Toward overcoming the concept of effective stiffness and damping in the dynamic analysis of structures with viscoelastic components
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In the current state-of-practice, the time-domain dynamic analysis of structures incorporating viscoelastic members is generally carried out through the Modal Strain Energy (MSE) method, or other ...
In the current state-of-practice, the time-domain dynamic analysis of structures incorporating viscoelastic members is generally carried out through the Modal Strain Energy (MSE) method, or other procedures somehow based on the quite simplistic idea of substituting the actual viscoelastically damped structure with an equivalent system featuring a pure viscous damping.
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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