Abstract

Beams formed by long fiber composite materials have certain internal damping torque. A mathematical model for the displacement of this type of beams in cantilever configuration is the following initial-boundary value problem of an integro-differential equation [1, 14]:

ρ(x)wtt(x,t)2(0Lh(x,y)[wtx(x,t)wtx(y,t)]dy)x+(EIwxx(x,t))xx=f(x,t),
(1)
w(0,t)=0,wx(0,t)=0,
(2)
wxx(L,t)=bl1(t),
(3)
(EIwxx(x,t))x|x=L+20Lh(L,y)[wtx(L,t)wtx(y,t)]dy=bl2(t),
(4)
w(x,0)=w0(x),wt(x,0)=w1(x),
(5)

where L is length of the beam, w(x, t) is the transverse displacement of the beam at time t and position x, ρ(x) is the mass density, EI is the stiffness parameter. The interaction integral kernel h(x, ξ) is introduced in this model by considering a restoring torque which comes from spatially variable bending of the beam. This kernel h(x, ξ) depends on the material properties of the beam. Choosing a different material (different h(x, ξ)) can realize a different damping effect for the beam.

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