KEYWORDS: Band Gap ; Periodic structure; Ladder frame;
Periodic structures are known to exhibit alternating pass band and stop band charac-teristics over the frequency range. The attenuating characteristics of the stop band fre-quencies is particularly beneficial for vibration isolation applications. In earlier works, this feature of periodic structures have been exploited for vibration isolation at discrete
and narrow band frequencies. The objective of the present work is to extend this idea for broadband excitations. Towards this end, we seek to synthesize periodic structure having the largest fraction of the frequencies falling in the attenuation bands of the structure. We synthesize a unit cell of the periodic structure which comprises of two
distinct regions having different inertial, stiffness and geometric properties.
In particular, periodic bar, beam, ladder frame and square grid structures are suit-
ably synthesized to achieve a maximal attenuating frequency band. A common geo-
metric framework, namely, that of Eigenvalue map is developed. This concept is shown

to be applicable for a wide class of periodic structures. Despite the difference in the
mathematical model of these different structures, the Eigenvalue map offers common
and insightful results regarding the transmission and attenuation characteristics of these

structures. For the periodic rod, beam and ladder frame, the development of the Eigen-
value map is based on the transfer matrix formulation for the corresponding unit cell.

While the transfer matrix formulation for rod and beam is well-known, the exact ana-
lytical transfer matrix formulation for the ladder frame is not available in the literature.

In this work, based on the wave dynamics of the individual members constituting the ladder frame, an exact analytical formulation of the transfer matrix is proposed. The entire formulation and the all the results are presented in a non-dimensional framework
for wider applicability. Design guidelines are presented for the ladder frame and square grid structure. These indicate the maximal attenuation attainable using geometric and/or material property variation across the semi-unit cells. Note, based on practical consid-erations material change is difficult to implement whereas cross-sectional change is
more feasible. It was found that substantial attenuation can be achieved by changing
the cross-sectional property alone. It is envisaged that such design guidelines will be of
aid to the designer in making a judicious choice of geometric and material properties
of the structure. It is shown that the finite structure designed on the basis of periodic
structure theory has superior isolation characteristics in comparison to a homogeneous
or an arbitrarily chosen periodic structure. The general conclusion from the Eigenvalue
maps developed for the different structures suggest that a greater property mismatch
across the semi-unit cells is more favourable for the attenuation characteristics.