RESONANCE VIBRATIONS FORMATION FEATURES OF REGULAR SYSTEMS WITH A BREATHING CRACK

Transcription

1 RESONANCE VIBRATIONS FORMATION FEATURES OF REGULAR SYSTEMS WITH A BREATHING CRACK Vadym Kruts, Anatoliy Zinkovskii, and Yevheniia Onyshchenko Department of Vibrations and Vibration Resistance, G.S. Pisarenko Institute for Problems of Strength of the National Academy of Sciences of Ukraine, Kyiv, Ukraine The paper presents the solution to the problem of diagnostics of the presence of a breathing crack based on the consideration of vibrations of the regular system consisting of two one-type elements, which models a set of blades. The discrete model of the nonlinear system under investigation is developed. The results of calculation experiments on the determination of vibrodiagnostic parameter at the subharmonic mode of vibrations for the regular system in question at different values of elastic coupling coefficient, the characteristics of energy dissipation and the crack parameters are presented. It is shown that the chosen vibrodiagnostic parameter of the presence of a breathing crack due to elastic coupling between subsystems is observed not only for damaged, but also for undamaged subsystems. Moreover, with an increase of elastic coupling value, its magnitude increases for undamaged subsystems, whereas for damaged subsystem it decreases. The comparison of the results with that ones obtained for an isolated subsystem as a model of single blade is also carried out.. Introduction and Problem Statement One of the most loaded turbine machine elements are blades, which are subected to a wide spectrum of dynamic and temperature loads as well as the influence of foreign bodies of different type (dust, ice, birds and etc.). The specified facts are the potential sources of damages of various type, which can be divided into two groups. The first group involves dent-type damages, nicks and open cracks, which do not cause any considerable change of the elastic-inertial characteristics of the obect under investigation in the course of its deformation cycle at different modes of vibrations. The so-called breathing fatigue cracks as typical representatives of the second group of damages, on the contrary, cause the change of the abovementioned characteristics, which is the reason of nonlinearity of vibrations. An undue diagnostics of such type of damages can lead to turbomachine malfunction. Recently, a great attention has been paid to studying vibrations of the obects of engineering with a breathing crack as demonstrated by a large number of published papers, particularly [ 5]. These investigations are mainly aimed at studying vibrations of the separate structural element. However, due to the presence of various constraints between the components of machine assemblies, which is typical of them, separate consideration of a structural element does not allow one to describe its dynamic state with a sufficient accuracy. This is only possible by taking into consideration its interaction with other elements of the system under investigation. Therefore, in the theory of vibrations, an increasingly greater attention is being paid to studying complex mechanical systems, among which regular systems involving series or parallel connections of equitype elements (subsystems) occupy a special place. Primarily, it is a set of blades and blade assembly as a special type of regular systems with rotation symmetry. The results of review of the published papers demonstrate that there is a limited number of papers on the investigation of vibrations of the systems under consideration with fatigue crack-type damages [6]. The papers on the computational-experimental investiga-

2 tions of the edge non-closing crack influence on the vibrations of the regular system such as tuning fork type specimen (in the form of linear system) [7] should be mentioned as well as numerical investigation of the model of turbomachine blade assembly with such damage [8]. Considering the above-said, the purpose of the present paper is the numerical investigation of the influence of the breathing fatigue crack parameters on the formation of vibrations of the regular system, which consists of two equitype elements, modeling a packet of two blades. 2. Obect of Investigation and Its Modeling A tuning fork type specimen with an edge crack with normal opening in one of its beams (Fig.,b), which is regarded as the beam model of a packet of turbomachine blades (Fig.,a), is chosen as an obect of investigation as in [6]. A discrete model with a structural regularity described in [6] in detail and presented in Fig.2 is chosen as the model of the system under investigation. a Figure : Packet of turbomachine blades (a) and tuning fork type specimen (b). b Figure 2: Discrete model of the tuning fork type specimen with a fatigue crack. A strongly regular system, for which the given values of (М and М 2 ), stiffness factors (k and k 2, where k = k o + k P and viscous friction (c and c 2 ) of the subsystems are identical, i.e. М = М 2 = М; k = k 2 = k; c = c 2 = c, where k o is the stiffness factor of the beam having an open crack (in this case at u 0), is taken to understand the influence of a breathing crack on the vibrations of the simplest regular system in its initial undamaged state. As in paper [6], the damage parameter a is introduced, and it takes the following form: 2 ICSV23, Athens (Greece), 0-4 July 206

3 = (k k o )/k. Considering the assumption that due to the presence of a breathing fatigue crack the regularity violation of the vibration system depends only on the difference between the elastic properties of its subsystems with their mass equality, the system of differential equations describing vibrations of the chosen discrete model can be reduced to the following form: 2 2 u 2hu p 0.5α signu u γp u u2 q0 cos νt; 2 2 u2 2hu2 p u2 γp u2 u q0 cos νt, where h c 2M and p k M are the damping coefficient and natural frequency of vibrations of the undamaged subsystem, respectively; γ ks k is the coefficient of elastic coupling between the subsystems; k s - stiffness factors of elastic coupling between the subsystems; q0 P M. At 0 the system of decoupled differential equations describing vibrations of the specimen beam regarded as an isolated subsystem is obtained. 3. Procedure of Computational Experiments The algorithm for the calculation of forced vibrations of the obects under investigation and the results of their approbation are presented in [6] in detail. The solution to the system of differential equations () describing forced vibrations of the tuning fork type specimen model was performed using the Runge-Kutta method with an automatic step size control, which was used to determine the displacement of masses ui () t of the obect under investigation. At the next stage of the algorithm for the calculation of forced vibrations of the chosen model of the regular system with the presence of nonlinearity due to a breathing crack the amplitudes of vibration harmonics, which are within the obtained solutions ui () t, were determined using FFT method. 4. Results of Computational Experiments Two modes of vibrations (in-phase and antiphase) are typical of the tuning fork type specimen with the violation of its regularity even during in-phase excitation. Therefore, for the amplitudes of excited vibration harmonics of the specimen beams with a fatigue crack let us introduce A iq, where i =, 2 is the number of vibration harmonic; q = I, II is the type of the specimen vibration mode: I in-phase; II antiphase; =, 2 is the subsystem number: damaged, 2 undamaged. The system nonlinearity due to a breathing fatigue crack under the action of harmonic driving force is observed in the excitation of sub- and superresonance modes. As it is shown in [6], the most representative vibrodiagnostic parameter of the crack presence can be the ratio of the amplitudes of 2q q q2, q, q the second A and the first A harmonics - æ = A A at the superharmonic resonance 2 of the second order ( = 0.5p r ) and the amplitudes of the first at the superharmonic resonance of the /2 order ( = 2p r ) - ICSV23, Athens (Greece), 0-4 July A and the second q q, q2, q æ = 2q () A harmonics A A, where p r is the main resonance frequency of vibrations of the system with damage under investigation. Thus the specified amplitudes of the constituents of vibration harmonics and their relations were determined in the process of computational experiments. Based on the results of the given computational experiments for the discrete model of the tuning I fork type specimen (see fig.2) the dependencies of the vibrodiagnostic parameter æ on the nonlinearity characteristics α were determined for different values of elastic coupling coefficient. Fig. 3 shows the specified dependencies for the regular system in question at the in-phase mode of vibrations and model of the damaged beam in the isolated state. It is seen that the nature of the

4 dependence of the vibrodiagnostic parameter of the subsystem, which models the damaged beam of the tuning fork type specimen, does not change in comparison with that one for the beam model in the isolated state over the entire range of variations of the elastic coupling coefficient. The presence of the second harmonic in the spectrum of the amplitudes of vibrations of the undamaged beam due to elastic coupling of the beams attracts the attention. Moreover, with an increase of the elastic coupling coefficient the vibrodiagnostic parameter for the damaged beam decreases, whereas it increases for the undamaged one. This is clearly illustrated by the dependencies of the vibrodiagnostic parameter on the elastic coupling coefficient for the damage parameter value α = 0. shown in Fig.4. I Figure 3: Dependence of the vibrodiagnostic parameter æ on the damage parameter α for the damaged (open symbols) and the undamaged (filled symbols) subsystems of the tuning fork type specimen model at the subharmonic resonance for h = sec - and elastic coupling coefficients equal to 0.0 (, ), 0.05 (, ), 0.02 (, ). Dashed line is the model of the damaged beam in the isolated state. I Figure 4: Dependence of the vibrodiagnostic parameter æ on the elastic coupling coefficient for the damaged ( ) and the undamaged ( ) subsystems of tuning fork type specimen model at the subharmonic resonance for the damage parameter α = 0. and h = sec -. 4 ICSV23, Athens (Greece), 0-4 July 206

5 The analysis of the obtained dependencies shows that for small values of the elastic coupling coefficient the vibrodiagnostic parameter æ I for the undamaged beam is significantly lower than for the damaged one, which makes its use in the diagnostics of the fatigue crack more difficult. However, with an increase of the elastic coupling coefficient the above-mentioned vibrodiagnostic pa- I I rameters for the damaged and undamaged beams converge. Thus, at 0.06 the relation æ æ 2 is insignificant and approximately equals to.5. This fact makes it possible to conclude that the above-mentioned vibrodiagnostic parameter for the undamaged beam can be used for the diagnostics of the fatigue crack in the considered vibration system. It should also be mentioned that at the considered mode of vibrations the excitation of antiphase mode of the beams vibration of the tuning fork type specimen model is observed. However, due to small amplitudes, which are consistent with the specified mode of vibrations, they are insignificant. 5. Conclusion At the subharmonic resonance, as in case of an isolated beam, the ratio of the amplitudes of the first and the second harmonics of vibrations of beams at the in-phase mode of the system vibrations (on the contrary to the antiphase, when the amplitudes of the specified harmonics are insignificant) can be used as the vibrodiagnostic parameter of the presence of a breathing fatigue crack in the tuning fork type specimen. Due to the presence of the elastic coupling between the beams the vibrodiagnostic parameter for the damaged beam of the specimen is smaller than that one for the beam in its isolated state. With an increase of the stiffness factor of the elastic coupling for the damaged beam it decreases, whereas for the undamaged beam it increases. REFERENCES Matveev, V. V. and Boginich, O. E. Approximate analytical determination of vibrodiagnostic parameters of the presence of a crack in an elastic body under superharmonic resonance, Strength of Materials, 42 (4), , (200). 2 Matveev, V. V., Boginich, O. E. and Yakovlev, A. P. Approximate analytical method for determining the vibration-diagnostic parameter indicating the presence of a crack in a distributed-parameter elastic system at super- and subharmonic resonances, Strength of Materials, 42 (5), , (200). 3 Matveev, V. V., Boginich, O. E., Yakovlev, A. P. and Sinenko, E. A. Approximate analytical determination of vibrodiagnostic parameters of the presence of a closing crack in bar elements under subharmonic resonance, Strength of Materials, 46 (3), , (204). 4 Sinenko, E. A. and Zinkovskii, A. P. Influence of the exciting force application point on the amplitude spectrum of flexural vibrations in a beam with a «breathing» crack, Strength of Materials, 47 (4), , (205). 5 Matveev, V. V., Boginich, O. E., Sinenko, E. A. and Yakovlev, A. P. On vibrodiagnostics of the presence of a closing edge crack in a beam with amplitude-dependent damping capacity under superharmonic resonance, Strength of Materials, 47 (5), , (205). 6 Kruts, V.A., Zinkovskii, A. P. and Sinenko, E. A. Influence of a fatigue crack on the vibrations of the simplest regular elastic system, Strength of Materials, 45 (3), , (203). 7 Tokar, I. G. and Zinkovskii, A. P. Influence of the parameters of a local defect in a regular system on the range of eigenfrequencies of vibrations and the level of vibration stresses in elements of the same type, Strength of Materials, 42 (2), 67 74, (200). 8 Huang, Bo-Wun and Kuang, Jao-Hwa Vibration in the stability of a rotating blade disk with a local crack defect, J. of.sound and Vibration, 294 (3), , (2006). ICSV23, Athens (Greece), 0-4 July 206 5