Modelling of a drifting oscillator using the modified linear by jlhd32


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									                                                                                   ENOC 2011, 24-29 July 2011, Rome, Italy

   Modelling of a drifting oscillator using the modified linear plastic and elastic plastic
                                    contact force models
                   O.K. Ajibose, M. Wiercigroch, E. Pavlovskaia, and A.R. Akisanya
         Centre for Applied Dynamic Research, School of Engineering, University of Aberdeen,
                                     Aberdeen, United Kingdom
Summary. The dynamic behaviour of a drifting oscillator is investigated using the modified linear plastic and elastic plastic contact
force models. The influences of the description of the contact model on the dynamic behaviour is presented. The general response of
oscillator was found to be qualitatively similar irrespective of the force penetration relationship describing the loading or unloading
phases of the contacting bodies. Consequently, the simpler the model is the more information there is to obtain for the system dynamics.


Some engineering systems involving impacts, such as ground moling, piling and percussive drilling have been modelled as
drifting oscillators [1]. It has also been shown that the dynamic behaviour of the oscillator predicted would be influenced
by contact model for certain values of damping and driving frequencies[2]. In order to study these systems, it is important
to identify a contact law that reflects actual force penetration relationships during the impacts. However, some of theses
impacts have contact laws that are significantly nonlinear. This result in a dependence on time consuming numerical
methods to analyse the system.
In this paper, a comparison between results obtained from a the modified linear–plastic force penetration relationship
proposed by [1] and the elastic–plastic law based on experimental observations is discussed. The global dynamics of both
systems are concisely presented for each model and the similarities and differences discussed. Some conclusions are then
briefly made and important recommendations are suggested.

                                                     Contact Force Models

Figure 1: The modified linear–plastic discrete model for impacts.        Figure 2: Typical experimental force–penetration relationships dur-
Adopted from [1]                                                        ing both phases of the impact. Adopted from [3]

The contact response of a drifting oscillator based on a modified linear–plastic model is shown in Fig.1. It is assumed that
during the loading phase, the impacted medium initial undergoes elastic formation until the yield strength of the material
was attained. During this period, the force could be described using the Kelvin–Voigt(KV) contact law. Afterwards, it is
suggested that the plastic deformation remains at the yield load until the velocity of the impacting surface became zero.
The unloading phase then proceeds with the restoring force also following the KV law.
However, experimental studies have observed that while the loading and unloading takes place during the impact, the force
penetration relationship is significantly different from that discussed earlier. Figure 2 shows a typical force penetration
response obtained from experiments. It is worth noting that both the loading and unloading phases of the impact can be
approximated by a power law type elastic-plastic contact law. Also, it is obvious that the elastic and plastic deformation
in loading phase is not easily distinguishable because the force continues to increase during the plastic deformation.

                                                       Physical Modelling

Figures 3 and 4 represent the physical model for the drifting oscillator using the simplified model and the experimental
models. A mass , m is subjected to a force, Ps , and a harmonic force of amplitude, Pd , at an angular frequency, Ω,
and phase angle, ϕ. The mass impacts massless slider intermittently. While the simplified model has a linear spring and
damper sandwiched between two sliders, the other model has just a single slider.
The experimental model has three phases of motion namely: no contact, loading and unloading. For linear plastic models
this loading phase consist of a contact without progression and contact with progression phases, while it unloading phase
is also another contact without progression phase.
As the name suggests, in no contact phase the mass is not touching the slider. The loading phase begins when the mass
                                                                                                                 ENOC 2011, 24-29 July 2011, Rome, Italy

comes in contact with slider. For the linear plastic model, only the top slider moves while the both slider is stationary
during the contact without progression phase.In the contact with progression phose both sliders move with the mass at
the same velocity. In the unloading phase, the mass moves in opposite direction at the same velocity with slider for the
experimental model, while only the top slider moves in this phase with bottom slider is again motionless. The end of the
unloading phase corresponds to when contact force becomes zero. This marks the beginning of a new no contact phase
and the process is repeated again as the slider progresses further away from its initial position.

                                                  F=P +Pd cos(Wt+j)


                                                                    G| t=0

                             K(X2-X3)              C(X3’) slider                  X1


Figure 3: Physical model of an an oscillator using the modified                                        Figure 4: Physical model of an oscillator using the elastic-plastic
linear-plastic law                                                                                    law


                           12000                                                                                                    250







                            4000                                                                                                    100

                            2000                                                                                                    50

                               0.00        0.05   0.10       0.15   0.20        0.25   0.30
                                                                                                                                          0.0   0.1        0.2         0.3   0.4

                                                   Static Force,     b                                                                                Static Load, b

Figure 5: Progression v during 300 cycles of forcing as a function                                     Figure 6: Progression z during 150 cycles of forcing as a function
of the static force b                                                                                  of the static force b

The equations of motion for two systems have being solved numerically for both models in Figs. 3 and 4 using the Runge-
Kutta method. It worth noting that while the time histories of the displacements of the bottom slider for the simplified
model shows progression when the material is plastically deforming, the displacement time histories for the other model
clearly distinguishes the loading and unloading phases of the motion. However both models confirm the existence of an
optimal static load, lower than the amplitude of the harmonic force, above which any increase in the values of this force
would not increase the progression rates. The progression diagrams in Figs. 5 and 6 reveal that, for parameter used in
the simulation, the maximum progression is attained at about 50% of the amplitude of the forcing when the linear plastic
model was used, while the value estimated for the other model was about 75%.


It is therefore possible to conclude that, whatever model was used to represent the system, there is always an optimal static
load beyond which any increase in it value would not improve the progression rates estimated. Consequently, the simpler
the model is the more information there is to obtain for the system dynamics provided the energy consumption during the
impact was the same as that obtained for actual experiments and the final deformation after each impact was identical.
[1] Pavlovskaia E., Wiercigroch M., Woo K-C., Rodger A.A. (2003) Modelling of Ground Moling by an Impact Oscillator with Frictional Slider.
    Meccanica 38 :85-97.
[2] Ajibose, O.K., Wiercigroch M., Pavlovskaia E., Akisanya A.R.(2010) Global and Local Dynamics of Drifting Oscillator for different contact
    models. Int. J. Non–Linear Mechanics 45 :850-858.
[3] Ajibose, O.K. (2009) Nonlinear Dyanmics and Contact Fracture Mechanics of High Frequency Percussive Drilling. PhD Thesis, University of
    Aberdeen, Aberdeen.

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