Nonlinear Spring Force Research Articles

Tailored nonlinear stiffness and geometric damping: Applied to a bistable vibration absorber

A novel device is proposed to obtain arbitrary restoring force characteristics and nonlinear geometric damping. It consists of linear springs and dampers that are compressed along a track. The obtained nonlinear damping and stiffness law depend on the track’s shape. The device is then applied to obtain a nonlinear energy sink (NES) to damp the vibrations of a host system. Both the case of transient vibrations, induced by shock loads, and sustained vibrations, induced by harmonic loads, are studied. For arbitrary stiffness and damping, slow flow dynamics and slow invariant manifolds (SIMs) are derived by applying harmonic balancing and multiple timescales techniques. Under transient vibrations, two performance measures are derived from the SIM, the relative residual energy and the pumping time, and are found for generic nonlinear spring force and nonlinear geometric damping. For sustained vibrations, a load-dependent frequency response (FR) is derived from the SIMs. This FR predicts the occurrence of the efficient strongly modulated responses but also the occurrence of the unfavorable detached responses called isolas, where the NES fails to mitigate and even amplifies vibrations.This research investigates the performance of a conventional cubic NES and a bistable NES (BNES) with nonlinear damping obtained from the proposed device. Especially the BNES with nonlinear damping shows attractive properties, with a high degree of robustness over a wide energy range and a low vibration threshold under transient loading. Under harmonic loads, the nonlinear damping helps reduce the effect of isolas. A novel tuning methodology is proposed to avoid isolas for a range of load magnitudes. The new device opens up a whole range of possibilities regarding energy dissipation through nonlinear damping.

Anti‐Swing control of the Pendubot using damper and spring with positive or negative stiffness

AbstractIn this article, we study the anti‐swing control of the Pendubot which is a 2‐link planar robot with a single actuator driving the first joint, whose control objective is to stabilize the Pendubot to the downward equilibrium point (DEP) with the two links in the downward position for all its initial states with the exception of a set of Lebesgue measure zero. To achieve this control objective, with respect to the angle of the first joint, we present a proportional‐derivative (PD) controller using a damper and a linear spring with positive stiffness, and a sinusoidal‐derivative (SD) controller using a damper and a nonlinear spring (force being sinusoidal function of the displacement) allowing even negative stiffness. We prove that the control objective is achieved if some conditions on the stiffness of linear or nonlinear spring are satisfied. We present analytical solutions to the optimal design of the derivative gain and the stiffness which minimizes the real parts of the dominant poles of the linearized model of the corresponding closed‐loop systems around the DEP for all Pendubots in terms of their five physical parameters without any constraint. Our discussion shows that the SD controller is superior to the PD controller. We provide simulation results for a Pendubot to show that the presented SD controller can stabilize the Pendubot to the DEP faster than the presented PD controller.

ANALISIS STRUKTUR MOORING DOLPHIN KAPASITAS KAPAL 2000 GT (STUDI KASUS PELABUHAN MUNSE SULAWESI TENGGARA)

Sea transportation is the main transportation used by residents in the islands. The development of marine transportation facilities is expensive to support these activities. Therefore, several alternatives are needed to streamline construction costs, one of which is the construction of a pier using a mooring dolphin. The purpose of this study is to calculate the structural strength of the mooring dolphin with a ship capacity of 2000 GT at Munse Port, Southeast Sulawesi Province. Structural analysis is carried out by analyzing pile capacity and joint displacement analysis. The calculation of the strength of the pile elements at the pier was analyzed using the SAP 2000 program. For soil which is modeled as an elastic support, the ability to support the load depends on the magnitude of the modulus of subgrade reaction from the soil. Embedded pile modeling is modeled with a nonlinear spring force. The results of the analysis by analyzing the capacity of piles with dimensions of 508 mm with a thickness of 12 mm resulted in a capacity ratio of 0.72. The results of the analysis of joint displacement in service or operational conditions are 37.09 mm and in earthquake conditions, they are 13.01 mm.

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.

Vibration Characteristics of Rolling Mill System under Constraints of the Nonlinear Spring Force and Friction Force from Hydraulic Cylinder

Considering the two kinds of nonlinear constraints of rolling mill hydraulic cylinder, spring force and friction force, the vibration model of rolling mill system is established. The amplitude frequency response equations are obtained by using the average method. Comparing the time history curves of vertical vibration displacement of rolling mill system under the nonlinear spring force and friction force, the amplitude frequency characteristic curves are simulated. The external excitation amplitude is viewed as the bifurcation parameter, and the system bifurcation response changing with the external excitation amplitude is analyzed. The influence of the external excitation amplitude on the system stability is studied. The results indicate that the increase of the nonlinear spring force makes the rolling mill system’s unstable area to become wider, and the influence on the rolling mill system of nonlinear friction force behaves as the damping characteristics; the vibration of rolling mill system is alternating between the periodic, period-doubling, and the chaotic motion. The research results provide a theoretical support for restraining the vibration of the rolling mill system in the actual production process.

Analysis of Stabilized Crank-Nicolson Time-Stepping Scheme for the Evolutionary Peterlin Viscoelastic Model

The Peterlin viscoelastic model describes the motion of certain incompressible polymeric fluids. It employs a nonlinear dumbbell model with a nonlinear spring force law making it more nonlinear than other viscoelastic models. In this paper, we propose and study a fully implicit stabilized Crank-Nicolson time stepping scheme for finite element spatial discretization of the non-stationary Peterlin viscoelastic fluid model with non-homogeneous boundary conditions. The proposed scheme adds a suitable stabilizing term to improve the structural and stability properties of the scheme. We prove that the scheme is almost unconditionally stable, i.e., stable when the time step is less than or equal to a constant. Further, with the help of the a priori error bounds of the Stokes and Ritz projections, optimal error estimates for the velocity, the conformation tensor and the pressure are presented in suitable norms. Numerical examples are presented that illustrate the accuracy and stability of the scheme.

Coupled Vibration Behavior of Hot Rolling Mill Rolls under Multinonlinear Effects

A coupled vibration model of hot rolling mill rolls under multiple nonlinear effects is established by considering the nonlinear spring force produced by the hydraulic cylinder, the nonlinear friction between the work rolls, the dynamic variation of rolling force, and the effect of external excitation as well as according to the structural constraints of a four-high hot rolling mill in the vertical and horizontal directions. The amplitude-frequency response equation of rolling mill rolls is determined by using a multiple-scale approximation method. Furthermore, use of actual data for simulation indicates that the internal resonance is the main cause of coupling vibration of the rolling mill rolls. In addition, changes in the movement displacement of the hydraulic cylinder and the coupling parameters strongly affect the coupling system of the rolling mill rolls. Finally, the study of the dynamic bifurcation characteristics of the rolling mill rolls indicates that, with varying external excitation amplitude, the vibration of rolls alternates between periodic motion, period-doubling motion, and chaotic motion in both vertical and horizontal directions. This is one of the reasons for the appearance of periodic light and dark stripes on the strip surface. Furthermore, the range of the external excitation amplitude (F0) at which the rolling mill roll system vibrates violently, that is, 5.68e5 N < F0 < 5.84e5 N and F0 > 6.12e5 N, must be avoided. The research results can provide a theoretical reference for further exploration of the coupling vibration mechanism of hot rolling mills.

Periodic, aperiodic and chaotic motions of harmonically excited SDOF and MDOF nonlinear dynamical systems

Responses of nonlinear dynamical systems with single degree of freedom (SDOF) or multiple degrees of freedom (MDOF) to periodic excitations are investigated in this paper using a numerical scheme. The scheme is developed on the basis of the effective mass, damping and stiffness matrices, the incremental generalized coordinates and the Newmark integration method to solve efficiently and accurately second-order nonlinear ordinary differential equations in the hundreds or more. Using the proposed numerical method, long-term behavior of SDOF and MDOF systems of any type of nonlinearities including the well-known van der Pol nonlinear damping forces, the Duffing type nonlinear spring forces and the time-delayed spring force at any strength level (weak, moderate and strong) can be accurately determined. With the help of an adequate mapping frequency, orders of periodicity of periodic responses can be easily and reliably identified. Numerical results, obtained for three oscillators – an SDOF van der Pol oscillator, a time-delayed SDOF Duffing oscillator, and a five-DOF Duffing oscillator, demonstrate that the proposed scheme is ideally suited for solving large scale nonlinear dynamical problems.

A Dimensionless Parameter Analysis of a Cylindrical Tube Electromagnetic Vibration Energy Harvester and Its Oscillator Nonlinearity Effect

This paper investigates a single degree of freedom oscillator in a cylindrical tube vibration energy harvester which is applicable in a low-frequency range. A duffing-type nonlinear dynamic differential equation of the oscillator was developed for the nonlinear analysis and solved by using the harmonic balance method in order to widen energy harvesting frequency bandwidth. The exploitation of nonlinear spring force interactions of the oscillator to enhance the generator’s performance is also presented in this paper. The dimensionless harvested power formula of the nonlinear mass-spring-damper system was derived, and a parameter study was conducted for the design optimization of the harvester. The main contribution of this paper is to establish a dimensionless performance analysis method of a nonlinear electromagnetic vibration energy harvester system and to disclose the effects of the parameters and nonlinearity of the system on the harvesting performance.

Non-linear limit cycle flutter of a plate with Hertzian contact in axial flow

This paper is aimed at the nonlinear flutter of a cantilevered plate with Hertzian contact in axial flow. The contact effect is modeled as a nonlinear spring force with both square and cubic nonlinearity. The fluid force is considered as the sum of two parts, one is the reactive fluid force due to plate motion and the other is the resistive fluid force independent on plate motion. The reactive fluid force is derived by solving the bound and wake vorticity with the help of Glauert’s expansions, and the resistive force is evaluated in terms of drag coefficient. The governing nonlinear partial differential equation of the system is discretized in space and time domains by using the Galerkin method. Results show that the plate loses its stability by flutter and then undergoes limit cycle motions due to the contact nonlinearity after instability. The present fluid model is reliable and shows good agreement with other theories archived. A heuristic analysis scheme based on the equivalent linearization method is developed for the analysis of bifurcations and limit cycles. The Hopf bifurcation is either supercritical or subcritical, which is closely dependent on the contact location. For some special cases the bifurcations are, interestingly, both supercritical and subcritical. When the plate experiences limit cycles, with the increasing dynamic pressure there firstly appear the lock-in motions; and then the quasi-periodic motions show up as a breaking of limit cycle by inclusion of a secondary significant frequency with an irrational value of 14π of the dominant limit cycle frequency. Finally the plate undergoes dynamic buckling characterized by quasi-periodic divergence when the dynamic pressure is relatively large.

A Novel Nonlinear Piezoelectric Energy Harvesting System Based on Linear-Element Coupling: Design, Modeling and Dynamic Analysis.

This paper presents a novel nonlinear piezoelectric energy harvesting system which consists of linear piezoelectric energy harvesters connected by linear springs. In principle, the presented nonlinear system can improve broadband energy harvesting efficiency where magnets are forbidden. The linear spring inevitably produces the nonlinear spring force on the connected harvesters, because of the geometrical relationship and the time-varying relative displacement between two adjacent harvesters. Therefore, the presented nonlinear system has strong nonlinear characteristics. A theoretical model of the presented nonlinear system is deduced, based on Euler-Bernoulli beam theory, Kirchhoff’s law, piezoelectric theory and the relevant geometrical relationship. The energy harvesting enhancement of the presented nonlinear system (when n = 2, 3) is numerically verified by comparing with its linear counterparts. In the case study, the output power area of the presented nonlinear system with two and three energy harvesters is 268.8% and 339.8% of their linear counterparts, respectively. In addition, the nonlinear dynamic response characteristics are analyzed via bifurcation diagrams, Poincare maps of the phase trajectory, and the spectrum of the output voltage.

Response of a Harmonically Forced Dry Friction Damped System Under Time-Delayed State Feedback

Nonlinear dynamics, control, and stability analysis of dry friction damped system under state feedback control with time delay are investigated. The dry friction damped system is harmonically excited, and the nonlinearities in the equation of motion arise due to nonlinear damping and spring force. In this paper, a frequency domain-based method, viz., incremental harmonic balance method along with arc-length continuation technique (IHBC) is first employed to identify the primary responses which may be present in such system. The IHBC is then reformulated in a manner to treat the dry friction damped system under state feedback control with time delay and is applied to obtain control of responses in an efficient and systematic way. The stability of uncontrolled responses is obtained by Floquet's theory using Hsu' scheme, and the stability of the controlled responses is obtained by applying a semidiscretization method for delay differential equation (DDE).

Nonlinear dynamics characteristics and influence factors analysis of servo hydraulic cylinder

Aiming at problems of vibration, impact, and noise, the nonlinear dynamic characteristics of servo hydraulic cylinder are studied by using combination of theoretical derivation and simulation experiment methods. The nonlinear effect rules of spring force and friction force are analyzed based on nonlinear dynamic theory; it can be derived that the coupling effect rules of nonlinear spring force and friction force can be described by Duffing–Van Der Pol equation. The dynamic simulation model of servo hydraulic cylinder is established by using SimHydraulics software. The numerical simulation and experiment results showed that nonlinear vibration of servo hydraulic cylinder is mainly caused by exciting force, spring force, and friction force; the system has different motion states when the parameters take different values, which lays a good foundation for the optimal designs and precise control of hydraulic cylinder.

Magnetically levitated autoparametric broadband vibration energy harvesting

Some of the lingering challenges within the current paradigm of vibration energy harvesting (VEH) involve narrow operational frequency range and the inevitable non-resonant response from broadband noise excitations. Such VEHs are only suitable for limited applications with fixed sinusoidal vibration, and fail to capture a large spectrum of the real world vibration. Various arraying designs, frequency tuning schemes and nonlinear vibratory approaches have only yielded modest enhancements. To fundamentally address this, the paper proposes and explores the potentials in using highly nonlinear magnetic spring force to activate an autoparametric oscillator, in order to realize an inherently broadband resonant system. Analytical and numerical modelling illustrate that high spring nonlinearity derived from magnetic levitation helps to promote the 2:1 internal frequency matching required to activate parametric resonance. At the right internal parameters, the resulting system can intrinsically exhibit semi-resonant response regardless of the bandwidth of the input vibration, including broadband white noise excitation.

A physical model of a highly nonlinear string and its use in the music composition Quartet for Strings

A real-time virtual string model is created that supports two modes of haptic force-feedback interaction. Using one haptic controller, the string can be plucked, and using a second haptic controller, the virtual string's pitch can be continuously varied by applying pressure to the string. Via both controllers, the vibrations of the virtual string can be felt while it is played. These functionalities are achieved by modeling the string as a finite sequence of interleaved masses and stiffening springs. For artistic purposes, a nonlinear spring force characteristic function is selected that limits the effective stiffness of each spring to a range between k and (k + ks). This function is F(x) = kx + (ksx3)/(β2+x> + x2), where x is the displacement of the nonlinear spring and β approximately adjusts the displacement at which the stiffness becomes significantly nonlinear. For small displacements, the string's undamped fundamental frequency is tuned by k, and for sufficiently large displacements, the string's un...

ADAPTIVE SPARSE GALERKIN METHODS FOR VIBRATING CONTINUOUS STRUCTURES

Adaptive reduced-order methods are explored for simulating continuous vibrating structures. The Galerkin method is used to convert the governing partial differential equation (PDE) into a finite-dimensional system of ordinary differential equations (ODEs) whose solution approximates that of the original PDE. Sparse projections of the approximate ODE solution are then found at each integration time step by applying either the least absolute shrinkage and selection operator (lasso) or the optimal subset selection method. We apply the two projection schemes to the simulation of a vibrating Euler-Bernoulli beam subjected to nonlinear unilateral and bilateral spring forces. The subset selection approach is found to be superior for this application, as it generates a solution with similar sparsity but substantially lower error than the lasso.

Nonlinear Vibration Mechanism of Electro-hydraulic Servo System and Its Experimental Verification

The electro-hydraulic servo system(EHSS) is a nonlinear dynamics system. When it is in the work, the anomalies are easy to happen, such as nonlinear vibration, noise, impact, crawl and so on. These anomalies seriously affect the stability of EHSS. Miserably, the incentives are not easy to determine. Based on the principle of the nonlinear dynamics, the nonlinear dynamical model of EHSS is established. Then, the influence laws of nonlinear hydraulic spring force(NHSF) and nonlinear friction(NF) etc on dynamic characteristics of EHSS are explored through theoretical study. Two conclusions are put forward. The effect of NHSF can be described by Duffing equation, and the effect of NF can be described by Van Der Pol equation. Moreover, the measured signals are deeply analyzed by means of the methods, such as time-course, stroboscopic sampling, power spectrum and so on. Then, the mechanism and incentive of nonlinear vibration of EHSS are revealed. Results indicated that the nonlinear vibration of the EHSS is mainly due to the interaction of the "jump phenomenon" caused by NHSF and the limit cycle oscillation caused by NF.

Fluctuations in the coil-stretch transition of flexible polymers in good solvents: A peak due to nonlinear force relation

Long flexible polymers undergo a coil to stretch transition (CST) in an elongational flow. Near the CST, a peak can be observed in the fluctuations of the size of a molecule (|R|). Solvent effects on the fluctuations are studied using Brownian dynamics simulations of a nonlinear spring force relation that can represent real molecules. Ignoring the influence of hydrodynamic interactions, a linear region in the spring force relation is known to cause the peak in |R| fluctuations. In contrast, we find that a peak in the fluctuations can be obtained even for the nonlinear spring force relation. We analyze the influence of hydrodynamic interactions on the fluctuations using a dumbbell model with a conformation-dependent drag coefficient.

미지 환경 탐색 및 감시를 위한 다개체 로봇의 경로계획

This paper presents a multi-robot path planner for environment exploration and monitoring. Robotics systems are being widely used as data measurement tools, especially in dangerous environment. For large scale environment monitoring, multiple robots are required in order to save time. The path planner should not only consider the collision avoidance but efficient coordination of robots for optimal measurements. Nonlinear spring force based planning algorithm is integrated with the spatial gradient following path planner. Perturbation/Correlation based estimation of spatial gradient is applied. An algorithm of tuning the stiffness for robot coordination is presented. The performance of the proposed algorithm is discussed with simulation results.

427 個別要素法を用いた粒状体ダンパの減衰予測

In this paper, a numerical modeling method for particle damping is presented. The numerical modeling is executed by using analytical results for the motion of the granular materials in the damper cavity under sinusoidal excitation. The damping effect for particle damping is modeled as equivalent additional mass, nonlinear spring force and viscous damping force by means of the discrete element method. The validity of the method is examined by a comparison of the experimental results. Analytical solutions are provided to estimate the response of the primary system under sinusoidal excitation.