Proportional Damping Research Articles

종동력을 받는 외팔기둥의 동적 안정성에 미치는 구조감쇠 효과

안정성 지도(stability map)을 이용하여 부분 종동력(sub-tangentailly follower force)를 받는 외팔 기둥의 동적안정성 이론을 요약한다. Rayleigh 감쇠를 가정하여 내적 및 외적 감쇠효과를 2개의 감쇠비를 통하여 반영하고, 감쇠비 변화에 따른 플러터하중의 변화와 관련된 매개변수 연구를 수행한다. 또한, 종동력을 받는 외팔기둥에 대한 진동수 방정식의 엄밀해를 유도하고, 특정 감쇠비 범위에 대한 안정성 지도를 유한요소 해석결과와 함께 비교/분석한다. A stability theory of a damped cantilever column under sub-tangential follower forces is first summarized based on the stability map. It is then demonstrated that internal and external damping can be exactly transformed to Rayleigh damping so that the damping coefficients can be effectively determined using proportional damping. Particularly a parametric study with variation of damping coefficients is performed in association with flutter loads of Beck's column and it is shown that two damping coefficients can be correctly estimated for real systems under the assumption of Rayleigh damping. Finally a frequency equation of a cantilever beam subjected to both a sub-tangentially follower force and two kinds of damping forces is presented in the closed-form and its stability maps are constructed and compared with FE solutions in the practical range of damping coefficients.

Evolution of a primary pulse in the granular dimers mounted on a linear elastic foundation: An analytical and numerical study.

In this exposition we consider the wave dynamics of a one-dimensional periodic granular dimer (diatomic) chain mounted on a damped and an undamped linear elastic foundation (otherwise called the on-site potential). It is very well known that periodic granular dimers support solitary wave propagation (similar to that in the homogeneous granular chains) for a specific discrete set of mass ratios. In this work we present the analytical investigation of the evolution of solitary waves and primary pulses in granular dimers when they are mounted on on-site potential with and without velocity proportional foundation damping. We invoke a methodology based on the multiple time-scale asymptotic analysis and partition the dynamics of the perturbed dimer chain into slow and fast components. The dynamics of the dimer chain in the limit of large mass mismatch (auxiliary chain) mounted on on-site potential and foundation damping is used as the basis for the analysis. A systematic analytical procedure is then developed for the slowly varying response of the beads and in estimating primary pulse amplitude evolution resulting in a nonlinear map relating the relative displacement amplitudes of two adjacent beads. The methodology is applicable for arbitrary mass ratios between the beads. We present several examples to demonstrate the efficacy of the proposed method. It is observed that the amplitude evolution predicted by the described methodology is in good agreement with the numerical simulation of the original system. This work forms a basis for further application of the considered methodology to weakly coupled granular dimers which finds practical relevance in designing shock mitigating granular layers.

Dynamic particle difference method for the analysis of proportionally damped system and cracked concrete beam

A new dynamic particle difference method (PDM) for the simulation of a proportionally damped system subjected to dynamic load and the fracture simulation of cracked concrete beam is presented. The dynamic PDM utilizes only node model without involving any mesh or grid structure to take advantage of the merits of strong formulation; it remarkably accelerates computational speed owing to the avoidance of numerical integration and also sophisticatedly handles awkward topological change due to the crack growth in node model. The proportional damping is successfully implemented in the dynamic PDM by adding both mass and stiffness proportional terms to the equation of motion and the constitutive equation, respectively. It provides extra stability in the dynamic fracture simulation by eliminating erroneous oscillations. Governing partial differential equation is straightforwardly discretized in time by using the central difference method. However, a novel modification is devised in the Newmark method implementation where the transient equations and update formulae for kinematic variables are newly formulated; this modification appropriately corrects the period and amplitude of kinematic variable responses. The stability and accuracy of the developed methods are verified by solving various dynamic problems involving transient loading. Furthermore, the fracture process of the cracked concrete beam under the impact loading is successfully simulated and the dynamic energy release rates are effectively evaluated during the simulation.

Synchronization of dynamic response measurements for the purpose of structural health monitoring

This paper presents a technique for offline time synchronization of data acquisition systems for linear structures with proportional damping. The technique can be applied when direct synchronization of data acquisition systems is impossible or not sufficiently accurate. The synchronization is based on the acquired dynamic response of the structure only, and does not require the acquisition of a shared sensor signal or a trigger signal. The time delay is identified from the spurious phase shift of the mode shape components that are obtained from system identification. A demonstration for a laboratory experiment on a cantilever steel beam shows that the proposed methodology can be used for accurate time synchronization, resulting in a significant improvement of the accuracy of the identified mode shapes.

Vibration and damping analysis of the bladed disk with damping hard coating on blades

For the blisk (integrally bladed disk) with damping hard coating on blades, an analytical approach is developed to calculate its free vibration characteristics and damping effect, based on the constitutive model of complex modulus and Rayleigh–Ritz method. Firstly, by using the Oberst beam theory, equivalent elastic modulus and loss factor of the coated blade are derived. Then energy equations of the bladed disk on basis of complex modulus are given. By employing a set of orthogonal polynomials as the admissible function, the Rayleigh–Ritz method is used to formulate the equations of motion. Solutions are obtained as complex eigenvalues, whose real parts are undamped frequencies and imaginary parts represent the damping coefficient. Further, frequency response function can be achieved, by using of proportional damping model to obtain damping matrix. An academic bladed disk made of stainless steel is taken as example to conduct numerical calculation, and compared with the experimental results by both natural frequencies and modal shapes. NiCrAlY coating is deposited on single side of the blades to investigate its effect on the natural frequencies, modal loss factors and frequency responses. At last, influence of the coating thickness on the variation of natural frequencies and damping capacity is discussed.

Offline synchronization of data acquisition systems using system identification

This paper presents a technique for offline time synchronization of data acquisition systems. The technique can be applied when real-time synchronization of data acquisition systems is impossible or not sufficiently accurate. It allows for accurate synchronization based on the acquired dynamic response of the structure only, without requiring a common response or the use of a trigger signal. The synchronization is performed using the results obtained from system identification, and assumes linear dynamic behavior of the structure and proportional damping of the structural modes. A demonstration for a laboratory experiment on a cantilever steel beam shows that the proposed methodology can be used for accurate time synchronization, resulting in a significant improvement of the accuracy of the identified mode shapes.

Issues with Proportional Damping

Analytical derivation of damping is extremely difficult and in most cases not possible. As a result, damping properties are typically specified on a mode-by-mode basis because of the availability of test data that can be used either directly, or can be used as a guide for similar systems for which data are not yet available. Occasionally there is need to develop from the modal damping properties damping matrices that correspond to the physical coordinate sets in which the finite element model mass and stiffness matrices were developed. Proportional damping, in which combinations of mass and stiffness matrices are used to develop the physical coordinate damping matrix, has been used extensively. It is the purpose of this work to discuss the appropriateness of including a mass proportional term. It is concluded that damping formulations that include a scaled mass term are fundamentally flawed and should not be used.

An Inverse Approach for the Determination of\\ Viscous Damping Model of Fibre Reinforced Plastic Beams using Finite Element Model Updating

Investigations have been carried out both numerically and experimentally to settle with a practically feasible set of proportional viscous damping parameters for the accurate prediction of responses of fibre reinforced plastic beams over a chosen frequency range of interest. The methodology needs accurate experimental modal testing, an adequately converged finite element model, a rational basis for correct correlations between these two models, and finally, updating of the finite element model by estimating a pair of global viscous damping coefficients using a gradient-based inverse sensitivity algorithm. The present approach emphasises that the successful estimate of the damping matrix is related to a-priori estimation of material properties, as well. The responses are somewhat accurately predicted using these updated damping parameters over a large frequency range. In the case of plates, determination of in-plane stiffness parameters becomes easier, whereas for beam specimens, transverse material properties play a comparatively greater role and need to be determined. Moreover, for damping matrix parameter estimation, frequency response functions need to be used instead of frequencies and mode shapes. The proposed method of damping matrix identification is able to reproduce frequency response functions accurately even outside the frequency ranges used for identification.

Multi-body modelling of single-mast stacker cranes

In the frame structure of stacker cranes during non-stationary phases of movement due to inertial forces undesirable mast vibrations may occur. This effect can reduce the stability and positioning accuracy of these machines. The aim of this paper is to introduce an accurate and quite simple dynamical model of single-mast stacker cranes, which is suitable for investigating the mast vibrations of these machines. The multi-body modelling approach is selected to generate the differential equations of motion for this model. The solution of these equations is performed by means of the so-called modal coordinate transformation or modal superposition method. In this model structural damping is taken into consideration by means of the so-called proportional damping (Rayleigh damping) approach. The main advantage of the presented multi-body model is that with this model the mast-vibrations can be investigated in various positions of the mast. Dynamic models with varying lifted load positions can also be generated in simple way by using the introduced modelling technique. The main properties, i.e., the state space representation of our model as well as time domain simulation results, are also introduced.

시뮬레이션과 실험을 통한 전개하는 보의 횡 방향 진동 분석

The transverse vibration of the deploying beam from rigid hub was analyzed by simulation and experiment. The linear governing equation of the deploying beam was obtained using the Euler-Bernoulli beam theory. To discretize the governing equation, the Galerkin method was used. After transforming the governing equation into the weak form, the weak form was discretized. The discretized equation was expressed by the matrix-vector form, and then the Newmark method was applied to simulate. To consider the damping effect of the beam, we conducted the modal test with various beam length. The mass proportional damping was selected by the relation of the first and second damping ratio. The proportional damping coefficient was calculated using the acquired natural frequency and damping ratio through the modal test. The experiment was set up to measure the transverse vibration of the deploying beam. The fixed beam at the carriage of the linear actuator was moved by moving the carriage. The transverse vibration of the deploying beam was observed by the Eulerian description near the hub. The deploying or retraction motion of the beam had the constant velocity and the velocity profile with acceleration and deceleration. We compared the transverse vibration results by the simulation and experiment. The observed response by the Eulerian description were analyzed.

The ‘damping effect’ in the dynamic response of stochastic oscillators

We consider the dynamics of linear damped oscillators with stochastically perturbed natural frequencies. When average dynamic response is considered, it is observed that stochastic perturbation in the natural frequency manifests as an increase of the effective damping of the system. Assuming uniform distribution of the natural frequency, a closed-from expression of equivalent damping for the mean response has been derived to explain the ‘increasing damping’ behaviour. In addition to this qualitative analysis, a comprehensive quantitative analysis is proposed to calculate the statistics of frequency response functions from the probability density functions of the natural frequencies. Firstly, single-degree-of-freedom-systems are considered and closed-form analytical expressions for the mean and variance are obtained using a hybrid Laplace's method. Several probability density functions, including gamma, normal and lognormal distributions, are considered for the derivation of the analytical expressions. The method is extended to calculate the mean and the variance of the frequency response function of multiple-degrees-of-freedom dynamic systems. Proportional damping is assumed and the eigenvalues are considered to be independent. Results are derived for several probability density functions and damping factors. The accuracy of the approach for both single and multiple-degrees-of-freedom systems is examined using the direct Monte Carlo simulation.

A fully adaptive rational global Arnoldi method for the model-order reduction of second-order MIMO systems with proportional damping

The model order reduction of second-order dynamical multi-input and multi-output (MIMO) systems with proportional damping arising in the numerical simulation of mechanical structures is discussed. Based on finite element modelling the systems describing the mechanical structures are large and sparse, either undamped or proportionally damped. This work concentrates on a new model reduction algorithm for such second order MIMO systems which automatically generates a reduced system approximating the transfer function in the lower range of frequencies. The method is based on the rational global Arnoldi method. It determines the expansion points iteratively. The reduced order and the number of moments matched per expansion point are determined adaptively using a heuristic based on some error estimation. Numerical examples comparing our results to modal reduction and reduction via the rational block Arnoldi method are presented.

Density of states in granular media in the presence of damping.

We consider the density of states of granular media in which each grain-grain contact is damped with a damping force proportional to the relative velocity of the two grains, in addition to the usual spring constant. Under the assumption that the so-called criterion of proportional damping is only weakly violated we are able to deduce the density of states for undamped frequencies from the measured complex-valued frequencies of damped oscillations. We deduce a quantitative estimate of the deviation from the proportional criterion. We consider, specifically, numerical simulations of cases in which the grains are frictionless spheres that interact via Hertz central forces and all the nonzero contacts are damped with the same damping constant. We show how these ideas can be applied to data on real granular systems.

Different Approaches to Modeling of Proportional Damping

This article is focused on the comparison of mathematical models, which are usable for description of damping in the linear proportional systems with single and multi degrees of freedom (SDOF and MDOF). The viscous and hysteretic models have been applied on the SDOF system. The hysteretic damping, Rayleigh's and Caughey's damping matrices have been mentioned and considered for modeling of damping in systems with more than one degree of freedom. The wellknown Rayleigh's model is proportional to the mass and stiffness matrices. Caughey's model is more general than Rayleigh's. This model is able to prescribe an arbitrary number of damping ratios, but on the other hand using odd number of members in Caughey's series should not be allowed due to the negative values of damping ratios in higher modes, which brings self exciting effect. The hysteretic model is defined only in the frequency domain therefore the frequency response functions of each model have been compared.

Real-valued modal response history analysis for asymmetric-plan buildings with nonlinear viscous dampers

Modal analysis, which is clear in concept and simple in computation, is widely applied to evaluate the seismic responses of elastic structures with proportional damping. Nevertheless, nonlinear viscous dampers commonly installed in real buildings result in non-proportionally damped structures, which impede the modal analysis in real number field for such structures. This study develops the approach of performing real-valued modal response history analysis for elastic two-way asymmetric-plan buildings with nonlinear viscous dampers. To this end, this study first constructs the nonlinear three-degree-of-freedom (3DOF) modal equations of motion for such structures. The nonlinear 3DOF modal equations of motion retain the non-proportional damping characteristics in the modal space. Hence, by solving the nonlinear 3DOF modal equations of motion, the total response history is effectively attained by arithmetically summing up the modal response histories. This study uses nine two-way asymmetric-plan buildings, each equipped with nonlinear viscous dampers subjected to three bi-directional ground motions to verify the proposed approach. The investigation results show that the proposed simplified analysis approach provides satisfactory estimation of the seismic responses of common asymmetric-plan buildings with nonlinear viscous dampers.

A micro-inertia gradient visco-elastic motivation for proportional damping

In this paper, a micro-inertia gradient visco-elasticity theory is proposed and implemented for the description of wave dispersion in periodic visco-elastic composites characterised by (stiffness-)proportional damping. An expression for the internal length parameter has been derived in terms of geometry and material properties. The theory has been validated through a numerical simulation of wave propagation in a one-dimensional periodic composite bar for two different heterogeneity levels, where the proposed theory has shown good agreement with the solution obtained by explicitly modelling the material heterogeneity. The effects of both gradient enrichment and viscosity on wave propagation as well as their interaction have also been analysed.

A combined method for computing frequency responses of proportionally damped systems

Frequency response analysis requires the evaluation of an associated function for a typically large number of frequencies. Direct method for performing these calculations is time-consuming. In this paper, a method is proposed for solving frequency responses of a mechanical system with proportional damping. The method combines modal superposition with a model order reduction. Only the modes corresponding to a frequency range which is a little bigger than that of interest are used for modal superposition. Complementary part of contribution of computed modes for frequency response is calculated by a model order reduction method. Basis vectors are obtained by applying preconditioned conjugate gradient method to a modified undamped system at the highest frequency of interest. The existing factorized stiffness matrix developed for partial eigensolutions is used as preconditioner. This computational methodology is illustrated by its applications to two frequency response problems. It is shown that the present method can remarkably reduce the CPU time required by the direct method to frequency response analysis.

Topology optimization and fabrication of low frequency vibration energy harvesting microdevices

Topological design of miniaturized resonating structures capable of harvesting electrical energy from low frequency environmental mechanical vibrations encounters a particular physical challenge, due to the conflicting design requirements: low resonating frequency and miniaturization. In this paper structural static stiffness to resist undesired lateral deformation is included into the objective function, to prevent the structure from degenerating and forcing the solution to be manufacturable. The rational approximation of material properties interpolation scheme is introduced to deal with the problems of local vibration and instability of the low density area induced by the design dependent body forces. Both density and level set based topology optimization (TO) methods are investigated in their parameterization, sensitivity analysis, and applicability for low frequency energy harvester TO problems. Continuum based variation formulations for sensitivity analysis and the material derivative based shape sensitivity analysis are presented for the density method and the level set method, respectively; and their similarities and differences are highlighted. An external damper is introduced to simulate the energy output of the resonator due to electrical damping and the Rayleigh proportional damping is used for mechanical damping. Optimization results for different scenarios are tested to illustrate the influences of dynamic and static loads. To demonstrate manufacturability, the designs are built to scale using a 3D microfabrication method and assembled into vibration energy harvester prototypes. The fabricated devices based on the optimal results from using different TO techniques are tested and compared with the simulation results. The structures obtained by the level set based TO method require less post-processing before fabrication and the structures obtained by the density based TO method have resonating frequency as low as 100 Hz. The electrical voltage response in the experiment matches the trend of the simulation data.

Stochastic Dynamic Analysis of Structures with Spatially Uncertain Material Parameters

This paper investigates the uncertainty quantification in structural dynamic problems with spatially random variation in material and damping parameters. Uncertain and locally varying material parameters are represented as stochastic field by means of the Karhunen–Loève (KL) expansion. The stiffness and damping properties of the structure are considered uncertain. Stochastic finite element of structural modal analysis is performed in which modal responses are represented using the generalized polynomial chaos (gPC) expansion. Knowing the KL expansions of the random parameters, the nonintrusive technique is employed on a set of random collocation points where the structure deterministic finite element model is executed to estimate the unknown coefficients of the polynomial chaos expansions. A numerical case study is presented for a cantilever beam with random Young's modulus involving spatial variation. The proportional damping constants are estimated from the experimental modal analysis. The expected value, standard deviation, and probability distribution of the random eigenfrequencies and the damping ratios are evaluated. The results show high accuracy compared to the Monte-Carlo (MC) simulations with 3000 realizations. It is also demonstrated that the eigenfrequencies and the damping ratios are equally affected from material uncertainties.

Implementation and application of the unconditionally stable explicit parametrically dissipative KR‐α method for real‐time hybrid simulation

SummaryIn real‐time hybrid simulations (RTHS) that utilize explicit integration algorithms, the inherent damping in the analytical substructure is generally defined using mass and initial stiffness proportional damping. This type of damping model is known to produce inaccurate results when the structure undergoes significant inelastic deformations. To alleviate the problem, a form of a nonproportional damping model often used in numerical simulations involving implicit integration algorithms can be considered. This type of damping model, however, when used with explicit integration algorithms can require a small time step to achieve the desired accuracy in an RTHS involving a structure with a large number of degrees of freedom. Restrictions on the minimum time step exist in an RTHS that are associated with the computational demand. Integrating the equations of motion for an RTHS with too large of a time step can result in spurious high‐frequency oscillations in the member forces for elements of the structural model that undergo inelastic deformations. The problem is circumvented by introducing the parametrically controllable numerical energy dissipation available in the recently developed unconditionally stable explicit KR‐α method. This paper reviews the formulation of the KR‐α method and presents an efficient implementation for RTHS. Using the method, RTHS of a three‐story 0.6‐scale prototype steel building with nonlinear elastomeric dampers are conducted with a ground motion scaled to the design basis and maximum considered earthquake hazard levels. The results show that controllable numerical energy dissipation can significantly eliminate spurious participation of higher modes and produce exceptional RTHS results. Copyright © 2014 John Wiley & Sons, Ltd.