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Wave Vibration Analysis of Classical Multi-story Planar Frames
C. Mei
Department of Mechanical Engineering, The University of Michigan - Dearborn,
4901 Evergreen Road, Dearborn, MI 48128, U.S.A.
Keywords: Wave Vibration, Multi-story Frame, Bending Vibration, Longitudinal Vibration.
Abstract: This paper concerns free vibration analysis of in-plane vibrations in classical multi-story planar frame
structures. An exact analytical solution is obtained using wave vibration approach. The coupling effects
between bending and longitudinal vibrations in frames are taken into account. Classical beam theories are
applied in modeling the flexural and longitudinal vibrations. Reflection matrices at “sliding” and “rolling”
boundaries, as well as reflection and transmission matrices at the “L” and “T” joints are obtained.
Numerical examples are presented along with comparisons to results available in literature.
1 INTRODUCTION
Due to their complexity, vibrations in multi-story
planar frame structures are often analyzed either
based on approximated discrete models such as
lumped mass/elasticity models, or using numerical
approach such as the Finite Element Analysis (FEA)
approach. There are very limited analytical studies
based on distributed models found in the literature.
Lumped mass/elasticity may be suitable for
finding the fundamental frequency of a multi-story
frame, but they are prone to large errors and are
therefore not suitable for identifying higher modes
of vibration. The FEA approach is usually applied in
modeling multi-story frames (Vertes, 1985, and
Meirovitch, 2001), its accuracy is dependent upon
the number of meshes or nodes per structural
element. In general, higher modes demand more
nodes, consequently the FEA approach is only
suitable for relatively low frequencies. A branch
mode method was developed for studying in-plane
vibrations in multi-story frames (Gladwell, 1964)
with longitudinal vibrations neglected in the
analysis.
In this paper, vibrations in multi-story planar
frame structures are obtained analytically from wave
vibration standpoint (Graff, 1975; Cremer et. al.,
1987, and Doyle, 1989). Reflection matrices at
“sliding” and “rolling” boundaries, as well as
reflection and transmission matrices at the “L” and
“T” joints are discussed. The coupling effects of
flexural and longitudinal motions at joints are taken
into account. This study is based on the classical
vibration theories, as a result, it is suitable to
relatively low frequencies.
2 EQUATION OF MOTION AND
WAVE PROPAGATION
Consider the forces and moments acting on a
uniform element of a beam lying along the x-axis.
When applying classical beam/rod related theories,
the equations of motion for bending and longitudinal
vibrations are (Inman, 1994; and Ginsberg, 2001)
),(
),(),(
2
2
4
4
txq
t
txy
A
x
txy
EI
),(
),(),(
2
2
2
2
txp
x
txu
EA
t
txu
A
where x is the position along the beam axis, t is time,
and
are the transverse and
longitudinal deflections of the centerline of the
beam;
and
are the externally applied
transverse and longitudinal forces; E and
are the
Young’s modulus and mass density; respectively. I
is the area moment of inertia of cross section, A is
the cross-sectional area.
The shear force
, bending moment
, and longitudinal force
at any
section of the beam are related to the transverse
488
Mei C..
Wave Vibration Analysis of Classical Multi-story Planar Frames.
DOI: 10.5220/0004035204880493
In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 488-493
ISBN: 978-989-8565-21-1
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)