Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 3, pp. 687-696, Warsaw 2013

AN ANALYSIS OF THE PRIMARY AND SUPERHARMONIC CONTACT

RESONANCES – PART 2

Robert Kostek

University of Technology and Life Sciences in Bydgoszcz, Poland

e-mail: robertkostek@o2.pl

This paper presents results of investigations of non-linear normal contact micro-vibrations
excited by a harmonic force in a system of two bodies in planar contact. The system mo-
dels, for instance, slide units ofmachine tools or their positioning systems. Themain aim of
the computational analysis is to present resonance graphs and time histories obtained with
numerical and perturbationmethods.Good agreement between the perturbation and nume-
rical results leads to the conclusion that the perturbation solution is correct. The obtained
perturbation solution describes well both the primary resonance and the superharmonic
resonances. Characteristic phenomena typical for non-linear vibrations are depicted, viz.
asymmetry of vibrations, multi-harmonic vibrations, non-elliptical phase portraits, loss of
contacts, bending resonance peaks, bi-stabilities, andmulti-stability.

Key words: non-linear contact, vibrations, superharmonic resonance, multi-stability

1. Introduction

The contact of rough surfaces is flexible, which affects behaviour of machines. The contact
flexibility particularly affects the static deflections and natural frequencies of machine tools;
because contact deflections of sliding guideways can be larger than distortions of the machine
parts (Chlebus and Dybala, 1999; Fan et al., 2012; Gutowski, 2003; Kaminskaya et al., 1960;
Marchelek, 1974; Shi and Polycarpou, 2005; Thomas, 1999). In consequence, contact flexibility
affects vibrations ofmachine tools (Dhupia et al., 2007; Fan et al., 2012; Gutowski, 2003; Huo et
al., 2010;Marchelek, 1974;Moradi et al., 2010; Neugebauer et al., 2007). In this context, contact
vibrations and contact resonances are important.

The primary contact resonance and superharmonic resonance were studied experimentally
(Chajkin et al., 1939; Hess and Soom, 1992; Grigorova and Tolstoi, 1966; Kostek, 2004; Rigaud
and Perret-Liaudet, 2003; Perret-Liaudet andRigad, 2007; Tolstoi, 1967). The obtained experi-
mental results show the non-linear nature of contact vibrations, bi-stability andmulti-harmonic
vibrations. Moreover, the contact vibrations were simulated with numerical methods (Grudziń-
ski andKostek, 2007; Kligerman, 2003; Kostek, 2004, 2013b; Rigaud andPerret-Liaudet, 2003).
These methods provide an opportunity to study superharmonic resonances, bifurcations, and
chaotic vibrations, which are difficult to observe experimentally.

Finally, the contact vibrations can be studied with the perturbation method. This method
provides an opportunity to study periodic vibrations, hence the primary contact resonance and
superharmonic resonances can be computed. Still, a fewworks present the perturbation solution
of contact vibrations or similar vibrations (Hess andSoom, 1991a,b;Kostek, 2013a;Nayak, 1972;
Nayfeh, 1985; Perret-Liaudet, 1998, Perret-Liaudet and Rigad, 2007). Usually the perturbation
solutions hold only for the primary contact resonance or one superharmonic resonance. Thus,
there is a need to test for awide range of excitation frequencies and amplitudes, theperturbation
solution of the fourth order, which is presented in the first part of this article (Kostek, 2013a).
The results obtained with the perturbation method are compared with numerical results.



688 R. Kostek

In addition, the second aim of this paper is to present and analyse the contact vibrations
excited for various amplitudes and frequencies of the excitation force.

2. Theoretical fundamentals

The studied system has been presented previously in the first part of this publication, thus
certain information is repeated briefly in this section. The considered system consists of two
bodies in planar contact – a rigid block (slider), resting on a massive rigid base (slideway)
(Fig. 1). The interface of the bodies, which represents the planar contact of rough surfaces, is
modelled with a non-linear spring and damper (Hunt andCrossley, 1975; Kostek, 2004;Martins
et al., 1990). Four forces act on the slider, they are: the exciting force P, spring force Fs,
damping force Fd, and terrestrial gravity force Q. These forces and motion of the slider are
described by the following equations

IF y> 0 THEN Fs =−Scny
m2, δ= y ELSE Fs =0, δ=0

IF y> 0 THEN Fd =−Shny
lẏ ELSE Fd =0

(2.1)

and

P =Pacos(2πfet) Q=Mg Fk =Fs+Q

ÿ=M−1
(

Fk(y)+Fd(y, ẏ)+P(t)
) (2.2)

where y denotes the displacement, Fs the spring force, Fd the damping force, S the apparent
contact area S = 0.0009m2, cn, m2, hn and l are the parameters of the contact interface
cn =4.52693E16N/m

4, m2 =2, hn =3.5E11Ns/m
4, l=1, δ is the normal contact deflection,

whereas P denotes the exciting force, Pa the amplitude of the exciting force, fe the frequency
of excitation, t time, Q the terrestrial gravity force, M mass of the slider M =0.2106kg, and
g the acceleration of gravity g = 9.81m/s2, and finally Fk is the conservative force. The sum
of the spring force Fs (Fig. 1b) and the terrestrial gravity force Q is the conservative force Fk.
The graph of the conservative force is non-linear (Fig. 1c). More information is presented in the
first article.

Fig. 1. Model of the considered dynamical system (a), characteristic of the spring force Fs (b),
characteristic of the conservative force Fk (c), y0 and δ0 denotes the equilibrium position

3. Contact resonances

The resonance characteristics obtained for the studied system are depicted in Fig. 2. In this
figure, local minima andmaxima of a time history are presented against excitation frequency. A
number of resonance peaks is observed, for the studied single degree-of-freedom non-linear sys-
tem.This is common for non-linear systems, becausemany resonances are excited (Awrejcewicz,



An analysis of the primary and the superharmonic contact resonances – Part 2 689

1996; Fyrillas and Szeri, 1998; Grudziński and Kostek, 2007; Kostek, 2004; Nayfeh and Mook,
1995; Parlitz and Lauterborn, 1985; Thompson and Stewart, 2002). The primary resonance and
the superharmonic resonances are visible in Fig. 2. The resonances take place when the exciting
frequency fe is in the following relation to the natural frequency fn0

fe ≈
1

m
fn0 (3.1)

where fn0 denotes the natural frequency of a linearised system, whereas m is a positive integer.
If m = 1, then the primary resonance is excited, whereas for m > 1 the superharmonic reso-
nances are excited. If the frequency of the m-th harmonic of vibrations approaches the natural
frequency, then the amplitude of the m-th harmonic is rising; this is the nature of superhar-
monic resonances. Thus for 1/2 superharmonic resonance the second harmonic is amplified,
fe ≈ (1/2)fn0.

The resonance characteristics obtained with numerical and perturbation methods are de-
picted in Fig. 2. The perturbation solution is simplified because the linear model of damping
is adopted; moreover, the loss of contact is neglected, and therefore the numerical solution is
more accurate than the perturbation solution. Nevertheless, the perturbation solution describes
well the contact vibrations. The contact vibrations are asymmetrical to the equilibrium posi-
tion y0, thus the resonance graphs are asymmetrical to the equilibrium position too. Moreover,
the natural frequency is a function of the vibration amplitude, hence the primary resonance
peak bends, and bi-stability area is visible in Fig. 2. Further, the perturbation solution describes
well the primary resonance, even for loss of contact – “gapping”. Furthermore, the 1/2, 1/3
and 1/4 superharmonic resonances are well reflected. Nevertheless, the solution of order higher
than fourth is necessary to reflect the well bending of the 1/2 superharmonic resonance peak
(Fig. 2c). Finally, the perturbation and numerical solutions show good agreement for under re-
sonant (quasistatic) vibrations. This issuewill be described in the next article andwas discussed
in Skrodzewicz (2003).

Summarising, good agreement between the perturbation and numerical solutions leads to
the conclusion that the perturbation solution is correct. The time histories of contact vibrations
are analysed in the next section, which provides valid information about contact resonances.

4. An analysis of contact vibrations

The primary resonance is sensitive to non-linearity. Its peak rises quickly, and with the ampli-
tude of excitation being Pa = 0.05Q, a bi-stability area appears (Fig. 2a). The contact micro-
-vibrations of the primary resonance excited at frequency fe =1410Hz are asymmetrical. The
feature is visible in both time histories of the displacement (Fig. 3a) and acceleration (Fig. 3c).
Accordingly, the phase portrait is not elliptical and has a slightly asymmetrical shape (Fig. 3b),
which is coupledwith themulti-harmonic character of the vibrations (Fig. 3d). The phenomena
result from the non-linearity of the system. Finally, the perturbation solution well describes the
time history of displacement (Fig. 3a).

The 1/2 superharmonic resonance excited by the external force of the frequency fe =743Hz
and amplitude Pa = 0.05Q is not very clear in the resonance graph (Fig. 2a) nor in the time
history of the displacement (Fig. 4a). Nevertheless, the phenomenon is visible in the spectrum,
where the second harmonic appears (Fig. 4d). Moreover, this resonance is visible in the time
history of acceleration (Fig. 4c), where two local minima and maxima take place during one
period of vibration. At the same time, the shape of the phase portrait is far from the ellipsoidal
one (Fig. 4b), which is typical for non-linear vibrations. The 1/2 superharmonic resonance is an
effect of the non-linearity and growswith the increase of the excitation amplitude. Summarising,



690 R. Kostek

Fig. 2. Graphs of the contact resonances; local extrema of time histories against the frequency of
excitation fe; plus signs (+) represent resonant frequencies fe ≈ fn0/m

theperturbation solution showsexcellent agreementwith thenumerical results for this excitation
(Fig. 4a).

The area of bi-stability becomes larger with a further increase of the excitation amplitu-
de. This is a result of bending of the primary resonance peak and an increase of the primary
resonance amplitude. Therefore, for the excitation amplitude being Pa =0.15Q, the primary re-
sonance and the 1/2 superharmonic resonance canbe excited at the same frequency, fe =740Hz



An analysis of the primary and the superharmonic contact resonances – Part 2 691

Fig. 3. Time histories of the displacement (a), and acceleration (c), phase portrait (b), and
spectrum (d) of the primary resonance vibrations excited by a harmonic force of the parameters

Pa =0.05Q, fe =1410Hz

Fig. 4. Time histories of the displacement (a), and acceleration (c), phase portrait (b), spectrum (d) of
the 1/2 superharmonic resonance vibrations excited by a harmonic force of the parameters Pa =0.05Q,

fe =743Hz

(Fig. 2b). Asymmetry and “gapping” of the primary resonance vibrations are visible both in the
resonance graph (Fig. 2b) and in the time history of displacement (Fig. 5a). The loss of contact
is coupled with high acceleration of the slider (Fig. 5c) and significant loading of the slideway,
which result from hitting the slider in the slideway. The phase portrait of the vibrations has a
characteristic shape, typical for the contact vibrations (Fig. 5b), whereas the spectrum (Fig. 5d)
clearly reveals the increase of higher harmonics amplitudes.



692 R. Kostek

Fig. 5. Time histories of the displacement (a), and acceleration (c), phase portrait (b), spectrum (d) of
the primary resonance vibrations excited by a harmonic force of the parameters Pa =0.15Q,

fe =740Hz

The afore-mentioned 1/2 superharmonic resonance grows with the increase of the excitation
amplitude, thus the two local minima andmaxima are observed during one period, both in the
time histories of displacement (Fig. 6a) and acceleration (Fig. 6c). Consequently, the second
harmonic is larger than previously shown (Fig. 6d), the phase portrait has amore sophisticated
shape (Fig. 6b), and four curves present the resonance in Fig. 2b. In spite of the fact that the
vibration amplitude is larger, the perturbation solution holds (Fig. 6a).

Fig. 6. Time histories of the displacement (a), and acceleration (c), phase portrait (b), spectrum (d) of
the 1/2 superharmonic resonance vibrations excited by a harmonic force of the parameters Pa =0.15Q,

fe =740Hz



An analysis of the primary and the superharmonic contact resonances – Part 2 693

Fig. 7. Time histories of the displacement (a), and acceleration (c), phase portrait (b), spectrum (d) of
the primary resonance vibrations excited by a harmonic force of the parameters Pa =0.50Q,

fe =490Hz

Fig. 8. Time histories of the displacement (a), and acceleration (c), phase portrait (b), spectrum (d) of
the 1/2 superharmonic resonance vibrations excited by a harmonic force of the parameters Pa =0.50Q,

fe =490Hz

The further increase of the excitation amplitude leads to bending of the 1/2 superharmonic
resonance peak, which in turn introduces multi-stability to the system. In this case, three solu-
tions are possible: the three resonances are excited, viz. the primary resonance, the 1/2 and 1/3
superharmonic resonances for Pa =0.50Q and fe =490Hz (Fig. 2c). Analysis of the vibrations
reveals the complexity of their kinematics. The time histories of the displacement and accelera-
tion show that a number of minima andmaxima during one period is linked with the integer m
(Figs. 7a, 7c, 8a, 8c, 9a, 9c). Consequently, for the primary resonance one minimum and maxi-



694 R. Kostek

Fig. 9. Time histories of the displacement (a), and acceleration (c), phase portrait (b), spectrum (d) of
the 1/3 superharmonic resonance vibrations excited by a harmonic force of the parameters Pa =0.50Q,

fe =490Hz; three local minima (1, 2, 3) are observed over one cycle of the excitation (a), (c)

mum are observed during one period, thus two curves are present in the primary resonance in
Fig. 2c. Whereas for the 1/2 superharmonic resonance, two minima and maxima are observed
during one period, thus four lines are present in the resonance in Fig. 2c. Finally, six curves are
presented in the 1/3 superharmonic resonance in Fig. 2c. In addition, analysis of the spectrums
shows that the amplitude A of the m-th harmonic is significant (Figs. 7d, 8d, 9d). Also, at the
same time, the shape of the phase portraits becomesmore sophisticated with the increase of the
integer m. For theprimary resonance, a point rounds the origin onceduringoneperiod (Fig. 7b);
while for the 1/2 superharmonic resonance one rounds the origin twice (Fig. 8b). Additionally,
for the 1/3 superharmonic resonance, two additional loops are observed in the phase portrait
(Fig. 9b). Summarising this, the contact micro-vibrations are complex and their non-linearity
can lead to “gapping”, superharmonic resonances, bi-stability and multi-stability. Finally, the
perturbation solution reflects the nature of the 1/3 superharmonic resonance (Fig. 9a).

5. Conclusions

The presented study has been focused on comparison of the results obtained with the per-
turbation and numerical methods. The perturbation solution well describes both the primary
resonance and the superharmonic resonances, which is an important feature. Good agreement
between the perturbation and numerical results leads to the conclusion that the perturbation
solution is correct.

The contact vibrations are non-linear, thus characteristic phenomena are observed. The
asymmetrical and non-linear conservative force introduces asymmetry of contact vibrations.
Hence, the notion of the amplitude should be precisely defined. Moreover, for large excitation
amplitudes, the loss of contact is observed. Beside, non-linearity of the conservative force intro-
duces higher harmonics, thus contact vibrations aremulti-harmonic. In consequence, kinematics
of contact vibrations is complex, shapes of the phase portraits are non-elliptical, and local mini-
ma andmaxima are observed in the time histories of displacement. Finally, the multi-harmonic



An analysis of the primary and the superharmonic contact resonances – Part 2 695

nature of contact vibrations introduces superharmonic resonances, which result frommagnifica-
tion of higher harmonics.

For the studied system, the natural frequency is a function of the vibration amplitude, which
introducesbendingof the resonancepeaks,bi-stability andfinallymulti-stability. Thebi-stability
is a result of bending of the primary resonance peak, whereas the multi-stability is a result of
bending of the superharmonic resonance peaks.

Summarising, the contact vibrations are farmore complex than linear vibrations. Thus, non-
-linearmodelsof contact shouldbeused inmachine tools dynamics.Moreover, non-linear systems
should be investigated under a wide range of excitation signals, because many resonances can
be excited.

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Manuscript received August 11, 2011; accepted for print November 26, 2012