09-10-2014, 04:17 PM
Cables form important structural components in many engineering systems. The
capacity of elastic cables to carry high tensile loads is well established by their
wide application in suspension bridges, power lines and mooring cables.
Stringent safety requirements of these systems necessitate rigorous analysis even
in the determination of natural frequencies and mode shapes. The distinctive
characteristics of a cable are the curvature and elasticity. While there are several
investigations on the in plane response of cables under direct periodic forces,
behaviour under seismic support excitations and non-planar response under
lateral loads have not been understood well. In addition, since cables are by their
very nature displacement sensitive, consideration of nonlinear effects would be
necessary in some circumstances. In this context, it may be required to consider
stochastic inputs also, since earthquakes and wind forces are essentially random.
The present thesis is concerned with some aspects of linear and nonlinear
phenomena associated with cables under both deterministic and nondeterministic
inputs. The thesis is divided into six chapters.
Chapter I is devoted to a brief review of literature. Important past studies
pertaining to linear and nonlinear dynamic analysis of cables are reviewed. This
is followed by a critical discussion. The chapter ends with a summary bringing
into focus the motivation for the present research work of the author.
In Chapter II, the effect of a lateral load on the linear dynamic behaviour of a
cable is studied. When a non-uniform lateral load acts on a sagged cable, the
resulting vibration pattern of the cable is non-planar. Numerical results on
natural frequencies and mode shapes of non-planar vibration are presented.
Non-planar behaviour can be viewed as a coupling between lateral and in plane
motions which gives rise to further interesting problems. In particular, the effect
of a periodic lateral load in inducing parametric instability is investigated.
Numerical results on instability regions in the parameter space are presented for
typical examples.
In Chapter III, the forced response of cables under seismic support excitation in
studied, Both vertical and longitudinal components of motion are included in the
analysis. The significant participation of the elastic modes of the cable in the
response analysis is highlighted. Also, the influence of the propagation time
between the supports is examined in detail and is shown to cause considerable
variation in cable response. A modal superposition procedure based on the
response spectrum is developed to estimate the peak additional cable tension
under the seismic input motions. Results are presented for typical cables under a
few real earthquake records.
The study of nonlinear behaviour of cables under external excitation is the
subject matter of the next two chapters. In chapter IV, the planar and non planar
responses of a cable to combined in plane and lateral excitation are studied. Due
to the possible coupling between various modes, the nonlinear behaviour of a
cable is quite complicated. As a first step, the interaction between the first
symmetric mode of vertical and transverse displacement sis investigated. The
input is an in plane harmonic forcing function acting along with a uniform lateral
load. This type of excitation could be a model for the flow induced forces due to
vortex shedding. The equations of motion contain both quadratic and cubic
nonlinearities and are solved by the method of multiple scales. The steady state
response near the condition of internal and external resonance is investigated in
detail. Cables with even moderate sag-to-span ratio are shown to exhibit typical
nonlinear resonant behaviour different from that of a system with cubic nonlinearities.
The complicating effect of internal resonance in producing possible
non-periodic steady state solutions under periodic excitation is brought into
focus.
In Chapter V, the response of cables under a stochastic excitation is studied. Two
excitation models are considered, (i) a narrow band input and (ii) a combined
periodic and white noise input. The equations are solved by the method of
equivalent linearization using gaussian closure. It is found that the linearlized
equations lead to multi-valued response moments in certain regions of the
parameter space. A similar situation is known to arise in the case of Duffing’s
oscillator when analysed through equivalent linearlization technique. This is in
fact a limitation the approximate procedure. However, the acceptability or
otherwise of the results can be studied through a stochastic stability analysis.
Such an analysis is performed for both the cases of excitations mentioned above.
Digital simulation results are presented to support the theoretical results.