# Modelling of a drifting oscillator using the modified linear by jlhd32

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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
phases of the contacting bodies. Consequently, the simpler the model is the more information there is to obtain for the system dynamics.

Introduction

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.
penetration relationship is signiﬁcantly different from that discussed earlier. Figure 2 shows a typical force penetration
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.
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)
S

M

G| t=0

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

X2
Pf
X3

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

Discussions

12000                                                                                                    250

10000
200

8000
v

z
Progression,

Progression,

150

6000

4000                                                                                                    100

2000                                                                                                    50

0
0
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
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%.

Conclusions

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.
References
[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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