Nonlinear Damping Forces Research Articles

Nonlinear dynamics of diamagnetically levitating resonators

The ultimate isolation offered by levitation provides new opportunities for studying fundamental science and realizing ultra-sensitive floating sensors. Among different levitation schemes, diamagnetic levitation is attractive because it allows stable levitation at room temperature without a continuous power supply. While the dynamics of diamagnetically levitating objects in the linear regime are well studied, their nonlinear dynamics have received little attention. Here, we experimentally and theoretically study the nonlinear dynamic response of graphite resonators that levitate in permanent magnetic traps. By large amplitude actuation, we drive the resonators into nonlinear regime and measure their motion using laser Doppler interferometry. Unlike other magnetic levitation systems, here we observe a resonance frequency reduction with amplitude in a diamagnetic levitation system that we attribute to the softening effect of the magnetic force. We then analyze the asymmetric magnetic potential and construct a model that captures the experimental nonlinear dynamic behavior over a wide range of excitation forces. We also investigate the linearity of the damping forces on the levitating resonator, and show that although eddy current damping remains linear over a large range, gas damping opens a route for tuning nonlinear damping forces via the squeeze-film effect.

Innovative nonlinear vibration control of beam structures using shear thickening fluid dampers

This paper investigates the use of shear thickening fluid (STF) dampers for reducing vibrations in beam-based structures. Designing and analyzing these systems is challenging because STF’s viscosity changes as the fluid’s vibration frequency changes. To understand the impact of key design parameters, the behavior of STF dampers in various configurations of supports is analyzed. The governing equations are derived, considering the STF dampers as nonlinear damping forces, and solved/validated using Galerkin/Hencky’s discretization methods. An energy reduction factor is introduced to compare the performance of different configurations in the frequency domain to obtain the optimal design. The results show that although the performance of STF dampers is highly dependent on the excitation frequency (with high efficiency in reducing vibrations near the resonance frequencies), the STF dampers fail to dissipate vibrations in some frequency regions. Surprising ranges of dynamic behavior, from Quasi-periodic to chaotic, are detected and analyzed. The findings also reveal that by analyzing only the shape of force–velocity curves of the STF damper, one can accurately predict the system’s performance, without the need for a full system analysis.

On a thermo‐visco‐elastic model with nonlinear damping forces and L1 temperature data

The aim of this paper is to prove the existence of weak solutions to thermo‐visco‐elastic system of equations. In considered problem, the temperature term is included in the elastic constitutive equations (generalized Hooke's law) as well as in describing the evolution of visco‐elastic strain. The main idea of presented proof is to analyze reformulated model, in which visco‐elastic strain (internal variable) is eliminated. For such system of equations, we propose a two‐level Galerkin approximation, which allows us to show a non‐negativity of temperature during the entire deformation process. In order to characterize the weak limits in the nonlinearities occurring in the system at different levels of approximation, the following tools are used: Truncation methods for heat equation proposed in the paper of D. Blanchard [Nonlinear Analysis, 21(10):725‐743, 1993], modified Boccardo‐Gallouët's approach, and Minty‐Browder trick.

Nano- and microparticle Nonlinear Damping Identification in quadrupole trap

We propose a Nonlinear Damping Identification method for charged nano- and microparticles in a linear quadrupole Paul trap under nonlinear damping forces. We analytically and numerically evaluate conditions for the formation of plane limit cycle trajectories, i.e. “extended orbits” in spherical particle approximation. We introduce numerical metrics for such orbits and define a functional relation between the metrics and physical parameters of a trapped particle: size, mass, and charge. The method proposed allows non-destructive, simultaneous and separate determination of the trapped particle parameters in a wide range of sizes.

Experimental parameters identification of a flexible rotor system equipped with smart magneto-rheological bearing

Magnetorheological (MR) fluid is one of the most effective smart fluids with controllable rheological and viscoelastic properties that is used in semi-active vibration control of systems and structures. MR squeeze-film dampers (MRSFDs) are categorized among smart bearings with variable stiffness and damping properties, which enable vibration mitigation of rotary systems. In this aspect, precise modeling of the rotor systems equipped with MRSFDs and identifying their parameters are of paramount importance for analyzing the vibration behavior of the system and developing control strategies. The present study aims to present the analytical tools and algorithms for identifying the parameters of flexible rotor systems equipped with MRSFDs. For this purpose, a rotor-bearing system test setup including a flexible rotor, two ball bearings, an MRSFD, and an unbalanced disk is designed and fabricated, and its governing equations of motions are derived using finite element (FE) methods. The parameters of the flexible rotor system, as well as the stiffness and damping coefficients of the bearings and nonlinear damping forces of the MRSFD, are then identified using the developed algorithm and the harmonic responses of the system as its input. The procedure is then validated by comparing the results of the FE model with identified parameters with those measured experimentally. It is shown that the developed algorithm and FE model are capable of accurately identifying the parameters of the flexible rotor-bearing system equipped with an MRSFD.

VIV Control Strategies Using Displacement-Based ‎Phenomenological Model

Linear and nonlinear feedback control of vortex-induced vibrations are assessed using a single degree-of-freedom phenomenological model of the uncontrolled response. The model is based on the role of linear and nonlinear damping forces in inducing and limiting the amplitude of these vibrations. First, the model prediction is validated using data from previously published high-fidelity direct numerical simulations. Then, linear and nonlinear control are applied to the validated model over a broad range of gain values. The predicted controlled responses are also validated against previously published results from high-fidelity numerical simulations. Based on this validation, it is shown that the single degree-of-freedom model is an effective alternative, in terms of computational cost, to high fidelity simulations in assessing control strategies over broad regions of control gains.

Experimental and numerical study of a two-body heaving wave energy converter with different power take-off models

Wave energy is one of the most difficult energies to be captured among marine renewables. With the technical progress, wave energy converters (WECs) are being tested in relatively deeper waters, which makes floating concepts almost the only choice. In this paper, a two-body heaving WEC where the wave energy is absorbed through the relative motion between the outer annular and the inner cylindrical buoys is studied. Both experimental and numerical studies are adopted for regular wave conditions. In the physical model test, a hydraulic system is used to achieve constant power take-off (PTO) damping force. Numerical simulations, validated against experimental data, are applied using both the frequency domain and the time domains analyses. Different types of PTOs, including constant, linear and nonlinear damping forces, are undertaken to evaluate the hydrodynamic and power absorption performance of such device.

Direct identification of nonlinear damping: application to a magnetic damped system

In the identification of mechanical systems through inverse receptance methods, a linear model is usually assumed, at least on a first approximation. However, most of the systems have a degree of nonlinearity in their elastic or dissipative properties. Usually, stiffness nonlinearities are identified, while damping nonlinearities are neglected or even not detected. Indeed, an equivalent linear damping model, correctly representing the nonlinear dissipation of the system under testing condition, can always be found, but it is test-specific and not generally valid.This paper addresses the identification of dissipative models for multi degrees of freedom mechanical systems with a single damping nonlinearity. Based on the direct damping matrix identification through inverse receptance methods, this research proposes an extension of the Stabilised Layers Method, valid for linear systems, to damping nonlinearities, resulting in the identification of the test-independent linear and nonlinear damping matrices of the system. The proposed method is theoretically derived for the identification of nonlinear damping forces depending on powers of displacement and velocity. The sinusoidal input describing function approximation of the nonlinear damping is exploited to identify the coefficients of the nonlinear damping force from the system receptances, measured under sinusoidal sweeps at constant amplitudes of oscillation across the nonlinearity.The presented method is applied to identify the linear and nonlinear damping matrices of a multi degrees of freedom system with a localised nonlinear magnetic damper. The coefficients of the nonlinear magnetic damping force are identified for two configurations of the magnetic damper.The proposed approach is a parametric method to identify the nonlinear damping models of mechanical systems using standard experimental techniques usually adopted for linear systems. The identified model is valid under general excitation forces, predicting the system behaviour in a broader range of operation than the test-specific linear equivalent model.

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.

Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control

The conditions for existence of solutions and stability, asymptotic and exponential, of a large class of boundary controlled systems on a 1D spatial domain subject to nonlinear dynamic boundary actuation are given. The consideration of such class of control systems is motivated by the use of actuators and sensors with nonlinear behavior in many engineering applications. These nonlinearities are usually associated to large deformations or the use of smart materials such as piezo actuators and memory shape alloys. Including them in the controller model results in passive dynamic controllers with nonlinear potential energy function and/or nonlinear damping forces. First it is shown that under very natural assumptions the solutions of the partial differential equation with the nonlinear dynamic boundary conditions exist globally. Secondly, when energy dissipation is present in the controller, then it globally asymptotically stabilizes the partial differential equation. Finally, it is shown that assuming some additional conditions on the interconnection and on the passivity properties of the controller (consistent with physical applications) global exponential stability of the closed-loop system is achieved.

Nonlinear response of flapping beams to resonant excitations under nonlinear damping

The effect of excitation and damping parameters on the superharmonic and primary resonance responses of a slender cantilever beam undergoing flapping motion is investigated. The problem is cast into mathematical form using a nonlinear inextensible beam model which is subjected to time-dependent boundary conditions and linear or nonlinear damping forces. The flapping excitation is assumed to be non-harmonic, composed of two sine waves with different amplitudes. We employ a combination of Galerkin and perturbation methods to arrive at the frequency–response relationships associated with the second- and third-order superharmonic and primary resonances. The resonance solutions of the spatially discretized equation of motion, which involves both quadratic and cubic nonlinear terms, are constructed as first-order uniform asymptotic expansions via the method of multiple timescales. The effect of excitation and damping parameters on the steady-state resonance responses and their stability is described quantitatively using approximate analytical expressions. The critical excitation amplitudes leading to bistable solutions are identified. For the second-order superharmonic resonance, the critical excitation amplitude is determined to be dependent on the first-harmonic amplitude in the case of nonlinear damping. The third-order superharmonic resonance is determined to be independent of the second-harmonic excitation amplitude regardless of the damping types considered. The perturbation solutions are compared with numerical time-spectral solutions for different flapping amplitudes. The first-order perturbation solution is determined to be in very good agreement with the numerical solution up to 5° while above this amplitude differences in the two solutions develop, which are attributed to phase estimation accuracy.

Effect of non-linear damping on the structural dynamics of flapping beams

In this paper we investigate, through experiment and simulation, the effects of non-linear damping forces on the large amplitude structural dynamics of slender cantilever beams undergoing flapping motion in air. The aluminum beams are set into flapping motion through actuation at the beam base via a 4-bar crank-and-rocker mechanism. The beam strain response dynamics are investigated for two flapping amplitudes, 15° and 30°, and a range of flapping frequencies up to 1.3 times the first modal frequency. In addition to flapping at standard air pressure, flapping simulations and experiments are also performed at reduced air pressure (70% vacuum). In the simulations, linear and non-linear, internal and external damping force models in different functional forms are incorporated into a non-linear, inextensible beam theory. The external non-linear damping models are assumed to depend, parametrically, on ambient air density, beam width, and an empirically determined constant. Periodic solutions to the model equation are obtained numerically with a 1-mode Galerkin method and a high order time-spectral scheme. The effect of different damping forces on the stability of the computed periodic solutions is analyzed with the aid of Floquet theory. The strain-frequency response curves obtained with the various damping models suggest that, when compared to the linear viscous and non-linear internal damping models, the non-linear external damping models better represent the experimental damping forces in regions of primary and secondary resonances. In addition to providing improved correlation with experimental strain response amplitudes over the tested range of flapping frequencies, the non-linear (external) damping models yield stable periodic solutions for each flapping frequency which is consistent with the experimental observations. Changes in both the experimental ambient pressure and flapping amplitude are determined to result in some variation in the non-dimensional parameters associated with each of the non-linear external damping models. This result likely indicates an incomplete description of the model parameter dependence and/or non-linear functional form of the damping force.

First-Order Nonlinear Pseudo-Beat Vibration Model Identification of High Rise Buildings

With vibration amplitude increasing, the super high-rise building appears obvious pseudo-beat vibration under the combination of vortex-induced force and nonlinear damping force. Therefore, based on Scanlan nonlinear empirical model the first-order nonlinear pseudo-beat vibration responses of the super high-rise building are deduced with KBM method. And then the aerodynamic parameters and vortex-induced force can be identified with least square method. By using the aeroelastic model wind tunnel test of a certain super high-rise building, the wind-induced displacement vibration responses are gathered. By the least square method, each aerodynamic parameter is identified. It shows by analysis that the pseudo-beat vibration force is mainly offered by linear and nonlinear damping forces but secondary offered by linear stiffness force and periodic excitation force. The pseudo-beat vibration force calculated by KBM approximation has the predominant frequency like that of beat vibration response as well as high-order harmonic wave.

Asymptotic stability for anisotropic Kirchhoff systems

We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative anisotropic Kirchhoff systems, involving the p ( x ) -Laplacian operator, governed by time-dependent nonlinear damping forces and strongly nonlinear power-like variable potential energies. This problem had been considered earlier for potential energies which arise from restoring forces, whereas here we allow also the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. The results are further extended to the more delicate problem involving higher order damping terms.

A practical approach to parameter identification for a lightly damped, weakly nonlinear system

Fluidelastic systems are often lightly damped and exhibit weak nonlinear damping. The weak linear and nonlinear damping forces have a significant effect on the long-term behaviour of the system. However, parameter identification methods tend to concentrate on identifying the strongest forces. In this paper the applicability of two identification methods to a single degree of freedom fluidelastic system is investigated using experimental data. The system is weakly nonlinear under flow conditions and is prone to fluidelastic instability (i.e. self-excited limit cycle behaviour). The crucial effect of the weak damping forces on the system stability and post-stable behaviour has been demonstrated. The nonlinearity detection and identification has been done using an analytic representation of the displacement signal (FREEVIB method) and subsequently a nonlinear decrement method was used for comparison. The identified models were used to predict the global behaviour of the system in the form of limit cycle amplitudes and this has been used as an indication of accuracy. It was found that the nonlinear decrement method yields superior predictions, but it suffers from the limitation that the functional form of the system must be known a priori. Therefore, it is concluded that employing both methods together provides a more powerful approach for parameter estimation in a lightly damped system where weak nonlinear damping plays an important role.

ボールネジを用いた制震装置の開発 : その2 制震チューブの減衰性能とその評価法

Much more buildings with seismic dampers are designed and constructed after The Hyougoken-Nanbu earthquake. Some basic tests of RD series were issued preceding in previous paper. Concerning more heavy clamping RD-tube and long stroke RD-tube tests are made a report. RD series have non linear damping force. This is prospect of non-Newton viscous theory. Reduced damping coefficient is modeling from nonlinear damping force to linear damping, equivalent of input energy. Reduced damping coefficient model is extremely similar to nonlinear damping model in earthquake response analysis.

Local asymptotic stability for dissipative wave systems

We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative wave systems, governed by time-dependent nonlinear damping forces and by strongly nonlinear potential energies. This problem had been considered earlier for potential energies which arise from restoring forces, whereas here we allow as well for the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. The conclusions are related to and supplement earlier work of Payne and Sattinger [7], who treated the nondissipative case, and of Hale [1], who showed the existence of connected global attractors.

Unified Second-Order Stochastic Averaging Approach

First-order stochastic averaging has proven very useful in predicting the response statistics and stability of dynamic systems with nonlinear damping forces. However, the influence of system stiffness or inertia nonlinearities is lost during the averaging process. These nonlinearities can be recaptured only if one extends the stochastic averaging to second-order analysis. This paper presents a systematic and unified approach of second-order stochastic averaging based on the Stratonovich-Khasminskii limit theorem. Response statistics, stochastic stability, phase transition (known as noise-induced transition), and stabilization by multiplicative noise are examined in one treatment. A MACSYMA symbolic manipulation subroutine has been developed to perform the averaging processes for any type of nonlinearity. The method is implemented to analyze the response statistics of a second-order oscillator with three different types of nonlinearities, excited by both additive and multiplicative random processes. The second averaging results are in good agreement with those estimated by Monte Carlo simulation. For a special nonlinear oscillator, whose exact stationary solution is known, the second-order averaging results are identical to the exact solution up to first-order approximation.

Asymptotic stability for nonautonomous dissipative wave systems

We study the problem of asymptotic stability, as time tends to infinity, of solutions of dissipative wave systems, governed by time-dependent nonlinear damping forces and subject to the action of strongly nonlinear potential energies. In the final part of the paper we also consider the case of polyharmonic wave operators and higher-order dissipation terms. © 1996 John Wiley & Sons, Inc.

Asymptotic stability for nonautonomous dissipative wave systems

We study the problem of asymptotic stability, as time tends to infinity, of solutions of dissipative wave systems, governed by time-dependent nonlinear damping forces and subject to the action of strongly nonlinear potential energies. In the final part of the paper we also consider the case of polyharmonic wave operators and higher-order dissipation terms. © 1996 John Wiley & Sons, Inc.