12.9 Periodic Structures
477
[8]. Early work concerning with engineering application began with simple periodic
beams [9]. Periodic structures are subjected to dynamic load of high intensity. As
such their vibration analysis is of great importance. Cremer and Leilich, from their
investigation of flexural motion of periodic beam structures concluded that waves
can propagate in some frequency bands and not in others, that is, only waves of
certain frequencies pass through the beam. These frequencies are called pass-bands
and the beam behaves as a pass-band filter. The wave propagation in periodic beams
is discussed in details by Mead [10]. Mead and his co-research workers have worked
extensively on the problem and have taken the subject to a mature stage [11–18].
Taking advantage of the spatially periodic property of the structures enables
carrying out the vibration analysis in a more convenient manner. This aids in catering
the whole problem with a much less computational effort. The computation is now
based on typical periodic element with boundary conditions at its extreme ends. For
example, for a structure having ‘N’ elements and each element has ‘i’ degrees of
freedom, a routine analysis will pertain to the solution of ‘N × i’ equations. Treating
the structure as a periodic structure, a maximum of ‘i’ equations have to be solved.
For the response analysis of the structure it becomes almost unnecessary to determine principal modes at all, as the vibrating characteristics of the periodic structure
is best understood in terms of propagating and non propagating free wave motion.
The natural free waves assist in understanding the vibration caused by moving
loads or pressure fields. The wave motion approach forms an important topic for
understanding the subject.
12.9.1 Different Types of Periodic Structures
A periodic structure consisting of a line of periodic elements joined together end-toend is referred to as one-dimensional periodic structure. Let us take the example of
the rail of a railway track supported over equidistant sleeper in an identical manner.
Each element is connected to its neighbour through two coordinates- the transverse
deflection and the bending rotation at the support. This is called a bi-coupled system.
If out of the two, one of the deformation is eliminated, such as the vertical deflection
by providing a vertical rigid support, then the system becomes mono-coupled. Similarly, multi-coupled system will have higher coupling coordinates between adjacent
elements.
A periodic structure consisting of plate type periodic elements joined together
end-to-end and side-by-side is called a two-dimensional periodic structure. This
will result in a multi-coupled periodic system. If a periodic plate structure is of
finite width and have equally spaced unidirectional stiffener, the problem as such is
multi-coupled, but can be reduced to bi-coupled periodic system if the plate edges
perpendicular to stiffeners are simply supported.
A periodic structure consisting of ‘layers’ of two-dimensional structures coupled
on one-on-top of another is known as three-dimensional period structure. Typical
477
[8]. Early work concerning with engineering application began with simple periodic
beams [9]. Periodic structures are subjected to dynamic load of high intensity. As
such their vibration analysis is of great importance. Cremer and Leilich, from their
investigation of flexural motion of periodic beam structures concluded that waves
can propagate in some frequency bands and not in others, that is, only waves of
certain frequencies pass through the beam. These frequencies are called pass-bands
and the beam behaves as a pass-band filter. The wave propagation in periodic beams
is discussed in details by Mead [10]. Mead and his co-research workers have worked
extensively on the problem and have taken the subject to a mature stage [11–18].
Taking advantage of the spatially periodic property of the structures enables
carrying out the vibration analysis in a more convenient manner. This aids in catering
the whole problem with a much less computational effort. The computation is now
based on typical periodic element with boundary conditions at its extreme ends. For
example, for a structure having ‘N’ elements and each element has ‘i’ degrees of
freedom, a routine analysis will pertain to the solution of ‘N × i’ equations. Treating
the structure as a periodic structure, a maximum of ‘i’ equations have to be solved.
For the response analysis of the structure it becomes almost unnecessary to determine principal modes at all, as the vibrating characteristics of the periodic structure
is best understood in terms of propagating and non propagating free wave motion.
The natural free waves assist in understanding the vibration caused by moving
loads or pressure fields. The wave motion approach forms an important topic for
understanding the subject.
12.9.1 Different Types of Periodic Structures
A periodic structure consisting of a line of periodic elements joined together end-toend is referred to as one-dimensional periodic structure. Let us take the example of
the rail of a railway track supported over equidistant sleeper in an identical manner.
Each element is connected to its neighbour through two coordinates- the transverse
deflection and the bending rotation at the support. This is called a bi-coupled system.
If out of the two, one of the deformation is eliminated, such as the vertical deflection
by providing a vertical rigid support, then the system becomes mono-coupled. Similarly, multi-coupled system will have higher coupling coordinates between adjacent
elements.
A periodic structure consisting of plate type periodic elements joined together
end-to-end and side-by-side is called a two-dimensional periodic structure. This
will result in a multi-coupled periodic system. If a periodic plate structure is of
finite width and have equally spaced unidirectional stiffener, the problem as such is
multi-coupled, but can be reduced to bi-coupled periodic system if the plate edges
perpendicular to stiffeners are simply supported.
A periodic structure consisting of ‘layers’ of two-dimensional structures coupled
on one-on-top of another is known as three-dimensional period structure. Typical