agnetic excitation
Zohreh Mohammadi, Azadeh MaroufMashat, Hassan Salarieh, Mehdi Maasoumy Haghighi,
Mohammad Abediny, Aria Alasty
Department of Mechanical Engineering, Sharif University of Technology, P.O.Box 11365-9567, Tehran, Iran

A
bstract
:
This paper studies the chaotic behavior of a doubly clamped Euler-Bernoulli beam under magnetic excitation via an
experimental approach. The Responses of vibrations of the beam under different magnetic excitations are being investigated.
The setup includes a doubly-clamped beam which is installed on two static bases and it is excited in the middle by an
electromagnetic exciter. The excitation frequency varies from 1 to 50 Hz with different bias and amplitude voltages. Using
various numerical methods such as Fast Fourier Transformation, Phase diagrams and Maximum Lyapunov Exponents, the
experimental results are examined to find regular and irregular responses. The experimental results show that there exists some
harmonic and super harmonic and in some cases chaotic responses in this system
.
Key words: Electromagnetic excitation, Doubly-Clamped Beam, Chaos.

1. Introduction:
Chaos is a dynamic property that appears in nonlinear
systems in some special conditions. It can be introduced as
a behavior that dose not follow the specific pattern in phase
diagrams, like periodic orbits [1].
The important point is that the differential equations of
these systems are non-linear. In addition, these equations
have order of 3 or higher in continuous form. In many
cases, lower order but discrete and non-linear equations
may introduce chaotic behaviors. The most important
property of a chaotic system is its intense sensitivity to
initial conditions, so a very small difference in initial
condition causes big differences in results [2].
Vibrations under electro-magnetic excitation may introduce
nonlinear phenomena such as sub-harmonic, super-
harmonic oscillations and chaos [3-5]. In [6] using extended
Kalman filtering method, the nonlinear force of an
electromagnet on a single clamped beam is identified. In
that work, it is shown that harmonic excitation on a
clamped elastic beam may result in super-harmonic
behavior and ir-regular response. Electro-magnetic
excitation have many applications in active magnetic
bearing systems [7,8]. In [9] using a nonlinear model of
electromagnetic force which can justify the chaotic
response of one-dimensional magnetic levitation systems,
has been proposed. Using an experimental setup, Chang
and Tung [9] showed the chaotic response for magnetic
bearing system in high frequency range (30 to 40 Hz).
Instead of a multiplicative form they have identified a
superlative form for electromagnetic force model [6,10]. In
[11] experimental and analytical studies are performed on
chaotic behavior of active magnetic bearings. Investigation
of chaos in magnetically levitated doubly clamped beams is
examined via analytical and experimental methods in [12].
In this paper, using an experimental setup the chaotic
behavior of a doubly-clamped beam under electromagnetic
excitation is investigated. It is shown that the nonlinear
response of the system consists of the n
th
order super-
harmonic responses of the excitation frequency.

A/D Card in Computer

Mediator Board

Fig.1. Prepared setup for experiments

2. Experimental Setup:
The prepared setup (Fig. 1) includes an elastic doubly-
clamped beam which is excited by an E-type
electromagnetic exciter with ferrite core. A non-contact
Doubly clamped
Beam
Electromagnetic
Exciter
Voltage to Current Converter
Sensor Output Amplifier
Non-contact
proximity sensor proximity displacement sensor with characteristic curve of
Fig. 2 and sensitivity of 0.1% is used to measure
displacement of the mid-point of the beam. The head
diameter of this sensor is 15mm.
Fig.2. Characteristic curve of the non-contact proximity
sensor
A code was written in Visual Basic that produces the
exciting voltage and saves the output data in computer. By
using this code, the amplitude and DC magnitude of
voltage, and also exciting frequency and time interval of
excitations can be set. After running the program and
exciting the beam, the program saves displacement data
taken by non-contact proximity displacement sensor. Also
the excitation current which is related proportionally to the
computer voltage is saved. The time interval for data
acquisition is 0.004 sec.
First, the output of displacement sensor enters an amplifier,
and then through a mediator board the analogue signal
enters to an I/O card to prepare digital

data needed for
computer analysis. On the other hand, the output voltage of
the computer enters to the mediator board, and then it enters
a linear voltage-current transducer. After that voltage is
converted to current and this current in the electromagnetic
exciter coil, causes the beam to vibrate.
Fig.3. shows a schematic of the setup. The bases of the
beam are installed on a polyamide flat plate, to avoid
vibrations of the bases. These bases are glued on the
polyamide plate. The electromagnetic exciter is glued to
other poly-amid plate too. To have better calibration of the
sensor and also changing the distance between the exciter
and the beam, the setup is prepared such that the vertical
position of the electromagnetic exciter stand can be
changed easily.

3. Experimental Results:
For investigating the non-linear behavior of the beam,
under electromagnetic exciting, the required voltage after
passes through I/O card, enters to the voltage-current
transducer, in the form of the
)
sin(
0
t
V
V
+
. (
V
is the
amplitude of the voltage and
0
V
is the bias voltage.) The
current in the magnetic coil causes the oscillations of the
beam in the midpoint. The required data are saved in the
computer and the required diagrams are obtained via these
data. The experiments are done for different ratio of
amplitude and bias voltages, and different exciting
frequency.
Some of the results are noted here:
First,
V
and
0
V
are set to be 5 a 4volt respectively. We
will see an increase in non-linear and unstable behavior and
also a decrease in periodic and harmonic behavior of the
system while increasing exciting frequency. In this case the
periodic and quasi periodic behaviors in the system are
observed for different exciting frequencies. Existence of the
2nd, the third and the 4th and higher order super-harmonic
responses is the most important phenomena in this system.
In addition, maximum Lyapunov exponent of the time
series is negative or very near to 0. If the exciting frequency
is set to 5Hz, the behavior of the system will be like what is
seen in Fig.4.
If the exciting frequency is 20Hz, the behavior of the
system will be almost quasi periodic and not periodic like
the previous case and the magnitude of maximum
Lyapunov exponent is positive but very near to 0 (Fig 5).

Fig.3. Schematic of the prepared setup for experiments (b)
(a)
(c)
Fig.4. Experimental results for sinusoidal excitation of
4 5sin(10 )
U
t
= +
and forcing frequency of 5 Hz., (a) time history,
(b) Fourier transform, (c phase plane

(b)
(a)
(d)
(c)
Fig.5. Experimental results for sinusoidal excitation of and
4 5sin(40 )
U
t
= +
forcing frequency of 5 Hz., (a) time history, (b)
phase plane, (c) Fourier transform, (d) maximum Lyapunov exponent, the stability is at
0.25
= (b)
(a)
(d)
(c)
Fig.6. Experimental results for sinusoidal excitation of
4 5sin(80 )
U
t
= +
and forcing frequency of 5 Hz., (a) time history, (b)
phase plane, (c) Fourier transform, (d) maximum Lyapunov exponent, the stability is at
0.204
=
If the exciting frequency is 40Hz, the stability of the
system behavior decreases and the phase diagram is not
regular according to Fig6.
For amplitude and bias voltage equal to 5 and 3 volt
respectively we will see instability and non-linearity in the
system while exciting frequency is increasing up to 20Hz.
After that, there is a periodic and harmonic behavior in
this system while increasing the exciting frequency up
to 50Hz and maximum Lyapunov exponent is negative
here. If the exciting frequency is 24Hz it is clear from FFT
diagram in Fig 7 that the response frequencies are integer
factors of exciting frequency.
If the exciting frequency is 30Hz (Fig 8) it is clear from
time history of the system that the behavior is not periodic
and the phase diagram is nearly chaotic and maximum
Lyapunov exponent is positive.
(b)
(a)
(d)
(c)
Fig.7. Experimental results for sinusoidal excitation of
3 5sin(48 )
U
t
= +
and forcing frequency of 5 Hz., (a) time history, (b) phase plane, (c) Fourier transform, (d) maximum Lyapunov exponent, the stability is at
0.27
=
(b)
(a)
(d)
(c)
Fig.8. Experimental results for sinusoidal excitation of
3 5sin(60 )
U
t
= +
and forcing frequency of 5 Hz., (a) time history,
(b) phase plane, (c) Fourier transform, (d) maximum Lyapunov exponent, the stability is at
0.27
=
After increasing the exciting frequency to 40Hz, again it is
observed that the response frequencies are integer
coefficients of excitation frequency and the system
behavior is periodic (Fig 9). In this case maximum
Lyapunov exponent is negative and one can not have a
deterministic decision about the chaotic behavior of the
system. Fig 10 shows the behavior of the system with
exciting frequency of 50Hz