Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 4, pp. 1049-1058, Warsaw 2011 APPLICATION OF SELF-EXCITED ACOUSTICAL SYSTEM FOR STRESS CHANGES MEASUREMENT IN SANDSTONE BAR Janusz Kwaśniewski, Ireneusz Dominik, Jarosław Konieczny, Krzysztof Lalik, Abdurahim Sakeb AGH – University of Science and Technology, Department of Process Control, Kraków, Poland e-mail: kwa j@uci.agh.edu.pl; dominik@agh.edu.pl A self-excited acoustical system is suggested formonitoring of the chan- ge of stress in a sandstone bar. Stress changes manifest themselves in small but detectable variations of frequency in the self-excited system. The suggested system can be applied for the continuous stress changes monitoring in elastic constructions and rock masses. Tests on a single sample of sandstone were performed. They examined the impact on the stress measurement such parameters as: position of sensors and the ac- tuator as well as the influence of geometrical shape and dimensions of the sample. A theoretical model whose parameters have been confirmed by performed experiments is also developed. Key words: self-excited system, stress monitoring 1. SAS system in stress change measurement 1.1. Introduction to self-excited systems By bringing a speaker closer to a microphone, the system is excited and it produces a hum. This effect also occurs in the radiotechnics. The autodyne lamps,which areused in the old radio sets, can excite themselves. This effect is called the autodyne effect. This effect consists in variation of such parameters of a self-excited system as: the amplitude, frequency and bias voltage. Autonomous non-conservative systems are characterized by the fact that their vibrations are associatedwith a gain of energy. If at the time of vibration of an autonomic system the flow of energy from the outside appears causing a gain of the amplitude of these vibrations or compensating the loss of energy 1050 J. Kwaśniewski et al. and supporting periodic oscillation, then the system is called self-excited and its vibration – self-excited vibration (Votoropin et al., 2008). The most characteristic feature of self-excited systems is the way they charge theenergy. It allows one todistinguishbetweenautonomous self-excited systems and non-autonomous systems. Energy flowoccurs in non-autonomous systems by the action of external, explicitly time-dependent forces (Nikolov et al., 2008). The time in self-excited systems occurs in equations not explicitly. The energy source is constant, not dependenton time.The energyflow is controlled bytheoscillating systemitself (which iswhywecall sucha self-excited system). So considering these systems, we can distinguish the following elements: a – constant source of energy, b – oscillatory system, c – control device supplyingpower to the vibrating system, controlling the flow of energy, d – positive feedback between the system and the control device, through which the system directs the vibrating energy. Elementsaandcmaybe linear, partb is non-linear (Fig.1). It results from dispersion equations. If any of the physical quantities (displacement, velocity or acceleration) occurs in these equations as non-linear, the oscillating system will also be non-linear (Bogusz et al., 1974). Fig. 1. Block diagram of a self-excited system (Bogusz et al., 1974) It can be assumed, according to a great number of tests conducted on several kinds of stones, that the dependence between the velocities of the wave and the change of the stresses in the sample is non-linear, in result the oscillating system as well. This non-linear dependence for the marble sample is shown in Fig.2. During the study on the acoustical systems for stress changes measure- ment, the authors did not find any solutionworking in the closed loop, neither in Polish nor foreign literature. However, there are a lot of open loop system Application of self-excited acoustical system... 1051 Fig. 2. Dependence between the wave propagating velocity in marble and the load solutions.Oneof themost effective open loop systemwas patented in theUSA [6]. The systempresented in this patentworks in an open loop anduses a criti- cally refracted longitudinal ultrasonic technique. Every open loop system has a little resistance to disturbances (Deputat et al., 2007) and worse sensitivity than in our closed loop system. In the paper the new, innovating method for stress changes measurement is presented together with experimental data. 1.2. Structure of SAS system During our research conducted atTheDepartment of Processes Control at TheUniversity of Science andTechnology, the Self-excited Acoustical System (SAS) has been developed. It is an assembly of devices which similarly to the speaker-microphone system or autodyne lamps exploit the phenomenon of self-excitation. The SAS system diagram is shown in Fig.3, where the receiving head is a piezoelectric sensor and the emitting head (shaker) is a piezoelectric actuator. The power amplifier, emitter (E) and receiver (R1) are formed in the feedback loop. As a result of positive feedback, there is a bilateral interaction between the control device and the vibrating system, which allows the self-excited system to control its own energy balance. As a result of that, despite the fact of losses of energy occurring in the system, there are non-fading periodic oscillations. Many applications of the self-excited phenomenon are known. These are: vibrations of cutting tools, turbineblade vibration, vibration of aircraftwings, vibration of bridge suspension which may cause destruction of the bridge (as happened to the Tacoma Narrows bridge), etc. In such systems, we try to eliminate these vibrations. 1052 J. Kwaśniewski et al. Fig. 3. The SAS diagram, where R1 is the receiver and E is the emitter, F – load The systemwas intended tomeasure a change of stress in elastic mechani- cal structures, constructions and rockmasses. The purpose of this study is to determine the possibilities of using this system for real objects such as brid- ges, dams, buildings, mines, etc. The sensitivity of this system, for small and large deformation, is higher than the sensitivity of othermeasurement systems (especially open systems). Tests on a single sample of sandstone were performed. They examined the impact on the stress measurement such parameters as: position of sensors, position of actuator, and the influence of geometrical shape and dimensions of the sample. Fig. 4. The test stand for endurance tests using the SAS System In the study a sample of sandstone compressed in the frame by a hydraulic press (Fig.4)was used. In addition, the systemwas equippedwith a force sen- Application of self-excited acoustical system... 1053 sor to calculate the compressive stress in the beam in real time. Accelerations were measured by three accelerometers for three reasons: • to determinewhat impact the distance from the emitter to the receivers has on the results and the position of resonances, • to avoid a situation where the sensor is located in the resonance node, which would result in incorrect test results, • to determine the velocity of wave propagation in the material with cor- relation methods. The study was conducted in three configurations of the position of emit- ter (E). For a sandstone sample with rectangular cross-section of 60×70mm, the emitting and receiving devices were mounted in the configuration shown in Fig.5. Fig. 5. Diagram of deployment of emitters on sandstone, E is the emitter and F is load 2. Test results The SAS System is designed for use in real objects such as bridges, dams, buildings,mines, etc. Therefore it is necessary to create a universal procedure to detect the change in the strain in these structures. The designers decided to use first the system to conduct tests in the open loop (Kwaśniewski et al., 2009). To the sample charged with various compressive forces, the chirp 1054 J. Kwaśniewski et al. signal emitted in the frequency range of 100Hz-20kHzwas initially given.This allowed one to obtain the amplitude-frequency characteristics (Fig.6) of the structure with visible resonances (Fig.7). Fig. 6. Comparison of vibration transmissibility functions for different sample loads in the open loop system Fig. 7. Selected and expanded resonance peaks believed to be related with the change of stress in the open loop system Application of self-excited acoustical system... 1055 Figure 7b presents themovement of the first resonance peak believed to be relatedwith the change of stress in the sample. Figure 7a – the third resonance peak.Thedamage of frequency respectively amounts to 150-250Hz. InFig.7a, the fourth peakwith the highest amplitude is also shown. It has been omitted, because it comes from the own resonance of the emitter, so this 4th peak is out of consideration. For the closed loop system, which is shown in Fig.8, the frequency change at the same load variation is approximately 450Hz. In addition, the amplitude of vibration is 20-30 times greater than in the open loop system, which makes them more visible and easier to identify. Fig. 8. Selected and expanded resonance peaks of the closed loop system believed to be related with the change of stress, where E is a position of the emitter and R is the position of the receiver A theoretical model needs still additional adjustment in order to confirm themodel datawith the experimental ones. Themathematical formulaswhich were used in themodel are described in the following (Bobrowski et al., 2004). The emitted signal reaches the receiver with a time delay τ = l/V , where V is the velocity of the elastic wave. The delayed signal, registered by the receiver R1, returns to the power amplifier by means of a positive feedback system. The frequency of oscillations in such a system ω follows the phase balance equation ωτ =2πm (2.1) 1056 J. Kwaśniewski et al. where m is a integer number, which is estimated as the number of waveleng- ths λ on the distance l: m≈λ. It follows from(2.1), that the m -th frequency of self oscillations is equal to ωm = 2πm τ = 2πmV l (2.2) Then the frequency shift ∆ωm, caused by a small variation ∆V of sound velocity, equals ∆ωm = 2πm l ∆V (2.3) The changes in stress in rocks cause variations in the sound velocity V , and thereby in the frequency of oscillations in the feedback system. The frequency of normal mode f is given by the equation of balance of the phase 2πfτ+ϕE(f−fE)= 2πm m=1,2,3, . . . (2.4) which requires that thephasedifference in the feedback loop is amultiplication of 2π. Expression ϕE(f − fE) is the phase difference from the emitter. As- suming that the difference ∆fE between the frequency of normal mode f and the frequency of the emitter fE is relatively small, the expression ϕE(f−fE) can be linearized as ϕE(f−fE)≈ 2πτE(f−fE) (2.5) where τE is the delay in the emitter, defined by the formula τE = 1 2π dϕE df (2.6) The time of transition of the wave through the stone sample τ is much longer than the time of the conversion of the voltage signal to mechanical vibrations in the shaker τE, so it can be assumed that the delay from (2.6) is small compared with the time of flight of the wave τ τE ≪ τ (2.7) where τ = l/v and v is the velocity of the wave. Substituting equation (2.5) with equation (2.4) and dividing both sides by 2π, we obtain fτ+ τE∆fE =m (2.8) According to inequality (2.7), and solving equation (2.8) in line with the theory of disorders, it is assumed that f = f0+∆f τ = τ0+∆τ (2.9) Application of self-excited acoustical system... 1057 where f0, τ0 corresponds to the initial stress, and ∆f and ∆τ correspond to the change of the stresses. According to formula (2.9), we have approximately f0 = m τ0 (2.10) then the first disorder is ∆f f0 =− ∆τ τ0 − τE τ0 ∆f0 E f0 (2.11) and ∆f0 E describes thedifferences between the frequency f0 and the resonance frequency of the emitter. The indicator m in formulas (2.9) and (2.11) appro- ximately shows the number of lengths of waves over the distance l between the emitter and receiver R m≈ l λ (2.12) 3. Conclusions During thework on theSAS system, the researchers conducted a great number of tests, whichwill facilitate the implementation of the system to real objects. The following algorithm was developed: • one header system works initially in the open loop system, • it will allow one to select the frequency range in which the resonance peak will be responsible for the change in stress – this is going to be the identification step, • secondly, the system operates with two headers, which will allow one to determine the speed of propagation of the acoustic wave in the investi- gated material. Using this algorithm for creation of a compact measuring apparatus and carrying out research on industrial constructions will be the priority in the development of the current project. Acknowledgements The research work was supported by the Polish government as a part of the research programmeNo. NN501234435 carried out in the years 2008-2010. 1058 J. Kwaśniewski et al. References 1. Bobrowski Z., Chmiel J., Dorobczyński L., Kravtsov Y.A., 2004, Ul- trasonic system formonitoring stress changes anddeformations in the ship hull, EXPLO SHIP, ISSN 0209-2069 2. Bogusz W., Eegel Z., Giergiel J., 1974, Drgania i szumy, Wydawnictwo Geologiczne,Warszawa 3. Deputat J., Mackiewicz S., Szelążek J., 2007, Problemy i techniki nie- niszczących badań materiałów – wybrane wykłady, GAMMA 4. Kwaśniewski J., Dominik I., Konieczny J., Kravtsov Y., Sakeb A., 2009, Experimental system for stress measurement in rock, 9th Conference on Active Noise and Vibration Control Methods, Kraków-Zakopane, Poland 5. Nikolov N., Kenarov N., Popov P., Gotszalk T., Rangelov I., 2008, All-digital PLL systems for self-oscillationmode of microcantilevers with inte- grated bimorph actuator and piezoresistive readout, Sensors and Transducers Journal, ISSN 1726-5479 6. US Patent 6424922 – Ultrasonic stress measurement using the critically re- fracted longitudinal (LCR) ultrasonic technique, US Patent Issued on July 23, 2002 7. Votoropin S.D., Noskov V.Ya., Smolskiy S.M., 2008, An analysis of the autodyne effect of oscillatorswith linear frequencymodulation,RussianPhysics Journal, 51, 6 Zastosowanie samowzbudnego akustycznego systemu do pomiaru zmian naprężeń Streszczenie W artykule przedstawiono koncepcję i wyniki badań nad samowzbudnym aku- stycznym systemem, który jest proponowany jako nowy sposóbmonitorowania zmian naprężeń w belce z piaskowca.W badanym rozwiązaniu zmiana naprężeń objawiała się małymi, aczkolwiek mierzalnymi, zmianami częstotliwości. Proponowany system pomiarowy może znaleźć zastosowanie do ciągłego monitorowania zmian naprężeń w konstrukcjach sprężystych orazmasach skalnych. W artykule opisano przeprowadzone badania doświadczalne z zastosowaniem próbki wykonanej z piaskowca. Podczas badań sprawdzono wpływ umiejscowienia nadajnika i odbiornika oraz wymiary geometryczne i kształt próbki na uzyskiwane wyniki. Dodatkowo został zbudowany model teoretyczny, którego parametry zostały pozyskane i potwierdzone na drodze eksperymentalnej. Manuscript received October 4, 2010; accepted for print December 20, 2010