Rayleigh Damping Model Research Articles

Effect of Damping on the Identification of Bridge Properties Using Vehicle Scanning Methods.

Vehicle scanning methods are gaining popularity because of their ability to identify modal properties of several bridges with only one instrumentation setup, and several methods have been proposed in the last decade. In the numerical models used to develop and validate such methods, bridge damping is often overlooked, and its impact on the efficacy of vehicle scanning methods remains unknown. The present article addresses this knowledge gap by systematically investigating the effects of bridge damping on the efficacy of vehicle scanning methods in identifying the modal properties of bridges. For this, acceleration responses obtained from a numerical model of a bridge and vehicle are used. Four different scenarios are considered where vehicle damping, presence of road roughness, and traffic on the bridge are varied. Bridge damping is modeled using mass-proportional, stiffness-proportional, and Rayleigh damping models. The impacts of ignoring bridge damping or considering one of these damping models on the modal frequencies and mode shapes identified using the vehicle response are investigated by comparing the results. The outcomes of the numerical analysis show that ignoring bridge damping in vehicle scanning applications can significantly increase the efficacy of these methods. They also show that the identifiability of the bridge frequencies and bridge mode shapes from the vehicle response decreases significantly when bridge damping is considered. Further, the damping model used impacts which bridge modes can be identified because different damping models provide different modal damping ratios for each mode. The results highlight the importance of correctly simulating damping behavior of bridges, which is often ignored, to be able to correctly evaluate the efficacy of vehicle scanning methods, and they provide an important stepping stone for future studies in this field.

Quantification of modeling uncertainty in the Rayleigh damping model

AbstractUnderstanding and accurately characterizing energy dissipation mechanisms in civil structures during earthquakes is an important element of seismic assessment and design. The most commonly used model is attributed to Rayleigh. This paper proposes a systematic approach to quantify the uncertainty associated with Rayleigh's damping model. Bayesian calibration with embedded model error is employed to treat the coefficients of the Rayleigh model as random variables using modal damping ratios. Through a numerical example, we illustrate how this approach works and how the calibrated model can address modeling uncertainty associated with the Rayleigh damping model.

Damping in masonry arch railway bridges under service loads: An experimental and numerical investigation

This article investigates the damping behavior of masonry arch bridges under service loads extracted from experimental data and provides guidelines on how to emulate this behavior in numerical analysis, particularly in discrete element model applications. First, an experimental campaign is undertaken and vibrations on three masonry arch railway bridges under train loads were monitored. The modal damping ratios from several sensors on each bridge were extracted by isolating the modal component of free decay vibrations which commence immediately after the train leaves the bridge. The modal damping ratios identified under service loads were compared with their counterparts identified under ambient vibrations. The suitability of mass-proportional, stiffness-proportional and Rayleigh damping models in emulating damping in masonry arch bridges was evaluated. In the numerical phase of the study, a single-arch masonry bridge was modeled using mixed discrete continuum approach and a moving load analysis was conducted without applying any additional viscous damping. The results of the numerical analysis indicate that the inherent damping in discrete element models provided by their nonlinear nature can be sufficient to emulate the damping behavior of masonry arch bridges under service loads. The research provided in this article is unique in the sense that it combines an experimental study and a numerical study on damping of masonry arch bridges under service loads. Unlike its counterparts in literature, which use either ambient vibrations or seismic action, damping values are computed under appropriate levels of vibration amplitudes for service loads, which is critical to estimate the modal damping ratios accurately under these loads.

Rayleigh Damping Modelling to Assess Viscous Behaviour in Actuated Breast Tissue

Breast cancer is a significant health problem worldwide. Emerging non-invasive methods assess tumor presence by its impact on breast tissue mechanics. This paper outlines a method to identify equivalent viscous damping in breast tissue and fit a model based on the Rayleigh damping (RD) model. Surface motion information of actuated breast tissue was captured using the Digital Image Elasto Tomography (DIET) system. The viscous damping was calculated for over 14,000 reference points using an ellipse fitted to the force-displacement hysteresis loop response data to calculate work done. A damping model based on RD is suggested and fit to median filtered data of viscous damping constant plotted against the major ellipse axis. This successfully described the trend for all 29 breasts (14 cancerous, 15 healthy) with average R2 values ranging from 0.79-0.88. One model coefficient, ‘a’, proportional to local mass, showed reasonable consistency among the subjects and decreased with increasing frequency. No significant difference in this ‘a’ coefficient between healthy and cancerous breasts showed indication of diagnostic potential. However, consistent values across all breasts suggest it is a fundamental tissue property. Further suggestions to normalise the major axis by frequency demonstrate the potential for the model to be developed to describe viscous behaviour of breast tissue in the form of storage and loss modulus.

Multi-objective optimum design of an aero engine rotor system using hybrid genetic algorithm

A working aero engine rotor system is subjected to multi-objective optimisation using Genetic Algorithm based optimisation. A Hybrid Genetic Algorithm (HGA) is introduced to reduce the weight and unbalance response of the rotor system with constrain on critical speed. The existing aero engine gear box casing vibration is found to be within the critical speed constraint and additional constraints are imposed to move the critical away from this zone. Bearing–pedestal model and Rayleigh damping model are used for accurate results. The optimisation resulted in Pareto optimal solutions and best solution selected using utopia point concept. The outcome of the paper is a comparative study which highlights the advantages of HGA over Controlled Elitist Genetic Algorithm (CEGA) and Goal Programming (GP).

Energy management through topology optimization of composites microstructure using projected gradient method

In this paper the projected gradient method is applied as an effective gradient-based topology optimization algorithm in order to direct energy propagation through the desired region of composites microstructure. Rayleigh Damping model is also used in order to take the effect of internal damping mechanisms into account and thus, to fill in the gap between the designed layouts and those in reality. The success of the proposed algorithm is illustrated through several numerical experiments by revealing a set of various designed optimal layouts besides their corresponding energy distributions.

Multi-frequency Rayleigh damped elastography: in silico studies

Rayleigh damping (RD) is commonly used to model energy attenuation for analyses of structures subjected to dynamic loads. In time-harmonic Magnetic Resonance Elastography (MRE), the RD model was shown to be non-identifiable at a single frequency data due to the ill-posed nature of the imaginary components describing energy dissipation arising from elastic and inertial forces. Thus, parametrisation or multi-frequency (MF) input data is required to overcome the fundamental identifiability issue of the model. While parametrisation allows improved accuracy of the identified parameters, simultaneous inversion using MF input data is a prerequisite for theoretical identifiably of the model. Furthermore, to establish good practical identifiability, frequencies should be separated over a wide range to produce different dynamic response. This research investigates the effects on practical identifiability of the RD model using MF data over different combinations of frequencies in noise-free heterogenous simulated geometry and compares the outcomes to reconstruction result based on single frequency input data. We tested eight frequencies in silico for a phantom type geometry comprises three independent material regions characterised by different mechanical properties. Combinations of two near or well separated frequencies are used to test the separation necessary to obtain accurate results, while the use of four or eight simultaneous frequencies is used to assess robustness. Results confirm expected non-identifiability of the RD model given single frequency input data. Practical identifiability of the RD parameters improved as more input frequencies were used for simultaneous inversion and when two frequencies were well separated. Best quality reconstruction results were achieved using full range data comprising eight available frequencies over a wide range. The main outcome is that high quality motion data over at least two frequencies over a wide range is required for establishing minimal practical identifiability of the model, while quality of the practical identifiability increases proportionally with more input frequencies used. Further simulation studies are required to determine acceptable signal-to-noise ratio (SNR) thresholds in motion data for accurate inversion of the RD parameters.

Multi-frequency inversion in Rayleigh damped Magnetic Resonance Elastography

Magnetic Resonance Elastography (MRE) is able to identify mechanical properties of biological tissues in vivo based on underlying assumptions of the model used for inversion. Models, such as the linearly elastic or viscoelastic (VE), can be used with a single input frequency data and can produce a reasonable estimate of identified parameters associated with mechanical properties. However, more complex models, such as the Rayleigh damping (RD) model, are not identifiable given single frequency data without significant a priori information under certain conditions, thus limiting diagnostic potential. To overcome this limitation, two approaches have been postulated: simultaneous inversion across multiple input frequencies and a parametric approach, when only single frequency data is available.This research compares simultaneous multi-frequency (MF) RD reconstructions using both zero-order and power-law (PL) models with parametric reconstructions for a series of tissue-simulating phantoms, made of tofu and gelatine materials, tested at 4 frequencies (50Hz, 75Hz, 100Hz and 125Hz) that are commonly applied in clinical MRE examinations. Results indicate that accurate delineation of RD based properties and concomitant damping ratio (ξd) using MF inversion is still a challenging task. Specific results showed that the real shear modulus (μR) can be reconstructed well, while imaginary components representing attenuation (μI and ρI ) had much lower quality. However, overall trends correlate well with the expected higher damping levels within the saturated tofu material compared to stiff gelatine in both phantoms. Depending on the phantom configuration, measured μR values within the tofu and gelatine materials ranged from 4.77 to 7kPa and 15.5 to 16.3kPa, respectively, while damping levels were 11–19% and 3.1–4.3%, as expected. Correlation of the μR and ξd values with previously reported result measured by independent mechanical testing and VE based MRE is acceptable, ranging from 48 to 60%. Both PL and zero-order models produced similar qualitative and quantitate results, thus no significant advantage of the PL model was noted to account for dispersion characteristics of these types of materials.The relatively narrow range of frequencies used in this study limited practical identifiability and can thus produce a potentially false assurance of identifiability of the model parameters. We conclude that application of multiple input frequencies over a wide range, as well as selection of an appropriate model that can accurately account for dispersion characteristics of given materials are required for achieving robust practical identifiability of the RD model in time-harmonic MRE.

Parametric-based brain Magnetic Resonance Elastography using a Rayleigh damping material model

The three-parameter Rayleigh damping (RD) model applied to time-harmonic Magnetic Resonance Elastography (MRE) has potential to better characterise fluid-saturated tissue systems. However, it is not uniquely identifiable at a single frequency. One solution to this problem involves simultaneous inverse problem solution of multiple input frequencies over a broad range. As data is often limited, an alternative elegant solution is a parametric RD reconstruction, where one of the RD parameters (μI or ρI) is globally constrained allowing accurate identification of the remaining two RD parameters. This research examines this parametric inversion approach as applied to in vivo brain imaging.Overall, success was achieved in reconstruction of the real shear modulus (μR) that showed good correlation with brain anatomical structures. The mean and standard deviation shear stiffness values of the white and gray matter were found to be 3±0.11kPa and 2.2±0.11kPa, respectively, which are in good agreement with values established in the literature or measured by mechanical testing. Parametric results with globally constrained μI indicate that selecting a reasonable value for the μI distribution has a major effect on the reconstructed ρI image and concomitant damping ratio (ξd). More specifically, the reconstructed ρI image using a realistic μI=333Pa value representative of a greater portion of the brain tissue showed more accurate differentiation of the ventricles within the intracranial matter compared to μI=1000Pa, and ξd reconstruction with μI=333Pa accurately captured the higher damping levels expected within the vicinity of the ventricles.Parametric RD reconstruction shows potential for accurate recovery of the stiffness characteristics and overall damping profile of the in vivo living brain despite its underlying limitations. Hence, a parametric approach could be valuable with RD models for diagnostic MRE imaging with single frequency data.

Parameter Identification in a Generalized Time-Harmonic Rayleigh Damping Model for Elastography

The identifiability of the two damping components of a Generalized Rayleigh Damping model is investigated through analysis of the continuum equilibrium equations as well as a simple spring-mass system. Generalized Rayleigh Damping provides a more diversified attenuation model than pure Viscoelasticity, with two parameters to describe attenuation effects and account for the complex damping behavior found in biological tissue. For heterogeneous Rayleigh Damped materials, there is no equivalent Viscoelastic system to describe the observed motions. For homogeneous systems, the inverse problem to determine the two Rayleigh Damping components is seen to be uniquely posed, in the sense that the inverse matrix for parameter identification is full rank, with certain conditions: when either multi-frequency data is available or when both shear and dilatational wave propagation is taken into account. For the multi-frequency case, the frequency dependency of the elastic parameters adds a level of complexity to the reconstruction problem that must be addressed for reasonable solutions. For the dilatational wave case, the accuracy of compressional wave measurement in fluid saturated soft tissues becomes an issue for qualitative parameter identification. These issues can be addressed with reasonable assumptions on the negligible damping levels of dilatational waves in soft tissue. In general, the parameters of a Generalized Rayleigh Damping model are identifiable for the elastography inverse problem, although with more complex conditions than the simpler Viscoelastic damping model. The value of this approach is the additional structural information provided by the Generalized Rayleigh Damping model, which can be linked to tissue composition as well as rheological interpretations.

Initial versus tangent stiffness‐based Rayleigh damping in inelastic time history seismic analyses

SUMMARYIn the inelastic time history analyses of structures in seismic motion, part of the seismic energy that is imparted to the structure is absorbed by the inelastic structural model, and Rayleigh damping is commonly used in practice as an additional energy dissipation source. It has been acknowledged that Rayleigh damping models lack physical consistency and that, in turn, it must be carefully used to avoid encountering unintended consequences as the appearance of artificial damping. There are concerns raised by the mass proportional part of Rayleigh damping, but they are not considered in this paper. As far as the stiffness proportional part of Rayleigh damping is concerned, either the initial structural stiffness or the updated tangent stiffness can be used. The objective of this paper is to provide a comprehensive comparison of these two types of Rayleigh damping models so that a practitioner (i) can objectively choose the type of Rayleigh damping model that best fits her/his needs and (ii) is provided with useful analytical tools to design Rayleigh damping model with good control on the damping ratios throughout inelastic analysis. To that end, a review of the literature dedicated to Rayleigh damping within these last two decades is first presented; then, practical tools to control the modal damping ratios throughout the time history analysis are developed; a simple example is finally used to illustrate the differences resulting from the use of either initial or tangent stiffness‐based Rayleigh damping model. Copyright © 2013 John Wiley & Sons, Ltd.

Non-identifiability of the Rayleigh damping material model in Magnetic Resonance Elastography

Magnetic Resonance Elastography (MRE) is an emerging imaging modality for quantifying soft tissue elasticity deduced from displacement measurements within the tissue obtained by phase sensitive Magnetic Resonance Imaging (MRI) techniques. MRE has potential to detect a range of pathologies, diseases and cancer formations, especially tumors. The mechanical model commonly used in MRE is linear viscoelasticity (VE). An alternative Rayleigh damping (RD) model for soft tissue attenuation is used with a subspace-based nonlinear inversion (SNLI) algorithm to reconstruct viscoelastic properties, energy attenuation mechanisms and concomitant damping behavior of the tissue-simulating phantoms. This research performs a thorough evaluation of the RD model in MRE focusing on unique identification of RD parameters, μI and ρI.Results show the non-identifiability of the RD model at a single input frequency based on a structural analysis with a series of supporting experimental phantom results. The estimated real shear modulus values (μR) were substantially correct in characterising various material types and correlated well with the expected stiffness contrast of the physical phantoms. However, estimated RD parameters displayed consistent poor reconstruction accuracy leading to unpredictable trends in parameter behaviour. To overcome this issue, two alternative approaches were developed: (1) simultaneous multi-frequency inversion; and (2) parametric-based reconstruction. Overall, the RD model estimates the real shear shear modulus (μR) well, but identifying damping parameters (μI and ρI) is not possible without an alternative approach.

Evaluation of Rayleigh damping and its influence on engineering demand parameter estimates

SUMMARYThe effects of Rayleigh damping model on the engineering demand parameters of two steel moment‐resisting frame buildings were evaluated. Two‐dimensional models of the buildings were created and response history analysis were conducted for three different hazard levels. The response history analysis results indicate that mass‐proportional damping leads to high damping forces compared with restoring forces and may lead to overestimation of floor acceleration demands for both buildings. Stiffness‐proportional damping, on the other hand, is observed to suppress the higher‐mode effects in the nine‐story building resulting in lower story drift demands in the upper floors compared with other damping models. Rayleigh damping models, which combine mass‐proportional and stiffness‐proportional components, that are anchored at reduced modal frequencies lead to reasonable damping forces and floor acceleration demands for both buildings and does not suppress higher‐mode effects in the nine‐story building. Copyright © 2012 John Wiley & Sons, Ltd.