Amplitude-dependent Stiffness Research Articles

Wire rope isolator identification and dynamic modeling for small amplitude vibrations

Wire rope isolators are used in variety of applications to protect sensitive equipment from vibration. The nonlinear hysteretic behavior of steel wires provides advantages when compared to linear vibration isolators. This study proposes an amplitude dependent stiffness and damping model for low amplitude vibrations under axial loading, where the parameters can be determined with an experimental procedure. Comprehensive experimental results of forced vibration tests with varying loading, frequency and preload were considered in the model identification. Amplitude dependent stiffness and loss energy models were determined from the test data, and the effect of the preload and loading frequency on the model parameters were studied. It is shown, that the effect of preload and frequency is not evidently clear, while the effect of vibration amplitude is more significant. The mathematical model was further verified against measurements from base excitation loading. The proposed model can be used to study the effectiveness of the selected wire rope isolator configurations in chosen application, and to effectively perform dynamic design studies.

Hysteretic behaviour of uniaxially thermoformed auxetic foams under 3-point bending low-frequency vibration

The work describes experiments and models related to auxetic (negative Poisson’s ratio) foams subjected to low-frequency and variable amplitude 3-point bending loading. A custom 3-point bending vibration test rig is designed and used to perform the dynamic test of auxetic PU foam beams within low-frequency range (1–20 Hz) and 5 different displacement amplitudes. The auxetic foams tested in this work are manufactured using a simplified and relatively low-cost uniaxially thermoforming compression technique, which leads to the production of foams with transverse isotropic characteristics. Auxetic foam beam samples with two different cutting orientations and different thermoforming compression ratios rc (20–80%) are tested and compared, also with the use of theoretical Euler–Bernoulli-based and finite element models. The dynamic modulus of the foams increases with rc, ranging between 0.5 and 5 MPa, while the dynamic loss factor is marginally affected by the compression ratio, with overall values between 0.2 and 0.3. The auxetic PU foam has a noticeable amplitude-dependent stiffness and loss factors, while the dynamic modulus increases but slightly decreases with the frequency. The dynamic modulus is also 20–40% larger than the quasi-static one, while the dynamic and static loss factors are quite close. A modified Bouc–Wen model is also further developed to capture the amplitude and frequency-dependent properties of the conventional and auxetic foams with different volumetric compression ratios. The model shows a good agreement with the experimental results.

Vibroacoustic response of steel–timber composite elements

With the increasing demand for sustainable construction and, at the same time, for larger spans in buildings and high-rise buildings, composite timber structures are gaining more attention, and steel–timber hybrid elements offer a promising approach. Therefore, this study analyzes a novel steel–timber composite floor structure experimentally and numerically. The investigations focus on the vibrational behavior of the structures while considering nonlinearities due to frictional effects in joints. First, eigenfrequencies, damping properties, and consequently, the vibrational characteristics of the systems are determined. Then, detection of nonlinearity follows, investigating amplitude-dependent stiffness and damping and the occurrence of higher harmonics. Results reveal nonlinearities indicated by the occurrence of higher-order harmonics and slight amplitude-dependent damping. Furthermore, the influence of parameters, such as steel thickness and fastener type, on the vibrational behavior of the structures is investigated, comparing specimens of various configurations. Finally, the experimental results validate a linear Finite Element model using the commercial software ANSYS as a first approach despite the nonlinearities that occurred during the measurements. Nonetheless, further investigations could deal with nonlinear models to get more detailed insight.

Reducing uncertainty in seismic assessment of multiple masonry buildings based on monitored demolitions

A significant part of the existing building stock in regions of low to moderate seismic hazard has been designed without modern seismic considerations and is, in the meantime, exceeding its design life span. The assessment of seismic performance poses an engineering challenge, due to unknown material properties, undocumented structural interventions and the scarcity of event-based information. Operational modal analysis has been applied in some cases to verify model assumptions beyond visual inspection. However, masonry buildings exhibit amplitude-dependent stiffness even at very low response amplitudes, raising questions about the validity of such methods. Planned demolitions provide engineers with the opportunity to leverage higher-amplitude vibrations generated during demolition activities to better understand the dynamic behaviour of existing buildings. This paper introduces a Bayesian model-updating framework, which aims at reducing uncertainty in seismic analysis, by fusing dynamic measurements with best-practice structural models. The proposed hybrid framework is applied to nine real masonry buildings, representative of existing residential buildings, as typically encountered in Switzerland, that have been monitored during controlled demolition. A vast reduction in prediction uncertainty is achieved through data-driven model updating, additionally exposing intra- and inter-typological differences in terms of seismic capacity and ductility. In addition, differences between updated model predictions and typical engineering assumptions and generic typological curves are discussed. Overall, this contribution demonstrates, applies and discusses the practical benefits of a straightforward methodology for fusing monitoring data into the seismic evaluation of existing masonry structures.

Numerical study on vortex-induced vibration of circular cylinder with two-degree-of-freedom and geometrical nonlinear system

The vortex-induced vibration (VIV) of circular cylinder in uniform flow with two-degree-of-freedom (2DOF) and geometrical nonlinear system is investigated numerically using the combined sliding and layering dynamic meshes. The characteristics of vibration amplitude, fluid force coefficients, branching behaviour, vibration trajectory, energy transfer and flow pattern of circular cylinder in VIV are studied with two-dimensional shear stress transfer (SST) k-ω turbulence model and three-dimensional large eddy simulation (LES). It is found that in the simulation with LES model the predicted vibration amplitude and fluid force coefficients show good agreement with the experimental data, while in the simulation with SST k-ω model these parameters are significantly underestimated. The two triplets (2T) vortex shedding mode happens in the super upper branch of circular cylinder with 2DOF linear system and leads the large-magnitude positive energy transferring from fluid to cylinder, which causes the maximum vibration amplitude. For the circular cylinder with 2DOF nonlinear system, the vibration simulated by LES model shows a galloping-like phenomenon that the amplitude keeps increasing with the reduced velocity Ur. The crescent-shaped trajectory and the 2T vortex shedding mode happen upon the critical Ur, which is similar to that in the super upper branch of circular cylinder with 2DOF linear system. This phenomenon is caused by the amplitude-dependent stiffness of nonlinear system and the modified reduced velocity U∗ reveals the universality of vibration amplitude for the circular cylinders with linear and nonlinear systems.

Steady state harmonic response of nonlinear soil-structure interaction problems through the Preisach formalism

Steady state harmonic response of nonlinear soil-structure interaction problems is addressed in this paper. Due to well-known stiffness degradation phenomenon, the steady-state study is referred to small-medium strain condition whereas no significant pore water pressure change is observed (e.g. clayey soils and sands in drained conditions). A novel cyclic hysteretic model based on the Preisach formalism is proposed to describe the cyclic behavior of soil-foundation interaction. Through a harmonic balance procedure, furthermore, the steady state response of nonlinear soil-structure interaction problems is determined. Equivalent amplitude-dependent stiffness and damping have been derived in closed form to reliably describe the nonlinear soil-structure interaction phenomenon. Those expressions are function of the elastic foundation soil stiffness, the maximum attainable horizontal force and the moment capacity of the foundation. Furthermore, handy expressions of the equivalent fundamental period and damping, also dependent from the level of the excitation, are also derived. The proposed hysteretic model, moreover, has the advantage to be easily calibrated to match the modulus reduction and damping curves determined either from numerical or experimental tests. Numerical applications and comparisons with more advanced models and experimental data available in literature are also presented.

Modal damping variations in nonlinear dynamical systems

For any linear dynamical system coupled with one or more nonlinear dynamical attachments, the effect of nonlinear energy content on modal damping variations of the entire system has not been clearly addressed in the literature. Accordingly, a novel method is employed here to formulate the amplitude-dependent modal damping matrix for such nonlinear dynamical systems using an amplitude-dependent stiffness approach. The proposed method is directly applied into the equations of motion where numerical and analytical solutions are not required to be known a priori. This advantage is highly desirable to study the dynamical behavior of nonlinear dynamical systems by direct application of methods into equations of motion. Accordingly, the modal damping content variations under the effect of amplitude-dependent stiffness are investigated here. The method is based on linearizing the nonlinear coupling stiffness where a scaled amplitude-dependent stiffness has been obtained to replace the original nonlinear coupling stiffness in the system. Accordingly, the amplitude-dependent modal damping matrix in modal coordinates is obtained and investigated. Consequently, new significant findings regarding modal damping content variations under the effect of the change in nonlinear energy during oscillation are achieved through this study. Furthermore, the nonlinear amplitude-dependent modal damping matrix of the equivalent system is found to be satisfying all matrix similarity conditions with the linear amplitude-independent modal damping matrix of the original system. These findings are expected to be of significant impact on passive nonlinear targeted energy transfer for shock mitigation and energy-harvesting fields.

A stochastic optimal control strategy for viscoelastic systems with actuator saturation

A nonlinear stochastic optimal control strategy for single degree-of-freedom viscoelastic system with actuator saturation is proposed based on the stochastic averaging method and stochastic dynamical programing principle. First, the viscoelastic system is converted into an equivalent nonlinear non-viscoelastic system by replacing the viscoelastic force with amplitude-dependent stiffness and damping. Second, the partially averaged Itô stochastic differential equation for total system energy as one-dimensional controlled diffusion process is derived by using the stochastic averaging method of energy envelope. For the semi-infinite time-interval ergodic control, the dynamical programming equation is established by applying the stochastic dynamical programming principle to the partially averaged Itô equation with a performance index. The optimal saturated control consisting of unbounded optimal control and bang-bang control is determined by solving the dynamical programming equation. Both the analytical and simulation results verify the higher efficiency and chattering attenuation of the proposed control strategy compared with the corresponding bang-bang control.

Nonlinear stochastic optimal control of viscoelastic systems

A nonlinear stochastic optimal control strategy for single degree-of-freedom viscoelastic systems with strongly nonlinear stiffness under Gaussian white noise excitation is proposed. First, the viscoelastic system is converted into an equivalent nonlinear non-viscoelastic system by replacing the viscoelastic force with amplitude-dependent stiffness and damping. Second, the partially averaged Itô stochastic differential equation for total system energy as one-dimensional controlled diffusion process is derived by using the stochastic averaging method of energy envelope. For the semi-infinite time-interval ergodic control, the dynamical programming equation is established based on the dynamical programming principle and is solved to yield the optimal control force. Finally, the variances of the responses of controlled and uncontrolled systems and that of the optimal control force are predicted analytically. Numerical results show that the proposed control strategy is very effective and efficient.

Non-linear dynamics of a whole vehicle finite element model using a harmonic balance method

The aim of this paper is to apply the Harmonic Balance Method (HBM) to a finite element model of a complete vehicle (body, engine and engine mounts) to calculate the non-linear response of the assembly. The non-linear effects come from the amplitude-dependent stiffness of the engine mounts. First, the HBM is presented. A condensation process on the non-linear degrees-of-freedom is also proposed. This process reduces the original non-linear system by focusing only on the solution of the non-linear equations associated with the system’s non-linear components. Second, the engine mount stiffness dependency with amplitude is measured on a test bench to estimate a polynomial stiffness law. Finally, the numerical analysis is performed to analyse the non-linear response of the whole vehicle using the HBM algorithm with appropriate condensation located only on the non-linear coordinates of the system to minimise computer time.

Nonlinear identifications using transmissibility: Dynamic characterisation of Anti Vibration Mounts (AVMs) with standard approach and nonlinear analysis

Anti Vibration Mounts (AVMs) are very common passive vibration isolation devices used in a variety of engineering applications. These are, in most cases, off-the-shelf items which engineers select on the basis of the manufacturer's specifications (in particular the dynamic stiffness). However, only seldom the nominal values of the stiffness of AVMs (especially that of elastomeric mounts) are close to those measured. It can be argued that the main reason for the disagreement is the nonlinear characteristic of the mounts. In practice, the difference between nominal and measured stiffness value can cause major problems to engineers. In fact, in many engineering applications, structures are required to be tested for clearance or certification purposes (e.g. hardware items on helicopters): if an AVM has been selected on the basis of its nominal stiffness its choice may present problems when these are excited at high amplitude of vibration, when the nonlinear behaviour becomes pronounced. In this paper an existing identification method based on Frequency Response Function (FRF) data is shown to be applicable also when transmissibility is measured. Transmissibility is a frequency-dependent, non-dimensional, function which is particularly useful especially when the test item is excited at the base. In this article, the applicability of the method is first demonstrated using numerical simulations. Subsequently, it is shown that the method allows extraction of amplitude-dependent stiffness characteristics of the mount from standard dynamic tests. This enables calculation of stiffness functions and building of a more refined numerical model of the system. The advanced model is used to predict the system's response which compares well with measured data.

The frequency, amplitude and magnetic field dependent torsional stiffness of a magneto-sensitive rubber bushing

A dynamic torsional stiffness model of a magneto-sensitive circular annular rubber bushing is presented where influences of frequency, amplitude and magnetic field dependence are included. This is achieved by employing a newly developed non-linear magneto-sensitive audio frequency constitutive equation in an engineering formula for the torsional stiffness of a rubber bushing. The engineering stiffness formula predicts the frequency and amplitude dependent stiffness in a simple way, based on geometric dimensions and the shear modulus. The shear modulus is provided by the rubber model. The results from these calculations predict and clearly display the possibility of controlling the magneto-sensitive rubber bushing stiffness over a large frequency range, through the application of a magnetic field.

On estimating system damping from frequency response bandwidths

As damping determines the maximum vibratory response of a system at resonance, reliable estimates of damping are critical both to the design and qualification of systems to be subjected to a vibratory environment and for the evaluation of the effectiveness of modifications or additions provided to increase damping. The sharpness of the frequency response at resonance is often used for this purpose, quantified by the width of the frequency range (bandwidth) for which the response is above some fraction of the maximum response. Several aspects of the use of bandwidth methods in interpreting test results are considered. It is shown that the use of an excessive rate of change of test frequency in a sine sweep leads to overestimates of system damping. A criterion is offered for the identification of the maximum sweep rate for which an observed frequency response function provides a true indication of system damping, rather than an erroneous value dominated by the sweep rate. Applicability of the criterion is demonstrated through the use of results from actual tests. Excessive sweep rates are shown to inflate estimates of system loss factors above the true values in proportion to the square root of the sweep rate. It is also demonstrated with a specific form for an amplitude-dependent stiffness that the resulting nonlinearity can lead to erroneous observations of bandwidth frequencies, as well as the need for further reductions in the sweep rate.

Rotordynamic Performance of a Rotor Supported on Bump Type Foil Gas Bearings: Experiments and Predictions

Gas foil bearings (GFBs) satisfy the requirements for oil-free turbomachinery, i.e., simple construction and ensuring low drag friction and reliable high speed operation. However, GFBs have a limited load capacity and minimal damping, as well as frequency and amplitude dependent stiffness and damping characteristics. This paper provides experimental results of the rotordynamic performance of a small rotor supported on two bump-type GFBs of length and diameter equal to 38.10mm. Coast down rotor responses from 25krpm to rest are recorded for various imbalance conditions and increasing air feed pressures. The peak amplitudes of rotor synchronous motion at the system critical speed are not proportional to the imbalance introduced. Furthermore, for the largest imbalance, the test system shows subsynchronous motions from 20.5krpm to 15krpm with a whirl frequency at ∼50% of shaft speed. Rotor imbalance exacerbates the severity of subsynchronous motions, thus denoting a forced nonlinearity in the GFBs. The rotor dynamic analysis with calculated GFB force coefficients predicts a critical speed at 8.5krpm, as in the experiments; and importantly enough, unstable operation in the same speed range as the test results for the largest imbalance. Predicted imbalance responses do not agree with the rotor measurements while crossing the critical speed, except for the lowest imbalance case. Gas pressurization through the bearings’ side ameliorates rotor subsynchronous motions and reduces the peak amplitudes at the critical speed. Posttest inspection reveal wear spots on the top foils and rotor surface.

Steady‐State Oscillation of Hysteretic Differential Model. I: Response Analysis

This study considers the periodic forced vibration of nonlinear hysteretic system. Multiharmonic steady-state responses of a general hysteretic system adopting the Bouc-Wec differential model, subjected to arbitrary periodic excitation, are analyzed by the Galerkin/Levenberg-Marquardt method. The frequency/time domain alternation and the fast Fourier transformation (FFT) techniques are introduced in this frequency domain solution procedure. The frequency-controlled and amplitude-controlled algorithms are simultaneously formulated to obtain complete, probably multivalued, frequency response curves. This approach gives possible harmonic, superharmonic, and subharmonic responses of the Bouc-Wen hysteretic system. The response characteristics of softening, hardening, and quasilinear hysteretic systems are studied through numerical computations. The proposed procedure is also extended to analyze the periodic vibration of degrading hysteretic system with amplitude-dependent stiffness deterioration. In addition, a verification is given by comparing with the results obtained by a numerical integration procedure.