476
12 Finite Element Method in Vibration Analysis
δ W =
¨
δ { ¨
X }
T
e [N ]
T
[m] [N ] { ¨
X } e dx dy
(12.164)
For virtual displacements at nodes, the work done by the nodal forces
δ W = δ { ¨
X }
T
e {P}
(12.165)
Equating δ W of Eq. (12.164) to that of Eq. (12.165), yields
{P} =
¨
[N ]
T
[m] [N ] dx dy
or,
[M] e { ¨
X } e = {P} e
(12.166)
where [M] e is the consistent mass matrix and is given by
[M] e =
¨
[N ]
T
[M] [N ] dx dy
or,
[M] e =
1
− 1
1
− 1
[N ]
T
[m] [N ] |J | dx dy
(12.167)
12.9 Periodic Structures
The type of structure which contains a series of identical structural elements
connected in an identical manner to one another is referred to as periodic structure.
This suggests that in this type of construction, there is a regularly repeating section.
There are many examples of such structures, mostly arising in civil, aerospace and
ocean engineering and naval architecture. A tall building has uniform and identical
stories, a ship or an aircraft consists of a uniform shell reinforced with periodically arranged stiffeners. A jack-up rig or a space station is constructed of truss-like
components, where truss structures are essentially assemblage of identical bays and
thus are spatially periodic. Other applications include rails, shafts with bearings and
discs and bridges which consist of identical subsystems connected in identical ways.
Other types of periodic structures are those which possess material periodicity, such
as composite laminated structures which is beyond the scope of this present section.
Considerable work has been carried out on the analysis of periodic structures
[7]. A good summary of the earlier work may be found in Brillounin’s classic work