ABSTRACT
FIGURE 6.18: Segment of a beam with a rectangular cross-sectional area.
Substituting equations (6.7) and (6.11), after some simple calculations we obtain
M = Mbending +Mdamping
≈ EI ∂ 2y
∂x2 + cDI
∂3y
∂x2∂t . (6.13)
It is seen from the above equation that for a viscoelastic material, the moment is the sum of two terms: the first term is the standard internal (bending) moment and the second term is due to the structural damping (which is also referred to as Kelvin-Voigt damping). There are other types of damping models that one encounters frequently in practice. For example, for composite fiberous materials
Mdamping(t, x) = ∫ l x
dξ
b(ξ, θ) [ ∂2y
∂ξ∂t (t, ξ)− ∂
2y
∂θ∂t (t, θ)
] dθ.
