Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 2, pp. 475-486, Warsaw 2013

ANALYSIS OF THE PRIMARY AND SUPERHARMONIC CONTACT

RESONANCES – PART 1

Robert Kostek

University of Technology and Life Sciences in Bydgoszcz, Poland

e-mail: robertkostek@o2.pl

This paper presents results of numerical and analytical investigations of non-linear normal
contactmicrovibrations excited by a harmonic force in a systemof twobodies in planar con-
tact. The systemmodels, for example, the slide unit ofmachine tools or positioning systems.
Themain aimof the computational analysis is to present the evolution of the resonancephe-
nomena under various amplitudes of the excitation force. The studies show that, beside the
primary resonance, a number of superharmonic resonances appear, which take place in the
single-degree-of-freedomnon-linear systemexcitedbyaharmonic force.Thus, in a resonance
plot, a number of peaks is observed. The superharmonic (ultraharmonic) resonances take
place at excitation frequencies being below the natural frequency, and becoming stronger
with the increase of the excitation amplitude. The resonances are coupled with complex
non-linear phenomena like: asymmetry of vibrations, bending resonance peak, bi-stability,
multi-stability and loss of contact, which are presented and described in this paper.

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

1. Introduction

In the modelling of mechanical systems, the essential fact that the contact of rough surfaces is
flexible and strongly non-linear should not be neglected. The contact phenomena are likely to
affect the behaviour of machines (Fan et al., 2012; Shi and Polycarpou, 2005; Thomas, 1999).
For instance, contact flexibility has influence on the static stiffness of machine tools, where
contact deflections can be larger than distortions of machine parts (Chlebus and Dybala, 1999;
Gutowski, 2003; Kaminskaya et al., 1960; Levina and Reshetov, 1971; Marchelek, 1974); thus
most parts of themachines can bemodelled as rigid bodies (Gutowski, 2003; Marchelek, 1974).
The phenomena refer, in particular, to precision machines, such as grinding machines. Thus,
in turn, the contact flexibility affects dynamic properties of machines and, consequently, their
precision, productivity, stability and vibrations (e.g. chatter) (Dhupia et al., 2007; Fan et al.,
2012; Gutowski, 2003; Huo et al., 2010; Marchelek, 1974; Moradi et al., 2010; Neugebauer et al.,
2007). It is known that frequencies of the excitation force (cutting force) should be out of the
resonancearea, in order to reduceamplitudeof vibrations andmakemachining stable (Gutowski,
2003;Marchelek, 1974). These are the reasonswhy the contact phenomena (deflection, vibration,
friction, damping and wear) have currently become one of themain areas of scientific activity.

The resonance is one of the most important and common issue in dynamics, thus a number
of articles has beenwritten on contact resonances.Most of themare focused only on the primary
contact resonance (Chajkin et al., 1939; Grigorova and Tolstoi, 1966; Hess and Soom, 1991a,b;
Kligerman, 2003; Rigaud and Perret-Liaudet, 2003; Tolstoi, 1967), while still a few are on the
superharmonic resonances (Grudziński and Kostek, 2007; Kostek, 2004; Perret-Liaudet, 1998;
Perret-Liaudet andRigad, 2007). Most of the papers present vibrations excited within a narrow
range of frequencies and amplitudes, hence there is a need to present an analysis of the contact
vibrations excited with a wide range of frequencies and amplitudes of the excitation force.



476 R. Kostek

The present study is focused on the evolution of primary and superharmonic resonances due to
various amplitudes of the excitation. The superharmonic resonances can amplify the amplitude
of vibrations being excitedwith an excitation frequencywhich is below the natural frequency. In
machine tools, the resonances can affect chatter (Moradi et al., 2010). The problems have been
found important in the context of the dynamics of precise machine tools where low amplitudes
of vibrations are required.

2. Theoretical fundamentals

The considered system consists of two bodies in planar contact (Fig. 1), which is modelled in
this section. The system can be viewed as a simplified model of a slide unit of a machine tool.
The system consists of a rigid block (slider) resting on a massive rigid base (slideway). The
interface of the bodies, which represents the planar contact of rough surfaces, ismodelledwith a
great number ofmicrosprings andmicrodampers (Fig. 1b). Themicrosprings andmicrodampers
represent the interacting roughness of asperities. It has been assumed that the spring–damping
properties of the interface are macroscopically identical over all the contact area. Thus, after
homogenisation (Fig. 1c), the contact zone is modelled with one non-linear spring and damper
(Figs. 1d,e).

Fig. 1. Scheme of the considered dynamical system (a), and its physical models (b)-(d), characteristic of
the spring force Fs (e), characteristic of the conservative force Fk (f)

The displacement of the vibrating body (slider) y is determined by the coordinate Y with
respect to the rigid base (slideway). The coordinate system is pointing downward (Fig. 1e), and
the origin is fixed on the level where the contact deflection δ begins; therefore, y = 0 implies
δ = 0. Thus the contact deflection δ = y, if y > 0, and δ = 0, if y ¬ 0. This convention
allows for simulation of “gapping” (loss of contact) in the case of large contact vibrations. A
very important parameter is the static contact deflection due to theweight of the slider δ0 = y0,
which determines the equilibrium position.
The contact force (reaction) R is the sum of the spring force Fs and the damping force Fd

(Fig. 1d). The spring force is a non-linear function of the displacemen y, whereas the damping
force is a non-linear function of the displacement y and velocity ẏ. The forces are described by
the following formulas (Hunt and Crossley, 1975; Kostek, 2004; Martins et al., 1990)

IF y> 0 THEN Fs =−Scny
m2 , Fd =−Shny

lẏ , δ= y

ELSE Fs =0 , Fd =0 , δ=0
(2.1)



Analysis of the primary and superharmonic contact resonances – Part 1 477

where S denotes the nominal (apparent) contact area, while cn,m2, hn, and l are parameters
of the contact interface, and δ is the normal contact deflection. Values of the parameters have
been identified from experimental results (Grudziński et al., 2000; Grudziński andKostek, 2007;
Kostek, 2002, 2004); thus reliable values of the parameters have been adopted to the simulations:
S =0.0009m2, cn =4.52693 ·10

16N/m4, m2 =2, hn =3.5 ·10
11Ns/m4, l=1. Moreover, the

adoptedmodel of contact was validated against the experimental results (Hess and Soom, 1992;
Hess andWagh, 1995; Kostek, 2004). For further information please see appendix.
Apart from the spring and the damping contact forces, two more forces act on the slider

(Fig. 1). They are: the exciting (driving) harmonic force P, and the terrestrial gravity force Q,
which are expressed by the following formulas

P =Pacos(2πfet) Q=Mg (2.2)

where Pa denotes the amplitude of the exciting force, fe – frequency of excitation, t – time,
M –mass of the slider M =0.2106kg, and g – acceleration of gravity g=9.81m/s2. Thema-
gnitude ofmass is adopted for a cubemade of steel, whosewidth, length and height are 30mm.

The sum of the spring force Fs and the terrestrial gravity force Q can be treated as the
conservative force

Fk =Fs+Q (2.3)

The conservative force Fk is a source of non-linearity in the system. The graph of the
conservative force Fk is asymmetrical (Fig. 1f). Therefore, different values of the force are
observed for the same displacement magnitude, depending on whether the displacement is up
or down from the equilibrium position. Finally, the contact vibrations of the adopted model
(Fig. 1d) are described by the following equation

ÿ=M−1(Fk(y)+Fd(y, ẏ)+P(t)) (2.4)

The formulated differential equation of motion is non-linear, because of the non-linearity
of the spring and the damping contact forces. The equation can be solved numerically using
the 4-th order Runge-Kutta method, which was used to obtain the resonant characteristics of
the system. The system was excited by a constant amplitude harmonic force. After the steady-
state response was attained, local minima andmaxima of the time history were noted. Finally,
the frequency of excitation fe was changed. The procedure allows for simulation of complex
resonance phenomena (Figs. 2 and 3), which are presented in the next sections.
Apart from the numerical methods, the contact microvibrations can be studiedwith pertur-

bation methods. In this case, the conservative force Fk can be described by the Taylor series,
around the equilibrium position, while the damping force Fd can be linearised around the equ-
ilibrium position, which leads to the following equations

Fk =Mg−Scny
m2
0 −Scnm2y

m2−1
0 (y−y0)

−
1

2
Scnm2(m2−1)y

m2−2
0 (y−y0)

2+O(y−y0)
3

Fd =−Shny
l
0ẏ

(2.5)

Finally, after dividing the equation ofmotionby M, thevibrations aredescribedby the following
formulas

ü+2cu̇+ω20u+εhu
2 =Gcos(2πfet) u= y−y0 (2.6)

where u denotes displacement, while



478 R. Kostek

2c=M−1Shny
l
0 ω

2
0 =M

−1Scnm2y
m2−1
0

εh=
1

2
M−1Scnm2(m2−1)y

m2−2
0 G=M

−1Pa

are parameters of the equation, and ε is the small parameter associated with the non-linear
term. The solution is formed in terms of an infinite series of the perturbation parameter as
follows (Awrejcewicz, 1996; Nayfeh andMook, 1995)

ω2 =ω20 +εb1+ε
2b2+ . . .

u=u0(t)+εu1(t)+ε
2u2(t)+ . . .

(2.7)

where ω0 denotes natural frequency of the undamped linearised system, ω natural frequency
of the non-linear system, while b1, b2, . . . are functions of amplitudes, and u0(t), u1(t), u2(t),
. . . are terms of the solution. Submitting Eqs. (2.7) into Eq. (2.6)1 leads to a system of linear
equations, which can be solved successively:
— ε0

ü0+2cu̇0+ω
2u0 =Gcos(2πfet) u0 =A0cos(2πfet+ϕ0) (2.8)

— ε1

ü1+2cu̇1+ω
2u1 = b1u0−hu

2
0 = b1A0cos(2πfet+ϕ0)−hA

2
0cos

2(2πfet+ϕ0)

= b1A0cos(2πfet+ϕ0)−
1

2
hA20cos(4πfet+2ϕ0)−

1

2
hA20

b1 =0

u1 =A1cos(4πfet+2ϕ0+ϕ1)+A2

(2.9)

— ε2

ü2+2cu̇2+ω
2u2 = b1u1+b2u0−2hu0u1 = b2A0cos(2πfet+ϕ0)

−2hA0 cos(2πfet+ϕ0)[A1cos(4πfet+2ϕ0+ϕ1)+A2]

= b2A0cos(2πfet+ϕ0)−2hA0A2cos(2πfet+ϕ0)

−hA0A1cos(2πfet+ϕ0+ϕ1)−hA0A1cos(6πfet+3ϕ0+ϕ1)

b2 =2hA2+hA1cosϕ1

u2 =A3cos(2πfet+2ϕ0)+A4cos(2πfet+2ϕ0)+A5cos(2πfet+2ϕ0+ϕ1)

+A6cos(6πfet+3ϕ0+ϕ1+ϕ2)=A7cos(2πfet+2ϕ0)

+A5cos(2πfet+2ϕ0+ϕ1)+A6cos(6πfet+3ϕ0+ϕ1+ϕ2)

(2.10)

where A0-A7 are amplitudes and ϕ0-ϕ2 are phase angles. Similar non-linear vibrations have
been previously studied with the perturbation method (Hess and Soom, 1991a,b; Nayak, 1972;
Nayfeh, 1983; Nayfeh and Mook, 1995; Perret-Liaudet, 1998; Perret-Liaudet and Rigad, 2007)
that shows the complexity of the issue. The obtained approximate solution to the Eq. (2.6)1 is
interpreted in the next section, while Appendix contains more information and equations.

3. The contact resonances

Thenormal contact vibrations,whichare common inmachine tools, are coupledwith oscillations
of the slider about the equilibrium position y0 = δ0. When the amplitude of the contact vibra-
tions is very small, then the vibrations are linear and the natural frequency of the considered
system equals fn0 = 1485Hz. The perturbation solution reflects the phenomena, because the
terms hA20, hA0A1, u1, u2 and b2 are close to zero in this case (Eqs. (2.9) and (2.10)). In linear
systems, the number of resonance peaks is not larger than the number of degrees-of-freedom.



Analysis of the primary and superharmonic contact resonances – Part 1 479

Thus, no more than one resonance peak is found, for a single degree-of-freedom linear system.
In contrast, many kinds of resonances may be found in a non-linear single degree-of-freedom
system (Awrejcewicz, 1996; Belhaq and Fahsi, 2009; Bogusz et al., 1974; Cunningham, 1958;
Fyrillas and Szeri, 1998; Nayfeh and Mook, 1995; Parlitz and Lauterborn, 1985; Tang, 2000;
Thompson and Stewart, 2002). Therefore, many peaks can appear in a resonance graph (Fyril-
las and Szeri, 1998; Grudziński andKostek, 2007; Kostek, 2004; Nayfeh andMook, 1995; Parlitz
and Lauterborn, 1985; Thompson and Stewart, 2002). The resonances can take place when the
exciting frequency fe is in the following relation to the natural frequency fn0

fe ≈
n

m
fn0 (3.1)

where m and n are positive integers. Depending on the values of these integers, the following
kinds of resonancesmay take place in the system (Awrejcewicz, 1996;Bogusz et al., 1974;Nayfeh
andMook, 1995):

• primary (main) resonance, if n=1,m=1,

• superharmonic (ultraharmonic) resonance, if n=1,m> 1,

• subharmonic resonance, if n> 1,m=1, and

• super-subharmonic resonance, if m 6=1, n 6=1 and n/m 6=1.

The contact resonances can be presented graphically in different ways. They can be shown
as a peak-to-peak amplitude App = ymax−ymin in the logarithmic scale against the frequency of
excitation fe. This allows the presentation of a wide range of amplitudes, and the figure is easy
to interpret (Fig. 2). More information can be provided if all local extrema in the time response
(over one period of vibration) are presented as a function of the excitation frequency fe (Fig. 3).
This figure presents the following: a complex structure of the superharmonic resonances, the
asymmetry of contact vibrations, and “gapping”. If the slider loses contact with the slideway,
then the minima reach negative values. The appearance of more than two extrema per period
leads to more than two curves in the resonance graph. The issues are described in the next
paper. Generally, the way of presentation of the resonances should be fitted to the aim of the
presentation.

3.1. The primary resonance

In the case of a very small amplitude of the excitation force Pa, the simulated vibrations
are linear and the resonance is linear (Kostek, 2004), and the nature of the primary resonance
is clear. The loss of energy caused by the damping force Fd is compensated with the work done
by the external harmonic force P. The process is particularly effective when the frequency of
excitation fe is near the natural frequency fn0. Therefore, the amplitude of the vibrations is
the largest near the natural frequency fn0.

Gradually, with the increase of the excitation amplitude Pa, the contact vibrations become
larger and non-linear. Thus, the phenomena typical for non-linear vibrations take place (Gru-
dziński and Kostek, 2007; Kostek, 2004). The non-linear conservative force Fk, the graph of
which is asymmetrical (Fig. 1f), introduces characteristic phenomena. The contact microvibra-
tions become asymmetrical to the equilibrium position (Fig. 3) (see A2 Eq. (2.9)3), and finally,
the slider can lose contact with the slideway (Figs. 3b-3j). If the slider loses contact with the
slideway, then a minimum of the time history ymin attains a negative value (Eq. (2.1)1), which
reflects Fig. 3b-3j (see light gray curves). In other words, negative values of minima reveal the
loss of contact. The natural frequency of the non-linear system is a function of the vibration
amplitude (see Eqs. (2.7)1 and (2.10)2), and this, significantly, has influence on the resonan-
ce graph because the resonance peaks follow the natural frequency. In this case, they bend to



480 R. Kostek

Fig. 2. Graphs of the contact resonances; peak-to-peak amplitude App = ymax−ymin against the
frequency of excitation fe; plus signs (+) represent resonant frequencies fe ≈ fn0/m, Eq. (3.1)

the lower frequencies (Figs. 2 and 3), because the natural frequency becomes smaller with the
increase of the vibration amplitude (see Eq. (2.10)2). The bending of the primary resonance
peak introduces bi-stability to the system; thus a response of the system is ambiguous at some
frequencies of excitation fe (Fig. 2c). Consequently, the obtained solution depends on initial
conditions.



Analysis of the primary and superharmonic contact resonances – Part 1 481

Fig. 3. Graphs of the contact resonances; local extrema of time histories against the frequency of
excitation fe; (light gray line – minima, dark gray line – maxima obtained with the numerical method)

3.2. The superharmonic resonances

Usually, analyses of dynamical systems are focused on the primary resonance, while the rest
of the excitations are treated as safe. Nevertheless, apart from the primary resonance, a number
of superharmonic resonances can take place, for frequencies of excitation beingbelow thenatural



482 R. Kostek

frequency fe < fn0. Consequently, non-linear systems should be tested under a wide range of
excitations, and the resonance phenomena should be studied as a whole.

Vibrations of the non-linear system excited by a harmonic force aremultiharmonic (see Eqs.
(2.9)3 and (2.10)3); thus they contain a number of harmonics.When the frequency of the m-th
harmonic of the vibrations approaches the natural frequency, then the amplitude of the m-th
harmonic is rising (see Eqs. (2.9)1 and (2.10)1). Thus, the frequencies of the superharmonic
resonances form the series fe ≈ (1/m)fn0. In other words, if the frequency of the 2-nd harmonic
is near the natural frequency fn0, then the frequency of excitation fe is near the half of the
natural frequency, fe ≈ (1/2)fn0. As a result, the amplitude of the 2-nd harmonic is amplified
(see Eq. (2.9)1), and the 1/2 superharmonic resonance is at this place (Fig. 2d). The next 1/3
superharmonic resonance is a result of the amplification of the 3rd harmonic (see Eq. (2.10)1),
the frequency of which is near the natural frequency fn0, fe ≈ (1/3)fn0 (Fig. 2i), which, in
turn, reflects the afore-mentioned relation Eq. (3.1). Summarising these, the nature of the su-
perharmonic resonances is the amplification of higher harmonics, the frequency of which is near
the natural frequency fn0.
The phenomena such as asymmetry of vibrations, “gapping”, and bending of the resonance

peaks, take place at the superharmonic resonances too (Figs. 2 and 3). However, the resonances
are far more complex than the primary resonance; that is, the nature of the superharmonic
resonances. For instance, the magnification of higher harmonics makes time histories of the di-
splacement, velocity and accelerationmore complex, and a number of local minima andmaxima
is observed during one period of vibration. Consequently, more than two curves can represent
super harmonic resonances, which affect the resonance graphs (Figs. 3c-3j). Finally, phase por-
traits, and spectrums, aremore sophisticated as well. The spectrums, phase portraits, and time
histories to be presented in the next paper clearly show the complexity of superharmonic reso-
nances and their sophisticated kinematics.

3.3. The bi-stability and multi-stability

Theprimary resonance is non-linear. Its peakbends to lower frequencies (Figs. 2 and 3), thus
a bi-stability area appears for Pa = 0.05Q (Figs. 2a and 3a). The area of bi-stability becomes
larger with a further increase of the excitation amplitude Pa. Therefore, for the excitation am-
plitude being Pa =0.15Q, the primary resonance and the 1/2 superharmonic resonance can be
excited at the same frequency fe =740Hz (Figs. 2c and 3c). Next, the 1/2 superharmonic reso-
nance growswith the increase of the excitation amplitude. 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 solutions are possible: the three resonances are
excited, viz. the primary rezonance, the 1/2 and 1/3 superharmonic resonances for Pa =0.50Q
and fe = 490Hz (Figs. 2j and 3j). The visualization of the contact resonances is available on
the Internet https://www.youtube.com/watch?v=cNRU-TUXCao.

4. Conclusions

The presented study has been focused on the evolution of contact resonances simulated for a
wide range of excitations. The studied phenomena have been found to be important in the
context of precision machining.
Thecontact vibrationsarenon-linear, thusanumberof characteristic phenomenatakesplace.

The vibrations are asymmetrical, thus a notion of the amplitude should be precisely defined, e.g.
as the peak-to-peak amplitude App. Furthermore, the natural frequency of the vibrations should
be treated as a function of the vibration amplitude.Moreover, the resonanceswhich take place in
non-linear systemsare farmore complex than in linear ones.Apart fromtheprimary resonance, a



Analysis of the primary and superharmonic contact resonances – Part 1 483

number of superharmonic resonances takes place for frequencies of the excitation fe beingbelow
the natural frequency fn0. Their kinematics is complex, which is a result of the magnification
of higher harmonics. The contact resonances become less intensive and more complex, with an
increase of the integer m. An increase of the excitation amplitude Pa leads to an increase of
their amplitudes, bending of the resonance peaks, and consequently, to bi-stability and finally
multi-stability. Summarising these, a number of resonances and other non-linear phenomena
influence the dynamics of the considered system. This shows that the contact vibrations and
resonances should be studied under a wide range of excitation signals.

Simplified linearmodels of contact rough surfaces,which are typically used tomodelmachine
tool dynamics, donot allow thepropermodelling of complex dynamical phenomena, e.g. chatter.
Thus, to analyse these phenomena, non-linear models of contact, together with the theory of
non-linear dynamics, should be used.

A. Appendix

Natural frequencies calculated for various heights of bodies hb were compared with the experi-
mental results published by Hess and Soom (1992), Hess and Wagh (1995). In spite of the fact
that various contacts were tested, natural frequencies are similar (Fig. 4) (Kostek, 2002). Finally,
the results of simulation obtained for the adopted model can be compared against the experi-
mental results (Fig. 5). According with the presented results, one may conclude that reliable
values of the parameters have been adapted to the simulations.

Fig. 4. Comparison of magnitudes of the natural frequencies fn obtained experimentally for various
heights of bodies hb made of steel (Hess and Soom, 1992; Hess andWagh, 1995) with results of
simulation obtained for the rigid body and the adoptedmodel of contact (Kostek, 2002)

Fig. 5. Comparison of the hysteresis loop of contact obtained experimentally (Kostek, 2004) against the
results of simulation carried out for fe =1600Hz and Pa =0.40Q; displacement y versus normal

contact pressure p



484 R. Kostek

Further equations which are a part of the perturbation solution are presented below:

— ε3

ü3+2cu̇3+ω
2u3 = b1u2+b2u1+ b3u0−2hu0u2−hu

2
1

= b2A1cos(4πfet+2ϕ0+ϕ1)+ b2A2+b3A0cos(2πfet+ϕ0)

−hA0A7cos(4πfet+3ϕ0)−hA0A7cos(ϕ0)−hA0A5cos(4πfet+3ϕ0+ϕ1)

−hA0A5cos(ϕ0+ϕ1)−hA0A6cos(8πfet+4ϕ0+ϕ1+ϕ2)

−hA0A6cos(4πfet+2ϕ0+ϕ1+ϕ2)−
1

2
hA21cos(8πfet+4ϕ0+2ϕ1)

−
1

2
hA21−2hA1A2cos(4πfet+2ϕ0+ϕ1)−hA

2
2

=A8+A9cos(4πfet+ϕ3)+A10cos(8πfet+ϕ4)

b3 =0

u3 =A11+A12cos(4πfet+ϕ3+ϕ1)+A13cos(8πfet+ϕ4+ϕ5)

(A.1)

— ε4

ü4+2cu̇4+ω
2u4 = b1u3+b2u2+ b3u1+ b4u0−2hu0u3−2hu1u2

= b2A7cos(2πfet+2ϕ0)+ b2A5cos(2πfet+2ϕ0+ϕ1)

+ b2A6cos(6πfet+3ϕ0+ϕ1+ϕ2)+ b4A0cos(2πfet+ϕ0)

−2hA0A11cos(2πfet+ϕ0)−hA0A12cos(6πfet+ϕ0+ϕ1+ϕ3)

−hA0A12cos(2πfet−ϕ0+ϕ1+ϕ3)−hA0A13cos(10πfet+ϕ0+ϕ4+ϕ5)

−hA0A13cos(6πfet−ϕ0+ϕ4+ϕ5)−hA1A7cos(6πfet+4ϕ0+ϕ1)

−hA1A7cos(2πfet+ϕ1)−hA1A5cos(6πfet+4ϕ0+2ϕ1)−hA1A5cos(2πfet)

−hA1A6cos(10πfet+5ϕ0+2ϕ1+ϕ2)−hA1A6cos(2πfet+ϕ0+ϕ2)

−2hA2A7cos(2πfet+2ϕ0)−2hA2A5cos(2πfet+2ϕ0+ϕ1)

−2hA2A6cos(6πfet+3ϕ0+ϕ1+ϕ2)=A14cos(2πfet+ϕ6)

+A15cos(6πfet+ϕ7)+A16cos(10πfet+ϕ8)

b4 =−A
−1
0

[

b2A7cos(ϕ0)+ b2A5cos(ϕ0+ϕ1)−2hA0A11cos(0)

−hA0A12cos(−2ϕ0+ϕ1+ϕ3)−hA1A7cos(−ϕ0+ϕ1)−hA1A5cos(−ϕ0)

−hA1A6cos(ϕ2)−2hA2A7cos(ϕ0)−2hA2A5cos(ϕ0+ϕ1)
]

u4 =A17cos(2πfet+ϕ6+ϕ0)+A18cos(6πfet+ϕ7+ϕ2)+A19cos(10πfet+ϕ8+ϕ9)

(A.2)

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Analiza rezonansu kontaktowego głównego oraz rezonansów ultraharmonicznych – Część 1

Streszczenie

W artykule tym przedstawiono wyniki badań numerycznych oraz analitycznych drgań kontaktowych
normalnych. Drgania te zostały wzbudzone przez silę harmoniczną w układzie składającym się z dwóch
ciał, które tworzą parę cierną.Głównymcelempracy było zbadanie zmian rezonansówkontaktowychpod
wpływem wzrostu amplitudy wymuszenia. Oprócz rezonansu głównego zaobserwowano także rezonanse
ultraharmoniczne, które sąwzbudzane dla częstotliwości będących poniżej częstotliwości własnej układu.
Amplituda rezonansów ultraharmonicznych staje się stopniowo większa wraz ze wzrostem amplitudy
wymuszenia. Ponadto zaobserwowano asymetrię drgań kontaktowych, odrywanie się ciał od siebie, wiele
atraktorówdla takiego samegowzbudzenia oraz zaginanie się pikówrezonansowych.Wymienione zjawiska
zaobserwowano dla nieliniowego układu o jednym stopniu swobody.

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