Civil Engineering Reference
In-Depth Information
10.3 MULTIPLE DEGREES-OF-FREEDOM
SYSTEMS
Figure 10.4 shows a system of two spring elements having concentrated masses
attached at nodes 2 and 3 in the global coordinate system. As in previous exam-
ples, the system is subjected to gravity and the upper spring is attached to a rigid
support at node 1. Of interest here is the dynamic response of the system of two
springs and two masses when the equilibrium condition is disturbed by some
external influence and then free to oscillate without external force. We could take
the Newtonian mechanics approach by drawing the appropriate free-body dia-
grams and applying Newton's second law of motion to obtain the governing
equations. Instead, we take the finite element approach. By now, the procedure of
assembling the system stiffness matrix should be routine. Following the proce-
dure, we obtain
1
1
3
k
2
m
U
2
2
k
3
m
U
3
3
k
−
3
k
0
Figure 10.4
A
spring-mass system
exhibiting 2 degrees
of freedom.
[
K
]
=
(10.30)
−
3
k
5
k
−
2
k
0
−
2
k
2
k
as the system stiffness matrix. But what of the mass/inertia matrix? As the masses
are concentrated at element nodes, we define the system mass matrix as
00 0
0
m
0
00
m
[
M
]
=
(10.31)
The equations of motion can be expressed as
+
=
U
1
U
2
U
3
U
1
U
2
U
3
R
1
mg
mg
[
M
]
[
K
]
(10.32)
where
R
1
is the dynamic reaction force at node 1.
Invoking the constraint condition
U
1
=
0,
Equation 10.32 become
m
U
2
U
3
5
k
U
2
U
3
mg
mg
0
−
2
k
+
=
(10.33)
0
m
−
2
k
2
k
which is a system of two second-order, linear, ordinary differential equations in
the two unknown system displacements
U
2
and
U
3
. As the gravitational forces
indicated by the forcing function represent the static equilibrium condition, these
are neglected and the system of equations rewritten as
m
U
2
U
3
5
k
U
2
U
3
0
0
0
−
2
k
+
=
(10.34)
0
m
−
2
k
2
k
As a practical matter, most finite element software packages do
not
include
the structural weight in an analysis problem. Instead, inclusion of the structural
