THE HARDEST MODE OF BROADBAND RANDOM VIBRATION ON THE OWN SPECTRUM BANDS OF MECHANICAL SYSTEMS

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1 THE HARDEST MODE OF BROADBAND RANDOM VIBRATION ON THE OWN SPECTRUM BANDS OF MECHANICAL SYSTEMS Igor N. Ovchinnikov Bauman Moscow State Technical University, -nd Baumanskaya, 5, , Moscow, Russia During the study of the impact of the width of the random vibration spectrum on the durability and vibroloading the existence of mechanical systems spectrum bands is discovered. These bands are similar to the resonance frequencies of mechanical systems, but they apparent under broadband random vibration. Own spectrum bands oscillation are the hardest loading mode as well as resonance oscillations. This result has been obtained experimentally, theoretically, and by means simulation. In the presented paper, the result was confirmed with the Rice formula, which is used in radio engineering for the analysis of random processes. 1. Introduction Not long ago the main part of the making construction and rocket, aviation, vechicle engine ready including strength, took place during natural testing. For the organizing of resources testing modern aircraft engine during 5000 hours 60 tanks of fuel are necessary. The price of such product was very high, so the range of the natural testing must be reduced and the importance of laboratory testing raises. There are two main problems in vibration testing: 1. Making authentic fatigue curves of material during operational loads, test of construction vibroloading and prediction durability of the tested object in environment on it.. Reliability evaluation and vibration survival of tested object in for a specified time under specified loads. Thus at the completion of testing there is no reliable information about the remaining resource of the object. To bring the design to destruction, many of which are unique, too expensive. Today for laboratory testing the following types of loading are used: harmonic excitation (also with scanning), polyharmonic excitation, band limited random vibration (BRV), working load vibration. Most effective the broadband random vibration testing. International Electrotechnical Commission (IEC) makes a demand for testing on random vibration two main demands: authenticity of the results and their repeatability in different laboratories with various personal, which is especially important in acceptance testing. The demand of IEC about repeatability of the results is the demand of standardization testing modes [1]. At the last years the system of international standards for the testing on broadband random vibration is made, including those which are introducing in our country. The using of simple rectangular shape of spectrum, getting after «levelling» dynamic of the object was developed. Such shape of spectrum must be more suitable for the mode repeatability. But nomenclature of objects is wide and they all have their own dynamic characteristics of the tested objects. With the help of the ICSV4, London,3-7 July 017 1

2 main recommendation the problem of repeatability of the testing results was not solved. It is possible the cause is: the broadband random vibration is characterizing by width of spectrum (boundary frequencies: initial f bd1 and terminal - f bd ), and units of spectrum width does not exist yet. For the comparison at the harmonic load there are no problems width repetability of the testing results, because resonant modes with their own object frequencies, that is object dynamic, are determine identically in the all laboratories. The necessity of object dynamics at the broadband vibration testing is obvious. At the development of the modern technique the testing of different purposes according their methods are made: vibration survival, reliability, durability, maintainability and acceptance testing. The problem of unification of the all types of testing became actual. At the long exploitation of technique there is problem of rapid testing. It can not be solved only by the increase of loading, because the process of accumulation of fatigue defects is changed. For the realization of the rapid testing at the broadband random vibration the solving of the problem of forming of the «hardest vibration mode» is necessary, which is described in []. The problem of reproducibility of the test results can only be achieved by improving the reliability of the results, first of all, the fatigue test (to failure), and accounting for the dynamics of the test object. Time to destruction of the test object (t destr ) is the only significant parameter characterizing the quality of test mode, in this case - the degree of danger the regime. And this time it is necessary to find a function of the spectral width f. No stress in the material (), or vibration velocity (v) of the vibrations of structural elements is not uniquely determine the degree of danger the regime. It is determined by both of these parameters together, and they (their dispersions) are known as the Rice formula [3], which is characterized by the number of intersections of a random process specified level. This number determines the degree of danger and longevity of the regime of the test object (t destr ). N D ( x a) exp D x D x x, (1) where N the number of positive crossings of normal random process of the fixed level; Dx dispersion of process vibrospeed; D x dispersion of process displacement; а expectation value of the process. By essence,nis number of cycles of loading. As the dispersion of parameter is proportional to the parameter itself, and the minimum meaning of console beam displacement in the dangerous cross-section, the maximum meaning N(minimal durability) can be reached at the maximum fraction of Rice formula. Rice formula is a fundamental result, which was obtained in 1950s for signal-noise ratio calculation in radio engineering. This formula has attracted attention of mechanical engineers due to its universality and simplicity. Thus, if we accept that «expectation value» a is the material endurance limit, we can calculate the number of corresponding intersections of this value. The special approach for reliability calculations based on the Rice formula is designed [4].The Rice formula has not been spread to durability calculations, despite the fact that such attempts have been made [5]. This formula will be used for qualitative analysis of vibrational loading. It is shown [6] that the velocity variance D x 1 under the broadband random excitation (Fig. 1) is always greater than the velocity variance D x under the arbitrary unimodal excitation (Fig. )providing that the power spectra of the two signals is equal. The case of mean displacement square minimization in condition of given input signal mean also leads to requirement of constant vibration spectrum. Such a signal has a maximum velocity. ICSV4, London,3-7 July 017

3 Thus, vibrational loading with uniform spectra leads to maximum variance, maximum velocity and minimize mean square of displacement. Figure1. The spectral density of white noise Figure. The spectral density of unimodal excitation After substituting these values in to the Rice formula the numerator turns to maximum while the denominator turns to minimum. So, the random process maximum number of cycles and hence minimal durability is realized under more broadband excitation. Real structure dynamics introduces some correction but it is obvious that the spectral width influence on the vibrational loading and durability so that the spectrawidening makes the vibrational mode more dangerous.. Investigation of oscillations of a beam In [7] is showed theoretically, that broadband random vibration loading parameters extreme exists in any mechanical system. They accord to extreme of power of strength, appear in the system at the broadband random vibration. With position of theory of dynamical systems the extreme of mutal information between input and output signal characterize correlation of the attainability field real mechanical system with modeling system. The model can considered successively as dualfrequency, three-frequency and so on. At that the number of extreme («own bands spectrum of the system») at one is smaller than number of allowed own frequencies of mechanical system. Investigation effect spectrum width f vibration on vibration speed and lifespan was conducted on cantilevered fixed beam in kinematic excitation (Fig.3). The trial was conducted with random modes for following values of f: 10 Hz (mode 5); 30 Hz (mode 6); 100 Hz (mode 7); 300 Hz (mode 8) (Fig. 4). Numbers of testing modes are given with correspondence to full series of testing modes [8]. The results of trials in harmonic mode 1 are introduced only for qualitative comparison. Figure 3. Three first forms of original oscillations of testing object (beam) in kinematic excitation Figure 4. Continuous spectrums of incoming vibroeffect Experiment on fatigue was conducted according to method, which does not simplify any random process, because replacement of random vibration loadingwith harmonic mode is not correct [8]. ICSV4, London,3-7 July 017 3

4 All measurements and, most importantly deformations were applied from the start of trial until the fracture of the object. As it common for lifespan investigations the results were introduced in a form of fatigue curves dependence of logarithm of time till the fracture on average tension in the material (Fig. 5). It is shown that with the widening of load range lifespan of them at aerial of the object is falling in the beginning and then raising. Existence of extremum of durability (resource) depending on boundary frequency fbd of spectrum of vibration loading during constant tension in the object is clearly shown in Fig. 6, which has been obtained by processing results of fatigue trials. Figure 5. Fatigue curves of modes 1(harmonic), 5-8 (random) Figure 6.Depending average vibration speedv av from f under the constant average stress av. As a limiting frequency of spectrum for harmonic mode (mode 1 [8]) logarithm of first original frequency of the object f 1 =7 Hz, and for random mode with continuous spectrum logarithm of sum of average spectrum frequency (first original frequency of the object) and half of spectrum width for chosen mode: 1 lg f lg bd f1 f () During random oscillations the object was significantly being exited on the second and the third original frequencies (f =177 Hz, f 3 =475 Hz). In order to raise informational content of the results of trials vibration loading curves, which show the dependence of logarithm of time until fracture on vibration speed in sample s dangerous cross-section (Fig. 7), have been drawn first time in practice. ICSV4, London,3-7 July 017 4

5 Figure 7. Depending t p of the samples of V av in a dangerous cross-section of the beam (curves of the vibro loading ) Figure 8. Depending average stress av from f under the constant vibration speedv av Restructuring of the results at the BRV ( modes) depending on the medium voltage ( ср ) and medium vibration velocity(v ср ) from the spectral width (f), expressed through the logarithm of theboundary frequency of spectrum lgf грn, where «n» - the amount of the natural frequencies of object considered with fluctuations (much excited), showed the presence of extrema in these dependencies (Fig. 6 and Fig. 8). Despite the fact that curvature of Fig. 6 and Fig.8in given coordinates is relatively small, there is no doubt in the existence of extremums, because the points on the curves correspond to high amount of fractured test objects. The veracity of the test results is high enough due to application more perfect methods of trials andcapacitive deformation sensors [8]. These results indicate the presence of mechanical systems "effective width spectrum" ( f n ef ). In[9,10] is made by modelling Galerckin method of broadband random vibration kinematic loading on cantilever beam in the range of first sixteen own frequencies. And discovered here, the detected extrema, the analogues of which are shown in Fig Results We consider the cantilevered beam; its amplitude-frequency response is shown in Fig. 9.On the beam acts the BRV of equal power with the expanding range covering initially two own frequency of the beams, then three own frequency of the beams and after n own frequency of the beams, where n (Fig.10).At some values of spectrum width of the load called "effective"( f n ef ), the beam can have extreme values of average voltageσ av and medium vibrationspeedv av in its dangerous section (Fig.11). Earlier, similar results were obtained theoretically [7], in experiment [8, 10] and modeling [9,10]. This result was generalized and the "effective width of the spectrum" was called "own bands spectrum of the system", similar to the natural frequency of the loading object. Own spectrum ICSV4, London,3-7 July 017 5

6 bands, as well as its own object frequencies depend on the dynamics of the object. Therefore the dynamics of the object should be considered when conducting the vibration tests. Figure 9. Amplitude frequency response of testing object Figure 10. The spectral density of external Influence Figure 11. The Figure reaction parameters of testing object ICSV4, London,3-7 July 017 6

7 3. Conclusion By ISO series standards the BRV tests are carried out without accounting of dynamics of the object. In experimental research of the effect of spectrum width on vibration speed and durability of a cantilever beam with kinematic excitation the extremes of durability in function of boundary frequency of vibration exposure spectrum were obtained at a constant stress in samples and at a constant vibration velocity at the same spectral width. This width of spectrum is called the "own" and the mode - "hardest". The similar extremes were obtained by modeling of the beam vibrations and confirmed by theory. In this way the Rice formula confirms the existence of the own spectrum bands of mechanical systems. The heaviest BRV mode is determined by the parameters of the mechanical system and solves the problems of reproducibility of the test results, the standardization of vibration tests and fatigue modes tests, the formation of the accelerated tests mode. It is possible to use it in acoustics. "Own spectrum width" Δf n, where n is amount of own mechanical system frequencies involved in the vibration. This is a new property of the mechanical systems, which is given the name "own spectrum bands" of mechanical systems (analogue to own frequencies). REFERENCES 1. GOST Tests on the effect of the random vibration with broadband digital vibration control system. (International Standard EC ).. Vibration in technique: Guide in 6 volumes / Editing V.N Chelomey (сhairman).vol. 5. Measuring and testing. Editing M.D. Genkin. Moscow, Engineering, (1981). 3. Franks L. E. Signal theory. Prentice-Hall, Englewood Cliffs, (1969). 4. Vibration in technique: Guide in 6 volumes / Editing V.N Chelomey (сhairman).vol. 1.The vibrations of linear systems. Editing V.V. Bolotin, Moscow, Engineering, (1981). 5. Rikher V.L. Hypothesis of spectral summation and its application for determination of the fatigue life under the influence of random loads. Works of TsAGI. Moscow, Publishing department. Vol. 1134, (1969). 6. Arutyunov S.K, Ovchinnikov I.N. About the influence of the width and shape of the spectrum of vibration impact on vibration loading of objects // Problems of strength., (1981). 7. Ovchinnikov I. N. The usage of transparency frequency bands of mechanic systems for standardization of tests to random vibration. Proceedings 1 International Congress on Sound and Vibration, Lisbon, Portugal, July, (005). 8. Ovchinnikov I.N., Brancevich P.Y. Experimental determination of the hardest condition of broadband random vibration. Proceedings 1 International Congress on Sound and Vibration, Beijing, China,13-17 July, (014). 9. Ovchinnikov I. N. Simulation of oscillations of a beam at random vibroloading. Proceedings 15 International Congress on Sound and Vibration, Daejeon, Korea, 6-10 July, (008). 10. Ovchinnikov I.N. A new property of the mechanical systems own spectrum bands.proceedings International Congress on Sound and Vibration. Florence, Italy, 1-16 July, (015). ICSV4, London,3-7 July 017 7