling is achieved through the progressive residual stress control of the deposition processes along the fabrication steps. Surface profile of the released device shows bistable buckling of 200μm which matches well with the amount of buckling designed. Dynamic testing demonstrates the energy harvester operates with 50% bandwidth under 70Hz at 0.5g input, operating conditions that have not been demonstrated by MEMS vibration energy harvesters before.

Bidirectional actuation of buckled bistable beam using twisted string actuator

A new actuation mechanism using the twisted string concept to trigger the snap-through of a bistable buckled beam to produce an effective on/off bistable actuator is proposed. The twisted string concept combined with a pin utilizes actuation moment to actuate a bistable beam. The required actuation loads are analytically formulated using the Euler–Bernoulli beam model and solved with the proposed solution algorithm. The actuation mechanism is fabricated to meet the 24.3 N mm actuation requirements. A prototype of the actuator was built, and its performance was evaluated. In a unidirectional actuation, an actuator response time of 104 ms was achieved. The overall response time of the actuator is affected by the length of the string. The twisted string mechanism was also placed in an antagonistic configuration to enable bidirectional actuation. The shape of the input voltage signal also affected the bidirectional performance of the actuator. The actuator produced an actuation bandwidth of 2 and 5 Hz with sine and square input voltages, respectively, while generating 10 mm output displacement.

Stochastic P-Bifurcation of a Bistable Viscoelastic Beam with Fractional Constitutive Relation under Gaussian White Noise

In this paper, we study the stochastic P-bifurcation problem for axially moving of a bistable viscoelastic beam with fractional derivatives of high order nonlinear terms under Gaussian white noise excitation. First, using the principle for minimum mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be simplified to an equivalent system. Second, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by stochastic averaging, and using singularity theory, we find the critical parametric condition for stochastic P-bifurcation of amplitude of the system. Finally, we analyze the types of the stationary PDF curves of the system qualitatively by choosing parameters corresponding to each region within the transition set curve. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order viscoelastic material model to adjust the response of the system.

Flow sensor based on the snap-through detection of a curved micromechanical beam.

We report on a flow velocity measurement technique based on snap-through detection of an electrostatically actuated, bistable micromechanical beam. We show that induced elecro-thermal Joule heating and the convective air cooling change the beam curvature and consequently the critical snap-through voltage (VST ). Using single crystal silicon beams, we demonstrate the snap-through voltage to flow velocity sensitivity of dV ST/du ≈ 0.13 V s m -1 with a power consumption of ≈ 360 μ W. Our experimental results were in accord with the reduced order, coupled, thermo-electro-mechanical model prediction. We anticipate that electrostatically induced snap-through in curved, micromechanical beams will open new directions for the design and implementation of downscaled flow sensors for autonomous applications and environmental sensors.

A nonlinear resonant mass sensor with enhanced sensitivity and resolution incorporating compressed bistable beam

A new piezoelectric actuated nonlinear mass sensor is proposed by using the snap-through phenomenon of a compressed bistable beam to enhance the sensitivity and resolution, which can be used to weigh or detect threshold mass by tracking the bifurcation frequency shift. According to the nonlinear finite element modeling technique, the nonlinear dynamic response of the combined nonlinear structure is numerically calculated, which shows that the bifurcation point can be accurately identified by the sharp and great amplitude change regardless of the damping effect, thus providing an effective way for tracking the bifurcation frequency. Hence, the nonlinear sensitivity depicted by the bifurcation frequency shift per unit mass can reach 3.3 times the sensitivity of linear mass sensors having the same size. Also, as a mass switch, the response amplitude jumps sharply when the added mass is greater than or equal to the threshold value, which is dependent on the excitation frequency. Meanwhile, the influences of the beam compression and excitation voltage on the sensitivity and minimum detectable mass were obtained for sensor optimization. For concept validation, a macro-sized nonlinear mass sensor was fabricated with the geometric size of 58.0 mm long and 4.0 mm wide, and the experimental results show that the sensitivity is around 575.0 Hz/g compared with the simulated sensitivity of 542.0 Hz/g. For a mass switch, the minimum threshold mass is 0.2 mg. The fair agreement between the simulation and experiments adequately validated the proposed nonlinear bistable mass sensor.

Exploiting the subharmonic parametric resonances of a buckled beam for vibratory energy harvesting

The potential of harvesting vibratory energy via a bistable beam subjected to subharmonic parametric excitations is investigated. The vibrating structure is a buckled beam with two stable equilibria separated by a potential barrier. The beam is subjected to a superposition of a static axial load beyond its buckling load and a harmonic axial excitation whose frequency is around twice the frequency of the buckled beam’s first vibration mode. A macro-fiber composite patch is attached to one side of the beam to convert the strain energy resulting from the beam’s oscillation into electricity. The study considers two regimes of excitations: an amplitude sweep and a frequency sweep. In the first regime, the amplitude of excitation is quasi-statically varied while the excitation frequency is tuned at twice the natural frequency of the first vibration mode. In the second regime, the excitation frequency is swept forward and backward around the subharmonic resonant frequency while the amplitude of excitation is kept constant. A theoretical model which governs the electromechanical coupling of the transverse vibrations of the beam and the output voltage is used to monitor the response as the excitation parameters are changed. An experimental setup is also built and a series of tests is performed to validate the theoretical findings. It is shown that, depending on the amplitude and frequency of excitation, the harvester can perform small-amplitude periodic intra-well motion, intra- and inter-well chaotic motions, as well as periodic inter-well motions. Experimental results also show that, as compared to the classical linear resonance, utilizing the sub-harmonic resonance of a bistable energy harvesters can result in a broadband frequency response.

Vibration energy harvesting from impulsive excitations via a bistable nonlinear attachment—Experimental study

Abstract A bistable nonlinear electromagnetic energy harvester coupled to an impulsively excited primary linear oscillator has been experimentally investigated. The design of the energy harvesting system was guided by a preliminary numerical study, which predicted a favorable dynamic regime for harvesting purposes. The harvesting system consisted of a lightweight permanent magnet, moving within a stationary coil, nonlinearly coupled to a grounded, weakly damped linear oscillator through a prebuckled, bistable slender steel beam which exhibited both cubic nonlinear and negative linear stiffness behaviors in transverse response along its weak axis. This paper provides the results of an experimental investigation, illustrating the capacity of the bistable element to dramatically enhance harvesting efficiency when subjected to broadband, low-amplitude vibration, executing large amplitude oscillations between the two stable equilibrium positions. Single and repeated impulses of varying amplitude applied to the LO are employed as system inputs, and it is shown, both experimentally and through numerical simulations, that robust harvesting efficiency is achieved primarily through periodic cross-well oscillations, particularly at low levels of input energy.

Design and Testing of a Flexure-Based Constant-Force Stage for Biological Cell Micromanipulation

This paper presents the design of a novel flexure-based precision positioning stage with constant output force for biological cell micromanipulation. One uniqueness of the proposed design is that it produces a constant force without using a force controller. Only a motion control is needed to produce a constant output force, which significantly simplifies the system design process. The stage is driven by a piezoelectric actuator through a displacement amplifier. Analytical models of the displacement amplifier and the zero-stiffness structure are established and verified by conducting finite-element analysis simulations. The structure parameters are optimally designed to guarantee the requirement on output force, motion range, and physical size. A prototype stage is fabricated by 3-D printing process and a series of experiments is carried out. Experimental results show that the developed positioning stage delivers a near constant output force with slight fluctuation in the reachable constant-force motion range of $138~\mu \text{m}$ . The applications of the developed constant-force stage in biological cell manipulation have been demonstrated through experimental investigations. Note to Practitioners —A constant-force stage can produce a constant output force without using a force control. It is attractive for biological micromanipulation. This paper presents the design and testing of a novel constant-force flexure stage. The constant force indicates a zero stiffness for the mechanism. The stage mechanism is devised using modified leaf flexure (MLF) to achieve positive-stiffness structure. Bistable beams are used to design negative-stiffness structure by making use of their postbuckling characteristics. Two bistable beams and two MLFs are combined together to construct a zero-stiffness structure. A conventional stage is also fabricated for comparison study. The performance of the proposed constant-force stage has been verified by simulation and experimental studies. Results indicate that the developed stage system has great superiority over conventional one in terms of reducing driving force, increasing motion range, and reducing force fluctuation. Experimental demonstration of bio-micromanipulation has been presented to reveal its potential applications.

Rotational energy harvesting using bi-stability and frequency up-conversion for low-power sensing applications: Theoretical modelling and experimental validation

Abstract Kinetic energy harvesting has drawn great attention in the past decade, but low-frequency and broadband operation is still a big issue which impedes this technology to be widely deployed in low-power Internet of Things applications. In this paper, theoretical modelling and experimental validation of a rotational harvester with bi-stability and frequency up-conversion is presented for harnessing low-frequency kinetic energy with a wide bandwidth. Piezoelectric transduction was adopted to convert the rotational kinetic energy into electricity. Distributed-parameter modelling was employed for analyzing the electromechanical dynamics of the bistable piezoelectric beam. Bistable and frequency up-converting behaviours were considered in the theoretical model by introducing two external input magnetic forces. Different oscillating modes were analyzed, showing the variation of power generation capability under different modes, and the advantage of operating in the periodic double-well mode. From the potential well study, we got a conclusion that for the same input magnetic force, periodic double-well mode is capable of achieving a larger vibration amplitude compared to a harvester without bi-stability. Asymmetric potential well shapes were investigated. This asymmetric shape provides a way to stabilize the initiation position of the beam for each plucking cycle, and eventually to stabilized the output. Key design factors to control the oscillating modes were studied, providing a guideline for future design. An experimental study was conducted to verify the theoretical results. A close match was achieved. This bistable harvester demonstrated a significant improvement (up to 2 × ) compared to a harvester without bi-stability over a wide bandwidth (from 1 to 11 Hz) at low frequencies, when operating in the periodic double-well mode. This paper presents a detailed theoretical model and in-depth analysis of a bistable frequency up-converting harvester, providing a fundamental understanding and guidance for low-frequency broadband energy harvester design.

Constrained-Energy Dynamic Cross-Well Motion of Bistable Structures Subjected to Noise Disturbance

Bistable structures have two stable equilibrium positions and can be utilized to maintain a static shape with no energy consumption. This paper focuses on the minimum energy required for performing snap-through of a bistable structure subjected to noise disturbance. This paper uses the Duffing–Holmes equation as a one-degree-of-freedom representative model of a bistable structure. This equation is numerically solved to calculate the energy required for cross-well oscillation under different system and forcing conditions. The paper shows how the energy required for cross-well transfer varies as a function of damping ratio and frequency ratio at specific harmonic force amplitude when the system is externally disturbed with a band-limited noise signal. A magneto-elastic bistable beam is fabricated and tested to validate the used mathematical model. Various nondimensional parameters are used to highlight interesting phenomena. The relationships between signal-to-noise ratio (SNR), dynamic-to-static force ratio, and damping ratio to the response behavior are shown. It is found that the domain of low energy regions decreases by increasing the level of noise. Additionally, underactuated bistable and linear systems behave similarly for high levels of noise. This paper specifically identifies the critical force ratio, which allows for snap-through as a function of critical nondimensional system parameters.