ABSTRACT
We consider the origin of the terms “weak or variational form” of Chapters 7 and 9 as opposed to strong or closed form of PDE’s. We use the beam equation to illustrate ideas.
Recall Example 6, the cantilever beam. This example, given in classical form (which can be derived in a straightforward manner using force and moment balance-see [BT]) is
ρ ∂2y
∂t2 + γ
∂y
∂t +
( EI
∂2y
∂ξ2 + cDI
∂3y
∂ξ2∂t
) = f(t, ξ) (11.1)
with boundary conditions
y(t, 0) = 0 ∂y ∂ξ (t, 0) = 0
(11.2)
( EI ∂
) |ξ=l = 0
[( EI ∂
)] |ξ=l = 0
(11.3)
and initial conditions
y(0, ξ) = Φ(ξ) ∂y ∂t (0, ξ) = Ψ(ξ).
