ABSTRACT
The vibration of plates represents an important problem in structural mechanics, the solutions being applicable to the design of many structural components such as bridge decks, building floors, platforms carrying machinery, aircraft partitions and vehicle panels. The equation of motion for the undamped free vibration of a rectangular plate may be written as [1, 2]
2 0 4
w x
w x y
w y D
w t
∂ ∂ +
∂ ∂ ∂ +
∂ ∂ +
ρ ∂ ∂ = (10.1)
where w is the transverse displacement at a point defined by the coordinates {x, y} at any given time t, D is the flexural rigidity of the plate and ρ is the mass per unit area of the plate. If the plate has constant thickness h and constant material properties E (Young’s modulus of elasticity) and ν (Poisson’s ratio), then the flexural rigidity is given by
D Eh
=
− ν12(1 )
2 (10.2)
Let us assume harmonic vibration. This allows us to express the displacement in the form
w(x, y, t) = W(x, y) sinω t (10.3)
where W(x, y) is a shape function satisfying the boundary conditions and describing the shape of the deflected middle surface of the vibrating plate, and ω is a natural circular frequency of the plate. Substituting for w in Equation (10.1), we obtain
W x
W x y
W y
W ∂ ∂ +
∂ ∂ ∂ +
∂ ∂ − η =2 0
4 (10.4)
where
D
η = ρ ω 2
(10.5)
Various finite-difference schemes for representing differential equations may be seen in standard texts on numerical methods. Adopting the centraldifference scheme, and taking equal mesh intervals d = Δx = Δy in the x and y directions, the finite-difference representation of Equation (10.4) at a pivotal point (m, n) of the mesh is as follows [2]:
20 8
W W W W W
W W W W
W W W W W
( ) ( )
− + + +
+ + + +
+ + + + − λ =
(10.6)
where
λ = η d4 (10.7)
Figure 10.1 shows the 12 mesh points around a pivotal point denoted by O. Application of Equation (10.6) to this arrangement yields the centraldifference equation for point O as
20 8 2
W W W W W W W W W
W W W W W
( ) ( )− + + + + + + + + + + + − λ =
(10.8)
For mesh points adjacent to boundaries, it is necessary to introduce what are called fictitious points. The deflection values to be used for the fictitious points are obtained by considering the boundary conditions along the edges of the plate. For instance, the condition of zero slope across a clamped edge (Figure 10.2(a)) requires the imaginary deflection at the fictitious point to be the same in magnitude and sign as that at the corresponding real mesh point, while the condition of zero moment and zero deflection across a simply supported edge (Figure 10.2(b)) requires the imaginary deflection at the fictitious point to be the same in magnitude but of opposite sign to that at the corresponding real mesh point.
