Question

Give first mode as initial condition in analytical solution of beam

Answer 1

Let us consider the problem of excited vibrations of
the Bernoulli-Euler beam with arbitrary boundary con-
ditions



E I ∂
4w(x,t)
∂x4 + ρ A ∂
2w(x,t)
∂t2 = f(x, t)
f(x, t) = q(x, t) + P
N
i=1
Fi δ(x − xi)
+
P
M
j=1
Mj
d
dx
δ(x − xj )
, (1)
where w(x, t) is the transverse displacement of a beam,
E is the Young modulus of material, ρ is the volume den-
sity of material, I is the moment of inertia of the beam
cross-section, A is the area of the beam cross-section,
Mj (t) is the j-th moment, xj is the point of action of j-
th moment, Fi(t) is the i-th concentrated force, xi
is the
point of action of i-th concentrated force, q(x, t) is the
distributed force.