Linear Stiffness Coefficients Research Articles

Delayed and fractional-order dynamics of a model for earthquake induced by stress perturbations

Fractional calculus and time delay provide a powerful tool to model complex systems with memory and fractal systems and, the viscoelastic systems. Earthquakes are both complex systems with long-memory and some of their faults have fractal properties and, the rocks constituting faults have viscoelastic behavior. In this paper, we examined the dynamics of the spring-block considering the fractional viscous damping force and interaction between the blocks. The shear stress response is studied using the harmonic balance method and the numerical simulations are performed through Adams-Bashforth-Moulton scheme. The effects of the fractional-order and time delay on the amplitude-frequency curves and on the transition between steady state and seismic regime are investigated. The system response shows the existence of the resonance and anti-resonance. It is appears that the resonance amplitude and resonance frequency are strongly dependent on the time delay and fractional-order. This resonance phenomenon results in an accumulation of energy which can lead to the destabilization of the fault system. At the anti-resonance the system response has minimum amplitude, and the blocks move without accumulated energy. The results show that the delay and fractional viscous can affect the properties of the rock, which is characterized by the modified linear stiffness coefficient, modified linear damping coefficient and the modified linear friction coefficient. The stability and hopf bifurcation are investigated and the critical value of the time delay for the occurrence of Hopf bifurcation is determined. The transition from stationary state to the periodic orbit and vice-versa through the hopf bifurcation is observed in the system.

Energy Transfer and Dissipation in Combined-Stiffness Nonlinear Energy Sink Systems

Abstract Nonlinear energy sinks (NES) are highly efficient vibration energy absorption and dissipation devices, and play an important vibration-suppression role in many types of structures. In this study, the influence of parameters on the combined stiffness nonlinear energy sink system is revealed from the perspective of energy, in which combined-stiffness terms are composed of piecewise linear stiffness and cubic stiffness. First, the slow-varying derivative of the combined-stiffness nonlinear energy sink system is calculated based on the complexification-averaging and multiscale methods. Second, an approximate expression for the extreme points on the slow-invariant manifold (SIM) of the system is derived by polynomial approximation, and the energy dissipation equation of the combined-stiffness nonlinear energy sink system is derived. The impacts of the stiffness gap, piecewise linear stiffness coefficient, and cubic stiffness coefficient on the system are analyzed by studying the energy transfer efficiency equation. Additionally, the relationship between the damping ratio of the primary structure and the dissipation time is analyzed.

Bifurcation and Chaotic Behavior of Duffing System with Fractional-Order Derivative and Time Delay

In this paper, the abundant nonlinear dynamical behaviors of a fractional-order time-delayed Duffing system under harmonic excitation are studied. By constructing Melnikov function, the necessary conditions of chaotic motion in horseshoe shape are detected, and the chaos threshold curve is obtained by comparing the results obtained through the Melnikov theory and numerical iterative algorithm. The results show that the trend of change is the same, which confirms the accuracy of the chaos threshold curve. It could be found that when the excitation frequency ω is larger than a certain value, the Melnikov theory is not valid for these values. Furthermore, by numerical simulation, some numerical results are obtained, including phase portraits, the largest Lyapunov exponents, and the bifurcation diagrams, Poincare maps, time histories, and frequency spectrograms at some typical points. These numerical simulation results show that the system exhibits some new complex dynamical behaviors, including entry into the state of chaotic motion from single period to period-doubling bifurcation and chaotic motion and periodic motion alternating under the necessary condition of chaotic occurrence. In addition, the effects of time delay, fractional-order coefficient, fractional order, linear viscous damping coefficient, and linear stiffness coefficient on the chaotic threshold curve are discussed, respectively. Those results reveal that there exist abundant nonlinear dynamic behaviors in this fractional-order system, and by adjusting these parameters reasonably, the system could be transformed from chaotic motion to non-chaotic motion.

Improved identification of stiffness coefficients of non intrusive nonlinear geometric reduced order models of structures

Significant, successful efforts have been devoted in recent years to the non intrusive construction from finite element software of reduced order models of the nonlinear geometric response of structures. A key assumption in these models is that the elastic restoring forces can be represented as cubic polynomials of their generalized coordinates. However, there has been only very limited discussion of whether this representation is appropriate, how deviations from this polynomial form affect the identification of its coefficients and ultimately the prediction of the structural response, and how current identification strategies can be improved.These questions are the focus of the present investigation which considers first a simple structural model that exhibits the potential for snap-through and thus represents an interesting validation case with strongly nonlinear geometric behavior. It is shown that the force–displacement relationship which can be derived in closed form is not exactly a cubic polynomial but can be very closely modeled by one. However, the match between exact curve and polynomial fit is very sensitive to small changes in the linear, quadratic, and cubic stiffness coefficients and thus the prediction of the response may range from excellent to significantly in error depending on how the identification of the coefficients is performed. In particular, it is found that a cubic approximation obtained by Taylor series around the undeformed configuration is not very accurate post snap-through.Based on these findings, two strategies are proposed to improve the identification of the stiffness coefficients of these reduced order models. The first one is a tuning procedure in which a limited number of key coefficients of the model are adjusted to fit additional static response data. The second approach performs the identification of the stiffness coefficients at a series of different levels, as opposed to one as currently performed, and seeks the most appropriate value of each coefficient based on a “small change” criterion.The applicability of these two novel strategies is demonstrated on several challenging structural models: a clamped–clamped curved beam undergoing snap-through as well as a curved panel and a hat stiffened one. It is found that the reduced order models identified via the tuning or multi-level strategies not only lead to more accurate predictions of the structural response as compared to the original finite element model but also are more computationally stable than their counterparts identified with the current single-level strategy.

Thermal-structural reduced order models for unsteady/dynamic response of heated structures in large deformations

This paper focuses on applications of recently developed thermoelastic reduced order models (ROMs) for the geometrically nonlinear response and temperature of heated structures. In these ROMs, both displacements and temperature fields with respect to the undeformed, unheated configuration are expressed in a reduced order modeling format, i.e., as modal-type expansions of the spatial and temporal variables with constant basis functions. Accordingly, the time varying generalized coordinates of the response and temperature expansions satisfy a generic set of coupled nonlinear differential equations derived from finite deformations thermoelasticity using a Galerkin approach. Finally, the coefficients of these governing equations, which characterize the structure considered and its loading conditions, are determined from structural and thermal finite element models non intrusively so that commercial finite element software can be used. This approach is considered here for the prediction of the displacements and stress fields in the presence of unsteady temperature distributions to enrich previous investigations limited to steady temperature distributions. Specifically considered here are: (i) a panel undergoing rapid heating and (ii) an oscillating flux on a panel. These problems not only demonstrate the extension of the thermal-structural reduced order framework to unsteady problems but also show the importance of the selection of the basis functions. It is also noted that the temperature dependence of the linear stiffness coefficients on temperature can induce in the unsteady situation the existence of a parametric-type excitation of the structure. This behavior is studied in the oscillating flux example and a strong sub-harmonic resonance is in particular found. The computational benefit of using ROMs is discussed and demonstrated.

The origin point of the unstable solution area of a forced softening Duffing oscillator

Each Duffing equation has an unstable solution area with a boundary, which is also a line of bifurcation. Generally, in a system that can be modeled by the Duffing equation, bifurcations can occur at frequencies lower than the origin point frequency of the unstable solution area for a softening system and at higher frequencies for a hardening system. The main goal of this research is to determine the analytical formulas for the origin point of the unstable solution area of a system described by a forced Duffing oscillator with softening stiffness, taking damping into account. To achieve this goal, two systems of softening Duffing oscillators that differ strongly in their nonlinearity factor value have been selected and tested. For each system, for three combinations of linear and nonlinear stiffness coefficients with the same nonlinearity factor, bistability areas and unstable solution areas were determined for a series of damping coefficient values. For each case, curves determined for different damping values were grouped to obtain the origin point curve of the unstable solution, ultimately developing the target formulas.

Passive nonreciprocity-induced directional wave scattering

Controllable nonreciprocal wave redirection in two dimensions is demonstrated by a monatomic lattice of masses and nonlinear springs. The key is a functional section with a spatially asymmetric arrangement of bilinear stiffness. Regardless of the external force driving frequency or the location of the source relative to the functional section, a stable effect is obtained showing scattered wave motion towards two opposite directions each with oppositely signed displacement offsets. Crucially, the bilinear nature of the springs, with linear response but different stiffness coefficients in compression and tension, makes the passive nonreciprocal redirection effect independent of signal amplitude. Consistent nonreciprocal scattering is demonstrated first for a lattice section with asymmetrically distributed bilinearity. Combinations of these fundamental lattice sections with modified bilinear stiffness and orientation of the asymmetric arrangement demonstrate a wide variety of directional scattering effects, illustrating an ability to control the preferred propagating directions and the signs of the dynamic displacement offsets. These results suggest a novel type of nonreciprocal 2D waveguide whose underlying nonlinear mechanism is fundamentally different from actively-achieved alternative methods such as topologically protected edge states.

Enhanced performance of airfoil-based piezoaeroelastic energy harvester: numerical simulation and experimental verification

This paper performs the detailed simulation analyses of airfoil-based piezoaeroelastic energy harvester with two degrees of freedom self-induced plunge-pitch motions, for exploring the flow field characteristic and enhancing the harvesting performance. The designing of the harvester for achieving flutter is first accomplished, finite element model is then built and simulation analyses are performed, and a prototype of the harvester system is finally fabricated. The obtained simulation results are in good agreement with the experimental and theoretical values. The effects of the key structural parameters of the harvester on flow field, aeroelastic vibration, and harvesting performance are numerically investigated. The results demonstrated that the structural parameters of the harvester determine primarily flutter onset of velocity, and consequently affect the dynamic behavior and harvesting performance. The harvester system takes place flutter and occurs limit cycle oscillations after flutter onset of velocity, which is suitable for harvesting energy. The smaller pitch structural stiffness coefficient and the more moderate both plunge stiffness and pitch damping coefficients are, the better aeroelastic vibration and output performance can be captured. A maximum output voltage of 29.08 V and output power of 3.382 mW can be harvested when the linear and cubic structural stiffness coefficients in pitch are 2.5 N·m and 100 N·m at 14 m/s, respectively, which corresponds to the power density of 21.138 mW/cm3 and demonstrates the superior harvesting performance over others. This work provides a significant guidance for designing the more efficient airfoil-based piezoaeroelastic energy harvester utilized in unmanned aerial vehicles.

Equivalent Beam Model for Spatial Repetitive Lattice Structures with Hysteretic Nonlinear Joints

This study presented an equivalent modeling method for analysis of the fundamental harmonic response of the spatial repetitive lattice structures with hysteretic nonlinear joints. Firstly, the joint was modelled as a spatial nonlinear spring-damper system with bilinear hysteresis, the equivalent linear stiffness and viscous damping coefficients of the hysteretic joint were derived using the describing function method. Subsequently, high-precision displacement and rotation fields that consider the warping and distortion of the cross section of the lattice structure were presented for the repeating element and the equivalent continuum modeling method based on the principle of energy equivalence was adopted to obtain an 8-DOFs spatial beam element for the repeating element of the lattice structure. Afterwards, the equations of motion of the equivalent beam model for the lattice structure were established in the frequency domain and solved by using the Newton-Raphson method. In the numerical studies, the joints with axial nonlinearity and rotational nonlinearity were considered, and the influences of joint parameters and excitation amplitude on the dynamic response of the lattice structure were investigated. The correctness of the presented modeling method was verified by comparison with the nonlinear finite element model of the original structure.

Numerical Analysis and Experiment of an Underwater Magnetic Nonlinear Energy Harvester Based on Vortex-Induced Vibration

This paper designs an underwater magnetic nonlinear piezoelectric energy harvester (PEH) based on the principle of vortex-induced vibration (VIV). The magnetic force between the permanent magnets provides the basis for the system frequency modulation. According to the positional relationship between the PEH’s vibrator and the magnet, the values of the linear stiffness coefficient and the nonlinear stiffness coefficient can be determined. According to the value of linear stiffness coefficient, the system can be divided into hard stiffness system, soft stiffness system, and negative stiffness system. The results show that the hard stiffness system’s open-circuit voltage (VRMS) increases from 9.43 V to 10.31 V, and the frequency band is widened by up to 36.8%. According to the simulation and experimental data, the hard stiffness system has a better output effect in a high flow rate. For a soft stiffness system, its initial vibration velocity decrease. When the water flow velocity is 0.2 m/s, the output voltage of the nonlinear PEH from experimental data is 10 times that of the non-magnetic PEH. Both numerical simulations and experiments prove that the soft stiffness system is more suitable for application in a low-speed water environment.

Исследование линейной жесткости в соединении LVL с когтевыми шайбами

The study considers the performance of double-cut dowel connections with the use of claw washers in the LVL beam. The authors present a method of testing the double-cut wood connection performance under compression, the angle of forces application in regard to the fibers being 45°. There has been carried out an experimental study of the connection with claw washers in the LVL beam for a 45°angle between the direction of the fibers in the outer and middle elements under the action compressive load. The testing results allow assessing the deformation value in the connection and the experimental values of the linear stiffness coefficient. There has been implemented an evaluation of results, and the nature of the connection`s fracture is considered.

Vibration energy harvesting using magnetic spring based nonlinear oscillators: Design strategies and insights

The rising technologies of wearable electronics resulted in urgent demand for developing eco-friendly power sources that can utilize free energies around us including ambient vibrations. Magnetic springs are amongst the most common techniques used to build vibration energy harvesting systems that convert vibrational energy into useful electric power. Here, we present an experimental and theoretical platform for design guidelines and analysis of magnetic springs encountered in vibration energy harvesting systems. An energy harvester prototype consisting of an oscillating solid magnet levitated between two stationary ring magnets is constructed and used for experimental evaluation. Results show excellent agreement between model and experiment. The use of the analytical force model to represent magnetic force nonlinearities is essential at high accelerations. While the magnetic damping coefficient varies during dynamic operation and is dependent on the position of the levitated magnet, it is shown that approximating this coefficient as a constant provides accurate prediction of the dynamic behavior of the system. Approximate analytical expressions for linear and nonlinear stiffness coefficients are obtained. Results suggest that linear and nonlinear stiffness coefficients are coupled. The outer diameter of the stationary ring magnet can be used to tune the nonlinearity of the energy harvesting system to obtain linear, hardening nonlinear, or softening nonlinear response. This work serves as a tool for designers to understand the behavior of magnetic spring based harvesting systems and evaluate their performance in light of their design parameters. This work also can serve other energy systems that utilize magnetic springs including energy sinks.

The influence of two kinds of time delays on the vibrational resonance of a fractional Mathieu–Duffing oscillator

Vibrational resonance is studied in a fractional Mathieu–Duffing oscillator with two types of time delays: fixed and distributed delays. The theoretical expression of the response amplitude is obtained by utilising the method of direct partition of slow and fast motions. Relative errors between the theoretical prediction and the numerical simulation are introduced to verify the validity of analytical approaches. The relative error of the displacement and the relative error of the response amplitude are calculated. Small relative errors show that the theoretical analysis is statistically correct. Therefore, the effects of fractional order, linear stiffness coefficient, low-frequency signal, time delay intensity and damping coefficient on the Mathieu–Duffing oscillator with distributed delay are studied successively. In order to better illustrate the impact of distributed time delay on the model, the case of fixed time delay is analysed and compared, and it can be found that the distributed delay has more significant influence than fixed delay on the system. In addition, the influence of distributed delay on the system is more significant than that of the fixed delay.

On the steady-state forward and backward whirling motion of asymmetric nonlinear rotor system

Nonlinear lateral vibrations and the corresponding whirling motions of asymmetric horizontally supported rotor system are investigated within this article. The rotating shaft is modeled as a two-degree-of-freedom nonlinear Jeffcott-rotor system. The nonlinear restoring force of the rotating shaft, asymmetry in both the linear and nonlinear stiffness coefficients, the disk weight, the disk eccentricity, and the eccentricity orientation angle are included in the studied model. The asymptotic analysis is utilized to derive the autonomous amplitude-phase modulating equations that govern the system lateral vibrations in the horizontal and vertical directions. Bifurcation diagrams of both the system vibration amplitudes and the corresponding phase angles are obtained. The main acquired results revealed that the existence of the asymmetry in the rotating shaft stiffness coefficients widens the spinning speed interval at which the system may have more than one stable solution. In addition, increasing the linear asymmetrical stiffness coefficient can eliminate the system backward whirling motion. Finally, numerical confirmations for the obtained analytical results are performed that illustrated their accuracy in the prediction of the system vibration amplitudes and the whirling direction whether forward, backward, or along a straight line.

МАТРИЦЯ ЖОРСТКОСТІ СТРИЖНЕВОГО СКІНЧЕННОГО ЕЛЕМЕНТА НА ПРУЖНІЙ ОСНОВІ З НЕЛІНІЙНОЮ ЖОРСТКІСТЮ УЗДОВЖ ЕЛЕМЕНТА

Розглянута задача моделювання підшипника суднового валопровода з неметалевим вкладишем як пружної основи методом скінченних елементів. Виконано аналіз існуючих матриць жорсткості стрижневих скінченних елементів на пружній основі. Розроблено матрицю жорсткості елемента на пружній основі Вінклеровського типу з параболічним законом зміни коефіцієнта жорсткості за довжиною елемента. Показана ефективність розробленої матриці порівняно з матрицями, отриманими з використанням постійного і лінійного розподілу жорсткості. Подані рекомендації щодо вибору кількості елементів, необхідних для адекватного моделювання неметалевих вкладишів підшипників суднового валопровода.

A single-spring model predicts the majority of variance in impact force during a fall onto the outstretched hand

The objective of this study was to validate a single-spring model in predicting measured impact forces during an outstretched arm falling scenario. Using an integrated force plate, impact forces were assessed from 10 young adults (5 males; 5 females), falling from planted knees onto outstretched arms, from a random order of drop heights: 3, 5, 7, 10, 15, 20, and 25 cm. A single-spring model incorporating body mass, drop height plus the estimated linear stiffness of the upper extremity (hand, wrist and arm) was used to predict impact force on the hand. We used an analysis of variance linearity test to test the validity of using a linear stiffness coefficient in the model. We used linear regression to assess variance (R2) in experimental impact force predicted by the single-spring model. We derived optimum linear stiffness coefficients for male, female and sex-combined. Our results indicated that the association between experimental and predicted impact forces was linear (P < 0.05). Explain variance in experimental impact force was R2 = 0.82 for sex-combined, R2 = 0.88 for males and R2 = 0.84 for females. Optimum stiffness coefficients were 7436 N/m for sex-combined, 8989 N/m for males and 4527 N/m for females. In conclusion, a linear spring coefficient used in the single-spring model proved valid for predicting impact forces from fall heights up to 25 cm. Results also suggest the use of sex-specific spring coefficients when estimating impact force using the single-spring model. This model may improve impact force to bone strength ratios (factor-of-risk) and prediction of forearm and wrist fracture.

On vibration behavior and motion bifurcation of a nonlinear asymmetric rotating shaft

This article investigates the nonlinear vibrations of asymmetric vertically supported Jeffcott rotor system. Asymmetry in both linear and nonlinear stiffness coefficients of the rotating shaft is considered. The disk eccentricity and its orientation angle are included in the system model. Asymptotic analysis is sought to obtain an analytical approximate solution for the considered system model in the primary resonance case. Bifurcation diagrams for the different system parameters are obtained to explore the system steady-state lateral vibrations. The main acquired results revealed that (1) the symmetric system can oscillate by one of three stable forward whirling amplitudes at the same rotational speed depending on the initial position of the rotating disk. (2) Asymmetry in the linear stiffness coefficient does not affect the symmetry of the whirling motion, but it may change the system natural frequency. (3) Asymmetry in the nonlinear stiffness coefficient is responsible for both asymmetrical and backward whirling motions. All obtained analytical results have been verified via solving the system original equations numerically, where the analytical and numerical results are in excellent agreement.

Changes in surge and pitch decay periods of floating wind turbines for varying wind speed

Recently published experimental results show variations in surge and pitch decay periods of floating wind turbines (FWTs) subjected to different incident wind velocities. This paper explores the external loads acting on a FWT, with special attention to nonlinearities which affect its low-frequency global motions. The period variations in surge and pitch are found to have different sources. In surge, the mooring system nonlinearities dominate, while for pitch the relative phase between the nacelle velocity and the thrust induced in the rotor leads to an “apparent inertia/damping” effect. Simplified 2-DOF models using linearized stiffness coefficients for surge and modified inertia and damping matrices for pitch are developed. Comparisons with state-of-the-art aero-hydro-servo-elastic time-domain simulations show excellent agreement for three distinct catenary-moored FWT designs.

Parameter-splitting perturbation method for the improved solutions to strongly nonlinear systems

A procedure named parameter-splitting perturbation method for improving the perturbation solutions to the forced vibrations of strongly nonlinear oscillators is proposed. The idea of the proposed procedure is presented in general first. After that, it is applied to optimize the solutions obtained by the multiple-scales method which is one of well-known perturbation methods. The harmonically forced Duffing oscillator, the harmonically forced oscillator with both nonlinear restoring force and nonlinear inertial force, the harmonically forced purely nonlinear oscillator and harmonically forced two-degree-of-freedom system with cubic nonlinearity are analyzed in various cases to show the advantages of the proposed method. The ratio of nonlinear stiffness coefficient to linear stiffness coefficient is chosen to be larger than one to highlight the validity of the propose method when dealing with strongly nonlinear oscillators. The validity of the proposed procedure is examined by comparing the frequency–response curves obtained by the proposed method, conventional multiple-scales method and numerical continuation method. Moreover, the errors corresponding to the results obtained by multiple-scales method are compared with those obtained by the proposed method to examine the performance of the proposed method. The results show that the proposed method can give much improved solutions in comparison with those obtained by multiple-scales method.

Mechanics of Fluid-Conveying Microtubes: Coupled Buckling and Post-Buckling

This paper investigates the coupled mechanics of a fluid-conveying microtube embedded inside an elastic medium and subject to a pretension. The fluid-structure interaction model of the microsystem is developed based on Lagrange’s equations for the open system of a clamped-clamped microtube. A continuation model is used to examine the nonlinear mechanics of this microsystem prior to and beyond losing stability; the growth and the response in the supercritical regime is analysed. It is shown that the microtube stays stable prior to losing stability at the so-called critical flow velocity; beyond that point, the amplitude of the buckled microsystem grows with the velocity of the flowing fluid. The effects of different system parameters such as the linear and nonlinear stiffness coefficients of the elastic medium as well as the length-scale parameter and the slenderness ratio of the microtube on the critical speeds and the post-buckling behaviour are analysed.