Viscous Damping Model Research Articles

Feasibility study of the physical implementation of rate-independent linear damping for the protection of low-frequency structures

Damage to nonstructural components or excessive displacements in low-frequency structures caused by recent major earthquakes, such as the 2011 Great East Japan Earthquake, highlighted the need to protect these structures against earthquake-induced damage. Rate-independent linear damping (RILD) has been found to be a viable option for reducing the excessive displacement of low-frequency structures because its control force is larger in the low-frequency region than that of conventional damping elements. Most of the studies on RILD have focused on theoretical and mathematical aspects rather than their practical application. The main objective of this study was to examine the feasibility of the physical implementation of RILD for the protection of low-frequency structures. In this study, a passive causal RILD (CRILD) device comprising a Maxwell element and a negative stiffness element was considered to mechanically realize RILD. Numerical analysis and real-time hybrid simulation (RTHS) on a single-degree-of-freedom (SDOF) system incorporated with the proposed device were performed to identify the challenges in the physical implementation of a passive CRILD device. The results confirmed that CRILD can achieve a similar control effect of low-frequency structures to the ideal RILD and can reduce structural dynamic responses more effectively than the commonly used linear viscous damping (LVD) model.

Effect of the elastic hysteresis term formulation and response to non-harmonic periodic excitations of a non-linear SDOF dynamical model with weak frequency-dependency in the time domain

We elaborate a SDOF time-domain model for elastic hysteretic damping, by modifying the viscous damping model to introduce an instantaneous correction factor that recursively depends on the state variables of the system, such that the response exhibits weak dependency on frequency, corresponding to a large array of engineering materials. The effect of different formulations for calculating the instantaneous correction factor on the predicted hysteresis loops and the potential manifestation of singularities is studied. Hysteresis loops anticipated by the model are plotted and forced vibration responses to harmonic and other periodic non-harmonic excitations are simulated and discussed, also in comparison to the conventional viscous and Reid’s damping models.

Effect of Viscous Damping Models on Displacement Ductility Demands for SDOF Systems

The viscous damping is a highly idealized but mathematically convenient way of representing the energy dissipation mechanisms not related to the hysteretic response. In this study, the tangent-stiffness proportional Rayleigh damping (TSPRD) appropriate for inelastic hysteretic structures outfitted with the supplemental damping devices is utilized in inelastic dynamic analysis of simple single-degree-of-freedom (SDOF) structures. A numerical procedure of dynamic equilibrium equation for SDOF systems with TSPRD model is derived for development of the general numerical solution scheme. By specifying various combinations of tangent-stiffness and mass proportional damping terms in TSPRD model, the influence of viscous damping models on displacement ductility demands is investigated. The results indicate that the difference in ductility demands for various damping models is considerable, depending on the periods of vibration, relative lateral strength, hysteretic models, stiffness deterioration and initial damping ratios. The constant-strength displacement ductility spectra, by considering both tangent-stiffness proportional and mass proportional damping, are developed in terms of site conditions, hysteretic rules and initial damping ratios.

Structural damping models for passive aeroelastic control

Aeroelastic qualification requirements are typically met by sizing aircraft to achieve adequate stability margins and keep peak gust responses below specified thresholds. A possible alternative approach is delaying flutter and alleviating gust response by embedding dissipative materials into structural components. This approach requires accurate damping models applicable to the analysis of complex configurations. In this paper, the effect of damping models in the evaluation of flutter boundaries and gust response of an aeroelastic test-bed is studied. In particular, three damping models completely different in frequency are considered to model the presence of skin patches for passive aeroelastic control: viscous damping, hysteretic damping and Biot damping models. In order to make them comparable the three models are tuned in order to dissipate the same amount of power at the flutter frequency by means of the introduction and definition of a generalized loss factor. The damping models are compared by evaluating their effect on flutter suppression and gust load alleviation. Finally, the sensitiveness of the stability and response aeroelastic analyses to the considered damping models are outlined and the obtained results are compared to provide modeling recommendations for passive flutter suppression and gust alleviation studies. The proposed methodology is applicable to any linear material model for which the related complex stiffness can be expressed in the frequency domain meeting the Hilbert restriction.

Modeling viscous damping in actuated breast tissue to provide diagnostic insight for breast cancer: A proof-of-concept analysis.

This study develops a viscous damping model (VDM) based on Rayleigh Damping (RD) with potential use in low cost, non-invasive breast cancer diagnostics using Digital Image Elasto Tomography (DIET). A clinical trial involving 13 subjects, each with a tumor in one breast, resulted in 13 cancerous and 13 healthy breasts. Displacement data following actuator induced steady state vibration in the breast tissue were captured using the DIET system. Over 14000 reference points on the breast surface were split into four segments and viscous damping constant calculated for each reference point. The VDM was fit to median-filtered data for each breast segment and VDM coefficients compared within each breast. One model coefficient, relating to stiffness, was hypothesized to differ in breast segments containing a tumor. Comparison of " " coefficients in different breast segments using percentage tolerances provided an unbiased, generalizable diagnostic method. Bootstrapping with replacement was used to upsample the data and create smooth receiver operator characteristic (ROC) curves. A total of 12 breast segmentation configurations were used to demonstrate the robustness of the method. Fitting the VDM to median-filtered data gave consistent results for one VDM coefficient (" ") across all breasts. The second VDM coefficient (" ") showed diagnostic potential with breast segments having consistent coefficients in healthy breasts. In cancerous breasts " " coefficients were found to be statistically different in segments containing and adjacent to the tumor compared with the segment furthest from the tumor with p<0.02 using the Student t-Test. Large discrepancies in " " coefficients were found to be indicative of a tumor with a 14.5% tolerance resulting in sensitivity and specificity of 76.9%. The optimal breast configuration resulted in an area under the ROC curve (AUC) of 0.81 with sensitivity and specificity at 77% and 72%, respectively. This VDM method enables a computationally simple diagnostic technique using DIET for comfortable breast screening for women of all ages. Regular screening potential allows for tolerance alteration based on age, prior subject-specific results, and other risk factors to manage false positives, reducing psychological harm while optimizing early detection for successful treatment and decreased mortality.

PhysEx: A novel procedure to estimate full-rank physical matrices of a structure from an incomplete modal model

This work is devoted to presenting and applying a novel methodology that estimates a linear physical model, represented by full-rank mass, stiffness and damping matrices, capable of replicating the dynamic behaviour of an incomplete modal model. No hypothesis is made with respect to the damping matrix, so the procedure is applicable assuming any linear viscous damping model. It is shown that, if the number of considered modes satisfies a certain restriction with respect to the number of degrees of freedom, the problem has infinite solutions that differ in their exogenous eigenvalues. By controlling their effect on the dynamic behaviour of the model in the frequency range of interest, a proper final solution is obtained. The methodology is applied to a three-degrees-of-freedom discrete model and a discretized cantilever beam. Comparisons are made in both frequency and time domains which reveal that the methodology provides physical models that accurately replicate the dynamic behaviour of the incomplete modal model. Even for complete models, where the solvability condition is not met, the methodology manages to provide meaningful results in some situations. To prove the usefulness of the estimated models, a mass is added on one DOF of both examples to show that they reproduce the same effects as the original modified ones. Finally, in the interests of reproducibility, all models and algorithms presented in this article have been made publicly available.

Damping of piston mode resonance between two fixed boxes

In this paper, the damping of wave-induced resonant oscillations of water column in the gap between two fixed boxes is investigated through physical model testing. A pair of rectangular fins with different geometries is attached to the sidewalls of the gap to suppress the resonant response. This study is an extension to Tan et al. [“A viscous damping model for piston mode resonance,” J. Fluid Mech. 871, 510–533 (2019)], where the damping of wave resonance in a plain gap without fins was investigated. We found that (1) the resonant wave height in the gap decreased substantially with the increasing fin thickness, but was not very sensitive to the variation of the fin height. This observation suggests that the fin thickness has a more significant influence on the damping of oscillations than the fin height; (2) the resonant frequency decreased as the thickness of the fins was enlarged, which is primarily attributable to the decrease in the natural frequency (due to increasing added mass) and partly ascribed to the increase in viscous damping. The viscous dissipation was found to be approximately 3–10 times larger than the radiation energy and explains the well-known over-estimations of inviscid potential solution in resonant amplitude. A modified potential flow model with a damping mechanism was employed to predict the wave response in the gap, wave reflections, and transmissions. Reasonable agreements were obtained between the experimental results and the modified potential flow solutions with respect to amplitude, whereas a difference of 6%–14% in resonant frequency was observed. This difference is ascribed to the effects of flow separation on the added mass of fins, which cannot be fully captured by modified potential flow models. In addition to the present geometries, the local loss coefficients for other appendage configurations were also calibrated based on the available experimental and computational fluid dynamics data in the literature.

Numerical modeling of DNA nucleotides binding process mechanics considering oscillations

The paper deals with the process of binding of two nucleotides, which is considered part of a more complex process, such as the transfer of telomerase to the telomere region of chromosomes during cell division. The binding process is evaluated using the originally developed attraction-related energy dissipation mechanism. Energy dissipation has also been observed using a viscous damping model. Viscous damping is considered to obtain dissipative oscillations during the interaction of nucleotides, which can be part of the binding process. The work presents the dynamics of nucleotides, indicating the values of force, distance, velocity for a certain period of time.

Study on nonlinear damping behavior of damaged recycled aggregate concrete beams

AbstractAn experimental study on the nonlinear damping behavior of recycled aggregate concrete (RAC) beams is conducted through subjecting dynamic loadings at the different damage stages. The effects of the loading frequency, loading amplitude, damage, recycled aggregate, and replacement percentage on the energy dissipation by damping and the equivalent viscous damping ratio (EVDR) of the RAC beams are analyzed. The experimental results show that the energy dissipation and the EVDR are not directly influenced by the loading frequency, and the EVDR rises with the increasing loading amplitude for the damaged beams. The damage stage is of great significance to the damping mechanism. The EVDR is firstly increased, peaks at the yielding stage, and finally declines with the damage accumulation, and the EVDR at the yielding stage can be increased to 2.6–3.1 times compared to the undamaged RAC beams. The EVDRs are higher for the beams with the incorporation of the recycled aggregates, because of their lower stiffness and higher energy dissipation. The experimental results demonstrate that the commonly applied viscous damping model is not capable of correctly describing the damping behavior of RAC beams, whereas the modified Coulomb damping model performs much better. A shift in the damping mechanism is observed with the damage evolution, and the relationships between the damage indices and the damping parameters are established.

Investigation of the Effect of Block Size, Shape, and Freeze Bond Strength on Flexural Failure of Freshwater Ice Rubble Using the Discrete Element Method

Abstract As part of the Mechanics of Ice Rubble project, recent experiments have been carried out to study the strength and failure behavior of ice rubble beams and the freeze bonds that form between individual ice blocks. In this study, we present results obtained from a newly developed model for the three-dimensional (3D) discrete element modeling (DEM) open-source code LIGGGHTS. The ice model contains normal and shear springs that operate between neighboring particles which are bonded or that overlap due to compressional stresses. Energy dissipation is accounted for by using a viscous damping model. Using this DEM model, medium-scale freshwater ice rubble punch tests have been simulated. Rubble specimens were generated by “raining” individual DEM ice pieces into a rectangular form and compacting the rubble mass to achieve the target porosity. Before the compacting pressure was removed, bonds between contacting blocks were introduced with parameter values based on representative freeze bond experiments. The rubble beam was then deformed by pushing a platen vertically downward through the center of the beam until failure occurred. Two types of block size and shapes have been simulated: cuboid blocks generated based on the size distribution of the actual rubble, and rubble blocks generated by image processing of actual blocks of broken ice used in the comparison experiments. The mechanism of flexural rubble failure for both cuboid block [s4.2] simulations and empirical block [s4.3] simulations is in line with experimental results; however, the empirical block simulations provide a significantly better estimation of the failure force.

Assessment of Reynolds-averaged Navier–Stokes method for modeling the startup regime of a Darrieus rotor

In this article, the suitability of the Reynolds-averaged Navier–Stokes (RANS) to predict the dynamic response in the startup process of small-scale Darrieus vertical-axis wind turbines (VAWT) was investigated. Using OpenFOAM, numerical simulations have been run to evaluate the regions of the rotor revolution where the prediction of the aerodynamic coefficients is expected to give unreliable results. The qualitative aspects of the prediction of the self-starting of the VAWT for tip speed ratio (λ) ranging from 0 to 3 have been evaluated and to model a more realistic condition, resistive forces acting on the rotor have been implemented. These resistive forces, which are the mechanical friction and the magnetic forces of the generator, were implemented numerically through a viscous damping model and, therefore, a new OpenFOAM library was developed adopting the object-oriented programming methodology as well as the C++ language. From the results, it has been observed that for a λ equal to 0.4 (starting speed), the aerodynamic forces are overestimated for a portion of 82% of the full revolution of the rotor. For a λ equal to 2.70 (steady state), this portion is reduced to 52%. This means that the aerodynamic forces and also the power output are overestimated by a large percentage in a full revolution of the rotor. Therefore, such results allow us to conjecture that RANS modeling may not seem to be an adequate approach to accurately assess the aerodynamic response of VAWTs.

VD-PQ; A Velocity-Dependent Viscous Damping Model for Wave-Structure Interaction Analysis

For the analysis and design of coastal and offshore structures, viscous loads represent one of the most influential parameters that dominate their response. Very commonly, the potential flow theory is used for identifying the excitation wave loads, while the viscous damping loads are taken into consideration as distributed drag type loads and/or as linear and quadratic damping loads approximated with the use of motion decay curves of the structure in specific degrees of freedom. In the present paper, is developed and proposed a numerical analysis method for addressing wave-structure interaction effects through a velocity-dependent viscous damping model. Results derived by a computational fluid dynamics model are coupled with a model that uses the boundary element method for the estimation of the viscous damping loads iteratively in every time-step of the analysis. The computational fluid dynamics model solves the Navier–Stokes equations considering incompressible flow, while the second model solves the modified Cummins Equation of motion of the structure in the time domain. Details about the development of the coupling method and the velocity-dependent viscous damping (VD-PQ) are presented. The coupling between the different models is realized through a dynamic-link library. The proposed coupling method is applied for the case of a wave energy converter. Results derived with the use of the developed numerical analysis method are compared against experimental data and relevant numerical analysis predictions. The importance of considering the instantaneous velocity of the structure in estimating the viscous damping loads is demonstrated. The proposed numerical analysis method for estimating the viscous damping loads provides good accuracy compared to experimental data and, at the same time, low computational cost.

基于Clough-Penzien谱激励的指数型非黏滞阻尼结构随机地震动响应简明封闭解

Compared with the traditional viscous damping model, the non-viscous-damping model can more accurately describe the energy dissipation behavior of structural materials, and its constitutive relationship is usually expressed in the form of convolution of exponential functions. In view of the complexity of the response to random ground motion obtained with the existent methods for structures with non-viscous damping, a simple closed-solution method for the analysis of 0~2nd-order spectral moments of structural responses was proposed based on the Clough-Penzien (C-P) spectrum. With this method, the exact equivalent differential constitutive relation of non-viscous-damping structures was first proposed and the ground motion equation of the structure was reconstructed with the C-P spectrum filtering differential equation. Then, based on the random vibration theory, the concise closed solution of the 0~2nd-order spectral moments of the structural random responses was obtained. Accordingly, the dynamic reliability of the structure was analyzed under the first excursion criterion and the Markov distribution rule. Finally, an example was given to demonstrate the accuracy and efficiency of the closed solution.

A non-linear model for elastic hysteresis in the time domain: Computational procedure

Hysteretic damping of a material or structure loaded within its elastic region is the dissipation of mechanical energy at a rate independent of the frequency of vibration while at the same time directly proportional to the square of the displacement. Generally, reproducing this frequency-independent damping can be computationally complex and requires prior knowledge of the system’s natural frequencies or the full time history of the system’s response. In this paper, a new model and numerical procedure are proposed whereby hysteretic material damping is achieved in the time domain. The proposed procedure is developed based on modifying the viscous model through a correction factor calculated exclusively using the local response. The superiority of the proposed approach lies in its ability to capture material hysteresis without any knowledge of the eigen- or modal frequencies of the system and without knowledge of the past time history of the system’s response or the characteristics of any excitation forces. A numerical procedure is also presented for implementing the proposed model in vibration analysis. The simplicity of the approach enables its generalisation to continuous systems and to systems of multi-degrees of freedom as demonstrated herein. The proposed model is presented as a correction to the viscous damping model which makes it attractive to implement into commercial finite element package using user-defined element subroutines as demonstrated in this study.

Bell-shaped proportional viscous damping models with adjustable frequency bandwidth

A proportional viscous damping model based on a bell-shaped basis function parameterized by the frequency and damping ratio at its peak has recently been proposed. The basis function in the frequency domain has a fixed frequency bandwidth and could not match a damping ratio curve that changes precipitously over a short frequency interval. This study proposes new parameters to control the bandwidth, while maintaining the sparsity of damping coefficient matrix and the order of computational efficiency. Three types of modification are introduced. Types 1 and 3 control the bandwidth while maintaining the symmetry of the bell shape on the logarithmic scale, and Type 2 allows the left and right bandwidths to be adjusted independently. Type 3 controls the bandwidth while maintaining the same number of nonzeros in the expanded damping matrix, but the other two have the number of nonzeros increased for a smaller bandwidth. These modifications make the model versatile in matching any smooth damping ratio curve, particularly for a curve that has damping ratios changing drastically over a short frequency interval. Several examples are given to showcase their matching performance and the sparsity of their expanded damping coefficient matrices.

Impact and Bandgap Characteristics of Periodic Rods With Viscoelastic Inserts and Local Resonators

Abstract A comprehensive theoretical and experimental study is presented of the bandgap behavior of periodic viscoelastic material (VEM) composites subjected to impact loading. The composites under consideration consist of an assembly of aluminum sections integrated with periodic inserts which are arranged in one-dimensional configurations. The investigated inserts are manufactured either from VEM only or VEM with local resonators (LR). A finite element model (FEM) is developed to predict the dynamics of this class of VEM composites by integrating the dynamics of the solid aluminum sections with those of VEM using the Golla-Hughes-Mctavish (GHM) mini-oscillator approach. The integrated model enables, for the first time, the accurate predictions of the bandgap characteristics of periodic viscoelastic composites unlike previous studies where the viscoelastic damping is modeled using the complex modulus approach with storage modulus and loss factor are assumed constants and independent of the frequency or the unrealistic and physically inaccurate Kelvin–Voigt viscous-damping models. The predictions of the developed FEM are validated against the predictions of the commercial finite element package ansys. Furthermore, the FEM predictions are checked experimentally using prototypes of the VEM composites with VEM and VEM/LR inserts. Comparisons are also established against the behavior of plain aluminum rods in an attempt to demonstrate the effectiveness of the proposed class of composites in mitigation of the structural response under impact loading. Close agreements are demonstrated between the theoretical predictions and the obtained experimental results.

Model Updating for Systems with General Proportional Damping Using Complex FRFs

In this paper, we propose a model updating method for systems with nonviscous proportional damping. In comparison to the traditional viscous damping model, the introduction of nonviscous damping will not only reduce the vibration of the system but also change the resonance frequencies. Therefore, most of the existing updating methods cannot be directly applied to systems with nonviscous damping. In many works, for simplicity, the Rayleigh damping model has been applied in the model updating procedure. However, the assumption of Rayleigh damping may result in large errors of damping for higher modes. To capture the variation of modal damping ratio with frequency in a more general way, the diagonal elements of the modal damping matrix and relaxation parameter are updated to characterize the damping energy dissipation of the structure by the proposed method. Spatial and modal incompleteness are both discussed for the updating procedure. Numerical simulations and experimental examples are adopted to validate the effectiveness of the proposed method. The results show that the systems with general proportional damping can be predicted more accurately by the proposed updating method.

The effect of energy dissipation on the dynamic response of reinforced concrete structure

Solving problems associated with protecting buildings from progressive collapse and minimizing resources is becoming increasingly important. In many countries, ensuring such protection is set in accordance with the requirements of national regulatory documents and, therefore, researches aimed at developing effective ways to protect constructive systems from progressive collapse under special actions are relevant. In this regard, the present work presents studies of the influence of energy dissipation in constructive reinforced concrete elements during their sudden structural rearrangement caused by the removal of one of the supporting elements. The hysteretic dispersion accompanying the nonlinear load-unload cycle of the cross-section during crack formation is taken into account. The damping mechanism of reinforced concrete elements of the constructive system is described by the equivalent viscous damping model, based on the hypothesis of proportionality of the damping force and the oscillation velocity. Approbation of the calculated model is carried out in relation to the reinforced concrete frame structures, the physical models of which were tested under the considered loading regime. A comparison of the experimental and calculated parameters shows the effectiveness of the proposed model for assessing the dynamic response of reinforced concrete structures with cracks under the static-dynamic loading regime.

Modeling viscous damping in nonlinear response history analysis of steel moment‐frame buildings: Design‐plus ground motions

AbstractInvestigated in this paper is the question: Can seismic demands on steel moment‐frame buildings due to maximum considered earthquake (MCER) design‐level ground motions (GMs) be estimated satisfactorily using linear viscous damping models or is a nonlinear model, such as capped damping, necessary? This investigation employs an enhanced model with several complex features of a 20‐story steel moment‐frame building. Considered are two linear viscous damping models: Rayleigh damping and constant modal damping; and one nonlinear model where damping forces are not allowed to exceed a predefined bound. Presented are seismic demands on the building due to two sets of GMs: MCER design‐level GMs (2% probability of exceedance [PE] in 50 years) and rarer excitations (1% PE in 50 years). Based on these results, we conclude that linear damping models are adequate for estimating seismic demands on steel moment‐frame buildings—designed to satisfy current story drift and plastic rotation limits due to MCER design‐level GMs. Between the two linear damping models, constant modal damping is preferred; it is available in commercial computer codes for earthquake structural analysis.

Investigation on residual displacements for SDOF systems with various initial viscous damping models

Nonlinear time history analysis (NTHA) is widely used in estimation of post-earthquake residual displacement demands. However, the choice of initial viscous damping models in NTHA has attracted insufficient attention in study of residual displacements. The more versatile viscous damping model similar to the tangent-stiffness proportional Rayleigh damping commonly used in multi-degree-of-freedom (MDOF) systems is utilized in NTHA of single-degree-of-freedom (SDOF) systems. With different combinations of mass proportional and tangent-stiffness proportional damping terms, the significance of the choice of initial damping models is examined in terms of energy dissipation and residual displacement demands. The residual displacement ratio, defined as the ratio of residual displacements to elastic spectral displacements, is computed for constant-strength SDOF systems when subjected to an ensemble of 240 earthquake ground motions. The results are statistically organized to investigate the effect of initial viscous damping models on residual displacement demands. The difference in residual displacement ratios for various damping models is found to be significant, and dependent on natural periods, relative lateral strength, post-yield stiffness, unloading stiffness degradation and initial damping ratios. The effect of these factors on residual displacement demands of SDOF systems with tangent-stiffness proportional damping, which is confirmed to be most appropriate for nonlinear analysis, is also examined. Finally, the constant-strength residual displacement response spectra are constructed for seismic assessment of structures.