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