Single Degree-of-Freedom System

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by a dynamically equivalent system, a single degree-of-freedom system for
example, and the spring of the system is designed such that under the
applied dynamic load, P(t), the maximum deflection of the mass, Xm, is
within acceptable limits.
The fundamental principles of dynamic analysis will not be discussed in
this section because there are several books and manuals which deal with
this subject, such as NAVFAC P-397, Introduction to Structural Dynamics by
Biggs, and Dynamics of Structures by R.W. Clough, et al.  The purpose of
this section is to present guidance in the application of the principles
used in the analyses of structures to resist the effects of blast (dynamic)
a.  Dynamically Equivalent System.  Structural elements in general have
a uniformly distributed mass and, as such, the elements can vibrate in a
unlimited number of modes, shapes or frequencies.  Usually, the fundamental
mode shape or frequency is of greater significance to the designer since it
dominates the response of the element.  Neglecting contributions from higher
modes introduces some errors, but these are insignificant.
(1) Single Degree-of-Freedom System.  To derive a single
degree-of-freedom system, the uniform mass of the system is considered
lumped at point of maximum deflection and a deformation pattern (usually the
fundamental mode) is assumed.  A load-mass factor is introduced in order to
simplify the design process and, by multiplying the actual mass of the
element by the load-mass factor, the element is reduced to an equivalent
single degree-of-freedom system which simulates the response (deflection,
velocity, acceleration) of the actual element.  Several references,
including NAVFAC P-397, present single degree-of-freedom approximations
for different structural configurations such as simply supported,
fixed-fixed beams, one- and two-way slabs.
(2) Period of Vibration.  The effective natural period of vibration
for a single degree-of-freedom system is expressed as:
EQUATION:
TN = 2[pi](KLMm/KE)1/2
(1)
where,
m = unit mass
KE = equivalent unit stiffness of the system
(3) Effective Mass.  As stated in paragraph 7.a.(1), a load-mass
factor is applied to the actual mass of the element so as to reduce it to a
equivalent single degree-of-freedom system.  The product of the mass, m, and
the load-mass factor, KLM, is defined as the effective mass.  Table 6-1
and Figure 6-5 of NAVFAC P-397 present load-mass factors for one- and two-
way elements of various support conditions.  The value of the load-mass
factor, KLM, as shown in Figure 2, depends in part on the range of
behavior; i.e., elastic, elasto-plastic or plastic.  Average values of KLM
are used for elasto-plastic, plastic, ad post-ultimate ranges of behavior.
2.08-5