Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 3, pp. 763-774, Warsaw 2013

INFLUENCE OF IDEAL AND NON-IDEAL EXCITATION SOURCES ON

THE DYNAMICS OF A NONLINEAR VIBRO-IMPACT SYSTEM

Fernando H. Moraes, Bento R. Pontes Jr, Marcos Silveira

UNESP – São Paulo State University, Department of Mechanical Engineering, Bauru, SP, Brazil

e-mail: brpontes@feb.unesp.br

José M. Balthazar

UNESP – São Paulo State University, Department of Mechanical Engineering, Bauru, SP, Brazil and

UNESP – São Paulo StateUniversity, IGCE,Department of AppliedMathematics andComputation, RioClaro, SP, Brazil

Reyolando M.L.R.F. Brasil

Federal University of ABC, Santo André, SP, Brazil

The objective of the present work is to analyze the dynamics of a vibro-impact system
consisting of two blocks of differentmass coupled by a springwith two stages. Two different
types of excitation sources for the system are used in the analysis: the first is an ideal
excitation in form of a harmonic force, and the second is a non-ideal excitation in form
of a DC electric motor with limited power supply which has an unbalanced rotor. The
control parameter for both situations is the excitation frequency of the system.The analysis
includes time histories of displacements and velocities, phase portraits and diagrams of the
displacement and frequency, used to show the Sommerfeld effect. For certain values of the
parameters of the system, the motion is chaotic. The mathematical model of the system is
used to obtain an insight to the global dynamics of the vibro-impact system. The model
with a non-ideal excitation ismore realistic, complete and complex than themodel with the
ideal excitation.

Key words: vibro-impact system, limited power supply, non-ideal excitation, Sommerfeld
effect

1. Introduction

There are many applications in engineering where it is necessary the use impacts to induce
vibrations in parts of the system in order to improve its performance or efficiency. The vibro-
-impact mechanisms deserve as much attention as well as vibration absorbers, since they are
present in a wide range of systems in mechanical, civil, and electrical engineering. Examples of
applications are vibratory conveyors, drillingmachines, compactingmachines, forging hammers,
impact hammers and powdermixers. Souza et al. (2008) considered vibro-impact systemswhich
have many implementations in applied mechanics, ranging from drilling machinery and metal
cutting processes to gearboxes. Moreover, from the point of view of dynamical systems, vibro-
-impact systems exhibit a rich variety of phenomena, including chaotic motion.
Pavlovskaia et al. (2001) conducted studies inavibro-impact systemof twodegrees of freedom

and stated that although it seems simple, the dynamics of the system is very complex, ranging
from periodic regimes to chaotic regimes. Ing et al. (2010) performed extensive experimental
investigations of an impact oscillator with a one-sided elastic constraint. Different bifurcation
scenarios have been shown for anumberof values of the excitation amplitude,with the excitation
frequency as the bifurcation parameter.
Lin and Ewins (1993) performed detailed numerical and experimental studies on chaotic

vibration of mechanical systems with backlash. The backlash, in such systems, arise in engine-
ering systems inwhich componentsmake intermittent contact due to the existence of clearances



764 F.H. Moraes et al.

(or gaps). The forcing parameter range for the existence of chaotic regime and the influence of
damping on the chaotic behavior were investigated.

Amajor application of vibro-impactmechanisms can be seen in soil drillingmachines, or oil
drilling, due to high costs involved in these types of projects, with a drill that may have more
than kilometers of length and has multiple vibration modes (Wiercigroch et al., 2005). In this
case, any kind of failure or maintenance is highly complex and expensive.

Very often the drilling operation does not allow error. Aguiar (2010) conducted studies
focused on oil drilling, where a condition of impact on the interaction between the drill and
the rock is able to facilitate the penetration by spreading cracks in the hard rock to be drilled.
The drilling of brittle materials requires a high rate of energy transfer to induce fracture of the
material to be drilled. Themost suitable transfer of energy between the drill bit and the rock is
by impact.

The energy sources can be classified as ideal and non-ideal. The ideal energy sources do not
consider that the acting forces on the system influence themotor dynamics, themodels that use
the ideal excitation are simplified models that disregard important phenomena. Thus, a model
that uses a non-ideal energy source is a significantly more complete and complex model.

Kononenko (1969) described the first type of the non-ideal problem, the Sommerfeld effect,
first seen in 1904. The Sommerfeld effect is due to the excitation source having limited power,
usually called a non-ideal source. Nayfeh and Mook (1979) obtained analytical solutions for
various non-ideal problems, which are composed of oscillators connected to this form of energy
source. Dimentberg et al. (1997) analised the dynamics of an unbalanced shaft interacting with
a limited power supply. Numerical and experimental solutions were obtained, and extensive
parametric studies were performed on the Sommerfeld effect.

The main influence of the non-ideal energy source on the system is notable as it passes
through resonance. As the power supplied to the source increases in the region before resonance,
the angular velocity of the energy source increases accordingly. Near the resonance, it appears
that additional power supplied to the motor will only increase the amplitude of the oscillating
system response,with little effect on the angular velocity of the energy source.As a consequence,
the non-ideal oscillating system cannot pass through the resonance frequency of the system, or
requires an intensive interaction between the vibrating system and the energy source to be able
to do so (Balthazar et al., 2003; Zukovic and Cvetićanin, 2007, 2009).

Souza et al. (2005a,b) investigated numerically the dynamical behavior of a non-idealmecha-
nical system comprised of an oscillating mass body excited by an unbalanced rotor driven by a
non-ideal motor, containing another mass which can oscillate back and forth, colliding with the
walls of themain mass. Themotor and the unbalanced rotor represent a limited energy source.
The dynamics of the oscillating system is largely influenced by the limited energy source.

Warminski et al. (2001) investigated vibrations of parametrically and self-excited systems
with ideal and non-ideal energy sources. Tsuchida et al. (2005) studied dynamics of a non-ideal
systemwith two coupled oscillators. The analysis of the responses showed that for certain values
of the parameters in the resonant regime, jump phenomena occur and chaos appears. Zukovic
andCvetićanin (2007, 2009) analysed dynamics of a non-ideal system comprised of an oscillator
connected with an unbalanced motor with clearance. The transient and steady state responses
and stability were studied. The Summerfeld effect was detected, and chaotic regimes were found
for a range of parameters.

The occurrence of chaos in non-ideal systems is associated with the presence of non-linear
terms, bifurcation points, instability regions and non-stationary regimes in the resonance region
aswell as abrupt transitions in stiffness. Therefore, it is of great importance to study the system
in those regions.

ADC electric motor with limited power supply is considered as a non-ideal source of energy
(Balthazar et al., 2005). The influence of the response of the flexible structure on theDCmotor



Influence of ideal and non-ideal excitation sources ... 765

causes the appearance of a jump phenomenon and the Sommerfeld effect. The DCmotor has a
family of static characteristic curves, each being associated with a torque constant and with an
angular velocity constant of themotor. Themanner of regulation,more usual, is the variation of
voltage applied to themotor terminals, which controls the angular velocity. The angular velocity
affects the motor torque.

This paper is organised as follows. Section 2 contains the mathematical models of the two
vibro-impact systems used in this work, the first uses an ideal excitation source, the second uses
a non-ideal excitation source. Section 3 contains the results from numerical simulations of both
models, including time histories of displacements and velocities, phase portraits and diagrams
of the displacement and frequency, used to show the Sommerfeld effect for the model with the
non-ideal excitation.

2. Mathematical modeling

Two cases are presented and analysed. The first (Case I) involves a vibro-impact model with
the ideal excitation. The excitation source is a harmonic force (Fh) acting on block 1, and the
control parameter is the excitation frequency (ω) related to this force. The second (Case II)
involves a vibro-impact model with a non-ideal excitation. The excitation comes from a DC
electric motor with limited power supply and an unbalanced rotor, and the control parameter is
the angular velocity constant of the motor (Ω0). This model differs from the previous one only
by the type of excitation source.

2.1. Model with ideal excitation

The model with ideal excitation has two degrees of freedom and consists of two blocks
(1 and 2) with mass m1 and m2, respectively, with m2 greater than m1. They are coupled by
a spring with elastic stiffness coefficient k1 and a viscous damper with damping coefficient c.
The displacements of blocks 1 and 2 are given by x1 and x2, respectively. The model is shown
in Fig. 1

Fig. 1. Vibro-impact model with ideal excitation

A harmonic force Fh provides a sinusoidal excitation on block 1. Block2 has an extension
subject to the dry friction force against a surface, with dry friction coefficient µ. This friction
generates a counter force to thedisplacement. In application todrillingmachines, thedry friction
represents the interaction between the drill and the surface.

A secondary spring (impact spring) exists between the two blocks, with a very high stiffness
compared to the first spring.The elastic constant k2 of the second spring represents the stiffness
of the material of block 2.

Springs 1 and 2 form together a spring system with two stages of stiffness. As the elastic
constant k2 is much greater than the elastic constant k1, there is an abrupt transition in
the stiffness as block 1 makes or loses contact with the second spring (Fig. 2). The gap G
is the distance between block 1 and the point of contact with the second spring.Whenever the



766 F.H. Moraes et al.

Fig. 2. Two-stage stiffness characteristic of the spring system (Zukovic and Cvetićanin, 2009)

relative displacement between the blocks is smaller than this gap, impact occurs. The system
behaves periodically when there is no abrupt stiffness transition in the two-stage spring, when
the relative displacement between the blocks is smaller than the gap G. Figure 2 shows the
stiffness characteristic of the two-stage spring system (Zukovic and Cvetićanin, 2009):
—while there is no impact on the system, meaning x2−x1 <G, the equations of motion are

m1ẍ1 =Fh−Fs1−Fd m2ẍ2 =Fs1+Fd−Ff (2.1)

— when the relative displacement between the blocks is greater than the gap, meaning
x2−x1 ­G, impact occurs, and the equations of motion are

m1ẍ1 =Fh−Fs1−Fs2−Fd m2ẍ2 =Fs1+Fs2+Fd−Ff (2.2)

From these equations, it is possible to see that the system is piecewise linear, consisting of
two regions definedby the relationship between the relative displacement x2−x1 and the gap G.
The harmonic force (Fh) acting on block 1 has the form

Fh =Asin(ωt) (2.3)

in which A is the amplitude and ω is the frequency of excitation.
The force of the first spring (Fs1) is given by

Fs1 = k1(x1−x2) (2.4)

in which k1 is the elastic constant of this spring, x1 and x2 are the displacements of blocks 1
and 2, respectively.
The force of the second, or impact, spring (Fs2) is given by

Fs2 = k2(x1−x2+G) (2.5)

in which k2 is the elastic constant of this spring and G is the distance between block 1 and the
point of contact with this spring.
The viscous damping force (Fd) is given by

Fd = c(ẋ1− ẋ2) (2.6)

in which c is the viscous damping coefficient, ẋ1 and ẋ2 are the velocities of blocks 1 and 2,
respectively.

The dry friction force is given by

Ff =µm2g sgn(ẋ2) (2.7)

in which µ is the dry friction coefficient, g is the acceleration of gravity and sgn(·) denotes the
signum function.



Influence of ideal and non-ideal excitation sources ... 767

This mathematical model has a kinematic compatibility condition: it is not valid when the
absolute position of block 1 is greater than the absolute position of block 2. This condition
ensures that the blocks do not reverse position. Also, there is no possibility of a block going
through the other. Otherwise, the model would be physically inconsistent.

Dimensional parameters (m1, m2, k1, k2, c, µ, G, A, ω) were based on the model from Ho
et al. (2010). Three simulations were performed for this case, with ω=3.5Hz, ω=5.5Hz and
ω = 6.5Hz. Values of the control parameter (ω) were obtained in a manner that the system
shows periodic, quasi-periodic and chaotic responses, with andwithout the occurrence of abrupt
transition of stiffness.

2.2. Model with non-ideal excitation

The model with non-ideal excitation has three degrees of freedom, with the third degree of
freedombeing the angular position φ of the unbalanced rotor of theDCmotor.All other aspects
of this model are the same as the model with ideal excitation. Figure 3 shows the vibro-impact
model with non-ideal excitation.

Fig. 3. Vibro-impact model with non-ideal excitation

Block 1 is now excited by amotorwhich has an unbalanced rotor, with unbalancedmass m3,
eccentricity e andmoment of inertia J. This motor is characterized as a non-ideal source. The
dimensional parameters (m1, m2, k1, k2, c, µ, G) of the model with the non-excitation source
are equal to the model with the ideal excitation source.

The torque provided by the motor, M(φ̇), is a function of the angular velocity of the rotor
axis with an unbalanced rotor, and is given by

M(φ̇)=M0
(

1−
φ̇

Ω0

)

(2.8)

in which M0 is the torque constant of the motor, φ̇ is the angular velocity of the rotor and
Ω0 is the angular velocity constant of the motor.

The equations of motion valid while there is no impact (x2−x1 <G) are

(m1+m3)ẍ1 =m3e(φ̈sinφ+ φ̇
2cosφ)−Fs1−Fd

m2ẍ2 =Fs1+Fd−Ff

(J+m3e
2)φ̈=m3eẍ1 sinφ+M(φ̇)

(2.9)

inwhich m3 is theunbalancedmass of the rotor, e is the eccentricity, φ is the angular position of
the unbalancedmass, J is themoment of inertia of themotor and M(φ̇) is the torque provided
by the motor.



768 F.H. Moraes et al.

When the relative displacement between the blocks is greater than the gap (x2 −x1 ­ G)
impact occurs, and the equations of motion are

(m1+m3)ẍ1 =m3e(φ̈sinφ+ φ̇
2cosφ)−Fs1−Fs2−Fd

m2ẍ2 =Fs1+Fs2+Fd−Ff

(J+m3e
2)φ̈=m3eẍ1 sinφ+M(φ̇)

(2.10)

The forces Fs1,Fs2,Fd,Ff and the torque M(φ̇) are defined in Eqs. (2.4), (2.5), (2.6), (2.7)
and (2.8), respectively.
The dimensional parameters of the non-ideal excitation source (m3, e, J,M0 and Ω0) were

obtained such that the force of the non-ideal excitation is similar to the force of the ideal
excitation, for a better comparison between themodels.
The angular velocity constant of the motor (Ω0) was adopted as the control parameter

because it is one of the easiest variables to change in a real experimentwithanon-ideal excitation
source (DC electric motor). Its variation is achieved by altering the current or voltage on the
electric motor.
Threevalues of theangular velocity constant areused for this case: Ω0 =3.5Hz, Ω0 =5.5Hz

and Ω0 = 6.5Hz. The control parameter values were obtained in a way that the first value
(3.5Hz) is below the natural frequency, the second (5.5Hz) and the third (6.5Hz) are above
the value of the natural frequency of the vibro-impact model (ωn =4.1Hz).
It is possible to analyze the influence of the angular velocity constant of themotor relative to

capture andpassage through the resonance frequency.To this end, three curveswere determined
representing the DC electric motor based on the angular velocity achieved at the steady state
of themotor. Each characteristic curve shows that themotor reaches equilibrium for the system
operating at different angular velocities (Fig. 4). The torque constant was kept fixed and the
angular velocity constant of the motor (Ω0) was varied.

Fig. 4. Motor characteristic curves used in Case II

The analysis includes time histories of displacements and velocities, phase portraits and
diagrams of displacement and frequency, used to show the Sommerfeld effect.
A relationship between the harmonic force of the model with ideal excitation and the force

provided by the unbalanced rotor has to be established to make a realistic comparison between
the models. Recalling Eq. (2.3) and relating it to the force generated by the unbalanced rotor
(

m3eω
2 sin(ωt)

)

, this relationship is expressed by

Asin(ωt)=m3eω
2 sin(ωt) A=m3eω

2 (2.11)

The left side of Eq. (2.11)1 corresponds to the ideal excitation force, and the right side
corresponds to the non-ideal excitation force. Thus, for each value of the unbalanced mass, the



Influence of ideal and non-ideal excitation sources ... 769

rotor generates a specific excitation force. Each characteristic curve of themotor determines an
angular velocity of the motor axis where the motor reaches the steady state.

The parameters common to the model with the ideal excitation and the model with the
nonideal excitation were used with the following values: m1 = 0.297kg, m2 = 2.94kg,
k1 = 200N/m, k2 = 12400N/m, c = 0.001Ns/m, µ = 0.001 and G = 0.001m. The para-
meters exclusive to themodel with the non-ideal excitation were usedwith the following values:
m3 =0.03kg, e=0.11m, J =0.001kgm

2 and M0 =0.005Nm.

Table 1 shows the dimensional parameters for the numerical simulations performed in Cases
I and II. The symbol ‘–’ used in the table indicates that the value does not exist.

Table 1.Parameter values of the excitation sources used in Cases I and II

Symbol
Case I Case II

A B C A B C

A [m] 0.0725 0.1140 0.1350 – – –

ω [Hz] 3.5 5.5 6.5 – – –

Ω0 [Hz] – – – 3.5 5.5 6.5

3. Numerical results

Thedisplacements x1 and x2 of the systemwith the ideal excitation source are shown inFigs. 5a,
6a and 7a,with the frequency of 3.5Hz, 5.5Hz and 6.5Hz, respectively. The overall displacement
rate of the system increases reasonably as the frequency is increased. Detailed views of these
displacements are shown in Figures 5b, 6b, 7b. In these figures, it is possible to see the points
where the impact occurs. As the frequency is increased and reaches 6.5Hz, the impacts aremore
frequent and start occurring at every cycle. This can be the cause of the perceived increase in
the displacement rate.

Fig. 5. Displacements of blocks 1 and 2 of the systemwith ideal excitation with the frequency of
3.5Hz (a) and a detailed view showing impacts (b)

Fig. 6. Displacements of blocks 1 and 2 of the systemwith ideal excitation with the frequency of
5.5Hz (a) and a detailed view showing impacts (b)



770 F.H. Moraes et al.

Fig. 7. Displacements of blocks 1 and 2 of the systemwith ideal excitation with the frequency of
6.5Hz (a) and a detailed view showing impacts (b)

Thedisplacements x1 and x2 of the systemwith the non-ideal excitation source are shown in
Figs. 8, 9 and 10, with the frequency of 3.5Hz, 5.5Hz and 6.5Hz, respectively. The displacement
rates are considerably higher than for the system with ideal excitation, and this happens for
all analysed frequencies. The displacement rate also tends to increase with higher excitation
frequencies. However, at 6.5Hz, the displacement increases rapidly until around 170s, and after
that it starts decreasing, although thenumberof impacts doesnot change considerably compared
to the situation with 5.5Hz, which has a higher displacement rate.

Fig. 8. Displacements of blocks 1 and 2 of the systemwith non-ideal excitation with the frequency of
3.5Hz (a) and a detailed view showing impacts (b)

Fig. 9. Displacements of blocks 1 and 2 of the systemwith non-ideal excitation with the frequency of
5.5Hz (a) and a detailed view showing impacts (b)

These results show that the models present very different results according to the type
of the excitation source, ideal or non-ideal, regarding the number of impacts and the rate of
displacement.
Figures 11, 12 and 13 show the phase portraits related to the frequencies of 3.5Hz, 5.5Hz

and 6.5Hz. The results show the large difference in displacement and velocities between the
models with different excitation sources. Notice that the scales of the figures vary greatly. The
displacements and velocities are much larger for the non-ideal excitation source model. For



Influence of ideal and non-ideal excitation sources ... 771

Fig. 10. Displacements of blocks 1 and 2 of the systemwith non-ideal excitation with the frequency of
6.5Hz (a) and a detailed view showing impacts (b)

Fig. 11. Phase portraits of the systems with ideal excitation (a) and with non-ideal excitation (b) with
the frequency of 3.5Hz

Fig. 12. Phase portraits of the systems with ideal excitation (a) and with non-ideal excitation (b) with
the frequency of 5.5Hz

Fig. 13. Phase portraits of the systems with ideal excitation (a) and with non-ideal excitation (b) with
the frequency of 6.5Hz



772 F.H. Moraes et al.

both types of excitation, the phase planes tend to become more regular with higher excitation
frequencies, with one impact at every cycle (period-1 limit-cycle) at 6.5Hz.
Figures 14a, 15a and 16a show the angular velocity of the motor. Figures 15a and 16a show

the influence of the system in the dynamics of the motor, called the Sommerfeld effect. In
Fig. 15a, the angular velocity constant of the motor is 5.5Hz, and the angular velocity of the
motor is captured by the resonance frequency of 4.1Hz, fromwhich themotor cannot escape. In
Fig. 16a, the angular velocity constant is 6.5Hz, and themotor is initially captured by resonance
at 4.1Hz, butmanages to overcome it and reaches the value of the angular velocity constant.
Figures 14b, 15b and16b show thedisplacementwith respect to themotor angular frequency.

InFig. 15b, it is possible to notice that themotorwas capturedby resonance (4.1Hz) and cannot
exceed this frequency, characterizing the Sommerfeld effect. In Fig. 16b, it is possible to notice
that the angular frequency of the motor is captured again by resonance at 4.1Hz, and after
approximately 100 seconds, the motor reaches a sufficient condition to overcome this frequency
and jumps to a frequency of approximately 6.2Hz.

Fig. 14. Angular velocity (a) and displacement versus motor frequency of the systemwith non-ideal
excitation with the frequency 3.5Hz (b)

Fig. 15. Angular velocity (a) and displacement versus motor frequency of the systemwith non-ideal
excitation with the frequency 5.5Hz (b)

Fig. 16. Angular velocity (a) and displacement versus motor frequency of the systemwith non-ideal
excitation with the frequency 6.5Hz (b)



Influence of ideal and non-ideal excitation sources ... 773

4. Conclusions

Thiswork analyzed the dynamics of a vibro-impact system.Themechanismwasmodeled in two
differentwayswith respect to the excitation source, an ideal sourcebymeans of aharmonic force,
and a non-ideal source by means of a DC motor with limited power supply and an unbalanced
rotor.
Although the models have similarities, the comparison between them has to be done care-

fully because the differences between the ideal and non-ideal excitation sources. A relationship
between the forces provided by each sourcewas definedwhich helped the comparison to bemore
realistic.

Graphswerepresented for thedisplacement, frequencyparameter, amplitudeof displacement
in relation to the motor rotation frequency.
The systemwith the non-ideal excitation source presents values of displacement and velocity

much larger in relation to the model with the ideal excitation source.

Regarding thenon-idealmodel, threedifferent situationswerepresented: in thefirst situation,
themotor has reached a steady state angular frequency similar to the angular velocity constant
of the motor, in the second and third situations, the motor exhibits the Sommerfeld effect. The
analysis of the dynamics of thesemodels showed the importance of using thenon-ideal excitation
source to obtain more realistic and consistent results.

Acknowledgements

The authors thank the Brazilian funding agencies FAPESP, CNPq and CAPES.

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Manuscript received October 31, 2012; accepted for print February 4, 2013