Advanced Methods of Structural Analysis

516 14 Dynamics of Elastic Systems

From mathematical point of view, the difference between the two types of sys-
tems is the following: the systems of the first class are described by ordinary
differential equations, while the systems of the second class are described by partial
differential equations.
Figure14.2a, b shows a massless statically determinate and statically indetermi-
nate beam with one lumped mass. These structures have one degree of freedom,
since transversal displacement of the lumped mass defines position of all points of
the beam. Note that these structures, from point of view of theirstatic analysis,
have the number degrees of freedomWD 0 for statically determinate beam (a) and
WD3D2HS 0 D 3  1  2  0  5 D 2 for statically indeterminate beam (b)
with two redundant constraints. It is obvious that a massless beam in Fig.14.2chas
three degrees of freedom. It can be seen that introducing of additional constraints
on the structure increases the stiffness of the structure, i.e., increase the degrees of
static indeterminacy, while introducing additional masses increase the degrees of
freedom.

f

C

Pontoon

d x

y

z

a

y 1

c

y 1 y 2 y 3

e x

y

b

y 1

Fig. 14.2 (a–f) Design diagrams of structures

Figure14.2d presents a cantilevered massless beam carrying one lumped mass.
However, this case is not a plane bending, but bending combined with torsion, be-
cause mass is not applied at the shear center. That is why this structure has two
degrees of freedom, such as the verticaldisplacement and angle of rotation iny-z
plane with respect to thex-axis. A structure in Fig.14.2e presents a massless beam
with an absolutely rigidbody. The structure has two degrees of freedom, such as the
lateral displacementyof the body and angle of rotation of the body iny-xplane.
Figure14.2f presents a bridge, which contains two absolutely rigid bodies. These
bodies are supported by a pontoon. Corresponding design diagram shows two abso-
lutely rigid bodies connected by hingeCwith elastic support. So this structure has
one degree of freedom.
The plane and spatial bars structures and plane truss are presented in Fig.14.3.
In all cases, we assume that all members of a structure do not have distributed
masses. The lumped mass (Fig.14.3a) can move in vertical and horizontal direc-
tions; therefore, this structure has two degrees of freedom. Similarly, the statically
indeterminate structure shown in Fig.14.3b has two degrees of freedom. However,
if we assume that horizontal member is absolutely rigid in axial direction (axial