Chapter 14
Dynamics of Elastic Systems
Structural dynamics is a special branch of structural analysis, which studies the
behavior of structures subjected to dynamical loads. Such loads develop the dynami-
cal reactions, dynamical internal forces, and dynamical displacements of a structure.
They all change with time, and maximum values often exceed static ones. Dynami-
cal analysis of structure is based on the free vibration analysis.
This chapter is devoted to linear free vibration analysis of elastic structures with
lumped and distributed parameters. The fundamental methods of structural analysis
(force and displacement methods) are applied for calculation of frequencies of the
free vibration and corresponding mode shape of vibration. They are inherent to the
structure itself and are called as the eigenvalues and eigenfunctions.
14.1 Fundamental Concepts.................................................
14.1.1 Kinematics of Vibrating Processes..........................
The simplest periodic motion can be written as
y.t/DAsin.! tC 0 /;whereAis the amplitude of vibration,' 0 is the initial phase of vibration,tis time.
This case is presented in Fig.14.1a. The initial displacementy 0 D Asin' 0 is
measured from the static equilibrium position. The number of cycles of oscilla-
tion during 2 seconds is referred to as circular (angular or natural) frequency of
vibration!D2 =T(radians per second or s^1 ),T(s) is the period of vibration.
Figure14.1b, c presents the damped and increased vibration with constant period.
14.1.2 Forces Which Arise at Vibrations............................
During vibration a structure is subjected to different forces. These forces have a
different nature and exert a different influence on the vibrating process. All forces
I.A. Karnovsky and O. Lebed,Advanced Methods of Structural Analysis,
DOI 10.1007/978-1-4419-1047-914,cSpringer Science+Business Media, LLC 2010
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