The Analysis of Distributed Systems with Nonlocal Damping
Lei, Y., Friswell, M. I. and Adhikari, S.,
SPIE Smart Structures and Materials Conference, San Diego, California, USA,
February-March 2006.
The dynamic response analysis of damped structures is important in
many areas of mechanical, civil and aerospace engineering, such as
the vibration isolation of precise equipment, aircraft noise, or
the vibration of cable stayed bridges. Although the damping model
plays a key role in the dynamic analysis, particularly for complex
structures, this model is often approximated by classical or
proportional damping distributions for convenience. In many
practical situations this simplified approach does not describe
the dynamics of the structure with sufficient accuracy because of
the complicated damping mechanisms that occur in practice.
Theoretically speaking, any model that makes the energy
dissipation function non-negative is a possible candidate for a
valid damping model. Research into the dynamics of structures with
viscoelastic materials has concentrated on the time dependence
using fractional derivative, GHM and other models. This paper
considers the effect of nonlocal damping on the response of a
structure. In a nonlocal model, the reaction force at any point is
obtained as a weighted average of state variables over a spatial
domain via convolution integrals with spatial kernel functions
that depend on a distance measure. In physical terms the non-local
damping or elasticity property often arises when two dimensional
structures are modelled as one dimensional.
In this paper, a nonlocal viscoelastic foundation model is used to
analyse the dynamics of beams and plates with a variety of
boundary conditions. The model yields an integro-differential
equation, and obtaining closed form solutions is only possible for
a limited range of boundary conditions by the transfer function
method. This paper will present approximate solutions using the
Galerkin method for beams and plates with typical spatial kernel
functions. This requires the approximation of the displacement to
be defined over the whole domain. To treat more complicated
problems with variable damping parameters, non-uniform section
properties, intermediate supports or arbitrary boundary
conditions, a finite element method for beams is developed.
However, in nonlocal damping models, nodes remote from the element
do have an effect on the energy expressions, and hence the damping
matrix is no longer block diagonal. The expressions for these
direct and cross damping matrices are obtained explicitly for
several common spatial kernel functions. The approach is
demonstrated on a range of examples. The form of the nonlocal
foundation model is shown to have a significant impact on the
dynamic characteristics of structures.
BiBTeX Entry
@INPROCEEDINGS{cp22,
AUTHOR={Y. Lei and M. I. Friswell and S. Adhikari},
TITLE={The analysis of distributed systems with nonlocal damping},
BOOKTITLE={SPIE Smart Structures and Materials Conference},
YEAR={2006},
Address={San Diego, California, USA},
Month={February-March},
Note={}
}
by Sondipon Adhikari