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This crude approximation in civil engineering applications is very often encouraged by manufacturers of the viscoelastic devices themselves, whose interest is to simplify as much as possible the design procedures for structures embedding their products. As an example, elastomeric seismic isolators are generally advertised and sold with a table listing the equivalent values of elastic stiffness and viscous damping ratio for different amplitudes of vibration. Unfortunately, many experimental and analytical studies confirm that the real dynamic behaviour of such devices is much more complicated, and cannot be bend to the interests of manufacturers and designers.

Despite the advances in the field made in the last two decades, two well-established beliefs continue to underpin use and abuse of the concepts of effective stiffness and damping for viscoelastically damped structures: first, MSE method and similar procedures are unconditionally assumed to provide good approximations, which are acceptable for design purposes; second, the implementation of more refined approaches is thought to be computationally too expensive, and hence suitable just for a few very important constructions.

In this presentation, as a further contribution to overcome these popular beliefs, a novel time-domain numerical scheme of dynamic analysis is proposed and numerically validated. After a brief review of the LPA (Laguerre’s Polynomial Approximation) technique for one-dimensional viscoelastic members of known relaxation function, the state-space equations of motion for linear structures with viscoelastic components are derived in the modal space. Aimed at making the proposed approach more general, the distribution of the viscoelastic components is allowed to be non-proportional to mass and elastic stiffness, in so removing the most severe limitation of previous formulations. Then, a cascade scheme is derived by decoupling in each time step traditional state variables (i.e. modal displacements and velocities) and additional internal variables. The joint use of modal analysis and improved cascade scheme permits to reduce the size of the problem and to keep low the computational burden. The illustrative application to the small-amplitude vibration of a cable beam made of different viscoelastic materials demonstrates the versatility of the proposed approach. The numerical results confirm a superior accuracy with respect to the classical MSE method, whose underestimate in the low-frequency range can be as large as 75%.

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- 1. Toward overcoming the concept of effective stiffness and damping in the dynamic analysis of structures with viscoelastic components<br />Alessandro Palmeri<br />School of Engineering, Design and Technology<br />University of Bradford, West Yorkshire, UK<br />CC2009, Funchal, 3rd September 2009<br />
- 2. a.palmeri@lboro.ac.uk<br />
- 3. Outline<br />Motivation of this study<br />Viscoelastic ≠ Elastic + Viscous<br />Linear viscoelastic solids<br />Generalized Maxwell’s model<br />Laguerre Polynomial Approximation<br />State-space equations of motion<br />Numerical scheme of solution<br />Validation<br />Elastic beam with VE strips (efficiency)<br />Cable beam made of VE material (versatility)<br />Concluding remarks<br />
- 4. Motivation of this study<br />1.<br />
- 5. Viscoelastic damping in... Wind Engineering<br />The 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<br />
- 6. Earthquake Engineering<br />Seismic 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<br />
- 7. Railway Engineering<br />In 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<br />
- 8. Motivation of this study<br />Current state-of-practice, for the time-domain dynamic analysis of structures incorporating viscoelastic members: <br />Substituting the actual viscoelastically damped structure with an equivalent system featuring a pure viscous damping<br />≠<br />
- 9. Motivation of this study<br />Manufacturers of viscoelastic devices encourage the use this crude approximation in civil engineering applications:<br />They are interested in simplifying as much as possible the design procedures for structures embedding their products<br />
- 10. Motivation of this study<br />As an example, elastomeric seismic isolators are generally advertised and sold with a table listing somehow equivalent values of elastic stiffness and viscous damping ratio<br />
- 11. Motivation of this study<br />Unfortunately, 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<br />
- 12. Motivation of this study<br />Two well-established beliefs continue to underpin use and abuse of the concepts of effective stiffness and damping for viscoelastically damped structures<br />This simplification always provides good approximations, which are acceptable for design purposes (FALSE!)<br />Palmeri et al (2004), J ENG MECH-ASCE 130, 1052<br />Palmeri et al (2004), WIND STRUCT 7, 89<br />Muscolino, Palmeri & Ricciarelli (2005), EARTHQUAKE ENG STRUC 34, 1129<br />Palmeri & Ricciarelli (2006), J WIND ENG IND AEROD 94, 377<br />Palmeri (2006), ENG STRUCT 28, 1197<br />Muscolino & Palmeri (2007), INT J SOLIDS STRUCT 44, 1317<br />More refined approaches are computationally too expensive, and hence suitable just for a few very important constructions (FALSE!)<br />
- 13. LINEAR viscoelastic SOLIDS<br />2.<br />
- 14. Linear viscoelastic solids<br />The term viscoelastic refers to a whole spectrum of possible mechanical characteristics<br />At one extreme we have viscous fluids, e.g. air and water<br />At the other end we have elastic solids, e.g. metals<br />Viscoelastic behaviour may combine viscous and elastic properties in any relative portion<br />
- 15. Linear viscoelastic solids<br />Two experimental tests can be used to reveal the viscoelastic behaviour of solids<br />CREEP 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)<br />Creep function<br />
- 16. Linear viscoelastic solids<br />Two experimental tests can be used to reveal the viscoelastic behaviour of solids<br />RELAXATION 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 determined<br />Relaxation function<br />
- 17. Linear viscoelastic solids<br />The Kelvin-Voigt model, made of a linear spring in parallel with a linear dashpot, is widely adopted in Structural Dynamics<br />Interestingly, the relaxation test is impossible for this model<br />
- 18. Linear viscoelastic solids<br />In the Standard Linear Solid (SLS) model the dashpot is substituted with a Maxwell’s element<br />This model allows describing (at least qualitatively) creep and relaxation processes of actual linear viscoelastic solids<br />
- 19. Linear viscoelastic solids<br />The 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)<br />pure elastic part<br />
- 20. Relaxation function(time domain)<br />j(t)<br />temperature<br />
- 21. Linear viscoelastic solids<br />The complex-valued dynamic modulus k(w) enables one to represent the viscoelastic behaviour in the frequency domain<br /> where<br />the REal part is the storage modulusk(w), which is a measure of the apparent rigidity at a given circular frequency w<br />the IMaginary part is the loss modulusk(w), which is proportional to the energy dissipated in a harmonic cycle<br />Dynamic modulus and relaxation function are interrelated as<br />
- 22. DYNAMIC MODULUS(Frequency domain)<br />k(w)<br />temperature<br />temperature<br />Storage modulus (rigidity)<br />Loss modulus (dissipation)<br />
- 23. Linear viscoelastic solids<br />The frequency-dependent behaviour of viscoelastic materials cannot be captured by the 2-parameter Kelvin-Voigt model<br />Kelvin-Voigt<br />Standard Linear Solid<br />storage<br />storage<br />loss<br />loss<br />
- 24. Linear viscoelastic solids<br />Dilemma<br />On the one hand, more refined models should be used to represent the dynamic behaviour of actual viscoelastic systems<br />On the other hand, convolution integrals in the time domain are computationally burdensome<br />Proposed approach:<br />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<br />
- 25. Linear viscoelastic solids<br /><ul><li>The Generalized Maxwell’s model is made of an elastic spring in parallel with Maxwell’s rheological elements
- 26. The relaxation function is the superposition of exponential functions having different relaxation times ti</li></ul>The time variation of the i-th internal variable is given by<br />
- 27. Linear viscoelastic solids<br />As an alternative, the Laguerre’s Polynomial Approximation can be used<br />The relaxation function is given by a single exponential function modulated by a polynomial of order <br />The evolution in time of the i-th internal variable is ruled by<br />t0 being a “characteristic” relaxation time of the system<br />
- 28. Linear viscoelastic solids<br />GM and LPA models have relative pros and cons<br />Palmeri et al (2003), J ENG MECH-ASCE 129, 715<br /> GM model is based on a classical chain of elastic springs and viscous dashpots<br /> The internal variable li(t) is ideally the strain in the elastic spring of the i-th Maxwell’s element<br />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 problem<br />Orbey & Dealy (1991), J RHEOL 35 1035<br />Mustapha & Phillips (2000), J PHYS D APPL PHYS 33, 1219<br /> The LPA techniques just require a relaxation test to obtain the +1 parameters characterizing this model<br /><ul><li>Both GM model and LPA technique enable one to accurately represent the constitutive law for linear viscoelastic solids</li></li></ul><li>State-Space equations of motion<br />3.<br />
- 29. State-space equations of motion<br />The 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 order<br />The 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)<br />Palmeri et al. (2004), WIND STRUCT 7, 89<br /> which requires the solution of a classical real-valued eigenproblem<br />
- 30. State-space equations of motion<br />In the reduced modal space the equations of motion take the form<br />The time-dependent modal relaxation matrix is given by the superposition of the relaxation functions of the r viscoelastic components<br />
- 31. State-space equations of motion<br />If 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 space<br />Although uncoupled, modal oscillators are viscoelastically damped<br />
- 32. State-space equations of motion<br />In 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 terms<br />For the LPA technique, i-th rigidity matrix Ri and i-th elementary relaxation function gi(t) particularize as<br />Rj,i being the j-thLaguerre’s rigidity of the i-th viscoelastic component<br />
- 33. State-space equations of motion<br />Analogously 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 variables<br />The i-th one (of size m) is ruled by<br />
- 34. State-space equations of motion<br />The modal equations of motion then become<br /> Interestingly, the modal coordinates in the array q(t) are coupled just by the Laguerre’s rigidity matrices Ri<br />Considering also the linear differential equations ruling the arrays li(t), the following (2+)m-dimensional state-space form can be obtained<br />
- 35. State-space equations of motion<br />Or, equivalently<br /> where<br />
- 36. Numerical scheme of solution<br />4.<br />
- 37. Numerical scheme of solution<br />Under the assumption that the external excitation varies linearly in each time step Dt, the exact time-domain response can be posed in the form<br /> where all the integration operators Q and G can be evaluated in closed form<br />
- 38. Numerical scheme of solution<br />Classical state variables<br />
- 39. Numerical scheme of solution<br />Additional state variables<br />
- 40. Numerical scheme of solution<br />The 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<br />
- 41. Numerical scheme of solution<br />Rearranging the equations, however, a new form can be derived, which leads to an alternative, and very effective, solution in cascade<br />The over-arc denotes the updated operators<br />In most of the cases E is diagonally dominant<br />
- 42. Validation:EFFICIENCY<br />5.1<br />
- 43. Validation / Efficiency <br />Proposed numerical solution vs. SIDE (System of Integro-Differential Equations) scheme base on the Newmark’sb method (constant average acceleration method)<br />Patlashenko, Givoli & Barbone (2001), COMPUT METHOD APPL M 190, 5691<br />FEM model of a slender cantilever beam, with no inherent damping, provided with a set of perfectly bonded viscoelastic strips<br />
- 44. Validation / Efficiency <br />Dimensionless relaxation function of the viscoelastic strips<br />Storage<br />Dimensionless dynamic modulus for the viscoelastic strips<br />Loss<br />
- 45. Validation / Efficiency <br />Convergence (percentage variation into parentheses)<br />M= number of elements in the transverse direction of the beam<br />Q = number of quadrilateral elements <br />r = number of viscoelastic springs in the FEM model<br />n = number of DoFs<br />W1 = undamped frequency of vibration<br />w1 = damped frequency of vibration<br />
- 46. Validation / Efficiency <br />Dynamic response of the slender beam for coarse (left) and fine (right) meshes<br />Proposed approach and Nodal SIDE (reference solution) are in excellent agreement<br />n = 40<br />n = 5,440<br />
- 47. Validation / Efficiency <br />Computational times required by different analyses<br />Numerical 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<br />
- 48. Validation:Versatility<br />5.2<br />
- 49. Validation / Versatility<br />Small-amplitude vibration of a symmetric concave cable beam made of viscoelastic materials<br />The structure is 10 m long and 1 m deep<br />Suspension cable , prestressing cable and vertical ties are made of different type-A, type-B and type-C viscoelastic materials<br />Main cables are connected by n-1 vertical ties<br />Four configurations are considered having a different values of n= 4, 8, 24, 96<br />Configuration at rest of the cable beam under investigation, with indication of input F1(t) and location of the selected outputs hi(t)<br />
- 50. Validation / Versatility<br />Relaxation functions<br />Storage moduli<br />Loss moduli<br />
- 51. Validation / Versatility<br />Modulus of the FRF with input F1 and output h1 for different numbers of DoFsn and modal coordinates m<br />The Modal Strain Energy (MSE) method substitutes the actual viscoelastically damped structure with an equivalent system featuring a pure viscous damping<br />
- 52. Validation / Versatility<br />Percentage error in the FRF with input F1 and output h1 for different numbers of DoFsn and modal coordinates m<br />
- 53. Validation / Versatility<br />Time histories of the vertical vibration at various locations of the objective cable beam with n=8 divisions under impulsive loading<br />The low-frequency vibrations are damped first The inaccurate prediction of the MSE method in this frequency range does not propagate along the cable beam<br />
- 54. Validation / Versatility<br />Time histories of the vertical vibration at various locations of the objective cable beam with n=8 divisions under sweep excitation<br />
- 55. Concluding remarks<br />6<br />
- 56. Concluding remarks <br />A novel time-domain numerical scheme for the dynamic analysis of structures with viscoelastic components has been proposed and numerically validated<br />The goal is to overcome the popular concepts of equivalent values of elastic stiffness and viscous damping<br />State-space equations of motion have been presented in the modal space<br />A non-proportional proportional distribution of viscoelastic components has been considered, in so removing the most severe limitation of previous formulations<br />A cascade scheme has been derived by decoupling in each time step traditional state variables and additional internal variables<br />Joint use of modal analysis and improved cascade scheme lead to reduced size of the problem and low the computational effort<br />Future investigations<br />Nonlinear effects through a convenient reanalysis technique<br />Semi-active control with the help of MR braces in series with VE dampers<br />

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