The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub- and super-resonance vibrations
The finite element model of an elastic body with a closing crack is presented. Using this model, we show that the non-linear distortions of vibrations at superharmonic and subharmonic resonances are the high sensitive indicators of crack presence. The intensity of these indicators manifestation subs...
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irk-123456789-478672013-08-03T15:35:30Z The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub- and super-resonance vibrations Bovsunovskii, A.P. Surace, P. Bovsunovskii, O.A. Научно-технический раздел The finite element model of an elastic body with a closing crack is presented. Using this model, we show that the non-linear distortions of vibrations at superharmonic and subharmonic resonances are the high sensitive indicators of crack presence. The intensity of these indicators manifestation substantially depends on the level of damping in the system and on the load application point. Представлено скінченноелементну модель пружного тіла з тріщиною, що закривається. Показано, що нелінійні спотворення коливань при супер- і субгармонічному резонансах е високочутливим проявом наявності тріщини. Інтенсивність цього прояву суттєво залежить від рівня демпфування в системі та від місця прикладення вимушеної сили. Представлена конечноэлементная модель упругого тела с закрывающейся трещиной. Показано, что нелинейные искажения колебаний при супер- и субгармоническом резонансах являются высокочувствительным проявлением наличия трещин. Интенсивность этого проявления существенно зависит от уровня демпфирования в системе и места приложения вынуждающей силы. 2006 Article The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub- and super-resonance vibrations / A.P. Bovsunovskii, C. Surace, O.A. Bovsunovskii // Проблемы прочности. — 2006. — № 5. — С. 61-67. — Бібліогр.: 16 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/47867 620.178; 620.179 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Bovsunovskii, A.P. Surace, P. Bovsunovskii, O.A. The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub- and super-resonance vibrations Проблемы прочности |
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The finite element model of an elastic body with a closing crack is presented. Using this model, we show that the non-linear distortions of vibrations at superharmonic and subharmonic resonances are the high sensitive indicators of crack presence. The intensity of these indicators manifestation substantially depends on the level of damping in the system and on the load application point. |
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Article |
| author |
Bovsunovskii, A.P. Surace, P. Bovsunovskii, O.A. |
| author_facet |
Bovsunovskii, A.P. Surace, P. Bovsunovskii, O.A. |
| author_sort |
Bovsunovskii, A.P. |
| title |
The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub- and super-resonance vibrations |
| title_short |
The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub- and super-resonance vibrations |
| title_full |
The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub- and super-resonance vibrations |
| title_fullStr |
The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub- and super-resonance vibrations |
| title_full_unstemmed |
The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub- and super-resonance vibrations |
| title_sort |
effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub- and super-resonance vibrations |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2006 |
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Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/47867 |
| citation_txt |
The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub- and super-resonance vibrations / A.P. Bovsunovskii, C. Surace, O.A. Bovsunovskii // Проблемы прочности. — 2006. — № 5. — С. 61-67. — Бібліогр.: 16 назв. — англ. |
| series |
Проблемы прочности |
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UDC 620.178; 620.179
T he E ffec t o f D am p in g a n d F o rce A p p lica tio n P o in t on th e N o n -L in ear
D ynam ic B eh av io r o f a C ra c k e d B eam a t Sub- a n d S up er-R eso n an ce
V ib ra tio n s*
A. P. Bovsunovskii,a C. Surace,b and O. A. Bovsunovskiia
a Pisarenko Institute of Problems of Strength, National Academy of Sciences of Ukraine,
Kiev, Ukraine
b Politecnico di Torino, Torino, Italy
УДК 620.178; 620.179
В п л и в д е м п ф у в а н н я і м ісц я п р и к л а д е н н я си л и н а н ел ін ій н у
д и н а м іч н у п о в е д ін к у с т р и ж н я з т р іщ и н о ю п р и суб- і суп ер-
р езо н ан сн и х к о л и в а н н я х
А. П. Бовсуновський^ Ч . Сураче6, О. А. Б овсуновський
а Інститут проблем міцності ім. Г. С. Писаренка НАН України, Київ, Україна
6 Туринський політехнічний інститут, Турин, Італія
Представлено скінченноелементну модель пружного тіла з тріщиною, що закривається.
Показано, що нелінійні спотворення коливань при супер- і субгармонічному резонансах е
високочутливим проявом наявності тріщини. Інтенсивність цього прояву суттєво зале
жить від рівня демпфування в системі та від місця прикладення вимушеної сили.
К лю ч о в і с ло в а : тріщина, що закривається, демпфування, нелінійні резонанси,
діагностика пошкодження.
Introduction. Fatigue cracks are the most widespread damage sources of
dynamically loaded structural elements. Such cracks periodically close and open
in the process of cyclic deformation of a body (and, for this reason, are often
called “closing” or “breathing” one), leading to the instantaneous change of its
stiffness. Usually the change of stiffness is modeled by the piece-wise linear
characteristic of the restoring force [1, 2] or by the specific modification of the
driving force [3].
A closing crack causes the dynamic behavior of vibrating system to be
significantly non-linear, creating a series of fundamental difficulties with regard
to determining analytical solutions. Numerical investigations of forced vibrations
of beams with a closing crack [4-7] have demonstrated that the main distinctive
features of such a vibration system are the manifestation of effects associated with
non-linearity, namely the presence of sub- and superharmonic resonances and
significant non-linearity of the vibration responses (displacement, acceleration,
* Report on International Conference “Dynamics, Strength, and Life of Machines and Structures”
(1-4 November 2006, Kiev, Ukraine).
© A. P. BOVSUNOVSKII, C. SURACE, O. A. BOVSUNOVSKII, 2006
ISSN 0556-171X. Проблемы прочности, 2006, № 5 61
A. P. Bovsunovskii, C. Surace, and O. A. Bovsunovskii
strain) at sub- and super-resonances. The sensitivity of these so-called non-linear
effects to the crack presence exceeds many times (one or even two orders of
magnitude) the sensitivity of natural frequencies and mode shapes, as well as the
sensitivity of non-linear effects at resonance. This opens up the prospect of using
sub- and super-resonance regimes for the diagnosis of fatigue cracks.
Furthermore, it has been shown in [8] that the manifestation of non-linear
effects depend not only on the crack parameters (size and location), but also on
the level of damping in a vibrating system. The data of direct experimental
investigations [9-13] attest that the fatigue crack growth is accompanied by a
considerable increase of damping characteristic of cracked specimens.
Consequently, the determination of the relationship between the crack parameters
and non-linear effects must be realized while taking into account the change of
damping in a vibrating system, rather than assuming constant damping which has
been the case in the past; if the increase in damping is neglected, the prediction of
damage magnitude will be in error.
The aim of the present work has been to develop the model of a beam with a
closing crack which takes into consideration the change of damping due to a
crack growth; based on this model, an investigation has been performed into the
relationship of non-linear effects on the crack parameters at sub- and super
resonance vibrations, as well as on the driving force application point.
Model of a Cracked Beam with Account for Damping in a Crack. The
presented mathematical model of the cantilever beam with an edge transverse
closing crack is based on the finite element model proposed in [5]. When the
crack is closed and its interfaces are completely in contact with each other, the
dynamic response can be determined directly as that one of the intact beam.
However, when the crack is opened, the stiffness matrix of the cracked element
should be introduced in replacement at the appropriate rows and columns of the
general stiffness matrix. Under the action of the excitation force, crack opening
and closing alternate in time, making the equations of motion of the cracked beam
non-linear. Since exact solution of these equations does not exist, it was suggested
that the system has the piecewise-linear characteristic of the restoring force. Then
the vibration of a cracked beam was described by two linear differential equations
in normalized coordinates:
J[I ] {q} + [A ] {q} + [a 2 ] {q} = { R } F ( t ),
V ] {q} + [A d ]{q} + [a 2 ] {q} = R } F ( t ), 0 )
where the matrices of a beam with a closed crack are shown without subscripts
and the matrices of a beam with an open crack are shown with subscript d
(damaged beam), [I ] is the mass matrix (unitary), [a ] is the stiffness matrix,
[A ] is the damping matrix, [R ] is the external load vector, and F ( t ) is the
external load function. The first equation describes the vibration of a beam with a
closed crack and the second one - of a beam with an open crack. These equations
were solved with the Runge-Kutta method proceeding step-by-step in time.
The mesh with a local concentration (Lc) in an area of crack is used for the
increase of the accuracy of a crack location determination (Fig. 1). To calculate
62 ISSN 0556-171X. npo6n.eMH npounocmu, 2006, N 5
The Effect o f Damping and Force Application Point
the additional strain energy caused by the crack, the Cherepanov formula was
used [14]:
4.2 M 3 3 1/?
K i = T 7 v ? [ ( 1 - r ) - ( 1 - y ) 3 ]1/2, (2)
bh
where M is the bending moment, b and h are the width and height of the beam
cross-section, and y = a/ h is the relative crack depth (a is the crack depth).
The results of estimation of the reliability of the beam model presented in
Table 1 indicate that the model can reliably predict the change of natural
frequencies of the cracked beam.
Fig. 1. The finite element mesh for the beam analyzed.
a
b
Calculation Results. Presented above model of a beam with a closing crack
was used for the investigations of non-linear effects arising at forced vibrations of
the beam. Geometrical and mechanical properties of the beam are the following:
L = 0.2 m, L c/ L = 0.1 or 0.5, h = 0.01 m, b = 0.01 m, E = 206 GPa (modulus of
elasticity), and p = 7850 kg/m (density of the material). All calculations were
performed for two levels of damping 6 = 1% and 10% (6 is logarithmic
decrement of vibration).
As can be seen from Fig. 2, one of the manifestation of the non-linearity of
the vibration response of the cantilever beam with a closing crack is the
appearance of the set of non-linear resonances, namely superharmonic resonances
of order 2/1 and 3/1 and subharmonic resonance of order 1/2. The amplitudes of
non-linear resonances are 20-160 times less than the amplitudes of principal
resonance at the damping level under study. This is why the reliable revealing of
the non-linear resonances by the change of amplitude of vibration may cause a
problem. in addition, the possibility of the detection of non-linear resonances
depends on the resonant width. The calculated results show that the resonant
width at the principal, subharmonic of order 1/2 and superharmonic of the odd
orders resonances are comparable by the value, but at superharmonic resonances
of even orders the resonant width is about by one order of magnitude less then at
principal resonance. Consequently, the excitation of accurate superharmonic
resonance of the order 2/1 is much more complex problem then the excitation of
principal or subharmonic resonances.
IS S N 0556-171X. npoôëeMbi npounocmu, 2006, N9 5 63
A. P. Bovsunovskii, C. Surace, and O. A. Bovsunovskii
T a b l e 1
Evaluation of the Model Reliability by the Test Results
Lc/L ah f c l f
(experiment)
f c l f
(prediction)
Distinction
(%)
Steel 15Kh2NMFA (220 X 13.8 X 4 mm) [15]
0.077 0.101 0.999 0.9980 -0.06
0.232 0.974 0.9810 0.67
0.362 0.942 0.9450 0.34
0.155 0.244 0.988 0.9810 -0.70
0.347 0.969 0.9580 -1.16
0.486 0.882 0.9050 2.59
0.277 0.101 0.997 1.0003 0.33
0.217 0.991 0.9910 0.03
0.312 0.983 0.9780 -0.55
Steel ATSTS-1018 (330X 25X 25 mm) [16]
0 0.200 0.924 0.9720 5.18
0.400 0.871 0.8820 1.32
0.600 0.725 0.7200 -0.71
0.182 0.200 0.947 0.9840 3.89
0.400 0.901 0.9280 2.99
0.600 0.830 0.8070 -2.81
Fig. 2. Frequency-response function of the cracked beam (a/h = 0.4, Lc/L = 0.1, and d = 1%).
The vibration responses at non-linear resonances are substantially non
harmonic. The reason of considerable non-linear distortions of vibration response
at non-linear resonances is the fact that at these regimes the vibration range
contains the harmonic, the frequency of which coincides with the frequency of the
principal resonance. The amplitude of this harmonic is comparable at accurate
non-linear resonance with the amplitude of the first harmonic. Therefore, below
the ratio of the amplitude of dominating harmonic in the vibration range to the
amplitude of the first harmonic is used as a characteristic of damage.
64 ISSN 0556-171X. npoôëeMbi npounocmu, 2006, N 5
The Effect o f Damping and Force Application Point
Two original procedures of damage diagnostics are proposed. They are based
on the non-linear distortions of vibration response at non-linear resonances. In the
first one, the force application point along the beam length is varied and the
spectral analysis of the beam end vibrations is executed (Fig. 1a). In the second
procedure, the driving force is applied only to the end of the beam and the data
from the set of sensors located along the beam length are used for the spectral
analysis (Fig. 1b).
As can be seen from Figs. 3 and 4, both procedures significantly respond to
the crack appearance by the increase of the amplitude of dominating harmonic.
Moreover, the first procedure distinctly reveals the crack location by the sharp
local change of the amplitude of dominating harmonic.
a2/ a1 A 1/2 ' A 1
a P (node of force application) b
Fig. 3. The amplitudes of dominant harmonics vs. the coordinate of the force application point. Here
and in Fig. 4: (a) super-resonance (2/1); (b) sub-resonance (1/2) (d = 1% - solid lines; d = 10% -
dashed lines).
a S,, m (sensors location) b
Fig. 4. The amplitudes of dominant harmonics vs. sensors location.
The damping suppresses the sensitivity of the presented damage
characteristics. The increase by one order of magnitude of the damping
characteristic leads to the drop harmonic at super-resonance up to 5 times and
sub-resonance - up to 8 times.
ISSN 0556-171X. Проблемы прочности, 2006, N 5 65
A. P. Bovsunovskii, C. Surace, and O. A. Bovsunovskii
As calculations showed, the qualitative distinction between the sub-resonance
and the super-resonance regimes of vibrations takes place (Fig. 5). At sub
resonance the damage characteristic shows up only if the crack reaches a certain
definite size. For instance, at damping level 6 = 1°% subharmonic in the vibration
range appears only in condition if the relative crack size will be more then
y = 0.07. Since that value the damage characteristic at subharmonic resonance
shows high sensitivity to the crack presence. The increase of damage
characteristic at super-resonance with the crack grows is less intensive but it takes
place after the minimal crack values.
Fig. 5. Crack depth dependence of the dominating harmonics of range of vibrations at non-linear
resonances of the beam (LcjL = 0.1 and d = 1%).
Conclusions. It was shown with the use of the FE model of a cracked beam
that the non-linear effects at sub- and super-resonance vibrations of a cantilever
beam are very sensitive to the presence of a closing crack. At the same time they
are strongly dependent not only on the crack parameters, but on the level of
damping in a vibrating system as well. The higher the level of damping, the lower
is the manifestation of non-linear effects.
Two procedures of crack detection based on the non-linear effects’ utilization
are proposed. The first procedure, in which the force application points are varied
and the sensor has only one location, makes it possible to determine both crack
size and location. The second procedure, in which the set of sensors located along
the beam is used and the driving force is applied to the beam end, makes it
possible to detect the crack presence, but cannot be used for the estimation of the
crack parameters.
Р е з ю м е
Представлена конечноэлементная модель упругого тела с закрывающейся
трещиной. Показано, что нелинейные искажения колебаний при супер- и
субгармоническом резонансах являются высокочувствительным проявлени
ем наличия трещин. Интенсивность этого проявления существенно зависит
от уровня демпфирования в системе и места приложения вынуждающей
силы.
66 ISSN 0556-171X. Проблемы прочности, 2006, № 5
The Effect o f Damping and Force Application Point
1. B. Zastrau, “Vibration of cracked structures,” Arch. M ech ., 37, No. 6,
731-743 (1985).
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asymmetrical piecewise-linear systems by incremental harmonic balance
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a cracked body,” Strength M ater., 33, No. 4, 370-379 (2001).
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produced by fatigue cracks,” Proc. of the 15th Int. Seminar on M o d a l
A nalysis, Leuven (Belgium, 19-21 Sept. 1990), pp. 791-805.
10. K. Jendoubi, N. Ranganathan, and N. Merah, “Effect of thickness on elasto-
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M o d a l P aram eters o f a C antilever, Fracture & Dynamics, Paper No. 37,
Department of Building Technology and Structural Engineering, University
of Aalborg, Denmark (1992).
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indicators by the results of laboratory tests,” Proc. of the 17th IMAC,
Kissimmee (Florida, 8-11 February, 1999), 2, pp. 1909-1915.
13. A. P. Bovsunovskii, “The mechanisms of energy dissipation in the non
propagating fatigue cracks in metallic materials,” Eng. Fract. M ech ., 71
(16-17), 2271-2281 (2004).
14. G. P. Cherepanov, M echanics o f B rittle F racture [in Russian], Nauka,
Moscow (1974).
15. A. P. Bovsunovskii, “On determination of the natural frequency of transverse
and longitudinal vibrations of a cracked beam. Part 2. Experimental and
calculation results,” Strength M ater., 31, No. 3, 253-259 (1999).
16. F. Ismail, A. Ibrahim, and H. R. Martin, “Identification of fatigue cracks
from vibration testing,” J. S o u n d V ibration, 140, No. 2, 305-317 (1990).
Received 12. 12. 2005
ISSN 0556-171X. npoôneMU npoHHocmu, 2006, № 5 67
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