Toward overcoming the concept of effective stiffness and damping in the dynamic analysis of structures with viscoelastic componentsAlessandro PalmeriSchool of Engineering, Design and TechnologyUniversity of Bradford, West Yorkshire, UKCC2009, Funchal, 3rd September 2009

a.palmeri@lboro.ac.uk

OutlineMotivation of this studyViscoelastic ≠ Elastic + ViscousLinear viscoelastic solidsGeneralized Maxwell’s modelLaguerre Polynomial ApproximationState-space equations of motionNumerical scheme of solutionValidationElastic beam with VE strips (efficiency)Cable beam made of VE material (versatility)Concluding remarks

Motivation of this study1.

Viscoelastic damping in... Wind EngineeringThe first application of viscoelastic materials in Civil Engineering was aimed at mitigating the wind-induced vibration of the “Twin Towers” in the World Trade Center

Earthquake EngineeringSeismic applications of viscoelastic dampers are more recent, e.g. through the use of elastomeric materials placed in the beam-to-column joints of semi-rigid steel frames

Railway EngineeringIn the innovative track of the Milan subway, a single elastomeric pad is placed under the base-plate, aimed at improving passengers’ comfort and extending components’ fatigue life

Motivation of this studyCurrent state-of-practice, for the time-domain dynamic analysis of structures incorporating viscoelastic members: Substituting the actual viscoelastically damped structure with an equivalent system featuring a pure viscous damping≠

Motivation of this studyManufacturers of viscoelastic devices encourage the use this crude approximation in civil engineering applications:They are interested in simplifying as much as possible the design procedures for structures embedding their products

Motivation of this studyAs an example, elastomeric seismic isolators are generally advertised and sold with a table listing somehow equivalent values of elastic stiffness and viscous damping ratio

Motivation of this studyUnfortunately, many experimental and analytical studies confirm that the real dynamic behaviour of viscoelastic devices can be very complicated, and cannot be bend to the interests of manufacturers and designers

Motivation of this studyTwo well-established beliefs continue to underpin use and abuse of the concepts of effective stiffness and damping for viscoelastically damped structuresThis simplification always provides good approximations, which are acceptable for design purposes (FALSE!)Palmeri et al (2004), J ENG MECH-ASCE 130, 1052Palmeri et al (2004), WIND STRUCT 7, 89Muscolino, Palmeri & Ricciarelli (2005), EARTHQUAKE ENG STRUC 34, 1129Palmeri & Ricciarelli (2006), J WIND ENG IND AEROD 94, 377Palmeri (2006), ENG STRUCT 28, 1197Muscolino & Palmeri (2007), INT J SOLIDS STRUCT 44, 1317More refined approaches are computationally too expensive, and hence suitable just for a few very important constructions (FALSE!)

LINEAR viscoelastic SOLIDS2.

Linear viscoelastic solidsThe term viscoelastic refers to a whole spectrum of possible mechanical characteristicsAt one extreme we have viscous fluids, e.g. air and waterAt the other end we have elastic solids, e.g. metalsViscoelastic behaviour may combine viscous and elastic properties in any relative portion

Linear viscoelastic solidsTwo experimental tests can be used to reveal the viscoelastic behaviour of solidsCREEP TEST: The specimen is subjected to a constant state of stress, and the resulting variation in strain e as a function of time t is determined (the strain variation after the stress is removed corresponds to the recovery test)Creep function

Linear viscoelastic solidsTwo experimental tests can be used to reveal the viscoelastic behaviour of solidsRELAXATION TEST: The specimen is subjected to a constant state of strain, and the resulting variation in stress s as a function of time t is determinedRelaxation function

Linear viscoelastic solidsThe Kelvin-Voigt model, made of a linear spring in parallel with a linear dashpot, is widely adopted in Structural DynamicsInterestingly, the relaxation test is impossible for this model

Linear viscoelastic solidsIn the Standard Linear Solid (SLS) model the dashpot is substituted with a Maxwell’s elementThis model allows describing (at least qualitatively) creep and relaxation processes of actual linear viscoelastic solids

Linear viscoelastic solidsThe reaction force r(t) experienced by a one-dimensional viscoelastic component can be expressed in the time domain through a convolution integral involving the time derivative of the associated deformation q(t)pure elastic part

Relaxation function(time domain)j(t)temperature

Linear viscoelastic solidsThe complex-valued dynamic modulus k(w) enables one to represent the viscoelastic behaviour in the frequency domain wherethe REal part is the storage modulusk(w), which is a measure of the apparent rigidity at a given circular frequency wthe IMaginary part is the loss modulusk(w), which is proportional to the energy dissipated in a harmonic cycleDynamic modulus and relaxation function are interrelated as

DYNAMIC MODULUS(Frequency domain)k(w)temperaturetemperatureStorage modulus (rigidity)Loss modulus (dissipation)

Linear viscoelastic solidsThe frequency-dependent behaviour of viscoelastic materials cannot be captured by the 2-parameter Kelvin-Voigt modelKelvin-VoigtStandard Linear Solidstoragestoragelossloss

Linear viscoelastic solidsDilemmaOn the one hand, more refined models should be used to represent the dynamic behaviour of actual viscoelastic systemsOn the other hand, convolution integrals in the time domain are computationally burdensomeProposed approach:Implementation of state-space models, in which a set of additional state variables li(t) takes into account the frequency-dependent behaviour of these systems

Linear viscoelastic solidsThe Generalized Maxwell’s model is made of an elastic spring in parallel with  Maxwell’s rheological elementsThe relaxation function is the superposition of  exponential functions having different relaxation times tiThe time variation of the i-th internal variable is given by

The Generalized Maxwell’s model is made of an elastic spring in parallel with  Maxwell’s rheological elements

The relaxation function is the superposition of  exponential functions having different relaxation times ti

Linear viscoelastic solidsAs an alternative, the Laguerre’s Polynomial Approximation can be usedThe relaxation function is given by a single exponential function modulated by a polynomial of order The evolution in time of the i-th internal variable is ruled byt0 being a “characteristic” relaxation time of the system

Linear viscoelastic solidsGM and LPA models have relative pros and consPalmeri et al (2003), J ENG MECH-ASCE 129, 715 GM model is based on a classical chain of elastic springs and viscous dashpots The internal variable li(t) is ideally the strain in the elastic spring of the i-th Maxwell’s elementThe experimental evaluation of the 2 parameters of the GM model is generally pursued with a non-linear regression based on the results of small-amplitude vibration tests, which unfortunately turns out to be an ill-posed problemOrbey & Dealy (1991), J RHEOL 35 1035Mustapha & Phillips (2000), J PHYS D APPL PHYS 33, 1219 The LPA techniques just require a relaxation test to obtain the +1 parameters characterizing this modelBoth GM model and LPA technique enable one to accurately represent the constitutive law for linear viscoelastic solids

Both GM model and LPA technique enable one to accurately represent the constitutive law for linear viscoelastic solids

State-Space equations of motion3.

State-space equations of motionThe dynamic equilibrium of a linear structure, having nDoFs and r linear viscoelastic components, is governed in the time domain by a set of n coupled integro-differential equations of second orderThe following modal transformation of coordinates can be used in order to reduce the size of the problem (the first m ≤ n modes of vibration will be retained in the analysis)Palmeri et al. (2004), WIND STRUCT 7, 89 which requires the solution of a classical real-valued eigenproblem

State-space equations of motionIn the reduced modal space the equations of motion take the formThe time-dependent modal relaxation matrix is given by the superposition of the relaxation functions of the r viscoelastic components

State-space equations of motionIf the distribution of the viscoelastic components is almost homogeneous along the structural system, then the out of diagonal terms in the modal relaxation matrix becomes negligible, and the equations of motion are decoupled in the modal spaceAlthough uncoupled, modal oscillators are viscoelastically damped

State-space equations of motionIn the general case where the modal relaxation matrix is sparse (non-classically non-viscously damped structure), it is always possible to rewrite this quantity as superposition of  termsFor the LPA technique, i-th rigidity matrix Ri and i-th elementary relaxation function gi(t) particularize asRj,i being the j-thLaguerre’s rigidity of the i-th viscoelastic component

State-space equations of motionAnalogously to the one-dimensional case, the matrix G appearing in the convolution integral in the modal equations of motion can be turned into a linear combination of  arrays of internal variablesThe i-th one (of size m) is ruled by

State-space equations of motionThe modal equations of motion then become Interestingly, the modal coordinates in the array q(t) are coupled just by the Laguerre’s rigidity matrices RiConsidering also the linear differential equations ruling the arrays li(t), the following (2+)m-dimensional state-space form can be obtained

State-space equations of motionOr, equivalently where

Numerical scheme of solution4.

Numerical scheme of solutionUnder the assumption that the external excitation varies linearly in each time step Dt, the exact time-domain response can be posed in the form where all the integration operators Q and G can be evaluated in closed form

Numerical scheme of solutionClassical state variables

Numerical scheme of solutionAdditional state variables

Numerical scheme of solutionThe two state arrays x(t) and y(t) are coupled, since the traditional state variables at the end of the time step, x(t+Dt), depend on the additional state variables at the same instant, y(t+Dt), and vice versa

Numerical scheme of solutionRearranging the equations, however, a new form can be derived, which leads to an alternative, and very effective, solution in cascadeThe over-arc denotes the updated operatorsIn most of the cases E is diagonally dominant

Validation:EFFICIENCY5.1

Validation / Efficiency Proposed numerical solution vs. SIDE (System of Integro-Differential Equations) scheme base on the Newmark’sb method (constant average acceleration method)Patlashenko, Givoli & Barbone (2001), COMPUT METHOD APPL M 190, 5691FEM model of a slender cantilever beam, with no inherent damping, provided with a set of perfectly bonded viscoelastic strips

Validation / Efficiency Dimensionless relaxation function of the viscoelastic stripsStorageDimensionless dynamic modulus for the viscoelastic stripsLoss

Validation / Efficiency Convergence (percentage variation into parentheses)M= number of elements in the transverse direction of the beamQ = number of quadrilateral elements r = number of viscoelastic springs in the FEM modeln = number of DoFsW1 = undamped frequency of vibrationw1 = damped frequency of vibration

Validation / Efficiency Dynamic response of the slender beam for coarse (left) and fine (right) meshesProposed approach and Nodal SIDE (reference solution) are in excellent agreementn = 40n = 5,440

Validation / Efficiency Computational times required by different analysesNumerical schemes have been implemented in Mathematica 6.0, and results have been obtained with a Microsoft Windows PC equipped with Dual-Core AMD Athlon 64 X2 processor at 3.01 GHz and with 1.96 GB of RAM. The selected time step was Dt= 0.1 ms, corresponding to 5,000 time steps for each analysis

Validation:Versatility5.2

Validation / VersatilitySmall-amplitude vibration of a symmetric concave cable beam made of viscoelastic materialsThe structure is 10 m long and 1 m deepSuspension cable , prestressing cable and vertical ties are made of different type-A, type-B and type-C viscoelastic materialsMain cables are connected by n-1 vertical tiesFour configurations are considered having a different values of n= 4, 8, 24, 96Configuration at rest of the cable beam under investigation, with indication of input F1(t) and location of the selected outputs hi(t)

Validation / VersatilityRelaxation functionsStorage moduliLoss moduli

Validation / VersatilityModulus of the FRF with input F1 and output h1 for different numbers of DoFsn and modal coordinates mThe Modal Strain Energy (MSE) method substitutes the actual viscoelastically damped structure with an equivalent system featuring a pure viscous damping

Validation / VersatilityPercentage error in the FRF with input F1 and output h1 for different numbers of DoFsn and modal coordinates m

Validation / VersatilityTime histories of the vertical vibration at various locations of the objective cable beam with n=8 divisions under impulsive loadingThe low-frequency vibrations are damped first  The inaccurate prediction of the MSE method in this frequency range does not propagate along the cable beam

Validation / VersatilityTime histories of the vertical vibration at various locations of the objective cable beam with n=8 divisions under sweep excitation

Concluding remarks6

Concluding remarks A novel time-domain numerical scheme for the dynamic analysis of structures with viscoelastic components has been proposed and numerically validatedThe goal is to overcome the popular concepts of equivalent values of elastic stiffness and viscous dampingState-space equations of motion have been presented in the modal spaceA non-proportional proportional distribution of viscoelastic components has been considered, in so removing the most severe limitation of previous formulationsA cascade scheme has been derived by decoupling in each time step traditional state variables and additional internal variablesJoint use of modal analysis and improved cascade scheme lead to reduced size of the problem and low the computational effortFuture investigationsNonlinear effects through a convenient reanalysis techniqueSemi-active control with the help of MR braces in series with VE dampers