Equivalent Viscous Damping Model Research Articles

A time-domain method for free vibration responses of an equivalent viscous damped system based on a complex damping model

The free vibration responses are divergent based on a complex time-domain damping model. The traditional step by step integration method cannot be used to calculate the time-domain responses. Based on the theoretic solution that eliminates the divergent term for the complex damping model, the equivalent viscous damping model is proposed and the corresponding average acceleration method is realized in this paper. The numerical cases show that the calculated results of the equivalent viscous damped system are convergent, which are equal to the ones of the deleted divergent term based on the complex damping model. The correctness of the proposed method is verified, and the computational efficiency is high.

Numerical Block-Based Simulation of Rocking Structures Using a Novel Universal Viscous Damping Model

Unreinforced masonry structures, particularly façade walls, are seismically vulnerable due to their weak connections with adjacent walls, floors, and/or roofs. During an earthquake, such walls formulate local mechanisms prone to out-of-plane collapse. This behavior has been largely investigated using classical rocking theory, which assumes the structure responds as a rigid body undergoing rocking motion, with energy dissipation at impact. Due to the complexity of the problem, however, e.g., number of degrees of freedom or boundary conditions, numerical block-based modeling is gaining momentum. However, numerical models lack a consistent and reliable treatment of the energy loss at impact. This paper bridges the gap between the well-established energy loss of classical rocking theory and the treatment of damping in numerical modeling. Specifically, it proposes an equivalent viscous damping model through novel ready-to-use predictive equations that capture the dissipative phenomena during both one-sided and two-sided planar rocking motion. The results reveal a satisfactory performance of the proposed model through comparisons with experimental results from literature and highlight its universality and robustness through applications of the model in fundamentally different block-based numerical modeling software.

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.

Time-domain analysis for dynamic responses of non-classically damped composite structures

A complex mode superposition method based on equivalent viscous damping model (CEV) is proposed for time-domain dynamic responses of non-classically damped composite structures. The damping matrix of hysteretic non-classically damped composite structures can be easily obtained and complex mode superposition method based on hysteretic damping model is directly realized in the physical space. However, hysteretic damping model is not causal, and is only applied in frequency-domain analysis. By the aid of constitutive model of Kelvin-Voigt model, viscous damping model which is equivalent to the hysteretic damping model is obtained based on single degree of freedom system. Combined with complex mode superposition method of hysteretic damping model and mathematical simplicity of equivalent viscous damping model, CEV can be realized. Structural vibration responses of shaking table test under random excitations are measured, which are compared with the corresponding calculation results using CEV and traditional complex mode superposition method based on viscous damping model (CR), respectively. The calculation results obtained by the two complex mode superposition methods are consistent with the test results. Compared with CR, the calculation results of CEV are unique, computationalefficiency of CEV is higher and the problem of choosing reasonable modes has been avoided.

Proposed SMA Tension Sling Damper for Passive Seismic Control of Building

Passive Shape Memory Alloy (SMA) based Tension Sling Damper (TSD) is proposed to control seismic response of building represented as dynamic SDOF system. Non-linear hysteretic behavior of SMA is characterized by Tanaka model with seismic excitations as input. It is converted to equivalent piece-wise linear elastic viscous damping model described by stiffness constant ‘keff’ and damping constant ‘ceff’ to implement it with linear dynamic system. Damping constant ‘ceff’ is evaluated for TSD for given seismic excitation at each instance of time defined as instantaneous damping, unlike constant damping used in other studies, to derive accurate seismic response of the dynamic system. Seismic response control parameters – peak and rms perfor-mance indices (PI) for displacement and acceleration, peak damper force, maximum damping ratio are evalu-ated for flexible, moderately stiff and stiff dynamic systems fitted with TSD under pulse and strong ground motion type seismic excitations with varied PGA. Substantial reduction in PI is observed for controlled systems through varying design parameters of TSD – cross-sectional area and length of SMA sling. TSD yields better seismic response control for moderately stiff and flexible system due to super-elastic properties of SMA. Fur-ther, it works efficiently for seismic excitations with low to moderate PGA.

Experimental and theoretical study on large amplitude vibrations of clamped rubber plates

In this paper, the large amplitude forced vibrations of thin rectangular plates made of different types of rubbers are investigated both experimentally and theoretically. The excitation is provided by a concentrated transversal harmonic load. Clamped boundary conditions at the edges are considered, while rotary inertia, geometric imperfections and shear deformation are neglected since they are negligible for the studied cases. The von Kármán nonlinear strain-displacement relationships are used in the theoretical study; the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model, which introduces nonlinear damping. An equivalent viscous damping model has also been created for comparison. In-plane pre-loads applied during the assembly of the plate to the frame are taken into account. In the experimental study, two rubber plates with different material and thicknesses have been considered; a silicone plate and a neoprene plate. The plates have been fixed to a heavy rectangular metal frame with an initial stretching. The large amplitude vibrations of the plates in the spectral neighbourhood of the first resonance have been measured at various harmonic force levels. A laser Doppler vibrometer has been used to measure the plate response. Maximum vibration amplitude larger than three times the thickness of the plate has been achieved, corresponding to a hardening type nonlinear response. Experimental frequency-response curves have been very satisfactorily compared to numerical results. Results show that the identified retardation time increases when the excitation level is increased, similar to the equivalent viscous damping but to a lesser extent due to its nonlinear nature. The nonlinearity introduced by the Kelvin-Voigt viscoelasticity model is found to be not sufficient to capture the dissipation present in the rubber plates during large amplitude vibrations.

Direct displacement based seismic design for single storey steel concentrically braced frames

The direct displacement based design (DDBD) approach is spreading in the field of seismic design for many types of structures. This paper is carried out to present a robust approach for the DDBD procedure for single degree of freedom (SDOF) concentrically braced frames (CBFs). Special attention is paid to the choice of an equivalent viscous damping (EVD) model that represents the behaviour of a series of full scale shake table tests. The performance of the DDBD methodology of the CBFs is verified by two ways. Firstly, by comparing the DDBD results with a series of full-scale shake table tests. Secondly, by comparing the DDBD results with a quantified nonlinear time history analysis (NLTHA). It is found that the DDBD works relatively well and could predict the base shear forces (<TEX>$F_b$</TEX>) and the required brace cross sectional sizes of the actual values obtained from shake table tests and NLTHA. In other words, when comparing the ratio of <TEX>$F_b$</TEX> estimated from the DDBD to the measured values in shake table tests, the mean and coefficient of variation (<TEX>$C_V$</TEX>) are found to be 1.09 and 0.12, respectively. Moreover, the mean and <TEX>$C_V$</TEX> of the ratios of <TEX>$F_b$</TEX> estimated from the DDBD to the values obtained from NLTHA are found to be 1.03 and 0.12, respectively. Thus, the DDBD methodology presented in this paper has been shown to give accurate and reliable results.

AP/VP specific equivalent viscous damping model for base-isolated buildings characterized by SDOF systems

In this paper, an AP/VP specific equivalent viscous damping model is proposed through regression analysis to predict the maximum inelastic displacement of base-isolated buildings characterized by SDOF system. In order to present comprehensive results, a large number of parameters are specified for bilinear SDOF systems, i.e., 8 initial natural periods between 0.1s and 1.5s, 16 ductility ratios ranging from 2 to 50 and 4 post-to-pre-yieldstiffnessratios: 0.0, 0.05, 0.1 and 0.2. A total of 45 earthquake ground motions are selected as seismic input and classified into three groups according to different frequency contents represented by peak ground acceleration to peak ground velocity ratios (AP/VP). Results show that the AP/VP ratio is of significant importance for the estimation accuracy of equivalent linearization method, especially for buildings with relatively small post-to-pre-yieldstiffnessratio. Finally, the estimation accuracy of the proposed model is verified through comparison with other models.

Sensitivity of the stochastic response of structures coupled with vibrating barriers

The sensitivity of the stochastic response of linear behaving structures controlled by the novel Vibrating Barrier (ViBa) device is scrutinized. The Vibrating Barrier (ViBa) is a massive structure, hosted in the soil, calibrated for protecting structures by exploiting the structure–soil–structure interaction effect. Therefore the paper addresses the study of the sensitivity of soil–structure coupled systems in which the soil is modelled as a linear elastic medium with hysteretic damping. In order to accomplish efficient sensitivity analyses, a reduced model is determined by means of the Craig–Bampton procedure. Moreover, a lumped parameter model is used for converting the hysteretic damping soil model rigorously valid in the frequency domain to the approximately equivalent viscous damping model in order to perform conventional time-history analysis. The sensitivity is evaluated by determining a semi-analytical method based on the dynamic modification approach for the case of multi-variate stochastic input process. The ground motion is modelled as non-stationary zero-mean Gaussian random process defined by a given evolutionary Power Spectral Density function. The paper presents the sensitivity of the response statistics of a model of an industrial building, passively controlled by the ViBa, to relevant design parameters. Comparisons with pertinent Monte Carlo Simulation will show the effectiveness of the proposed approach.

Equivalent Viscous Damping Model for Short-Period Reinforced Concrete Bridges

This paper investigates the effect of spectral shape (intensity and width of the constant acceleration region) and postyield stiffness ratio on equivalent viscous damping for short-period RC bridge columns (effective period<1s). The modified Takeda degrading stiffness hysteretic model, with parameters appropriate to bridge columns (often termed thin Takeda in the literature), is used for analysis. Insight regarding the importance of these parameters is provided, and a new equivalent viscous damping model is proposed that includes the effect of spectral shape and postyield stiffness ratio, as well as effective period and ductility. The proposed damping model is compared with two existing models. The results indicate that significant improvement is achieved in predicting the peak displacement using the proposed damping model when compared with existing models.

Equivalent Viscous Damping for Steel Moment-Resisting Frames with Cross-Laminated Timber Infill Walls

AbstractIn the direct displacement-based design method, energy dissipative capacity of structures can be represented by an equivalent viscous damping (EVD). A number of studies have been reported in the formulation of EVD for different structural systems and hysteretic models. In this paper, an EVD model is developed and calibrated for steel-timber hybrid structures, where cross-laminated timber (CLT) shear panels are used as an infill in steel moment-resisting frames (SMRFs). To develop the EVD model, 243 single-story, single-bay CLT-infilled SMRFs analytical models are subject to semi-static cyclic analysis. Different model parameters of the hybrid structure are varied: gap between CLT panel and steel frame, bracket (connection) spacing, CLT panel thickness and strength, and postyield stiffness ratio of steel members. The EVD of each model was computed from the hysteretic responses based on an area-based approach. The design of computer experiments and response surface methodology were utilized to formu...

Improved equivalent viscous damping model for base-isolated structures with lead rubber bearings

Nowadays, seismic isolation system has been widely applied in the world to mitigate damage risk of structures. Although maximum displacement demand can be obtained through nonlinear time history (NLTH) analysis, many approximate methods are frequently recommended in structural specifications to reduce the required computational time. One of the best-known methods is the equivalent linear (EL) method, in which the nonlinear response of isolator can be adequately modeled using a fictitious viscously damped elastic structure. In this paper, a comparison between existing expressions supplying the state of research is carried out and then, an improved expression is presented for equivalent linearization of structures supported on lead rubber bearings (LRB). Based on the concept of secant stiffness, the optimal damping ratios, which minimize the errors of maximum displacement between EL analysis and NLTH analysis, are calculated and averaged over 12 ground motions. Then, a rational model to estimate equivalent damping ratio is derived through statistic analysis of the optimal damping ratios. To examine the prediction accuracy of the proposed model, mean ratios of approximate to exact maximum displacement and root mean square error for different isolated period are calculated as evaluation indicator. Compared with other EL models, the newly proposed model predicts a displacement that is in better agreement with the one obtained through NLTH analysis.

Anti-Seismic Energy Dissipation of Structures with Stress-Dependent Complex Damping with Consideration of Zero Amplitude Damping

Under most cases, the non-liner energy dissipation is approximately replaced with Rayleigh damping model which belongs to Maxwell-Kelvin type, the method is a fuzzy evaluation method of damping. Based on complex damping theory, equivalent dynamics equation of complex damping model is derived and loss factor is discussed, the accuracy of the equation is theoretically confirmed in this paper. Based on the existing damping theories, a new damping model is proposed, which is a zero amplitude damping model combing with stress dependent complex damping model. The model is used in seismic resistance analysis. Taking a steel beam for an example, the relationship between response amplitude and damping ratio is analyzed with the new damping model. It shows that the method of stress-dependent damping with consideration of zero amplitude damping can precisely describe the energy input and energy dissipation principles and the dynamic response under seismic loads can be precisely obtained. A solid foundation is laid for the further study of complex damping theory, its equivalent viscous damping model and engineering application.

Bidirectional Cyclic Loading Experiment on a 3D Beam–Column Joint Designed for Damage Avoidance

A near full-scale three-dimensional jointed precast prestressed concrete beam-to-column connection designed and constructed in accordance with an emerging damage avoidance design philosophy is tested under displacement-controlled quasistatic reverse cyclic loading. The performance of the subassembly is assessed under unidirectional loading along both orthogonal directions as well as under concurrent bidirectional loading. The specimen is shown to perform well up to a 4% column drift with only some minor flexural cracking in the precast beams, while the precast column remains uncracked and damage free. This superior performance is attributed to steel armoring of the beam ends to mitigate the potential for concrete crushing. Under bidirectional loading a tapered shear-key layout is used to effectively protect the beams against adverse torsional movements. A three-phase force – displacement relationship is proposed that gives due consideration to the prerocking flexural deformation of the beam; the rigid body kinematics during the rocking phase; and the yielding of the external dissipaters and posttensioning tendons. Good agreement is demonstrated between the proposed theoretical model and experimental observation. An equivalent viscous damping model is also proposed to represent both change in the prestress force in the subassembly and yielding of the supplemental energy dissipaters in the rocking connection.

Computing the dynamic response of an axially moving continuum

The aim of this study was to apply an approximate Galerkin finite-element method to solve the initial boundary-value problem of a viscously damped axially moving pre-tensioned beam including arbitrary support excitations. In order to validate the Galerkin finite-element method the results were compared with exact solutions when available and with solutions obtained using the finite-difference and Galerkin methods. The energy dissipation of the system was considered in the form of an equivalent viscous-damping model. It was also shown that for certain values of the parameters, especially at high velocities, the Galerkin method using stationary string eigenfunctions can give a poor prediction of the dynamic response.

Influence of harmonically varying normal load on steady-state behavior of a 2dof torsional system with dry friction

A nonlinear dry friction problem with harmonically varying normal load is formulated, in the context of a two-degree of freedom torsional system, since virtually all of the prior literature focuses on the topic of time-invariant normal load. First, pure stick, pure slip and stick–slip motions are computationally and analytically determined when excited by a sinusoidal torque, in the presence of harmonically varying saturation torque; mean terms are included in both. These analyses yield both transient and steady-state time histories under various conditions. Second, the effects of time-varying normal load on steady-state responses have been investigated and nonlinear spectral maps (including super-harmonics) are developed. Results show that the actuation system parameters could affect steady-state stick–slip motions in different ways over the lower and higher frequency regimes, as a result of time-delay in slip motions with respect to the torque excitation. In particular, the negative slope characteristics in the friction law exaggerate the stick–slip vibration problems, and it is the major cause of bifurcations and quasi-periodic or chaotic motions. Around the super-harmonic peak frequencies, the nonlinear system tends to lose stability as abrupt jumps in the spectral maps take place. An equivalent viscous damping model is considered to analytically investigate the instability mechanism. Further, the periodicity of the system response under harmonically varying actuation is conceptually by employing the harmonic balance method. Finally, steady-state behavior is examined for the nonlinear, time-varying dry friction problem.

An Efficient Technique for Design of Hydraulic Engine Mount via Design Variable-Embedded Damping Modeling

For an efficient design of hydraulic mounts, it is most important to have a good mathematical model available, which must be simple yet capable of representing dynamic characteristics of the hydraulic mounts accurately. Under high amplitude excitations in the low-frequency range, the hydraulic mounts show strongly frequency-dependent stiffness and damping characteristics, which are related with so-called inertia track dynamics. Since nonlinear damping models based on fluid mechanics are typically used to predict the dynamic characteristics of the hydraulic mounts, relations between various design variables, such as geometry of the inertia track, and resultant stiffness and damping characteristics are understood only by tedious numerical computations. In this paper, the use of an equivalent viscous damping model—derived from a nonlinear model and represented in terms of design variables in an explicit manner—is proposed and, based on the equivalent linear model, are presented simple as well as very useful formulas for an efficient design of the hydraulic mounts.

Time-domain response of linear hysteretic systems to deterministic and random excitations

The causal and physically realizable Biot hysteretic model proves to be the simplest linear model able to describe the nearly rate-independent behaviour of engineering materials. In this paper, the performance of the Biot hysteretic model is analysed and compared with those of the ideal and causal hysteretic models. The Laguerre polynomial approximation (LPA) method, recently proposed for the time-domain analysis of linear viscoelastic systems, is then summarized and applied to the prediction of the dynamic response of linear hysteretic systems to deterministic and random excitations. The parameters of the LPA model generally need to be computed through numerical integrals; however, when this model is used to approximate the Biot hysteretic model, closed-form expressions can be found. Effective step-by-step procedures are also provided in the paper, which prove to be accurate also for high levels of damping. Finally, the method is applied to the dynamic analysis of a highway embankment excited by deterministic and random ground motions. The results show that in some cases the inaccuracy associated with the use of an equivalent viscous damping model is too large. Copyright © 2005 John Wiley & Sons, Ltd.

IDENTIFICATION OF DAMPING: PART 2, NON-VISCOUS DAMPING

In a companion paper (see pp. 43–61 of this issue), it was shown that when a system is non-viscously damped, an identified equivalent viscous damping model does not accurately represent the damping behaviour. This demands new methodologies to identify non-viscous damping models. This paper takes a first step, by outlining a procedure for identifying a damping model involving an exponentially decaying relaxation function. The method uses experimentally identified complex modes and complex natural frequencies, together with the knowledge of the mass matrix for the system. The proposed method and several related issues are discussed by considering numerical examples of a linear array of damped spring-mass oscillators. It is shown that good estimates can be obtained for the exponential time constant and the spatial distribution of the damping.

Damping characterization of unidirectional fibre reinforced polymer composites

The main concern of this paper is the modelling of material damping in composite structures. A method is derived to predict the homogenized damping properties analytically. The procedure is based on the micromechanical theory of a representative fibre-matrix cell. Since the dissipative phenomena cannot generally be defined by a unique mathematical model, an equivalent viscous damping model is suggested. On the structural level damping can be analysed by the finite element method. Modal damping is illustrated by numerical applications.