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yielding metallic dampers, and friction dampers are among the common energy dissipation systems
that have been used in seismic design applications. This study is focused on some analytical issues
primarily pertaining to the viscoelastic dampers.
MODELS OF VISCOELASTIC DAMPERS
For structural designs with an energy dissipation device, it is important to have the correct forcedeformation
model of the device to represent the actual behavior of the material used in the device.
The model should also be convenient for use in dynamic analysis to predict dynamic response as
accurately as possible to examine different design alternatives. The viscoelastic dampers use a
polymeric material that dissipates energy by shear deformations applied cyclically. These dampers
contribute both to the damping and stiffness of a structure. To represent the load-deformation
characteristics of these dampers, models with increasing degree of complexity have been used. One of
the simplest models used is a classical Kelvin model with frequency-independent damping and
stiffness elements arranged in parallel. The analysis of structures with such dampers is quite
straightforward. However, this simple model does not capture the frequency dependence of the
damping and stiffness characteristics of the material that are observed in the experiments. To capture
the frequency dependence, the complex modulus relating the strain with stress has been used; it
describes the relationship accurately for a purely harmonic motion. The real part of the complex
modulus (called the storage modulus) represents the stiffness property of the material. The imaginary
part (called as the loss modulus), on the other hand, is associated with the energy dissipation and thus
damping. The linear structures installed with these dampers can be analyzed using the discrete Fourier
transforms in frequency domain. For the response spectrum analysis of a structure installed with these
dampers, the modal strain energy approach (linger and Kerwin, 1962) can be used (Soong and
Dargush, 1997) to incorporate the frequency dependency of the parameters approximately.
Equation Section 3
Another model used to incorporate the frequency dependence is the fractional derivative model. It is a
generalization of the classical Kelvin model but with a fractional derivative. It is defined by the
following differential equation:
f=ku + cD a (u}, 0