Generally, hysteresis phenomenon that occurs in dynamic systems is due to dependence of the input-output parameters on the time history. The relation between the output and the input variables
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forms a loop and such loop occurs due the dynamic lag between input and output parameters. This phenomenon can be represented using the phenomenological model and with a resultant loop formation as shown in figure (4.11). This phenomenological behaviour can be represented by a means of the classic hysteretic models. The smooth hysteretic model presented herein is a variation of the model originally proposed by Bouc [20] and modified by Wen [21]. Documented in [82, 83]
were the application of the Bouc-Wen hysteresis models in different areas of structural dynamics.
The Bouc–Wen model as used for the smooth hysteresis representation has received an increasing interest in the last few years. This is due to the ease of its numerical implementation and its ability to represent a wide range of hysteresis loop shapes. The Bouc-Wen Hysteretic model is capable of simulating stiffness degradation, strength deterioration.
Figure 4.10: The Bouc-Wen Model
Generally, this model consists of a first-order nonlinear differential equation that contains some parameters that can be chosen, using identification procedures, to approximate the behaviour of given physical hysteretic system. By implementing the Bouc-Wen model to the dynamic systems as shown in figure (4.10), one obtains a single non-linear first order equation which can describe the evolution of the damping force developed by a device for almost any loading pattern (periodic, aperiodic or random). The Bouc-Wen model is defined as:
Kz Ku
F 1 ..……….……. (4.33)
Where F F1F2 and z is a displacement parameter that controls the response of the non-linear of restoring for and it is governed by the differential equation:
n
n z uz
z u u A
z 1 ..……….……. (4.34)
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n
n z z
u z A u du
dz 1
..……….……. (4.35) Where A, n, β and γ are the set of parameters that controls the hysteretic response of the nonlinear element.
In the formulation of the smooth hysteresis loop, the hysteretic characteristic is a symmetric loop formed between the restoring force (–Fmax ≤ F (u) ≤ Fmax,) obtained for a periodic motion between the displacement (−umax (t) ≤ u (t) ≤ umax (t)), and the displacement is a function of the excitation.
Most often, the force-displacement characteristic of most systems is of hysteretic type is shown in figure (4.11).
The hysteresis behaviour of the conductor is a time-dependent process where the output variable is a function of the past inputs.
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Figure 4.11: Bouc-Wen hysteresis loop
To implement this model for the conductors, under the excitation process, the vortex shedding induces a periodic oscillation resulting in the restoring force at the inter-strand interfaces which causes it to displaying the hysteresis phenomenon. This time-rate dependent phenomenon results in the restoring forces trying to counter the strands movements and that leads to dissipation of energy. The incremental constitutive relation of moment-curvature conforms to the hysteretic behaviour which can be modelled using the Bouc-Wen model.
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Figure 4.12: The hysteresis Model in terms of bending moment-curvature relation
To achieve the hysteretic modelling, the use of such a hysteretic constitutive law is necessary for the effective representation of the behaviour of the conductor under cyclic loading, since as the conductor structures undergo inelastic deformations, this cyclic behaviour weaken the conductor structure and so there is a loss in stiffness and strength.
The application of Bouc-Wen model to model hysteresis behaviour of conductor as related to the energy dissipation was done as function of the bending moment and the curvature, denoted M and κ respectively. The form of the Bouc-Wen model as related to the moment-curvature relation, and the model was considered as the coupling of a linear and non-linear element in parallel as shown in figure (4.12).
The relation between bending moments and curvatures can be expressed as:
M z
M
Y
Y
1 ..……….……. (4.36)
Where MY is the yield moment; κ is the yield curvature; α is the ratio of the post-yield to the initial elastic stiffness and z is the hysteretic component defined as:
M
u
z
M Y 1 ..……….……. (4.37)
The yield moment can be evaluated by:
min secmax EI EI
EI
MY ..……….……. (4.38)
To model the hysteretic behaviour of conductor with the two parallel springs; one linearly elastic and one elasto-plastic spring with changing stiffness upon yielding. The combined stiffness is given as:
hysteresis
elastic EI
EI
EI ..……….……. (4.39)
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In relation to equation (4.33), where the post-yielding stiffness of the linear elastic spring and the stiffness of the hysteretic spring can now be expressed as:
du K dz du K
dF 1 ..……….……. (4.40)
du z zn
du k
dF 0 1 1
sgn sgn
..……….……. (4.41)The bending moment-curvature relation in terms of the Bouc-Wen model for the conductor can be obtain by comparing the F and u in the force-displacement in equation (4.33) as exactly as M and κ in the moment-curvature relation. This can then be used to model the hysteretic response of the conductor and upon the substitution this yields:
d z zn
d EI
dM
max 1 1 sgn sgn ..……….……. (4.42)
n
n z z
z A
z 1 ..……….……. (4.43)
In this context of evaluating damping, the energy liberated is frequency dependent and the hysteretic behavior of the conductor exhibits stiffness degradation. The stiffness degradation occurs as the conductor moves from stick to slip states.
The analytical expressions for the dissipated energy, taking into account the frequency of vibration can be achieved with respect to the steady-state response under symmetric wave periodic input.
This input can be expressed by the sine waves. Thus, the bending moment as a function of the periodic excitation frequency is defined as:
t EI
M sin ..……….……. (4.44)
Where 𝜔, is the frequency of excitation.
Under this periodic excitation, the response varies asymptotically with periodic input and the hysteretic loop is traced repeatedly. The characterization of the hysteresis loop boundary conditions was done using the initial and final curvature as define by equations (4.25) and (4.26) for the single layer conductors and equation (4.29) and (4.30) for the multiple layer conductors.
Thus, the energy dissipation as a function of friction at the contact points, defined in relation to the bending moment and curvature generates the loop as shown in figure (4.13).
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Figure 4.13: The Bouc-Wen hysteresis loop (M vs k)
To implement the Bouc-Wen model, the hysteresis parameters in equation (4.43) has to be defined.
A developed matlab code was used to implement various combinations of these hysteretic response parameters. Based on various curves that were developed as compared with the optimized Bouc- Wen model parameters the following parameters, A = 1, β = 0.75, n = 0.75, and ɣ = 0.25, were used to implement the hysteresis model in this study
κ
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