Dynamical Response of a Beam in a Centrifugal Field …
107
where the matrix [R] expresses the change of the component of a vector from the local
coordinate system Oxyz to the fixed (global) reference system O
XYZ. The velocity
is obtained by differentiation:
v M,G = ˙
r M ,G = ˙
r o,G + ˙
R
⎧
⎨
⎩
x
0
0
⎫
⎬
⎭
+ ˙
R N δ e + R N ˙
δ e
(25)
The kinetic energy expression is:
E c =
1
2
L
0
ρ
A
˙
r M ,G
T ˙
r M ,G
+
ω
L
T [I ]
ω
L
dx
(26)
where:
[I] =
⎡
⎣
I x 0 0
0 I y 0
0 0 I z
⎤
⎦
(27)
I yy and I zz represent moments of inertia of the beam cross section about coordinate
axis Oy and Oz, respectively, of a reference system with its origin in the mass center
of the element dm = ρAdx (ρ-density); I xx is the inertia moment about the co-ordinate
axis Ox. We have chosen y and z as principal directions of inertia I yz = 0, we have:
ω
L
=
⎧
⎨
⎩
ω 1L
ω 2L
ω 3L
⎫
⎬
⎭
+
⎧
⎨
⎩
˙
α
˙
β
˙
γ
⎫
⎬
⎭
=
⎧
⎨
⎩
ω 1L
ω 2L
ω 3L
⎫
⎬
⎭
+
N
∗
{δ e }
(28)
here ω 1L , ω 2L , ω 3L are the components of the vector angular velocity refer to the
local coordinate system.
The Lagrangian for one is:
L = E c − E p − E a + W + W
c
.
(29)
Applying the Lagrange’s equations [21–24]:
d
dt
∂ L
∂ ˙
δ e
−
∂ L
∂δ e
= 0.
(30)
the motion equations for a single element in a centrifugal field can be obtained in the
form: