Magnetorheological damping and semi-active control of an autoparametric vibration absorber
Krzysztof Kecik
0
1
Andrzej Mitura
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1
Danuta Sado
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1
Jerzy Warminski
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1
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D. Sado Institute of Machine Design Fundamentals, Warsaw University of Technology
, Warsaw,
Poland
1
K. Kecik (&) A. Mitura J. Warminski Department of Applied Mechanics, Lublin University of Technology
, Lublin,
Poland
A numerical study of an application of magnetorheological (MR) damper for semi-active control is presented in this paper. The damper is mounted in the suspension of a Duffing oscillator with an attached pendulum. The MR damper with properties modelled by a hysteretic loop, is applied in order to control of the system response. Two methods for the dynamics control in the closed-loop algorithm based on the amplitude and velocity of the pendulum and the impulse on-off activation of MR damper are proposed. These concepts allow the system maintaining on a desirable attractor or, if necessary, to change a position from one attractor to another. Additionally, the detailed bifurcation analysis of the influence of MR damping on the number of periodic solutions and their stability is shown by continuation method. The influence of MR damping on the chaotic behavior is studied, as well.
1 Introduction
Pendulum-like systems are commonly used in many
practical applications, including special dynamical
dampers or energy harvesters [1]. Dynamics of such
systems can exhibit extremely complex behaviour.
Especially, if the system is nonlinear and includes the
inertial coupling, among strange attractors, multiple
regular attractors may co-exist for some values of
system parameters [2]. The presence of the coupling
terms can lead to a certain type of instability which is
referred to as autoparametric resonance. This kind of
phenomenon takes place when the external resonance
and the internal resonance meet themselves, due to the
coupling terms. The multiple solutions, evolution of
the solution due to variations in parameters or initial
conditions play a very important role in system
dynamics. The small perturbation of initial conditions
or systems parameters may transit the response to
dangerous motion, like a full rotation of the pendulum
or chaotic motion [3]. This problem is essential if the
pendulum plays role of a dynamical vibration absorber
or the energy harvesting device [4]. An autoparametric
system has been intensively analysed for three last
decades. The different responses in the autoparametric
pendulum vibration absorber for a linear mass-spring
damper system have been studied by the harmonic
balance method in Hatwal et al. [5]. A nonlinear
frequency analysis using the multiple scales method
has been presented in Cartmell et al. [6, 7]. The two
mode autoparametric interaction and robustness,
against variations on the excitation frequency, are
improved on the overall system by direct application of
an onoff servomechanism, controlling the effective
pendulum length and validating also the theoretical
results in an experimental setup. A similar pendulum
dynamic vibration absorber, with time delay in the
internal feedback force, is used to illustrate the
realtime application of a dynamic substructuring technique
in Kyrichko et al. [8] and in paper [9].
In this paper authors propose application of the MR
damper, installed between the oscillator and the
ground to provide controllable damping for the
system. The model of a damper takes into account
the hysteretic effect. The closed-loop control
algorithms offer possibility to move the system between
selected stable solutions. Moreover, we show, that MR
damping practically does not reduce the vibration
suppression effect and MR damping can cause a shift
of chaotic regions.
2 An autoparametric pendulum system
2.1 Model of magnetorheological damper (MRD)
The magnetorheological devices provide modern and
elegant solutions for semi-active control in a variety of
applications, offering several advantages: simplicity of
a structure, small number of mobile components,
noise-free fast operation and low power demands. The
MRD is a nonlinear component with dissipative
capability used in the control of semi-active
suspensions, where the damping coefficient varies according
to the applied electric current. MR damper is usually
characterized by the displacement and/or velocity of
the piston, the electric current applied to the coil as
inputs and the force generated on the piston as output.
The relationship between damping force and velocity
shows hysteresis loops whose shapes vary according to
the applied current. Hysteresis in dampers is due to the
difference between the accelerating and decelerating
paths of the force-velocity curve [10], thus imposing a
delay in the changes of internal pressures and
ultimately forces. Therefore, we propose the nonlinear
MR damping force (FMR) approximated by a
hyperbolic tangential function of the velocity and the
displacement of the oscillator based on the papers
[11, 12]. In terms of mathematical expressions, the
model makes use of a hyperbolic tange (...truncated)