Rayleigh Damping Research Articles

Dynamics and potential applications of a lunar space tethered system

Nonlinear dynamics and potential applications of a lunar space tether system, with one end connected to the Moon's surface and the other with a large floating counterweight, are investigated in the paper. Dynamical equations of the system with a massless viscoelastic tether considering the planar libration and elastic elongation within the elliptical Earth-Moon restricted three-body (EEMRTB) system are established. The equilibria and stability of the dynamical system are explored for the system at L1 side without considering small perturbations caused by eccentricity to facilitate the rough selection of the length and elastic coefficient of the tether. The potential applications including harvesting the Rayleigh Damping Dissipation power and accessible regions within the Earth-Moon system will be studied based on the dynamic analysis of the full nonlinear dynamics. The allowed parameter regions (the tether system will neither collide the Moon/Earth nor rotate about the Moon) are identified exactly using numerical analysis, moreover, the performance of the potential applications and other related references for building a tether system are explored at L1 and L2 numerically. It is suggested that one should design a space tether based energy harvester at L1 side, release an unpowered spacecraft with high mechanical energy at L2 side respectively. Some conclusions on selecting the parameters and evaluating the performance of the space tether based platform are obtained based on numerical calculations. The research will be useful for the Moon infrastructure constructions.

An adaptive hybrid expansion method (AHEM) for efficient structural topology optimization under harmonic excitation

One challenge of solving topology optimization problems under harmonic excitation is that usually a large number of displacement and adjoint displacement vectors need to be computed at each iteration step. This work thus proposes an adaptive hybrid expansion method (AHEM) for efficient frequency response analysis even when a large number of excitation frequencies are involved. Assuming Rayleigh damping, a hybrid expansion for the displacement vector is developed, where the contributions of the lower-order modes and higher-order modes are given by the modal superposition and Neumann expansion, respectively. In addition, a simple (yet accurate) expression is derived for the residual error of the approximate displacement vector provided by the truncated hybrid expansion. The key factors affecting the convergence rate of the truncated hybrid expansion series are uncovered. Based on the Strum sequence, the AHEM can adaptively determine the number of lower-order eigenfrequencies and eigenmodes that need to be computed, while the number of terms that need to be preserved in the truncated Neumann expansion can be determined according to the given error tolerance. The performance of the proposed AHEM and its effectiveness for solving topology optimization problems under harmonic excitation are demonstrated by examining several 2D and 3D numerical examples. The non-symmetry of the optimum topologies for frequency response problems is also presented and discussed.

Ritz Solution for Transient Analysis of Variable-Stiffness Shell Structures

The dynamic response of thin-walled structures is driven by mass and stiffness distribution. As such, variable-stiffness (VS) composites offer opportunities to tune structural dynamic responses. To this extent, efficient analysis tools become increasingly important for structural analysis and design purposes. In this work, an efficient and versatile Ritz method for free vibrations and linear transient analysis of VS doubly curved shell structures is presented. VS shell structures are modeled as an assembly of shell-like domains. The shell kinematics is based on the first-order shear deformation theory, and no further assumption is made on the shallowness or on the thinness of the structure. The description of the shell is provided by a rational Bézier surface representation, and general surface geometries can be represented. Legendre polynomials are employed to approximate the displacement field, and penalty techniques are used to enforce displacement continuity and kinematical boundary conditions. Classical Rayleigh damping is considered, and solutions are obtained through the Newmark integration. The resulting model allows a wide range of configurations and load cases for multicomponent, VS composite structures to be solved, providing the same levels of accuracy as finite element analysis, yet with a reduced number of variables and simpler data preparation.

Rayleigh Damping Modelling for Tumor Detection using Digital Image Elasto Tomography (DIET)

This study develops a model based on Rayleigh Damping (RD) with potential use in breast cancer diagnostics. Displacement data of over 14,000 reference points on the breast surface from 14 breasts was captured using the Digital Image Elasto Tomography (DIET) system. The reference points were split into four segments and an ellipse fit utilized to calculate the work done and consequent viscous damping constant for each reference point. Fitting a model based on RD to median filtered data gave consistent results for one model coefficient across all breasts. The other coefficient was seen to have diagnostic potential when the model was fit to unfiltered data, and is the focus of this paper. The coefficient value was compared between breast segments adjacent to and containing the tumor (locations given from X-ray mammography) to those opposite the tumor. A total of 11 out of 14 breasts had a higher coefficient found in the tumor segment and all breasts had a higher coefficient in at least one adjacent segment. This method showed potential for breast specific diagnosis and tumor localisation using the DIET system.

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.

Dynamic Performance and Anti-earthquake Analysis of Cable-stayed Arch Bridge

The strength, stiffness, and stability check calculations and the effect of earthquakes should be considered in the design of cable-stayed arch bridges with collaborative systems. This study aims to investigate the dynamic performance and structural response of cable-stayed arch bridges under seismic action. The space analysis model is enhanced of the Xiang Feng River Bridge using finite element software Midas Civil, whose lower foundation considers the effects of piles and soil. Firstly the vibration period, vibration frequency, and modal characteristics are computed, thus the dynamic performance is summarized of the bridge. Then, a proper seismic wave is selected according to engineering conditions and in terms of three orthogonal directions: inputting the adjusted El Centro seismic wave, considering Rayleigh damping, and calculating via the Newmark method. Furthermore, a time-history response analysis under the action of one-dimensional and multidimensional earthquake is performed. Lastly, the results of the response analysis is compared and the behavior characteristics of arch bridge is summarized under seismic action. The results show that the transverse stability problem of bridges is prominent and should be the focus of antiearthquake fortification, the inclined cable tower of this bridge is not conducive to the earthquake resistance of the structure in comparison with the vertical cable tower. and the influence of horizontal and vertical earthquake actions should be considered in antiearthquake designs.

On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic

Abstract In this article, the study of qualitative properties of a special type of non-autonomous nonlinear second order ordinary differential equations containing Rayleigh damping and generalized Duffing functions is considered. General conditions for the stability and periodicity of solutions are deduced via fixed point theorems and the Lyapunov function method. A gyro dynamic application represented by the motion of axi-symmetric gyro mounted on a sinusoidal vibrating base is analyzed by the interpretation of its dynamical motion in terms of Euler’s angles via the deduced theoretical results. Moreover, the existence of homoclinic bifurcation and the transition to chaotic behaviour of the gyro motion in terms of main gyro parameters are proved. Numerical verifications of theoretical results are also considered.

Geometrically nonlinear dynamic analysis of FG graphene platelets-reinforced nanocomposite cylinder: MLPG method based on a modified nonlinear micromechanical model

The present paper outlined a procedure for geometrically nonlinear dynamic analysis of functionally graded graphene platelets-reinforced (GPLR-FG) nanocomposite cylinder subjected to mechanical shock loading. The governing equation of motion for large deformation problems is derived using meshless local Petrov-Galerkin (MLPG) method based on total lagrangian approach. In the MLPG method, the radial point interpolation technique is employed to construct the shape functions. A micromechanical model based on the Halpin-Tsai model and rule of mixture is used for formulation the nonlinear functionally graded distribution of GPLs in polymer matrix of composites. Energy dissipation in analyses of the structure responding to dynamic loads is considered using the Rayleigh damping. The Newmark-Newton/Raphson method which is an incremental-iterative approach is implemented to solve the nonlinear dynamic equations. The results of the proposed method for homogenous material are compared with the finite element ones. A very good agreement is achieved between the MLPG and FEM with very fine meshing. In addition, the results have demonstrated that the MLPG method is more effective method compared with the FEM for very large deformation problems due to avoiding mesh distortion issues. Finally, the effect of GPLs distribution on strength, stiffness and dynamic characteristics of the cylinder are discussed in details. The obtained results show that the distribution of GPLs changed the mechanical properties, so a classification of different types and volume fraction exponent is established. Indeed by comparing the obtained results, the best compromise of nanocomposite cylinder is determined in terms of mechanical and dynamic properties for different load patterns. All these applications have shown that the present MLPG method is very effective for geometrically nonlinear analyses of GPLR-FG nanocomposite cylinder because of vanishing mesh distortion issue in large deformation problems. In addition, since in proposed method the distributed nodes are used for discretization the problem domain (rather than the meshing), modeling the functionally graded media yields to more accurate results.

Rayleigh Damping Coefficients of Laminated Rubber Bearing

In isolating the ground structure and the above ground structure from seismic loads, a significant device called laminated rubber bearing is usually found in structure. The complexity of the material which is made up from a combination of rubber and steel shim plates in alternate layer, has made it difficult to measure damping value. Damping is a dissipation of energy or energy losses in the vibration of the structure. Measuring the accurate amount of damping is fundamental as damping plays a crucial role in fixing the borderline between stability and instability in structural systems. Therefore, to determine the damping value including dynamic properties in any materials, modal analysis can be used. Hence, the main objective of this research is to determine the Rayleigh’s damping coefficients α and β and to evaluate the performance of the laminated rubber bearing using finite element and experimental modal analysis. Finding shows that, the finite element modal analysis with the addition of Rayleigh’s damping coefficients α and β, shows a good agreement with the experimental modal analysis in term of natural frequencies and mode shapes. Findings show that, the values of natural frequencies reduced when precise Rayleigh’s damping coefficient added in the finite element modal analysis. It can be concluded that both finite element and experimental modal analysis method can be used to estimate the accurate values of damping ratio and to determine the Rayleigh’s damping coefficients α and β as well.

Effects of the improper modeling of viscous damping on the first‐mode and higher‐mode dominated responses of base‐isolated buildings

SummaryIn many finite element platforms, a classical global damping matrix based on the elastic stiffness of the system (including isolators) is usually developed as part of the solution to the equations of motion of base‐isolated buildings. The conducted analytical and numerical investigations illustrate that this approach can lead to the introduction of unintended damping to the first and higher vibration modes and the spurious suppression of the respective structural responses. A similar shortcoming might be observed even when a nonclassical damping model (ie, an assembly of the superstructure and isolation system damping sub‐matrices) is used. For example, the use of Rayleigh damping approach to develop the superstructure damping sub‐matrix can lead to the undesired addition of damping to the isolated mode arising from the mass‐proportional component of the superstructure damping. On the other hand, the improper use of nonclassical stiffness‐proportional damping (eg, determining the proportional damping coefficient, βk, based on the first mode) can result in assigning significant damping to the higher‐modes and the unintended mitigation of the higher‐mode responses. Results show that a nonclassical stiffness‐proportional model in which βk is determined based on the second modal period of a base‐isolated building can reasonably specify the intended damping to the higher modes without imparting undesirable damping to the first mode. The nonclassical stiffness‐proportional damping can be introduced to the numerical model through explicit viscous damper elements attached between adjacent floors. In structural analysis software such as SAP2000®, the desired nonclassical damping can be also modeled through specifying damping solely to the superstructure material.

Dynamic behaviour of composite panels under acoustic loading

PurposeThere are structural elements on the aircraft that may be exposed to high-intensity sound levels. One of them is an air inlet duct of the jet engine. To prepare data for the air duct damage tolerance analysis, flat panels were tested under acoustic loading. The paper aims to discuss this issue.Design/methodology/approachThe acoustic fatigue test equipment for grazing wave’s incidence was designed based on the FE analyses. Flat composite panels were designed and manufactured using the Hexply 8552/AGP193-PW prepreg with the simulation of production imperfections or operational damage. The dynamic behaviour of panels has been tested using three regimes of acoustic loading: white noise spectrum, engine noise spectrum and discrete harmonic frequencies. The panel deflection was monitored along its longitudinal axis, and the ultrasonic NDT instruments were used for the monitoring of relevant delamination increments. The FE model of the panel was created in Abaqus to study panel dynamic characteristics.FindingsNo delamination progress was observed by NDT testing even if dynamic characteristics, especially modal frequency, of the panel changed during the fatigue test. Rayleigh damping coefficients were evaluated for their use in FE models. Significant differences were found between the measured and computed panel deflection curves near the edge of the panel.Originality/valueThe research results underscored the signification of the FE model boundary conditions and the element type selections when the panel works like a membrane rather than a plate because of their low bending stiffness.

A Bayesian model updating with incomplete complex modal data

Abstract Finite element (FE) model updating using measured dynamic data has gained much attention in view of its contribution towards important areas like model-correction, structural health monitoring. Usually model updating is observed to be performed using measured modal data like modal frequencies and mode shapes – indeed very few works are observed to provide scope for using damping measurements in FE model updating. The present work proposes a Bayesian probabilistic FE model updating using multiple sets of complex modal data (viz. complex modal frequencies and complex mode shapes) and thereby enabling the scope for incorporating damping measurements in updating. Moreover there remains the opportunity for using incomplete complex modal data and avoiding the requirement of mode matching. In this paper, a computationally efficient model updating approach based on maximum a posteriori (MAP) is formulated where all the mass, stiffness and damping parameters are updated in a sequential iterative manner. Detailed formulations are obtained considering damping to be contributed by classical damping (Rayleigh damping) and non-classical viscous damping. Besides, formulations for uncertainty estimation and probabilistic damage detection are also developed. The proposed approach is validated using two numerical examples while utilizing incomplete complex modal data. Performance in updating is evaluated for both undamaged and damaged cases. The proposed approach is observed to be successful for updating of FE model as well as for detection of changes/damages in probabilistic manner, while being computationally efficient as well.

A METHOD FOR DETERMINING RAYLEIGH DAMPING PARAMETERS OF COMPLEX FIELD

Rayleigh damping model, which is still adopted by many general finite element programsand widely used in analysis of engineering, leads to the inconformity of the calculating modaldamping ratio with the actual modal damping ratios. The appropriate Rayleigh damping matrix issignificant for accurate dynamic response analysis of complicated field. This paper establishes adual parameter optimization theory for calculation of Rayleigh damping coefficients. The functionalrelation between the relative error of dynamic response and Rayleigh damping coefficients isestablished based on CQC method. By taking the square sum of the errors of peak value ofdisplacement and the error of peak value of acceleration at the multiple points (DOF) of the surfaceof the complex site as the control objective, the equations for solving Rayleigh damping coefficientsare obtained based on the principle of minimizing the control objective. Then, as an example, theseismic response of a valley under the excitation of 28 representative seismic waves which arerandomly selected is calculated and the error due to Rayleigh damping model is analysed. Thenumerical result verifies the accuracy and applicability of the proposed method.

Response of micropiles in different seismic conditions

In recent years, due to the growth of the world population, suitable lands for construction and building are being reduced, gradually; hence, researchers have continuously sought to increase the bearing capacity and resistance of soil and improve its properties. One of the effective methods for improvement of the soil bearing capacity and reduction in foundation settlement is the implementation of micropiles below them. In this study, the seismic response of micropiles under real seismic loading was performed using finite element software. The stabilized soil behavior was modeled as elastoplastic along with Rayleigh damping, and micropiles were modeled as elastic beam elements by considering the interaction between soil and structure. The parametric study aims to focus on how to connect micropiles with foundation and effect of earthquake intensity. The results of the analysis show that decreased stiffness of the micropiles foundation system against the applied forces affects the values of lateral displacement and bending moment. Also, it is found that the Arias intensity of an earthquake is important parameter in characterizing of an earthquake nature influence on pile performance in soil.

Rayleigh damping parameters estimation using hammer impact tests

Structural Health Monitoring (SHM) systems for aerospace applications are becoming more prevalent and its potential cost-benefit relation is rapidly improving as more structures that were traditionally made with metallic materials are gradually being replaced by composites. To support these systems, simulations can play a crucial role; however, in the case of low energy impacts, the structural damping becomes a very significant element in the simulation. This damping is usually not well known, or is based in estimates with very low reliability. Therefore, in order to improve the reliability of these values this paper presents an attempt to estimate the Rayleigh damping parameters with the application of time-frequency analysis methods to transient signals, specifically, the reduced interference distribution. These parameters are estimated based in the results of a large number of structural tests done in a square carbon-fiber reinforced plastic panel, and are then validated by correlation of an explicit Finite Element Method (FEM) simulation. A discussion on how to apply the reduced interference distribution to the signals measured by piezoelectric sensors under an impact scenario and other possible usages of these results is also presented in this document.

Theoretical and experiment analysis on the sound absorption characteristics of a layer of fine lightweight powder.

In this study, the sound absorption characteristics of lightweight powders with a particle size on the order of tens of micrometers have been investigated. Herein, to theoretically determine the sound absorption coefficient of a fine lightweight powder in the longitudinal vibration mode, it was assumed that the powder was continuum and could be treated as a multiple degree-of-freedom damping vibration system. In addition, the acoustic impedance was calculated by assuming proportional viscous damping to determine the sound absorption coefficient. Finally, the theoretical and experimental values were compared. With the presented theoretical analysis, values close to the experimental results were obtained. Among the three calculated values based on proportional viscous damping, i.e., Rayleigh damping, rigidity proportional damping, and mass proportional damping, the results for Rayleigh damping were the closest to the experimental values.

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).

Calculating the Rayleigh damping coefficient under seismic dynamic simulation of a quayside container crane

ABSTRACT According to the Rayleigh damping coefficients based on the dynamic simulation analysis of a quayside container crane (QCC), the calculation and frequency selection for QCC type structures are improved through comparative analysis of similar models and simulation results; these models consider the boundary constraints of QCC and the calculation method of damping coefficients, but there is no consensus on the best methodology to use. The dynamic response of the QCC structure under different Rayleigh damping coefficients is compared and analysed by simulation and shaking table tests. The results show that the damping coefficient calculation method and frequency selection on the seismic response of the QCC structure increases with increased seismic acceleration. When the acceleration peak is 0.4 g and 0.62 g, the average error between the simulation result and the test value obtained by the existing damping coefficient calculation method exceeds 20%. In contrast, the error between the simulation result and the test value is within 10% when using the improved damping coefficient calculation method.

Numerical artifacts associated with Rayleigh and modal damping models of inelastic structures with massless coordinates

SummaryThe paper presents a detailed reexamination of the effects of three damping models on the inelastic seismic response of structures with massless degrees of freedom. The models considered correspond to (a) Rayleigh damping based on current properties (tangent stiffness), (b) Rayleigh damping based on initial properties, and (c) modal damping. The results suggest that some nonzero damping forces/moments at massless DOFs obtained in multistory frames for the case of Rayleigh damping with tangent stiffness may be numerical artifacts rather than a deficiency of the damping model. The results also indicate that significant artificial numerical oscillations in the velocities of the secondary components of MDOF structures are introduced when modal damping or mass‐proportional damping is used.

Size-dependent nonlinear analysis and damping responses of FG-CNTRC micro-plates

This paper presents a nonlinear numerical model, which is based on the modified couple stress theory (MCST), and trigonometric shear deformation theory coupled with isogeometric analysis. The present approach captures the small scale effects on the geometrically nonlinear behaviors of functionally graded carbon nanotube reinforced composite (FG-CNTRC) micro-plate with four patterns distribution. The equations of motion are established based on a Galerkin weak form associated with von-Kármán nonlinear strains. The MCST utilizes only one material length scale parameter to predict the size effect in FG-CNTRC micro-plate, for which its material properties are derived from an extended rule of mixture. The solutions of nonlinear static equation are obtained by using the Newton–Raphson technique and the Newmark time iteration procedure in association with Picard method is assigned to get responses of the nonlinear dynamic problems. In addition, the Rayleigh damping is applied to consider the influence of damping characteristic on the oscillation of FG-CNTRC micro-plates. Comparisons are performed to verify the proposed approach. Afterward, the numerical examples are used to show the effects of the distribution of carbon nanotubes (CNT), their volume fraction, the material length scale parameter and the boundary conditions on the nonlinear static and dynamic behaviors of FG-CNTRC micro-plates.