Method of Static Characteristics Calculation of Vibroinsulators with Rectilinear and Circular Working Sections in a Non-Linear Approach

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1 Modern Appied Science; Vo. 9, No. ; 05 ISSN E-ISSN 9-85 Pubished by Canadian Center of Science and Education Method of Static Characteristics Cacuation of Vibroinsuators with Rectiinear and Circuar Working Sections in a Non-Linear Approach Aexander S. Gvozdev & Vadimir S. Meentjev Samara State Aerospace University (SSAU), 4, Moskovskoe shosse, 44086, Samara, Russia Correspondence: Aexander S. Gvozdev, Samara State Aerospace University (SSAU), 4, Moskovskoe shosse, 44086, Samara, Russia. E-mai: mtrushin@mai.ru Received: Juy 8, 04 Accepted: August 4, 04 Onine Pubished: February 6, 05 doi:0.559/mas.v9np7 URL: Abstract An impementation of fu-meta vibroinsuators is one of the most effective measures for an eimination of harmfu effects of vibration in mechanica systems. However, in a contemporary iterature an issue of an anaytica cacuation of vibroinsuators of that type, considering a non-inearity of the characteristics, is addressed insufficienty. In the presented study, by an impementation of exact theory of bending of eastic rods, the methodoogy of oad characteristics cacuation of fu-meta vibroinsuators with eastic eements of rectiinear and circuar shape of initia axia ines was deveoped. A groundessness of Euer equation's impementation was demonstrated in a case of arge dispacement of inear rods, commensurabe with their ength. The method aows to increase designer's capabiities, to improve an accuracy of the cacuation and to reduce time spent for a fina adjustment of a product, which wi utimatey ead to cost reduction fina products. Keywords: fu-meta vibroinsuator, static characteristic, noninear easticity theory, easticay damping eement, eiptic parameters, eastic rod, eastic ring. Introduction A number of papers is devoted to probems of vibration protection and energy dissipation (Ao et a., 006; Popov, 986; Beousov and Faaeev, 989; Beоusоv et a., 009; Chegodaev and Ponomarev, 99; Jiang et a., 008). Nowadays, one of the most widey spread methods of vibration protection is an impementation of rope-type vibroinsuator based systems. Those vibroinsuators are more stabe in an operation because contacts of their friction pairs are distributed aong ines and areas (Pоnоmаrev, 00; Uаnоv et a., 009; Popp et a., 00). Periodic structure of eastic eements faciitates a creation of more precise cacuation modes of vibroinsuators during their oading with various combinations of forces. Moreover, such vibroinsuators have advantage in contexts of cost and abor-intensity during production, in comparison with vibroinsuators of other types, for exampe, MR materia based made from wire pressed together (Barry, 04; Uаnоv et a., 009; Ao et a., 006). In studies of researchers who study probems of energy dissipation using structura damping methods (Pоnоmаrev, 00; Hakan, 0; Jiang et a., 008; Uаnоv et a., 009) with an impementation of simiarity theory and dimensionaities a comprehensive anaysis of possibiities for energy dissipation by various systems of vibration protection is presented: rope-type shock absorbers, shock absorbers made from pressed wire MR materia, mutiayer dampers with different diagrams of pressure between ayers. It is estabished, that muti-ayer structures with an instantaneous sip with a uniform pressure between ayers have the best damping properties, which aowed the authors to deveop a number of highy effective damping devices structures for aviation and machine buiding (Pоnоmаrev, 00). Modern fu-meta vibroinsuators beong to devices that sustain significant strain during their operation, which are commensurabe with a size of eastic eements (Pоnоmаrev et a., 009; Beоusоv et a., 009). In those circumstances there is a substantiay non-inear dependence of arge dispacements from externa forces, athough a materia of eastic eements of vibroinsuator behaves easticay because of sma strains (Liu et a., 00; Jiang et a., 008; Uаnоv et a., 009). Therefore, inear theory of eastic rods bending gives a great inaccuracy in a case of vibroinsuators' design (Pоnоmаrev, 00; Popp et a., 00; Jiang et a., 008). Different authors propose different theories of eastic rods' behavior under strains that cause significant strains, which are commensurabe with their sizes (Zhdаnоv et a., 0; Hakan, 0). On a basis of theoretica 7

2 Modern Appied Science Vo. 9, No. ; 05 position's data (Popov, 986; Pоnоmаrev, 00; Barry, 04; Ao, et a., 006), in the presented study engineering methodoogy for a cacuation of vibroinsuators static characteristics with rectiinear and circuar working sections in a non-inear approach, which aows to greaty simpify a design for vibroinsuators of that type.. Materias and Methods Designating expressions, describing bending of easticay damping eement (EDE) in a form of a rectiinear eastic rod in a non-inear approach, using the dimensioness parameters (Note ): P p η =, ς =, () EJ where P - externa oad, characteristic size (in that case, ength of rod), E modue of easticity of materia in bending, J moment of inertia of cross-section of the rod, p dispacement (strain) due to externa oad. From the cassic strength of materias theory it is known that a strain of rectiinear rod is subjected to inear aw. Experiment data shows that, in fact, a functiona dependence of dispacements from externa forces, especiay in an area of arge strain, is consideraby non-inear. To consider that factor L. Euer proposed foowing equation: 6EJ P = p. () ( p) By a reduction of () to dimensioness from, taking into account (), obtaining 6ς η =. () ς However, it was determined experimentay that in Euer equation () non-inear nature of oading curves isn t described accuratey enough. In this regard, the most reiabe is a method of a cacuation based on eiptic parameters proposed by E.P. Popov. According to that method, for a determination of rectiinear rod s static characteristics, a system of equations must be soved using tabes of eiptic parameters' functions: F( ψ) F( ψ0) = β k sinψ 0 = / k cosψ = 0, (4) where k eiptic moduus, ψ 0, ψ [psi] initia and fina ampitudes of Legendre eiptic integras of first type F( ψ0), F( ψ ). Coordinates of fina point of an eastic rod are determined by foowing expressions. x y = k cosφ0 β where Ek ( ), Eφ ( 0 ) Legendre eiptic integras of second type, parameter φ 0 =ψ 0. Then, an equation of rod s eastic ine wi take foowing form, (5) = [ Ek ( ) E( φ0 )] β x k(cos 0 cos ) = φ φ β, (6) y s = [ E( φ) E( φ0 )] β where s arc ength to a desired point (s max =0, s min =). A soution for a system of equations based on Legendre integras of first and second kind aows to obtain an equation of rod s eastic ine. Then a non-inear oading characteristic P=f(p) wi take a form, presented in figure, and in that case p=y. In order to bring desired parameters to dimensiona type, using inverse transformation of the equation (): 7

3 Modern Appied Science Vo. 9, No. ; 05 η EJ P =, p = ς (7) Cacuation research conducted by the authors and carried out in accordance with theory proposed by E.P. Popov, showed, that an actua oading characteristic ies in an area between cassica inear strength of materias theory and L.Euer hypothesis () and the most accuratey described by method of eiptic parameters. However, because of its high abor intensity that method has not yet been widey impemented. The authors had the objective- to find an aternative repacement for theory of E.P. Popov. On the basis of L. Euer equation structure for an approximation of function η=f(ς) using east-squares method the foowing anaytica expression was obtained, which is correct for any rectiinear EDE having cantiever type support (8): By substituting () in (8), obtaining: 6ς η =. (8) 4 + 0,05ς,ς + 0,79ς ς 6 ( p/ ) ( ) ( ) ( ) ( ) EJ P = + 0,05 p/, p/ + 0,79 p/ p/ 4 The probem of strain of EDE in a shape of a ring is discussed further. Designating expressions, describing bending of EDE in a shape of a ring in a noninear approach. The equation (), in that case, becomes: Q R q π η =, ς =, (0) 8EJ R where Q= P compressing (stretching) forces, R radius of midde ine of circuar EDE, q - dispacement (strain) due to force Q. According to cassis theory of strength of materias, deformation of ring is described by the foowing expression: π 4 PR q =. () π EJ In a noninear approach, for a determination of static characteristics of discussed EDE, according to E.P. Popov, it is sufficient to anayze two forms I, II of eastic ine of contnuous thin ring with radius R in a case of its bending by two mutuay equipoise radiay compressing forces Q. Considering that form I can pertain to infection (form I ) and noninfection (form I ) type, obtaining a system of equations, expressed through eiptic parameters, which is describing forms of eastic ine of anayzed ring in axes', y', where axis x' is directed aong the externa force P=Q/, of a oca coordinate system: x s = E( φ) ( ), I : = x s = E( φ), R β0 R I : y = k ( cos φ ), R β0 R kβ0 k R y ( sin k φ ), R kβ0 (9) () 74

4 Modern Appied Science Vo. 9, No. ; 05 x s = E( φ), R β0 R y = k ( cos φ ) when 0 s sip., R β0 II : x s = [ Ek ( ) E ( φ )], R β0 R y π R = k ( + cos φ ) when sip. s R β0 Where s i.p infection point of eastic ine. By changing to xcy coordinate systems, which are reated with a ring s center C, for forms I and II obtaining: I and II: x = x, y = y y. (4) Creating a static characteristic (see Figure ) in the form of a reationship between Q=f(q), where q is a dispacement of force Q, at that 0 () Q = β ( EJ / R ), (5) where β0 = ( / π ) β For forms I, II obtaining: q = R x = R x. (6) In order to bring desired parameters to a dimensiona form, using inverse transformation of the equation: 8η EJ Q =, (7) π R q = ς R. (8) Determining power coefficient of simiarity η [eta] dependence from dimensioness strain ς [zeta], using the methodoogy proposed by the authors, in a form of a poynomia expression: ki i ς (9) i= η = At the same time, in order to ensure a succession of inear theory (), presenting coefficient near first member 4π of expansion in the form π 8. In that case (9) wi take foowing form: 4π η = ς 5,69ς +,98ς. (0) π 8 By substituting expressions (0) in the expression (0), finay obtaining expression, which is correct for any circuar EDE: EJ 4π q 5,69 q P, 98 q = + R π 8 R R R In order to iustrate the proposed engineering methodoogy the authors determined oading characteristic in a vertica direction of vibroinsuator, presented in Figurea. That vibroinsuator consists of four EDE in a form of cabe segments (rods), which ends are anchored in intermediate eements, made in the a form of two brackets, having two camps each for instaation of cabes in them, and in one bracket cabe is instaed using rigid fastening, in another it is movabe and mounted on the brackets in such a manner that every cabe has one fastening point with each bracket. () 75

5 Modern Appied Science Vo. 9, No. ; 05 а) b) c) Figure. Vibroinsuator a vibroinsuator with rectiinear EDE; b equivaent cacuation scheme; c - cacuation scheme of vibroinsuator. Cacuation scheme of vibroinsuator is shown in Figureb. By defining infection of a frame's end as y/, obtaining tota dispacement of points A and B reative to each other equa to y. Considering that the ange of rotation in points A and B is equa to zero, cacuation diagram of cantiever beam wi be equivaent to zero (Figure c). For one frame, the basic parameters are presented as δ = y, = L, ς = y = y, () L L obtaining P I EJ 6ς = ( L /) + 0,56ς 4,76ς +,6ς () or 6 ( y/ L) ( ) ( ) ( ) 4EJ PI = L + 0,56 y/ L 4,76 y/ L +,6 y/ L Because two frames are working in a vibroinsuator simutaneousy, then 6 ( y/ L) ( ) ( ) ( ) 8EJ PΣ = PI = L + 0,56 y/ L 4,76 y/ L +,6 y/ L (4) (5) 76

6 Modern Appied Science Vo. 9, No. ; 05 Since in vibroinsuator two frames operate simutaneousy, an equation that aows to create static characteristics of any vibroinsuators of that type, have a foowing form (0). With an impementation of deveoped cacuation methods parameter series of rope-type vibroinsuators are created for oads from 0. to 00 kg with rectiinear, radia and spatia shapes of eastic ines. Range of oads is divided on areas according to aw of geometric sequence. Inside each area, optima oads were seected according to aw of geometric sequence. Rigidity of vibroinsuators was determined with a condition of a provision of system s natura frequency of 0 Hz, according to recommendations for devices and equipment of airspace engineering. Standard arrangements were used of transversa cross-sections of ropes. A determination of geometric parameters and physica characteristics of vibroinsuators was carried out on a basis of obtained criterion expressions, in which vaues of dimensioness criterion were seected according to conditions of a provision of reiabiity, manufacturabiity, minima weight and dimensions.. Resuts The presented study conducted by the authors aowed to obtain foowing resuts.. Foowing anaytica expression of oading characteristics was obtained, which is correct for any rectiinear EDE having cantiever-type support in a dimensioness form: 6ς η = (6) 4 + 0,05ς,ς + 0,79ς ς and in dimensiona form: 6 ( p/ ) ( ) ( ) ( ) ( ) EJ P = + 0,05 p/, p/ + 0,79 p/ p/ 4 In Figure a comparison of oading characteristics is presented, which is created on the basis of the authors methodoogy, cassica inear easticity theory and the methods of E.P. Popov and L. Euer.. The expression is obtained, which is correct for any circuar EDE in dimensioness form and in dimensiona form (7) 4π η = ς 5,69ς +,98ς. (8) π 8 EJ 4π q 5,69 q P, 98 q = + R π 8 R R R In Figure a comparison of oading characteristics is presented, created on the basis of the authors methodoogy, cassica inear easticity theory and the method of E.P. Popov. (9) Figure. Diagram of static characteristic of rectiinear eastic rod 77

7 Modern Appied Science Vo. 9, No. ; 05 Figure. Diagram of static characteristic of eastic ring. Loading characteristic is obtained 6 ( y/ L) ( ) ( ) ( ) 8EJ PΣ = PI = L + 0,56 y/ L 4,76 y/ L +,6 y/ L which aows to create static characteristics of any vibroinsuators of a form factor, presented in Figure. 4. Discussion The fact that it is impossibe to use a cassica strength of materias theory and hypotheses of L. Euer for a cacuation of oading characteristics of EDE, subjected to strain, which is commensurabe with their size was estabished and experimentay confirmed (Pоnоmаrev, 00; Ao et a. 006; Jiang et a., 008; Barry, 04). In that case, the most accurate is the method of cacuation based on eiptic parameters and considering significant noninearity of strain process (Popov, 986). However, abor-intensiveness of that method does not aow to freey use it in engineering practice (Pоnоmаrev et a., 0). The proposed engineering method is based on the same principes, but it is much easier to impement, therefore, it can be used as an aternative repacement. The discussed method has a number of advantages as compared to finite-eement method. First, it is an increased speed of cacuation, because a creation of detaied anaytica mode isn t required. Second, there is a possibiity to directy determine a rationa ratio for a designed product, by anayzing obtained criterion equation without a search of various variants. It was experimentay confirmed that discussed method aows you to increase designer's capabiities, increase scientific and technoogica eve of design process, reduce time spent on fina adjustments of a product (Beоusоv et a., 009; Uаnоv, 009), as we as to use more precise modes of vibroinsuators in dynamica cacuations (Jiang et a., 008; Yurddаs et a., 04), which wi utimatey ead to a cost reduction of finished products. 5. Concusion It shoud be noted that the information provided in the presented paper does not cover the entire range of objectives, which a researcher accompishes in the process of a creation of the method for a cacuation of static characteristics of fu-meta vibroinsuators. A further direction of the research, in the authors opinion, is connected with a study of more compex forms of eastic eements in order to anayze phenomena such as, for exampe, quasi zero rigidity in structures, equa frequency of rigid eement by coordinate axes and a reguation in vibroprotective systems. Another important direction of research can be a deeper investigation of damping properties of discussed structures. It is necessary to note, that currenty damping is studied to a sufficient degree ony in eements with periodic structure, such as aminar structures. That probem is more topica in a fied of study of vibroprotective devices with rope-type eements. Nowadays, simiar to functiona reationships, which are true for pressed wire MR materia, for rope-type eements averaged damping coefficients are used, which are obtained by (0) 78

8 Modern Appied Science Vo. 9, No. ; 05 experimenta data processing. Apart from the fact, that damping coefficient is changing with time, it can be infuenced by many parameters (rope coiing ange, vaue of wire s tightening, manufacturing coatings and ubricants, etc.), which are hard to account. That, in turn, doesn t aow to create ogicay compete method, which incudes mass production technoogy of precision fu-meta vibroinsuators of that type. For vibroprotective systems acceptabe soutions in dimensioness form haven t been found so far. Acknowedgements The presented study was supported by the Ministry of education and science of the Russian Federation in a framework of Program for an increase of competitiveness of SSAU among the word's eading scientific and educationa centers for 0-00 years. References Bayakin, V. B., Faaeev, S. V., & Novikov, D. K. (00). Air-tightness of secondary gas end sea assemby. Gazovaya Promyshennost, 8, Barry, O., Zu, J. W., & Oguamanam, D. C. (04). Anaytica and experimenta investigation of overhead transmission ine vibration. Journa of Vibration and Contro. WOS: Beousov, A. I., & Novikov, D. K. (986). Seection of GTE rotor bearing hydrodynamic damper type. Soviet Aeronautics (Engish transation of Izvestiya VUZ, Aviatsionnaya Tekhnika), 9(), 6-0. Beousov, A. I., & Faaeev, S. V. (989). Gasostatic face sea with eastic working surface. Soviet Journa of Friction and Wear (Engish transation of Trenie i Iznos), 0(), 4-8. Beousov, A. I., Novikov, D. K., & Eskin, I. D. (98). Theory of hydrodynamic dampers with cyindrica and end sots. Soviet Aeronautics (Engish transation of Izvestiya VUZ, Aviatsionnaya Tekhnika), 4(), -7. Beousov, A. I., Bayakin, V. B., & Novikov, D. K. (985). Experimenta study in pressure distribution in the oi fim of short hydrodynamic damper. TRENIE & IZNOS, 6(4), Beousov, A. I., Bayakin, V. B., & Novikov, D. K. (985). Experimenta study of pressure distribution in ubricant ayer of 'short' hydrodynamic dampers. Soviet Journa of Friction and Wear (Engish transation of Trenie i Iznos), 6(4), Beousov, A. I., Ponomarev, Y. K., Yu., Pronichev, N., & Krypaev, D. G. (009). Theory of an annuar corrugated damper in the vibrator precession motion. Russian Aeronautics, 5(). Date Views Chegodaev, D. E., & Ponomarev, Y. K. (99). Mutiayer shock absorbers with controabe eastic hysteresis characteristics. Izvestiya Vysshikh Uchebnykh Zavedenij. Aviatsionnaya Tekhnika,, Hakan, Y., Seçuk, M., & Sadettin, K. (00). Hybrid input shaping to suppress residua vibration of fexibe systems. Journa of Vibration and Contro, 8,. Hongrui, A., Jiang, H. Y., & Uanov, A. M. (006). Estimation of the fatigue ifetime of meta rubber isoator with dry friction damping. Experimenta Mechanics in Nano and Biotechnoogy, Parts and, Vo. 6-8, WOS: Hongrui, A., Jiang, H. Y., Wei, W., & Uanov, A. M. (006). Study on the damping characteristics of MR damper in fexibe supporting of turbo-pump rotor for engine. ISSCAA 006: st Internationa Symposium on Systems and Contro in Aerospace and Astronautics, Vo. -, WOS: Jiang, H. Y., Zhang, R. H., Zhao, K. D., &. Novikov, D. K (005). Anaysis of damping characteristics of squeeze fim damper with meta rubber. Tuijin Jishu/Journa of Propusion Technoogy, 6(), Jiang, H. Y., Hao, D. G., Xia, Y. H., Uanov, A. M., & Ponomarev, Y. K. (008). Damping characteristics cacuation method of meta dry friction isoators. Journa of Beijing Institute of Technoogy (Engish Edition), 7(), Liu, T. Y., Chiang, W. L., Chen, C. W., Hsu, W. K., Lu, L. C., & Chu, T. J. (00). Identification and monitoring of bridge heath from ambient vibration data. Journa of Vibration and Contro, 7, Marinin, A. V., Fedotov, V. M., Ostapenko, V. I., Serbin, V. A., Kuchkin, V. I., & Ponomarev, Yu. K. (978). Producing stee Stsp for pates. Metaurgist, (), Mihir, C. M. (0). Free vibration of tapered isotropic rectanguar pates. Journa of Vibration and Contro, 8,

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