Piecewise Stiffness Research Articles

A novel optimal design method for tuned mass dampers with elastic motion‐limiting stoppers

AbstractTuned Mass Dampers (TMDs) are commonly used passive control devices in practical engineering applications. However, motion‐limiting stoppers are usually installed to control the excessive TMD displacement due to the building space limitation, resulting in piecewise nonlinearity and detuning of TMD. This paper studies the influence of elastic motion‐limiting stoppers on the optimal design of TMDs through a piecewise stiffness TMD (PSTMD) model. Performance of a PSTMD with classical design is first investigated and proven to be ineffective. To optimize the PSTMD parameters, the motion of PSTMD is decoupled from the controlled structure, and the frequency response equation of PSTMD is obtained analytically through the averaging method. Subsequently, the solution of the optimal design frequency for PSTMD is transformed into the solution of the jump frequency in the frequency response equation. With the optimal frequency of PSTMDs, the optimal damping and control performance of PSTMDs are discussed and analyzed compared with classical linear design, which fully showcases the effectiveness of the novel design method. Finally, the effectiveness of the novel design method is verified using a nine‐story benchmark frame structure, and the results demonstrate that the control performance of the optimal PSTMD can be improved by nearly under specific seismic excitation, compared to the PSTMD with classical linear method. In summary, the novel design method can effectively take into account the influence of piecewise nonlinearity caused by elastic motion‐limiting stoppers and improve the optimal control performance of TMD in a more realistic engineering environment.

A quad-stable nonlinear piezoelectric energy harvester with piecewise stiffness for broadband energy harvesting

A quad-stable nonlinear piezoelectric energy harvester with piecewise stiffness for broadband energy harvesting

Development of a continuum manipulator with variable bending length and piecewise stiffness for transoral laryngeal surgery.

Continuum manipulators (CMs) show great potential in transoral laryngeal surgery due to their flexibility. However, CMs for transoral surgery face several issues: large size, which reduces practicality; intersegment coupling, which causes undesired deflection; and a lack of versatility that limits their applicability across different patient groups. This work combines a rod-driven proximal segment and a cable-driven distal segment to achieve piecewise stiffness, alleviating the issue of intersegment coupling. A rigid constraint tube is integrated into the proximal segment to diversify its bending behavior. Preliminary experiments are conducted to validate the design concept. The proposed CM has an overall diameter of only 6.5mm. The proximal segment can achieve a 90° bending with various curvatures. At the working configuration, the coupling error between the proximal segment and the distal segment is less than 1mm. The effectiveness of the proposed CM is successfully validated using a human model. The proposed continuum manipulator possesses the desirable characteristics of small size, low coupling, and high versatility, indicating its great potentialities for the diagnosis and treatment of laryngeal lesion.

Equivalent continuous method based novel design of a nonlinear TMD with piecewise stiffness

Tuned mass Dampers (TMDs) are extensively used passive control devices for high-rise buildings to mitigate undesirable responses. In practice, limit devices are frequently applied due to the large displacement of TMD, resulting in the emergence of piecewise nonlinearity. In this study, a TMD with piecewise stiffness (PSTMD) is developed to investigate the piecewise nonlinearity, including piecewise linear stiffness (PLS) and piecewise nonlinear stiffness (PNS). Initially, a novel equivalent continuous method (ECM) is proposed based on the principles of the least squares method (LSM) to approximate the PSTMD using a nonlinear TMD with Duffing stiffness. The equivalent nonlinear TMD is proven to be accurate enough to approximate the PSTMD. Subsequently, frequency response analysis is conducted on the PSTMD designed by linear method, results indicate that the linear method fails to design PSTMD. Thirdly, the optimal parameters of PSTMD are determined based on the ECM and the improved design method for nonlinear TMD proposed by the author. Results indicate that this novel design method effectively enhances the performance of PSTMD and mitigates the effect of piecewise nonlinearity. Finally, the influence of parameters of PSTMD on the optimal tuning frequency is investigated. The influence of damping ratio of PSTMD is explored by numerical simulation, with results indicating that employing the damping ratio suggested by traditional linear design method as the optimal parameter is reasonable. In summary, the novel method proposed in this study holds significance for the engineering design of TMDs with piecewise characteristics.

Approximating piecewise nonlinearities in dynamic systems with sigmoid functions: advantages and limitations

In the industry field, the increasingly stringent requirements of lightweight structures are exposing the ultimately nonlinear nature of mechanical systems. This is extremely true for systems with moving parts and loose fixtures which show piecewise stiffness behaviours. Nevertheless, the numerical solution of systems with ideal piecewise mathematical characteristics is associated with time-consuming procedures and a high computational burden. Smoothing functions can conveniently simplify the mathematical form of such systems, but little research has been carried out to evaluate their effect on the mechanical response of multi-degree-of-freedom systems. To investigate this problem, a slightly damped mechanical two-degree-of-freedom system with soft piecewise constraints is studied via numerical continuation and numerical integration procedures. Sigmoid functions are adopted to approximate the constraints, and the effect of such approximation is explored by comparing the results of the approximate system with the ones of the ideal piecewise counter-part. The numerical results show that the sigmoid functions can correctly catch the very complex dynamics of the proposed system when both the above-mentioned techniques are adopted. Moreover, a reduction in the computational burden, as well as an increase in numerical robustness, is observed in the approximate case.

Frequency spectrum analysis of the rotor system with bolted joint: Numerical and experimental verification

The bolted joint is commonly used to connect several parts of aero-engine rotor system. Several papers have studied the nonlinear vibration responses of bolted rotor systems. However, the frequency spectrum characteristics of bolted rotor systems are not fully revealed. Aiming at this issue, this study develops a dynamic model of the bolted rotor system, where the piecewise stiffness characteristic is considered. A solution methodology is presented by incorporating the piecewise stiffness into the incremental harmonic balance method. The effects of unbalance and preload on the dynamic characteristics are discussed. Moreover, the stiffness and damping of the support structure and bolted joint are identified based on the particle swarm optimization algorithm and experimental results. Furthermore, the experimental study on a bolted rotor test rig is conducted to verify the accuracy of the identified parameters and the model. The simulation and experimental results indicate that 3X and 5X of the rotating frequency can be observed. Besides, the rotating speed corresponding to 5X's peak is different from the critical speed, and the peak of 5X precedes the primary resonance peak.

A Two-Segment Continuum Robot With Piecewise Stiffness for Maxillary Sinus Surgery and Its Decoupling Method

With the use of tendon-driven continuum manipulators, it is possible to reach deep-seated small lesions in a constrained anatomical space. This circumvents the difficulties encountered by existing straight endoscopes and surgical instruments, which require direct line of sight access during inspection and operation. However, maintaining adequate stiffness while ensuring smoothness and flexibility during operation for a continuum robot with a small diameter is difficult. In this article, we present the design, modeling, and validation of a new two-segment continuum robot for maxillary sinus surgery. It addresses the coupling effects that can affect the safety and precise operation within the maxillary sinus. To overcome potential distal-to-proximal deviations, a piecewise stiffness structure for the continuum manipulator is proposed. Modified kinematics, based on variable curvature continuum kinematics, is used to resolve motion coupling, and a model of the continuum manipulator with varying stiffness is proposed, with due consideration of tension coupling. Experiments are carried out to validate the proposed design, modeling, and compensation schemes. Results show that the proposed manipulator can flexibly and accurately reach the target positions in the maxillary sinus model. Comparisons of continuum robots with and without the proposed compensation schemes demonstrate the practical value and potential clinical use of the proposed method.

Design of compliant mechanisms with piecewise stiffness variation using self-contact

Compliant mechanisms with stiffness variation are required in engineering applications. But most studies in the field of compliant mechanisms have only focused on a mechanism with a single stiffness. This paper presents a design method of compliant mechanisms with multi-level stiffness variation using internal self-contact between rigid parts. The contact changes the reconfigurations of compliant mechanisms to vary their stiffness for a specific motion range. We avoid the need for an external control to achieve piecewise stiffness variation. We derived the general design guideline by which the desired multiple stiffness and motion ranges can be acquired through rational design of the structural parameter. To illustrate the effectiveness of the design idea, we fabricated proof-of-concept prototypes for experimental investigations, and the experimental results demonstrated that the designed compliant mechanism has a multiple-motion range corresponding to different stiffness. The design ideas can also be extended to the multi-axis spatial compliant mechanisms. This design idea enriches the theory and application of compliant mechanisms.

Theoretical and experimental analysis of vibration reduction for piecewise linear system by nonlinear energy sink

The vibration reduction of a piecewise linear system coupled with a nonlinear energy sink (NES) is studied. Piecewise stiffness can restrain the large vibration while the NES enhances the control effect. An experimental device of the piecewise system coupled with the NES is deigned, and the governing equation is established with Newton’s second law. Parameters of the system are obtained based on the experimental data. Especially, the nonlinear stiffness and the damping of the NES are identified by the restoring-force surface method. For a better accuracy during the analytical processing, a hyperbolic tangent function is introduced to fit the piecewise restoring force, instead of the Taylor series expansion. Considering the strong nonlinearity of the NES, the steady-state response of the coupled system is analyzed by the harmonic balance method (HBM), including the stability analysis. The numerical simulation and the experimental study verify that the result obtained with the HBM has a good accuracy. Compared with the response of the piecewise system without the NES, the NES has a significant damping performance for the piecewise structure. The NES even can restrain the vibration of the main system to a small region, in which the auxiliary spring in the piecewise system does not work. Based on the detailed investigations on the cubic nonlinear stiffness, the damping, and the mass of the NES, the control performance is optimized. A fork-shaped phenomenon exists at the top of the response curve of the main structure under optimized parameters. The discussion suggests that, a better control performance can be obtained with the NES which consists of a small mass, a strong nonlinear stiffness, and a well-designed damping.

Dynamic properties of piecewise linear systems with fractional time-delay feedback

The forced vibration of a single-degree-of-freedom piecewise linear system containing fractional time-delay feedback was investigated. The approximate analytical solution of the system was obtained by employing an averaging method. A frequency response equation containing time delay was obtained by studying a steady-state solution. The stability conditions of the steady-state solution, the amplitude–frequency results, and the numerical solutions of the system under different time-delay parameters were compared. Comparison results indicated a favorable goodness of fit between the two parameters and revealed the correctness of the analytical solution. The effects of the time-delay and fractional parameters, piecewise stiffness, and piecewise gap on the principal resonance and bifurcation of the system were emphasized. Results showed that fractional time delay occurring in the form of equivalent linear dampness and stiffness under periodic variations in the system and influenced the vibration characteristic of the system. Moreover, piecewise stiffness and gap induced the nonlinear characteristic of the system under certain parameters.

Nonlinear energy sink with limited vibration amplitude

In the research of applying nonlinear energy sinks for vibration reduction, usually the vibration amplitude of nonlinear energy sinks is not limited. Since the linear stiffness of the nonlinear energy sink is zero, the vibration amplitude of the nonlinear energy sink may be very large. This is obviously not acceptable in many mechanical engineering fields. Based on theoretical and experimental investigations, this paper investigates a limited nonlinear energy sink by using a piecewise spring device. Thus, the vibration amplitude of the nonlinear energy sink can be restricted from being too large. For free vibration, the effects of the piecewise stiffness and the gap width on the vibration responses of the nonlinear energy sink and the primary system, i.e. the linear oscillator, are profoundly examined. The numerical results show that the vibration of the nonlinear energy sink is effectively suppressed. Nevertheless, the vibration damping effect of the nonlinear energy sink on the linear oscillator is weakened after the introduction of the piecewise spring. However, after improving parameters reasonably, the nonlinear energy sink can still achieve considerable damping effects. In addition, the experimental results are conducted to verify the theoretical results. Moreover, the particle swarm optimization algorithm is used to optimize the piecewise stiffness and the gap width so that the vibration of the linear oscillator is suppressed most efficiently. The optimized results are verified with the differential evolution algorithm to illustrate that the particle swarm optimization is effective and accurate. In conclusion, this work provides useful information for the design of the nonlinear energy sink and can promote the engineering application of the nonlinear energy sink.

Steady-state responses of mechanical system attached to non-smooth vibration absorber with piecewise damping and stiffness

The steady-state dynamic characteristics of non-smooth vibration absorbers are investigated. The complexification-averaging method is used to obtain the steady-state response equation of a harmonic excited primary system attached to the non-smooth absorbers, with the equation solved using a Matlab program based on the least square method. Research results indicate that the traditional purely nonlinear absorber loses its efficacy after the excitation amplitude reaches a certain value. The non-smooth absorber with piecewise linear damping, by contrast, can suppress vibration of the primary system within a larger range of excitation amplitude than the purely nonlinear absorber. Then, the cubic stiffness component is substituted by a piecewise stiffness component to further enhance the performance of the above non-smooth absorber and good results are obtained. The non-smooth absorber with both piecewise damping and stiffness shows the stronger vibration absorption performance. In addition, the differences of higher branches of response which induced by the three absorbers are analyzed and discussed.

Vertical–Horizontal Coupling Vibration of Hot Rolling Mill Rolls under Multi-Piecewise Nonlinear Constraints

This study establishes a vertical–horizontal coupling vibration model of hot rolling mill rolls under multi-piecewise nonlinear constraints considering the piecewise nonlinear spring force and piecewise nonlinear friction force constraints of the hydraulic cylinder in the vertical direction of the rolls, the piecewise stiffness constraints in the horizontal direction, and the influence of the nonlinear dynamic rolling force in the rolling process. Using the average method to solve the amplitude–frequency response equation of the coupled vibration system and taking the actual parameters of a 1780 mm hot rolling mill (Chengde Steel Co., Ltd., Chengde, China) as an example, we study the amplitude–frequency characteristics of the mill rolls under different parameter settings. The results show that the amplitude and resonance region can be reduced by appropriately reducing the external disturbance force and the nonlinear spring force of the hydraulic cylinder, appropriately increasing the nonlinear friction force, and eliminating the gap between the bearing seat and the mill housing, to avoid the amplitude jump phenomenon due to piecewise variation. Furthermore, using the singularity theory to study the static bifurcation characteristics of the coupled vibration system, we establish a relationship between the vibration parameters and the topological bifurcation solution of the coupled system. The transition sets and their corresponding bifurcation topological structure in three cases are given, and the steady and unsteady process parameter regions of the rolls are obtained. The dynamic behavior of the coupled vibration system can be controlled by varying the bifurcation parameter. This study provides a theoretical basis for restraining the vibration of hot rolling mill rolls and optimizing the process parameters.

Electromechanical Tuning of Piecewise Stiffness and Damping for Long-Range and High-Precision Piezoelectric Ultrasonic Transducers

This paper presents a novel technique for quickly deactivating piezo-electric micromachined ultrasonic transducers (PMUT) in order to increase their potential for acoustic imaging. The proposed PMUT is anchored by four arms and surrounded by two mechanical dampers. The dampers are moved using electrostatic actuators in order to come into contact with the PMUT. By doing so, the resonant frequency of the PMUT can be shifted and the amplitude at the working frequency can be reduced. Measurement results show that this technique can effectively dampen the displacement of the PMUT. [2019-0123]

A tri-stable nonlinear energy sink with piecewise stiffness

A tri-stable nonlinear energy sink (NES) with negative stiffness and positive piecewise linear stiffness is presented for broad band vibration suppression of small vibrations of host structural system. The negative stiffness of the NES is produced by permanent magnets and the piecewise linear stiffness is produced by a combination of elastic beams. The structure and the mechanism of the NES is firstly introduced, and then the dynamic model of the main system with the NES is built. The vibration suppression ability of the considered NES under transient and steady state excitations are numerically studied and experimentally verified. The results show that the proposed NES has efficient vibration suppression ability, the dissipating speed increases 6.89 times in transient vibration suppression and vibration suppression rate reaches to 71% in steady state vibration suppression. And also, the NES has the ability to attenuate vibrations less than 1 mm.

Dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative

In this paper, the dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative is studied. First, a mathematical model of the single-degree-of-freedom system is established, and the approximate steady-state solution associated with the amplitude–frequency equation is obtained based on the averaging method. Then, the amplitude–frequency response equations are used for stability analysis, and the stability condition is founded. To validate the correctness and precision, the approximate solutions determined by the analytical method are compared with the solutions based on the numerical integration method. It is found that the approximate solutions and the numerical solutions are in excellent agreement. Finally, the effects of system parameters, such as fractional-order coefficient, order, clearance, and piecewise stiffness, on the complex dynamical behaviors of the piecewise linear single-degree-of-freedom oscillator are studied. The results show that the system parameters not only influence resonance amplitude and resonance frequency but also affect the size of the unstable region.

Atherosclerotic plaque mechanical characterization coupled with vector Doppler imaging in atherosclerotic carotid arteries in-vivo.

Methods used in clinical practice to diagnose and monitor atherosclerosis present limitations. Imaging the mechanical properties of the arterial wall has demonstrated the potential evaluate plaque vulnerability and assess the risk for stroke. Adaptive Pulse Wave Imaging (PWI) is a non-invasive ultrasound imaging technique, which automatically detects points of spatial mechanical inhomogeneity along the imaged artery and provides piecewise stiffness characterization. The aims of the present study are to: 1) demonstrate the initial feasibility of adaptive PWI to image the mechanical properties of an atherosclerotic plaque 2) demonstrate the feasibility to combine adaptive PWI with vector Doppler in a single imaging modality in order to simultaneously obtain information plaque mechanical properties and plaque hemodynamics. The common carotid arteries of 1 healthy subject and 2 carotid artery disease patients were scanned in vivo. One of the patients underwent carotid endarterectomy and a plaque sample was retrieved. In this patient, a higher compliance value of the stenotic segment was estimated by Adaptive PWI as compared with the adjacent arterial wall, and the healthy carotid artery. This was corroborated by histological staining of the plaque sample, which revealed the presence of a large necrotic core and a thrombus, characteristics associated with reduced stiffness. Moreover, the same sequence demonstrated the feasibility to obtain both stiffness maps and vector flow information, showing promise in atherosclerosis diagnosis and patient care.

Statistical Linearization Method for the Piecewise Nonlinear Systems

The Monte-Carlo numerical simulation method is applied to obtain the approximate probability distributions of the response of the piecewise nonlinear systems to random excitation. The expression of the equivalent linearization parameters for the piecewise nonlinear system is derived by the proposed statistical linearization method. To illustrate the accuracy of the linearization method, the present paper has compared the statistical characteristic value obtained by both the Monte Carlo numerical simulation method and the Statistical linearization method, while the time history analysis also shows a good agreement. Linearization of piecewise stiffness nonlinear model is displayed.

Programmable Self-Locking Origami Mechanical Metamaterials.

Developing mechanical metamaterials with programmable properties is an emerging topic receiving wide attention. While the programmability mainly originates from structural multistability in previously designed metamaterials, here it is shown that nonflat-foldable origami provides a new platform to achieve programmability via its intrinsic self-locking and reconfiguration capabilities. Working with the single-collinear degree-4 vertex origami tessellation, it is found that each unit cell can self-lock at a nonflat configuration and, therefore, possesses wide design space to program its foldability and relative density. Experiments and numerical analyses are combined to demonstrate that by switching the deformation modes of the constituent cell from prelocking folding to postlocking pressing, its stiffness experiences a sudden jump, implying a limiting-stopper effect. Such a stiffness jump is generalized to a multisegment piecewise stiffness profile in a multilayer model. Furthermore, it is revealed that via strategically switching the constituent cells' deformation modes through passive or active means, the n-layer metamaterial's stiffness is controllable among 2n target stiffness values. Additionally, the piecewise stiffness can also trigger bistable responses dynamically under harmonic excitations, highlighting the metamaterial's rich dynamic performance. These unique characteristics of self-locking origami present new paths for creating programmable mechanical metamaterials with in situ controllable mechanical properties.

Experimental and theoretical investigation of an impact vibration harvester with triboelectric transduction

There has been remarkable interest in triboelectric mechanisms because of their high efficiency, wide availability, and low-cost generation of sustainable power. Using impact vibrations, we introduce piece-wise stiffness to the system to enlarge frequency bandwidth. The triboelectric layers consist of Aluminum, which also serves as an electrode, and Polydimethylsiloxane (PDMS) with micro semi-cylindrical patterns. At the bottom of the PDMS layer, there is another Al electrode. The layers are sandwiched between the center mass of a clamped-clamped beam and its base. The center mass enhances the impact force on the triboelectric layers subjected to external vibrations. Upon impact, alternating current, caused by the contact electrification and electrostatic induction, flows between the Al electrodes. Because of the impact, the equivalent stiffness of the structure increases and as a result, the frequency bandwidth gets wider. The output voltage and power reach as large as 5.5V, 15μW, respectively at 0.8g vibrational amplitude. In addition, we report how the surface charge density increases with the excitation levels. The analysis delineates the interactions between impact vibrations and triboelectric transductions. The ability of the system to achieve wider bandwidth paves the way for efficient triboelectric vibrational energy harvesters.