Vibration measure as information On machine technical condition

MARCIN ŁUKASIEWICZ

University of Technology and Life Sciences

Summary

Well-known vibroacoustic methods of object diagnosing were applied in ma-chine diagnostics. A significant part of mama-chine reliability is the vibration level that could be dangerous for machine technical state. The disturbance of the balance state is the reason for vibration formation in machine parts. This vibration can exist and propagate even after their source expiration. Vibrations draw ahead in every me-chanical object. Vibrations could be essential just after crossing a certain threshold marked by amplitude and frequency of the phenomenon. They can be harmful to an object after crossing this threshold of vibration, or to its surroundings (e.g.: a de-crease in the durability of the material). An engineering application "Symptoms gen-eration" allows to process the vibroacoustic signal from the vibration amplitude of time domain to chosen estimators of the vibration process. Applicability possibilities of the application to estimator obtainment of the signal were introduced in this pa-per.

Keywords: diagnostic inference, vibration measure, vibrodiagnostics 1. Introduction

Every technical device at a given moment is in a certain definite state. Generally, a technical state of an object, machine, vehicle can be described as a set of all the parameter values that define a given object at a given moment of time t. Time sequences of this state could be considered the time of the device existence. It leads inevitably to destructive influences of external extorting and internal factors on machine condition change. The use of technical diagnostic methods makes it possible to qualify the current technical condition of a studied object, machine and vehicle [3,4,5].

The necessity of technical condition estimation is conditioned by the possibility of making de-cisions connected with object exploitation and the procedure of the next advance with an object. Present development of automation and computer science in the range of technical equipment and software creates new possibilities of the implementation of diagnosing systems and monitoring technical condition of more complex mechanical constructions. These new possibilities are con-nected with new constructions of intelligent sensors, module software and the modules of transport and data exchange [1,4].

2. The basis of machine vibrodiagnostics

Modern machines are described by such features as: functionality, safety, reliability, readi-ness, mobility and exploitative susceptibility. The main aim of applying machine diagnostics is the qualification of object condition on the basis of generated diagnostic signals (symptoms) and comparing them with nominal values.

Vibrodiagnostics is one of the machine condition description methods – understood as an or-ganized set of methods and means of technical condition estimation (its causes, evolution and consequence) of technical systems, with the utilization of vibration processes or the noise signal. Looking synthetically at possible uses of the vibration diagnostics in the next phase of object existence, one should distinguish the need for knowledge about an object, about signals, syn-dromes and symptoms and theory elements of the decision in the range of diagnostic inference, indispensable to the correct estimation of the technical condition of the object [1,3,4].

Vibroacoustics – is a discipline of knowledge concerning all vibration processes, acoustic and pulsating processes set in nature, technique and machines. Vibroacoustic processes are not only harmful but also parasitic. Applied suitably, they can be a good carrier of energy which can be used to the implementation of various technological processes.

The main tasks of vibroacoustics include:

• The identification of vibroacoustic energy sources – which depends on the location of each individual source in the range of object, machine or environment, the qualification of profiles and correlation between individual sources, the qualification of the vibration power, and also identification of the character of vibrations and sounds,

• The study of propagation ways of vibroacoustic energy in real constructions and envi-ronment (buildings, machines, objects), the theory studies the transfer and transformation of energy, passive and active methods of control of phenomena, analysis methods and in-vestigations,

• The study of vibroacoustic energy control methods (emission, propagation) in machines and environment, and also the study of the methods of controlling these phenomena along with active methods developed in the whole world,

• utilization of vibroacoustic signals for the aims of technical diagnostics, because they are good carriers of information about the condition of an object and implemented techno-logical process (vibroacoustic diagnostics),

• vibroacoustic synthesis of machines and objects, led for the obtainment of an optimum vibroacoustic activity. Including:

a) the synthesis of parameters describing the acoustic range, or the synthesis of vi-broacoustic sizes applied in active methods of vibration and noise reduction and the synthesis of sounds in speech acoustics;

b) machines and objects synthesis that is understood as structural synthesis, kinematic and dynamic synthesis leading to the obtainment of the optimum vibroacoustic ac-tivity;

• active uses of vibroacoustic energy – starting with supersonic welding, supersonic clean-ing, through materials and machine element transportation on technological lines, to the thickening of the moulding sands, knocking out and cleaning the casts as much as to thicken soils and concrete.

Vibroacoustics processes are dynamic phenomena set in the environment of a technical device with frequency from zero to ten thousand hertz. They are abundant in information about the threats to the environment or about machine condition, comparatively easy to observe and register, proc-ess and identify. We can introduce vibration procproc-esses in the following aspects:

• the generation of strength variables in time, influencing the structure and surrounding en-vironment,

• the propagation and transformation of energy in structures and liquids and other elements of the environment,

• sound radiation through the elements of the environment and mechanical structures. The investigation of vibroacoustic processes in many cases is very complicated, especially when vibration processes step out in real physical arrangements. Process investigation conforms to theory, with the use of various analysis methods and then on model investigation. During the analysis of the vibroacoustic process, we focus on such aspects as:

• temporary and spatial schedule of energy coming from the source, • answer arrangement and transfer by the propagating media, • the correlation between sources.

The investigation of the vibroacoustic process uses energy methods, discrete methods, and, especially, the finite element method, modal analysis methods and the analysis method of the acoustic power flow [1,3,4,5].

In diagnostic use, vibroacoustic signals independent of the mold in which they were introduced to the diagnostic system could be processed to the sets of parameters whose values are the basis of diagnostic objects condition description. The choice of these parameters fundamentally influences the efficiency of diagnosis processes that are often deficient in circumstances for this choice optimization. The general essence of vibroacoustic diagnostics bases on the search for relation-ships among the condition of machine Xn, and generated vibroacoustic signals Sm, with the omis-sion of different external influences, as it is introduced on figure 1.

Fig. 1. Observation model of machine condition X based on symptoms S Source: [4].

Basing on the observation model formulated above, we could create a matrix of observation in which possible damages (n) are taken down, on the one hand, represented by technical condition that imitate the evolution of damage, and, on the other hand, we receive an aggregate of symp-toms (m), characteristic for the condition of damage development at the moment of symptom measurement. The observation matrix of measured symptoms is introduced on figure 2.

Symptomy

Cechy Sm WartoĞci mierzonych symptomów

stanu obiektu Xn C K I Psk ... F0 ... Hv ... m 1. Bicie 2. Luz 3. Zacisk ... Symptomy diagnostyczne ... w dziedzinie czasu, amplitud, czĊstotliwoĞci. ... ( wymiarowe, bezwymiarowe )

n. IloĞü pracy

Fig. 2. The observation matrix of measured symptoms Source: [4].

On the basis of the observation matrix, we can affirm that many symptoms can inform about one failure, and a solution to the diagnostic problem requires the fulfilments of the following condition: m • n. After applying operator A, binding the sight of object condition X and its symp-toms S after identification, object condition X on the basis of measured sympsymp-toms can be inferred. 3. Engineering application "symptom generation"

MATLAB is a high-level language and interactive environment that enables you to perform computationally intensive tasks faster than with traditional programming languages such as C, C++, and Fortran. MATLAB provides a number of features for documenting and sharing your work. You can integrate your MATLAB code with other languages and applications, and distribute your MATLAB algorithms and applications. You can use MATLAB in a wide range of applica-tions, including signal and image processing, communicaapplica-tions, control design, test and measure-ment, modelling and analysis. Add-on toolboxes (collections of special-purpose MATLAB func-tions, available separately) extend the MATLAB environment to solve particular classes of prob-lems in these application areas.

On the basis of the Matlab environment, an application "SYMPTOMS GENERATION" was worked out for vibroacoustic process analysis, serving the processing and analysis of the investiga-tive data. The software enables the processing of vibroacoustic signal from an amplitude of vibra-tion in time domain to chosen estimators of vibravibra-tion process. The composivibra-tion of these estimators and their mathematical meaning with a short description was introduced in table 1.

Table 1. Estimators and their mathematical meaning generated in the application „Symptoms generation”

Estimator Symbol Mathematical meaning Interpretation

Average value _x ave

³

∞ →

=

T 0 t ave

x

(

t

)

dt

T

1

lim

x

Average value of signal – infor-mation centre value. High hesita-tions of nominal value shows that a system is in an abnormal state. Root-Mean-Square R.M.S. Xrms

=

→∞

³

T 0 2 T rms

x

(

t

)

dt

T

1

lim

x

Interpretation of a necessitating _{signal as a continuous signal.}

Maximal

value xmax xmax = max {x(t)} Maximal value of data set

Minimum

value xmin xmin = min {x(t)} Minimum value of data set.

Kurtosis _β kurt 4 RMS N 1 i 4 1 kurt

x

)

s

s

(

N

1

_¦

=

−

=

β

Kurtosis of distribution function is a measure of the "peakedness" of the probability distribution of a real-valued random variable.

Skewness _s 3 N 1 i 3 ave i

)

1

N

(

)

x

x

(

s

σ

−

−

=

¦

=

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. Standard deviation σ σ= ₋

¦

₌ − N 1 x 2 ave) x x ( 1 N 1

Standard deviation of a data set is a square root of its variance. Standard deviation is a widely used measure of the variability or dispersion. Form factor _s f ave rms f

x

x

s

=

Inform about dispersal of signal.

Peak factor _c f rms max f

_x

x

c

=

Peak factor informs about the relation between peak value and root-mean-square of a signal. Peak factory value rises when we notice angular frequency from rotatory components beating.

Impulse factor _I

ave

x

x

I

₌

max Establishes the relationship

between an average value and the peak value of a signal.

Clerance factor hf p max f

x

x

h

=

Value of a factor rises with energy storage or exists when data dispersion changes.

Estimator Symbol Mathematical meaning Interpretation 2 N 1 i i p

N

x

x

_¸

¸

¹

·

¨

¨

©

§

=

¦

= Rice fre-quency Fu ~ ) 1 ( ~ u

)

t

(

y

2

)

t

(

y

F

π

=

)

(

~

t

y

- resultant function.

Rice frequency gives a value of frequency that is the most repre-sentative in the signal spectrum.

Autocorrela-tion Rxx

τ

=

_→∞

³

+

τ

T 0 T xx

x

(

t

)

x

(

t

)

dt

T

1

lim

)

(

R

Autocorrelation is the

cross-correlation of a signal with itself.

Cross-correlation Rxy

τ

=

³

+

τ

∞ → T 0 T xy

(

)

lim

x

(

t

)

y

(

t

)

dt

R

Cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them.

Covariance _{Cov(x,y) Cov(x,y) = R}

xyσxσy

This is a measure defining dependence or independence among random variables.

Coherence function γ2xy

)

f

(

G

)

f

(

G

)

f

(

G

)

f

(

yy xx 2 xy 2 xy

=

γ

The function which from two points marks the one that delivers larger quantity of information about the technical condition of an object. Spectral density

C

2x { } ( ) 2

( )

2

( )

j x x

c

c

e

ωτ τ

ω

+∞

τ

− = −∞

=

¦

Spectral density captures the frequency content of a stochastic process and helps identify peri-odicities. Bispectrum x 3

C

{ 1 1 2 2} 1 2 ( ) 3

( , )

1 2 3 1 2

( , )

j x x

C

c

e

ωτ ωτ τ τ

ω ω

+∞ +∞

τ τ

− + =−∞ =−∞

=

¦ ¦

Bispectrum is a statistics used to search for nonlinear interactions. It may be applied to the case of non-linear interactions of a continuous spectrum of propagat-ing waves in one dimension

Bicoherence

(

1

,

2

)

2

f

f

b

1 2 2 1 2 _{2 2} 1 2 1 2

( , )

( , )

( ) ( )

(

)

xxx

S f f

b f f

E X f X f

E X f f

=

_ª

_{º ª}

_º

+

¬

¼ ¬

¼

Sxxx – spectra density for three

frequen-cies, E{} – expected value, X - ampli-tude

Bicoherence is a squared normal-ised version of the bispectrum. The bicoherence takes values bounded between 0 and 1, which makes it a convenient measure for quantifying the extent of phase coupling in a signal.

Estimator Symbol Mathematical meaning Interpretation Wigner Ville WVD( ft, ) WVD

( )

t,f x t τ₂ x* t τ₂ e−j 2πfτdτ ∞ ∞ ¸ ¹ · ¨ © § − ¸ ¹ · ¨ © § +

=

_³

Time-frequency distribution of a _{signal with very high time and} frequency resolution. Wavelet

_C

_{( s}

_τ

_,

₎

³

+∞ ∞ −

Ψ

=

f

t

t

dt

s

C

(

τ

,

)

(

)

*τ,s

(

)

Ψt,s – continuous wavelet function

In mathematics, a wavelet series is a representation of a square-integrable (real – or complex – valued) function by a certain orthonormal series generated by a wavelet.

Equivalent

Sound Level LAeq,T

( )

»

»

¼

º

«

«

¬

ª

¸¸

¹

·

¨¨

©

§

=

³

T 0 2 0 t A T , Aeq

dt

p

p

T

1

lg

10

L

Equivalent Sound Level – quantifies the noise

environment as a single value of a sound level for any desired duration. 4. Conclusion

Vibrations draw ahead in every mechanical object. Vibrations could be essential just after crossing certain threshold marked by an amplitude and frequency of the phenomenon. They can be harmful to an object after crossing this threshold of vibration, or for its surroundings (e.g.: a decrease in the durability of the material).

In the practice of diagnostic investigations, the utilization of vibrations enables the description of machine dynamic condition by a set of estimators from various vibration symptoms. The application "SYMPTOMS GENERATION" enables the processing of a vibroacoustic signal from an amplitude of vibration in a time domain to chosen estimators of vibration process. This soft-ware enables estimator obtainment from a vibration signal and then it makes it possible to describe the technical state of a machine in a matrix observation. A model of a machine in a good condition and a model of the same machine after a certain period of usage gives an inference base about object waste, vibration predominant sources, which allows construction modernization in the next step.

Bibliography

1. Cempel Cz.: Podstawy wibroakustycznej diagnostyki maszyn, WNT Warszawa 1982. 2. Cempel C.: Fundamentals of vibroacoustic condition monitoring, Handbook of condition

monitoring. Londres, Inglaterra: Chapman and Hall, 1998. 1ª Ed. 325–333 p. ISBN: 0-412-61320-4.

3. Uhl T.: Komputerowo wspomagana identyfikacja modeli konstrukcji mechanicznych, WNT 1997.

4. ĩółtowski B.: Elementy dynamiki maszyn, ATR Bydgoszcz 2002.

5. ĩółtowski B., Cempel Cz.: InĪynieria diagnostyki maszyn, Polskie Towarzystwo Diagnosty-ki Technicznej, Warszawa, Bydgoszcz, Radom 2004.

6. ĩółtowski B., ûwik Z.: Leksykon diagnostyki technicznej, Bydgoszcz 1996. 7. Signal Processing Tolbox for use with Matlab, Match Works, 2000.

8. Matlab – The language of Technical Computing, The Math Works, 2000.

POMIARY DRGAē W OCENIE STANIU TECHNICZNEGO MASZYN Streszczenie

W diagnostyce maszyn zastosowanych jest wiele znanych metod wibroakustycz-nych słuĪących do oceny stanu technicznego obiektu. Znaczący wpływ na niezawod-noĞü maszyny ma poziom drgaĔ, który po przekroczeniu wartoĞci dopuszczalnych moĪe byü destrukcyjny dla maszyny. Zakłócenie stanu równowagi jest podstawową przyczyną generacji drgaĔ w obiekcie mechanicznym. Drgania te mogą istnieü i pro-pagowaü po całym obiekcie technicznym nawet po wygaĞniĊciu Ĩródła ich generacji. W niniejszym artykule przedstawiono moĪliwoĞü wykorzystania aplikacji inĪynier-skiej „Symptomy”, która umoĪliwia generowanie estymatorów sygnału drganiowego na podstawie jego przebiegu czasowego wykorzystują transformacje Fouriera. Na podstawie zmian wartoĞci tych estymatorów moĪliwe jest wnioskowanie o aktualnym stanie technicznym obiektu.

Słowa kluczowe: wnioskowanie diagnostyczne, drgania mechaniczne, wibrodiagnostyka

*This paper is a part of WND-POIG.01.03.01-00-212/09 project. Marcin Łukasiewicz

University of Technology and Life Sciences Faculty of Mechanical Engineering