Spring Softening Research Articles

Rational design of CMUTs with annular electrodes for high ultrasonic emission via ESSE enabled stiffness adjustment

Capacitive micromechanical ultrasonic transducers (CMUTs) with high transmitting acoustic pressure are in urgent demand in the rapidly growing field of air-coupled and therapeutic ultras ound. However, most current CMUTs can rarely balance the performance improvement and batch fabrication capacity, which severely impedes their practical applications. This paper proposes novel CMUTs with annular electrodes that can implement significant improvement in multiple performances while featuring a simple structure and batch fabrication feasibility. The annular electrode configurated between the membrane edge and center areas can effectively soften the corresponding-area membrane stiffness through electrostatic spring softening effects, and leave the stiffness of the membrane central area unchanged, finally enabling the membrane to produce a piston-like deformation, thus improving average displacement and output acoustic pressure. A finite element method was employed to analyze the effect of annular electrodes on the CMUT main performance. The results demonstrated that the novel structure could achieve prominent enhancement in multiple performances, such as an average to maximum displacement rate of 0.46 (about 0.32 for conventional CMUTs), maximum improvements of 300%, 255%, and 11% in average displacement, acoustic pressure, and electromechanical coupling coefficients compared to those of conventional ones. Deep analyses of the variation of the main performances including acoustic pressure, receiving sensitivity, and collapse voltage suggested that an optimal electrode coverage range of 36% ∼ 55% can be used to achieve relatively high comprehensive performances. Meanwhile, the proposed CMUTs feature a simpler structure and fabrication process in comparison with previous ones, showing great promise in air-coupled and therapeutic ultrasound applications.

The Post-Buckling Behavior of a Beam Constrained by Nonlinear Springy Walls

Abstract The post-buckling behavior of a beam that is subjected to lateral constraints is of relevance to a range of medical and engineering applications, such as endoscopic examination of internal organs, the insertion of a guidewire into an artery in-stent procedures, root growth, deep-drilling, and more. In this paper, we address a disconnect between the existing literature and the reality of these systems, in which the lateral constraints are flexible and experience nonlinear deformations. As a step towards bridging this gap, we consider a beam undergoing planar deformations that is laterally constrained by a nonlinear springy wall, i.e., a wall that is laterally pushed by the beam against a nonlinear spring. Based on a simplified mathematical model, we obtain closed-form analytical solutions, which provide valuable insights and intuition. For example, we show that important features of the behavior, such as the transition from point contact to line contact and switching to the next mode, are dictated solely by a non-dimensional force, regardless of all other parameters of the system, and that the full description of the behavior is possible by means of two non-dimensional quantities that describe the relative stiffness of the nonlinear spring compared to that of the beam. The results also highlight the fundamental differences between the behavior with a stiffening spring or with a softening spring, such as the number of attainable modes and the monotonicity of the overall force–displacement relation. These results are then validated by experiments.

Hysteresis phenomenon of the sloshing-mode resonance in a moonpool induced by rolling motion excitations

Sloshing-mode resonance in the moonpool induced by roll motion excitations is investigated numerically using the OpenFOAM® package. Nonlinear characteristics of the sloshing-mode resonance are the main focus of the present study. When the roll motion excitation has a time-invariant amplitude, a typical softening spring behavior can be observed in the variation of free surface amplitude against the excitation frequency, including the decreased resonant amplitude and the jump frequency. Furthermore, the dimensionless jump frequency is independent of the moonpool breadth and draft in this situation. The hysteresis phenomenon is clearly observed under the roll excitation with varying amplitudes, where the phase locked-in mechanism is the essential reason for this phenomenon. The hysteresis loop is located between two jump frequencies by the accelerating and decelerating excitations that generate the lower and upper branches, respectively. With the increase in moonpool breadth, the decreased upper branch frequency and unchanged lower branch frequency are observed, leading to the increased region of the hysteresis loop. The variation of the moonpool draft has an insignificant effect on the region of the hysteresis loop.

Acoustic frequency atomic spin oscillator in the quantum regime

Quantum noise reduction and entanglement-enhanced sensing in the acoustic frequency range is an outstanding challenge relevant for a number of applications including magnetometry and broadband noise reduction in gravitational wave detectors. Here we experimentally demonstrate quantum behavior of a macroscopic atomic spin oscillator in the acoustic frequency range. Quantum back-action of the spin measurement, ponderomotive squeezing of light, and virtual spring softening are observed at oscillation frequencies down to the sub-kHz range. Quantum noise sources characteristic of spin oscillators operating in the near-DC frequency range are identified and means for their mitigation are presented.

Non-linear Isolator for Vibration and Force Transmission Control of Unbalanced Rotating Machines

PurposeThe objective of this paper is to investigate with simulations how non-linear spring and non-linear damper components of isolators can be employed to effectively reduce both the oscillations and the force transmitted to ground in the whole spinning range of unbalanced rotating machines.MethodsThe principal goal of this paper is twofold. First, to present a concise and consistent formulation based on the harmonic balance approach for the vibration response of spinning machines mounted on linear/non-linear, softening/hardening, un-tensioned/pre-tensioned springs and linear/non-linear dampers. Second, to provide a comprehensive overview of the vibration and force transmission control with non-linear isolators specifically tailored to unbalanced machines.ResultsThe study has shown that, the best vibration isolation is provided by a pre-tensioned linear and cubic softening spring combined with a linear and negative quadratic damper. The pre-tensioned spring should be designed in such a way as it holds the weight of the machine and thus produces on the vibrating machine a symmetric elastic restoring force proportional to the linear and cubic powers of the displacement. The cubic softening stiffness should then be tuned to minimise the frequency, and thus the amplitude, of the resonant response of the fundamental mode of the machine and elastic suspension system, while preserving stability and a desired static deflection. In parallel, to reduce the force transmission to ground above the fundamental resonance frequency, the negative quadratic damping effect should be tailored to maximize the energy absorption at higher frequencies.ConclusionThe study has shown that non-linear spring and non-linear damper components can be effectively employed to control the vibration and force transmission in the whole spinning range of the machine. In particular, a pre-tensioned softening cubic non-linear spring can be used to mitigate the vibration and force transmission at low frequencies, close to the fundamental natural frequencies of the elastically suspended machine. Also, a negative quadratic non-linear damper can be used to tailor the energy dissipation of the isolator in such a way as to have high damping at low frequencies and low damping at higher frequencies, which enhances the vibration and force transmission control at low frequencies and, rather importantly, mitigates the force transmission at higher frequencies.

Effect of external electric potential on the mechanical resonance of MEMS cantilever resonator

This work presents an analytical model to predict the natural frequency of electrostatically actuated micromachined cantilever beam under the application of DC voltage. The analytical modelling uses an energy based method with a sinusoidal vibration assumption. The electric field between the cantilever electrodes is assumed to be vertical and fringing field effects are neglected. The behaviour of natural frequency and a closed-form expression for pull-in voltage are evaluated. The electrostatic spring softening effect of DC bias on the resonant natural frequency is studied in particular. Results are compared with simulation in ANSYS.

A Monolithic CMOS-MEMS Reconfigurable/Tunable Capacitive Accelerometer with Integrated Sensing Circuits

MEMS accelerometers have been widely used in various applications with a wide range of signal levels and bandwidth. Therefore it is desired to have a sensor whose characteristics such as sensitivity and bandwidth can be reconfigured/tuned depending on specific applications. This paper presents a reconfigurable/tunable z-axis accelerometer whose mechanical resonant frequency, sensitivity and sensing bandwidth can be tuned by the electrostatic spring softening effect. The proposed accelerometer was designed using a commercial 0.35 μm CMOS foundry service. The tuning electrodes were implemented in the metal and polysilicon layers in the standard CMOS process. An on-chip chopper stabilized readout circuit was designed to convert the capacitance signal to a voltage readout. The sensing structure was released by post-CMOS wet metal etching. Measurement results showed the proposed accelerometer had a sensitivity of 12 mV/g in the range of 0–5 g. The sensitivity and bandwidth tuning ranges are 50% and 18%, respectively, for an applied tuning voltage of 25 V. The demonstrated device is the first reconfigurable/tunable CMOS-MEMS accelerometer in the literature to our best knowledge.

On the Performance of a Nonlinear Position-Velocity Controller to Stabilize Rotor-Active Magnetic-Bearings System

The performance of a nonlinear position-velocity controller in stabilising the lateral vibrations of a rotor-active magnetic-bearings system (RAMBS) is investigated. Cubic nonlinear position-velocity and linear position-velocity controllers are introduced to stabilise RAMBS lateral oscillations. According to the proposed control law, the nonlinear system model is established and then investigated with perturbation analysis. Nonlinear algebraic equations that govern the steady-state oscillation amplitudes and the corresponding phases are derived. Depending on the obtained algebraic equations, the different frequency response curves and bifurcation diagrams are plotted for the studied model. Sensitivity analysis for the linear and nonlinear controllers’ gains is explored. Obtained analytical results demonstrated that the studied model had symmetric bifurcation behaviours in both the horizontal and vertical directions. In addition, the integration of the cubic position controller made the control algorithm more flexible to reshape system dynamical behaviours from the hardening spring characteristic to the softening spring characteristic (or vice versa) to avoid resonance conditions. Moreover, the optimal design of the cubic position gain and/or cubic velocity gain could stabilise the unstable motion and eliminate the nonlinear effects of the system even at large disc eccentricities. Lastly, numerical validations for all acquired results are performed, where the presented simulations show accurate correspondence between numerical and analytical investigations.

A simple closed-form model to accurately calculate the electromechanical coupling coefficient of CMUTs

Abstract A simple highly accurate closed-form model to calculate the electromechanical coupling coefficient of a capacitive micromachined ultrasonic transducer (CMUT) is presented. The model exploits the electrostatic spring softening phenomenon to derive an expression for the energy converted from electrical to mechanical domain and includes the nonlinear change of the CMUT diaphragm stiffness during large deflections. The model has been validated by comparing the model predicted values with experimental results published elsewhere along with the results predicted by some existing models. The comparison shows that the model predicted values are in excellent agreement with experimental results. The model also enables one to quantify the effects of residual stress, bending, and nonlinear stretching of the diaphragm on the transduction efficiency of CMUTs without any computationally intensive finite element analysis method. The model resolves the ambiguity of the absence of electrostatic spring softening effect in some existing models.

Phase lag predicts nonlinear response maxima in liquid-sloshing experiments

Mass-spring models are essential for the description of sloshing resonances in engineering. By experimentally measuring the liquid's centre of mass in a horizontally oscillated rectangular tank, we show that low-amplitude sloshing obeys the Duffing equation. A bending of the response curve in analogy to a softening spring is observed, with growing hysteresis as the driving amplitude increases. At large amplitudes, complex wave patterns emerge (including wave breaking and run up at the tank walls), competition between flow states is observed and the dynamics departs progressively from Duffing. We also provide a quantitative comparison of wave shapes and response curves to the predictions of a multimodal model based on potential flow theory (Faltinsen & Timokha, Sloshing. Cambridge University Press, 2009) and show that it systematically overestimates the sloshing amplitudes and the hysteresis. We find that the phase lag between the liquid's centre of mass and the forcing is the key predictor of the nonlinear response maxima. The phase lag reflects precisely the onset of deviations from Duffing dynamics and – most importantly – at resonance the sloshing motion always lags the driving by $90^{\circ }$ (independently of the wave pattern). This confirms the theoretical $90^{\circ }$-phase-lag criterion (Cenedese & Haller, Proc. R. Soc. A, vol. 476, no. 2234, 2020, 20190494).

Using the Nonlinear Duffing Effect of Piezoelectric Micro-Oscillators for Wide-Range Pressure Sensing

This paper investigates the resonant behaviour of silicon-based micro-oscillators with a length of 3600 µm, a width of 1800 µm and a thickness of 10 µm over a wide range of ambient gas (N2) pressures, extending over six orders of magnitude from 10−3 mbar to 900 mbar. The oscillators are actuated piezoelectrically by a thin-film aluminium-nitride (AlN) layer, with the cantilever coverage area being varied from 33% up to 100%. The central focus is on nonlinear Duffing effects, occurring at higher oscillation amplitudes. A theoretical background is provided. All relevant parameters describing a Duffing oscillator, such as stiffness parameters for each coverage size as well as for different bending modes and more complex modes, are extracted from the experimental data. The so-called 2nd roof-tile-shaped mode showed the highest stiffness value of −97.3∙107 m−2s−2. Thus, it was chosen as being optimal for extended range pressure measurements. Interestingly, both a spring softening effect and a spring hardening effect were observed in this mode, depending on the percentage of the AlN coverage area. The Duffing-effect-induced frequency shift was found to be optimal for obtaining the highest pressure sensitivity, while the size of the hysteresis loop is also a very useful parameter because of the possibility of eliminating the temperature influences and long-term drift effects of the resonance frequency. An reasonable application-specific compromise between the sensitivity and the measurement range can be selected by adjusting the excitation voltage, offering much flexibility. This novel approach turns out to be very promising for compact, cost-effective, wide-range pressure measurements in the vacuum range.

Effect of Parameters on the Design of a Suspension System with Four Oblique Springs

This paper presents the fundamental static and dynamic characteristics of a suspension system consisting of four linear springs arranged in an X-shaped configuration to achieve geometric nonlinearity. The particular interest is towards the design of a softening spring geometry realizing a quasi-zero stiffness behaviour at large deflections, and the influence of the system parameters is investigated. The static performance is studied in terms of the force-deflection curve and the dynamic performance in terms of the frequency response curve. The softening-hardening behaviour of the suspension leads to a frequency response which bends to the lower frequencies reaching a well-defined minimum. It is found that both the static and dynamic behaviours may be described in terms of a single parameter, and a simple closed-form expression is determined which links the damping in the system to the excitation amplitude to achieve the lowest possible resonance frequency.

A theory for the coexistence of coiled and stretched configurational phases in the extensional flow of entangled polymer melts.

It has recently been demonstrated via nonequilibrium molecular dynamics (NEMD) simulation [M. H. Nafar Sefiddashti, B. J. Edwards, and B. Khomami, J. Chem. Phys. 148, 141103 (2018); Phys. Rev. Lett. 121, 247802 (2018)] that the extensional flow of entangled polymer melts can engender, within a definite strain-rate regime [expressed in terms of the Deborah number (De) based on the Rouse time], the coexistence of separate domains consisting primarily of either coiled or stretched chain-like macromolecules. This flow-induced phase separation results in bimodal configurational distributions, where transitions of individual molecules between the coiled and stretched states occur very slowly by hopping over an apparent energy activation barrier. We demonstrate that the qualitative aspects of this phenomenon can be described via the single-mode Rolie-Poly model including Convective Constraint Release (CCR) and finite extensibility of the chain-like macromolecules. This analysis reveals the physical mechanism for the configurational coexistence, namely, the nonlinear rate of change of the average entropic restoring force of a given entangled chain with extension. Under conditions of significant flow-induced disentanglement, the rate of change of the effective restoring force initially decreases with extension (effective spring softening) and then increases (hardens) as the maximum chain length is approached. When balanced by flow-induced chain stretching, we find that there can be two configuration states within the same De regime, as covered by the NEMD simulations; therefore, a region of conformational coexistence can indeed exist. However, we demonstrate that this coexistence of configurational microstates is only possible when the magnitude of the CCR parameters is consistent with the rate of flow-induced disentanglement, as observed in the NEMD simulations.

Finite strain elasticity based cohesive zone model for mechanoluminescent composite interface: Part II. Interface damage characteristics

The need to estimate the micromechanical properties of elastico-mechanoluminescent (EML) composites was established in Part A of this paper. The particle–matrix interface was identified to be poorly understood towards building accurate EML models and to predict functional life estimates of the EML devices. Towards this, Part A of this paper defined the mechanical behavior of the interface as a softening spring through a bilinear cohesive zone model (CZM). Macroscale stress measurements were combined with material properties to estimate undamaged stiffness of the interface spring (). In Part B of this paper, the damage characteristics of the interface spring are estimated through an experimental approach to complete the CZM. This is achieved by first establishing the finite strain behavior of the macroscopic EML–PDMS composite from uniaxial tensile testing. Secondly, a controlled tear is induced in the composite and the microscale displacement fields around the tear tip are captured using digital image correlation technique. Tear tip displacements and finite strain behavior are combined to estimate cohesive stresses in the tear-tip region of the composite. The contribution of particle–matrix interfaces towards resisting propagation of the tear is isolated from experimental results to obtain interface-CZM damage parameters. The interface CZM parameters are subsequently utilized in an FEM model to simulate macroscopic composite behavior to find good correlation. The experimental framework established in this paper can be utilized to estimate interface parameters of any particulate-elastomer composites that exhibit finite strain behavior.

Using Electrostatic Spring Softening Effect to Enhance Sensitivity of MEMS Resonant Accelerometers

Micromachined resonant accelerometers (MRAs) are very attractive for high-precision measurement applications due to their high sensitivity, frequency output, and large dynamic range. The sensitivity is one of critical metrics in MRA sensors. In order to enhance this metric, a micro-lever is usually used. However, the micro-lever may suffer from the attenuation of effective amplification ratio and increasing the structure complexity. Lowering spring stiffness of suspension structure and enlarging the proof mass are alternative methods, but these methods are restricted in real devices due to the trade-offs for the device sizes, cross-axis sensitivity, robustness and fabrication yield. This paper presents a novel method to improve sensitivity of MRAs by using electrostatic spring softening effect without sacrificing the mechanical robustness or enlarging footprint size of the device. Besides, this method can on-line adjust the sensitivity of MRAs, which is another potential advantage for specific applications. The prototype has been designed, fabricated and characterized in both open-loop and close-loop ways. In agreement with the proposed device model, the experiment results demonstrated that the sensitivity of the prototype device increased from 492.7 Hz/ ${g}$ ( ${g}$ being the gravity acceleration) to 2277 Hz/ ${g}$ by using a sensitivity enhancing bias voltage of 67V in a linear scale range of $\pm 1~{g}$ . Moreover, the prototype’s equivalent input acceleration noise reduces down from 5.5 ug / $\surd $ Hz to 1.6 ug / $\surd $ Hz within 1Hz bandwidth, and the bias-instability is down from 6.1 ug to 2.2 ug .

Models for analyzing squeeze film air damping depending on oscillation modes of micro/nano beam resonators

Models for analyzing squeeze film air damping depending on the oscillation mode of cantilever and double-clamped micro/nanobeam resonators are presented. The models are obtained from the damping theory of a rigid rectangular plate moving parallelly to a nearby object in viscous and molecular flow regime and electrostatic theory. The analysis results based on the models show that the influence of oscillation mode on squeeze film air damping in viscous regime is remarkably large compared to that in molecular flow regime. The damping in the first-order mode is larger than that in the high-order modes, while there is little deference in damping among the high-order modes. The damping of oscillation modes depends strongly on the oscillation amplitude and electrostatic spring softening. The obtained damping analysis models are useful for the optimal design of micro/nanobeam resonators and verifying the discrepancy between the theoretically calculated results and experiment data.

Finite element modeling and characterization of a magnetoelastic broadband energy harvester

Piezoelectric-based vibration energy harvesters have been extensively developed to the end of replacing low-power batteries. However, matching the frequency of the ambient vibration is not always possible and to broaden the frequency response of the harvesters, different proof-of-concept devices have been developed. This work presents a miniaturized device intended for magnetoelastic broadband vibration energy harvesting. The device consists of a silicon beam where ferromagnetic foils act as proof mass and interacts with an external pair of permanent magnets. The interaction is first simulated using a Finite Element Method (FEM) model for different distances between the magnets and beam (a) and between the magnets (b). This is done for both an attractive and a repulsive magnet configuration and the calculation is performed for a set of a and b values. Both spring softening and hardening effects are observed for the two magnet configurations. The attractive configuration has a monostable potential energy landscape with an associated spring softening for a large range of a and b values which makes this configuration very useful for energy harvesting applications. The attractive configuration is experimentally investigated by impedance measurements. These measurements are performed for a ∈ [400 μm, 2500 μm] and b ∈ [320 μm, 3140 μm] and a region that allows for broadband harvesting is found experimentally. Compared to the linear case the largest spring softening effect yielded a decrease in the effective spring constant of 74%, a decrease in the resonant frequency of 49%, and the coupling coefficient was increased with a factor of 2.6.

A vibration‐based electromagnetic and piezoelectric hybrid energy harvester

This work presents the fabrication and testing of a vibration-based single energy hybrid harvester (VSEHH). Electromagnetic and piezoelectric transduction mechanisms are utilized for energy extraction in the developed harvester. Electromagnetic portion of the device composed of a permanent magnet, planar coil and wound coil, however, for piezoelectric portion polyvinylidene difluoride (PVDF) membrane is used. In the harvester the PVDF membrane having the magnet is kept loose in order to exploit the spring softening nonlinearity. At lower base excitation levels (less than 0.5g) the response of the VSEHH is linear, however, from 0.5 to 2.5g the harvester exhibits spring softening nonlinearity and at acceleration levels greater than 2.5g, the spring hardening nonlinearity is invoked. Because of nonlinear behavior of the harvester, the shift of resonant frequency, the sudden jump-up and jump-down phenomena result in the enhancement of the harvester's frequency bandwidth. Under sinusoidal excitation and at 123 Hz frequency and 4g acceleration, the electromagnetic portion of the harvester produced 40.6 mV load voltage and 212.7 μW with planar coil and 73.5 mV load voltage and 319.1 μW power with wound coil. Moreover, under the same vibrations condition a load voltage of 2930 mV and power of 57.6 μW is generated by the piezoelectric portion of the harvester. Collectively, the harvester is capable of producing a power of 589.4 μW and a power density of 334.13 μW/cm3. Furthermore, when subjected to broadband random vibrations, two central frequency peaks are produced, one is due to spring softening and the other corresponds to the spring hardening of the membrane.

Nonlinear multi-modal energy harvester and vibration absorber using magnetic softening spring

This paper presents a nonlinear multi-modal energy harvester and vibration absorber (EHVA) for both energy harvesting and vibration suppressing in low-medium frequency band. Two different methods are applied to broaden the effective bandwidth. Not only the multi-modal shapes of EHVA are designed, but also the hysteresis property of nonlinear softening spring is realized by a novel permanent magnets structure proposed in this paper. The simulation results illustrate that the magnetic repelling force reduces significantly when the oscillator magnet is close to the stator magnet, causing the nonlinearity of the softening spring. The linear and nonlinear stiffness parameters, natural frequencies and damping values of the oscillator are experimentally identified. The analytical model of flux density in the space is established to obtain the induced voltage. Then, the identified parameters are used to obtain the analytical nonlinear frequency response. It indicates that the effective bandwidth is widened by the nonlinear softening spring. The experiments demonstrate that the designed EHVA can help attenuate the vibration more than 15 dB and harvest 47 mV maximum voltage in the bandwidth from 11 Hz to 24 Hz. Comparing with the harvester using hard spring, the size of EHVA proposed in this paper can be decreased theoretically.

Magnetostatic Spring Softening and Stiffening in Magneto-Mechanical Resonator Systems

Integrating magnets into resonant mechanical systems allows for intriguing capabilities, such as the ability to tune the mechanical resonance frequency or induce coupling between resonators without any physical contact. Here, we present analytical models as well as the experimental study of an integrated magneto-mechanical system. Using a point dipole approximation, we explore the magneto-static spring effect, which can either soften or stiffen a spring depending on dipole orientation and spatial position of the magnets. We use translational and rotational resonance as commonly encountered demonstrative cases and, experimentally, demonstrate both the spring softening and stiffening effects.