Abstract
The objective of this paper is the study of the dynamics of damped cable systems, which are suspended in space, and their resonance characteristics. Of interest is the study of the nonlinear behavior of large amplitude forced vibrations in three dimensions. As a first-order nonlinear problem the forced oscillations of a system having three-degrees-of-freedom with quadratic nonlinearities is developed in order to consider the resonance characteristics of the cable and the possibility of dynamic instability. The cables are acted upon by their own weight in the perpendicular direction and a steady horizontal wind. The vibrations take place about the static position of the cables as determined by the nonlinear equilibrium equations. Preliminary to the nonlinear analysis the linear mode shapes and frequencies are determined. These mode shapes are used as coordinate functions to form weak solutions of the nonlinear autonomous partial differential equations.
In order to investigate the behavior of the cable motion in detail, the linear and the nonlinear analyses are discussed separately. The first part of this paper deals with the solution to the self adjoint boundary-value problem for small-amplitude vibrations and the determination of mode shapes and natural frequencies. The second problem dealt with in this paper is the determination of the phenomena produced by the primary resonance of the system. The method of multiple time scales is used to develop solutions for the resulting multi-dimensional dynamical system with quadratic nonlinearity.
Numerical results for the steady state response amplitude, and their variation with external excitation and external detuning for various values of internal detuning parameters are obtained. Saturation and jump phenomena are also observed. The jump phenomenon occurs when there are multi-valued solutions and there exists a variation of kinetic energy among solutions.
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Abbreviations
- A=diag(a i ,i=1, 2, 3):
-
amplitude matrix (diagonal)
- A n,A :
-
undeformed area, deformed area
- B :
-
span of hanging cables
- D :
-
sag for static conditions
- E :
-
Young's modulus
- \(F = \frac{1}{m} V^1 G^{ 1} \int_{1 2}^{1 2} {R\bar f ds} \) :
-
vector of external force
- \(G = \int_{1 2}^{1 2} {R^2 ds} \) :
-
diagonal matrix
- \(\begin{gathered} H = \zeta {}^1(\zeta ^{1 2} - 1)I \hfill \\ + \zeta {}^2(2 - \zeta {}^{1 2})y'y'^1 \hfill \\ \end{gathered} \) :
-
symmetric coefficient matrix
- H * =HR′ I :
-
unit matrix
- \(K = - \int_{1 2}^{1 2} {RH' ds} \) :
-
diagonal matrix
- L :
-
original length of cables before hanging
- M :
-
the symmetric stiffness matrix
- N :
-
integer
- P :
-
damping constant matrix (diagonal)
- R :
-
linear mode shape matrix (diagonal)
- S :
-
sway of hanging cables
- T :
-
tension of cables
- T o :
-
tension of cables for static conditions
- T o(0):
-
tension of the lowest point for static conditions
- V:
-
eigenfunction matrix
- b=y′T R′ :
-
coefficient vector
- b:
-
\(\frac{f}{{m g}}\)
- c,c 1,c 2,c 3 :
-
vector, and the components in thex 1,x 2,x 3 directions respectively, in terms of cosine functions.
- e, e o :
-
strain, and static strain of elongation
- e 1 :
-
time-dependent perturbation ine
- f :
-
wind force in the sway direction
- f, f 0,f 1 :
-
vector of external force
- g :
-
gravity constant
- h :
-
time-dependent amplitude vector
- m :
-
mass density per unit length of the undeformed cable
- r=(R 1,R 2,R 3)T :
-
vector of modal shapes
- s :
-
undeformed arc length
- t :
-
time
- u 1 :
-
linear scalar in z
- u 2 :
-
quadratic scalar in z
- v 1,v 2,v 3 :
-
eigenfunctions inx 1,x 2, andx 3 directions, respectively
- x=(x 1,x 2,x 3)T :
-
Cartesian position vector and components
- y=(y 1,y 2,y 3)T :
-
static position vector and components
- ε:
-
error vector
- ℒ:
-
matrix operator
- Ω=diag[Ω1, Ω2, Ω3]:
-
internal frequency matrix and components
- \(\bar \Omega \) :
-
excitation frequency
- ϕ:
-
global matrix of coordinate functions
- α:
-
T o(0)/mgL
- β:
-
mgL/EA o
- ζ:
-
y′y′ T
- ξ:
-
s/L
- τ = diag[τ1, τ2, τ3]:
-
phase angle matrix and components of characteristic modes
- \(\bar \tau \) :
-
phase angle of excitation force
- η1, η2 :
-
time-dependent amplitude vectors in timet o and timet 1
- φij,i=1, 2...N,j=1, 2, 3:
-
theith coordinate function of thejth component
- φi = diag[φi1, φi2, φi3]:
-
theith matrix of coordinate functions
- γ:
-
global vector of modal amplitudes
- σ1 :
-
external detuning parameter
- σi,i=2, 3:
-
internal detuning parameter
- σi,i=1, 2, 3:
-
phase angles
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Tadjbakhsh, I.G., Wang, YM. Wind-driven nonlinear oscillations of cables. Nonlinear Dyn 1, 265–291 (1990). https://doi.org/10.1007/BF01865276
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DOI: https://doi.org/10.1007/BF01865276