Natural Oscillations of Cylindrical Bodies with External Friction on the Boundary

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1 Applied Mathematics, 5, 6, Published Online Mach 5 in SciRes. Natual Oscillations of Cylindical Bodies with Extenal Fiction on the Bounday Safaov Ismail Ibahimovich, Akhmedov Maqsud Shaipovich, Boltaev Zafa Ihteovich Bukhaa Engineeing-Technological Institute, Bukhaa, Republic of Uzbekistan maqsud.axmedov.985@mail.u, maqsud.axmedov.985@mail.u, lazizbek.axmedov.@mail.u Received 9 Febuay 5; accepted 5 Mach 5; published 6 Mach 5 Copyight 5 by authos and Scientific Reseach Publishing Inc. This wok is licensed unde the Ceative Commons Attibution Intenational License (CC BY). Abstact In this pape we conside of natual oscillations cylindical bodies with extenal fiction. Complex ates changes fom fiction paametes ae shown. Rate equations ae solved numeically by method of Mulle. Keywods Extenal Fiction, The Natual Oscillations, Cylindical Body, Flat Swing, Ant Plane Oscillation Fequency, Damping Facto. Intoduction Simulation of vibations of bodies located in the defomable medium is studied with many scientists and by vaious methods []-[4]. Study of dynamic stess-stain state of pipelines in soil medium, efes to the complex task of solid mechanics. In some eseaches [5] [6] defomable suounding in the pipe eplaced by spings and consideed as emeging (linea and nonlinea) educing foce. In this pape, vibations of pipelines as a cylindical body with adiuses and R at defomable suoundings ae modeled (Figue ). Medium was eplaced with viscous damping in the adial and tangential diection. The main aim of the wok is to study the oscillations of a cylinde with extenal fiction on the edge and to compae the esults of the body located in an elastic medium [3].. Statement of the Poblem Study fluctuations pipeline located in an elastic medium ae consideed diffeent methods [6]-[9]. In this pape, fluctuations pipelines ae modeled as a cylindical body with a adius and R, located in a defomable medium (Figue ). Medium was eplaced with viscous damping in the adial and tangential diec- How to cite this pape: Ibahimovich, S.I., et al. (5) Natual Oscillations of Cylindical Bodies with Extenal Fiction on the Bounday. Applied Mathematics, 6,

2 Figue. Design model of cylindical bodies with a viscous extenal fiction. tion. The main goal of the wok is to study the natual oscillations a cylinde with extenal fiction. In the study mentioned above, the optimal values of damping coefficients, in which the oscillations ae, damped pipelines as possible. Conside the poblem of the oscillations of an infinite elastic cylinde with extenal fiction at the inteface (Figue ). Closed system of equations of small oscillations of the fee elastic cylindical body has the fom: u µ u + ( λ + µ ) gad div u = ρ ; t () = λθδ + µε ; ij ij ij whee u displacement vecto; λ, µ lame coefficients; ρ the density of the cylinde; ε ij stain tenso. On the pat of the bounday ij the stess tenso; n = u As t given movement: u = ; apat R-Voltage:. well as the initial conditions: u u t= = ; t= = t Conside the poblem in cylindical coodinates (, θ, z). Assuming that z coodinate does not affect the oscillation, we obtain a system of equations splits into two independent tasks [7]: z θ z z uz + + ρ = f ; θ t uz u z = µ + ; z (а) uθ uz θ z = µ + ; z θ θ θθ u + + ρ = f ; θ t θ θθ uθ + + θ ρ = f ; θ θ t u u uθ u = µ + λ ; + + θ u uθ uθ θ = µ + ; θ uθ u u uθ u θθ = µ + + λ + +. θ θ 63

3 With bounday conditions: at = R: Figue. The dependence of the eal pat of the eigenvalues of α (n = ); The dependence of the imaginay pat of the eigenvalues of α (n = ). uz z = α ; t u = α ; t uθ θ = α ; t whee R-adius of the cylinde; αα,, α-paametes of fiction at = ε : < z < θ < u < u < z uθ < Then we call the bounday value poblem -ant plane and -flat o plana poblem of oscillations of a cylinde. 3. Methods fo Solving the Poblem of Natual Oscillations We call the natual oscillations of an elastic cylinde solution of and (d) (f = ) fo ant plane case types: (с) (d) 63

4 poblem (b and d) fo the plana case types: whee,, v, v z cos nθ θz = θ sin nθ e u z v cos nθ iωt cos nθ sin nθ = e u v cosnθ uθ vθ sin nθ θ θ iωt θ θ -unknown function of the adial vibation modes; ω = ωr + iωi -complex natual fequency of the cylinde, the eal pat of which chaacteizes the fequency of oscillation of the cylinde, and the imaginay pat ( ω I ) the ate of decay, n =,,, [8]. Substituting the epesentation (3a), (3b) in Equations and and the bounday conditions (p), (q), we obtain the spectal poblem fo the (two) odinay diffeential equations solved fo the fist deivatives of the adial coodinate : Fo the case of plane vibations Ant plana n λ λ = θ + k ( nvθ + v ) + ω v k k = ( + ) + ω k k λ V = ( nvθ + v ) k k Vθ = θ + ( nv + vθ ) µ λ λ θ θ k n vθ nv n vθ v = µ µ n = ρω v It can be shown that the conditions fo the finiteness of all the unknowns in the cente fo a solid cylinde equivalent to the following bounday conditions-fo flat cylinde oscillation: at n = : v =, v θ = ; at n = : =, = ; At n > can be set equal to zeo any set of two unknowns. Indeed, if the conditions of the limb in the cente of the cylinde, then, θ, v, vθ can be epesented as: = a + a + a + θ = b + b + b + (5а) v = c + c + c + vθ = d + d + d + whee ai, bi, ci, di, i =,,, unknown coefficients. Then substituting (5a) to (4a) we obtain a system of fou equations. Equating them coefficients of like powes of we obtain the ecuence elation: θ (3a) (3b) (4а) (4b) 63

5 λ λ λ nb a + a + knd + kc nd c = k k k λ λ λ b + k n d k nc na : + + = k k k λ λ nd c = k k nc + d = λ λ λ ( i + ) ai+ = ai+ nbi+ + kndi+ + kci+ ndi+ nci+ + ω ci k k k λ λ λ ( i + ) bi+ = bi+ + k n di+ + k nci+ + nai+ + ω di i : i =,, k k k λ λ ( i + ) ci+ = ai ndi+ ci+ k k k ( i + ) di+ = bi + nci+ + di+ µ At = ε we find that fo n =, θ finite, U, U θ tend to zeo; at n = U, U θ finite,, θ tend to zeo; fo n > tend to zeo all unknown. If ant plana cylinde oscillation: the n = = ; at n = : U =, at n > eithe of the two unknowns can be set equal to zeo. Indeed, if the conditions of the limb in the cente of the cylinde at the time: = v = m + m + m + whee i, mi, i =,,, unknown coefficients. Substituting (5b) in (4b), we obtain a system of two equations. Equating them to obtain coefficients of the same ecuence elation: : = µ nm : = µ nm i = imi i =,, µ = i + µ nm ρω m i i+ i i Then fo = ε at n = v of couse, tend to zeo; at n = of couse, v tends to zeo fo n > tend to zeo, both unknown. The geneal solution of the system of Equations (a,, c) can be expessed in tems of Bessel and Neman functions n the ode: ω ω ν = AIn + BYn cs cs ω ω ω ω = µ A I n + B Y n cs cs cs cs whee A, B abitay constants; I n Bessel function of ode n; Y n Neman function of ode n. When substituting (6) into (c and d), we obtain the chaacteistic equation fo ω: ω I ω n i Y ω µ = αω n cs cs cs (5b) (6) (7) 633

6 Theoem. Let the Eigen values ω k bounday value poblems µ u + u + ω u ; C ; 8, a = = c ρ a: u = a = const > R: u + iωαu = 8, б uk Simple. Then the system уk =, whee u k Eigen functions of the poblem (8), coesponding to iωkuk the eigenvalues ω k, Rises basis in the space L W [8]. Poof. To pove the theoem using the definition fom [8]. A bounday value poblem can be epesented as: whee ( n) ( n ) у, λ = у + p x, λ у + + pn x, λ у = n ( k ) ( k U, ( ) ) j у λ = a jk λ у + bjk λ у = j =,, n k = s ν ( λ) λ p x, = p x, p x = const, s =,, n, p ajk, b jk abitay polynomials λ. Then Definition is the numbe of s ν s ss nn ν = χ ode linea fo U (, ) j ν ( k ) ν ( k λ у ( ) o λ у ) atν + k = χ j and contains such vaiables in ν k χ j U ( у, λ ) = will be called the ode of the coesponding linea fom (, ) j (9) у λ fo (9), if the fom contains vaiables + >. Ode bounday condition j U j у λ. Numbe χ = χ + + χn called a total ode of the bounday conditions (9). Definition will be called the bounday conditions (9) nomalized if any n bounday conditions, they ae equivalent, i.e., eceived (9) ae linea combinations of not less than the total ode. Given the total ode of the bounday conditions (9) is called a total ode of the bounday conditions, we obtain fom (9) afte nomalization. Rewite Equation (8a) and bounday conditions (8b) in the fom (9). To make the eplacement is necessay, so that the new unknown changed fom zeo to unity. a x = = a x+ a. Equation (8) will become: a We intoduce ( ) d u a du + λ ( a) u = dx a x+ a dx We intoduce the coefficients p = p = ; p = a a x+ x+ a a Bounday conditions: ( ) λ ; ; ( ) = = = = p a p p p a U = u( ) = u U = λαu + = a t x= a = a = b = λα b = a () () 634

7 χ = total ode. The chaacteistic equation fo (9) has the fom: n n ω + pω + + pn h ω+ pnn = Its oots ae denoted ω,,. ω n substituting the coefficients we obtain: () ( a), a, ( a) ω ω ω = = = (3) If the oots of the chaacteistic Equation () is simple, and the coefficients p W L. Then fom [8] ν s s+ ν + that the complex plane can be divided into sectos h s,, sh h n, in each of which Equation (9) has a fundamental system of solutions у ( x, λ),, уn ( x, λ) whee these solutions and thei deivatives up to ode n shall be pemitted λ in these sectos asymptotic expansions: ( s ) s s ωk k ( λ) ωk λ ( ηks,, λ ηks,, ( λ )) у x, = x+ + x+, ks, =,, n (4) ν + whee ηks,, ν ( x) W ν =,, and the fist expansions-function ηks,, θ ( x) ηks, ( x) In [8] it is also noted that the functions ( x) ks,, ν, of secto and the sequence of the system of ecuence equations. We epesent the solution of ou poblem in the fom (4): do not depend on s. η involved in the expansions (4) do not depend on the choice ( a) xi ( a) xi у ω ω = η ( x) + ; у = η ( x) + ω ω, ( a) xiω у = ( a)( iωη ) ( x) + ; ω ω ( a) xiω = ω η + у i a The eigenvalues of the poblem (9), () ae detemined by the zeos of the chaacteistic deteminant ( λ ), which has the fom: ( λ ) = Expanding the deteminant, we obtain: U у U у U у U у n n n χ λµ jk ( λ) λ [ Fj ] = jk whee µ jk = ωα ; jk, k =,, n, abitay set of distinct positive integes anging fom to n. Fo k = we set α jk. µ = a, µ = a, µ =. Then µ = In this case j ( ai ) ω a ( ai ) ω U ( у ) = η + ; U ( у ) i = = ωα η + iω η +. ω a ω F η η α k = F η η α = ( ai ) ω ( ai ) ω ( iω) = iω η η ( α ) + η η ( α ) + ω (5) We intoduce the definition of [8]. Definition 3. Bounday value poblem (9), () is said to be egula if all the coefficients p ( x) n. ν s in Equa- 635

8 jk ik tion (9) summing functions and numbes F in the expansions F coesponding cone points µ jk diffeent fom zeo. Let M denote the smallest convex polygon containing the point µ jk ou case M-segment. Point µ jk that wee on the bounday of the polygon M, called a bounday, and the points that lie at the vetices of M-the cone. Definition 4. Regula bounday value poblem is said to be stongly egula if the zeos of the chaacteistic λ asymptotically simple and sepaated fom each othe by a positive numbe δ >. deteminant F if α, F if α. And obtained Equation (9), the coefficients p ( x) ν s -enteable functions. Hence when α и α ou poblem is egula. We show that it is a egula had task. We find the asymptotic appoximation of zeos of the ( ai ) ω deteminant ( iω ). To do this, we intoduce the notation z =. The (5) can by witten as: ( iω) = iω η ( ) η ( α ) + η η ( ) z( α ) + = z ω We assume without loss of geneality that η η β = η η ( ) The equation becomes: η η. Then we denote. α α β α β α z = z = ± At α thee exists a countable set of eigenvalues detemined by a multi-valued function of the logaithm: α π ni α > iωn = λn = ln + n =, ±, ± (6) a α + π( n+ ) i α < Hence the zeos of the chaacteistic deteminant ( iω ) asymptotically simple and sepaated fom each othe by a positive δ thei identical eal pats and imaginay spaced on π. So the poblem (8a) and (8b) inceasingly egulaly. Fom the coollay of the theoem in [8] fo a egula had task gets basis function uk уk = in space iωku L W. k 4. The Numeical Results of the Poblem of Thei Own Ant Plana Oscillations of the Cylinde The esults of calculations ae given in dimensionless system of units in which the value of the shea modulus µ, density cylinde ρ, adius of the cylinde Requal to one. Poisson's atio ν is assumed to by.5. In the case of α = oots of Equation (7) coespond to the natual fequencies of vibation of the fee elastic cylinde. Fo small α solution of the chaacteistic equation was found method of a small paamete, i.e., it was assumed that ω= ω + αω + (7) Then, afte the stand (7) (7) have: ω ω ωα = iαω Y Y cs cs (8) whee ω solution of the poblem on thei own ant plana oscillations of elastic cylinde with α =. In the case whee α cannot be consideed small paamete, a diect solution of the tanscendental chaacteistic equation is solved by Mulle [9]. Fo small α numeical solution coincides with the solution by the method of 636

9 the small paamete, see Figue : dotted line-solution to the poblem of the small paamete method of, the solid line-solution to the poblem method of Muelle; Figue shows the elationship α ; Figue the imaginay pat numbes ae maked on the chats ooms mod. At α the oots of the chaacteistic Equation (7), obtained by numeical method, tend to the natual fequencies of elastic vibations, fixed on the oute suface of the cylinde. In all of these cases, the solution of the spectal poblem (7), in contast to the composite semi-infinite od, exists fo any of the paametes of fiction. Figue and Figue 3 show the dependence of the eal and imaginay pats of the eigenvalues of the spectal poblem (9), (8) the paamete α and n =.4 espectively. The numbes maked on the chats numbe of oscillations in the ascending ode of the eal pat of the Eigen values. In all cases, except fo the fist mode and the second mode (n = 4) fo the eal pats of these cuves have the fom of smooth deceasing steps with a maximum angle of inclination of the tangent to the segment [.85;.] α coesponding imaginay pats have a chaacteistic maximum. With the gowing numbe of fashion maximum value inceases, the value of α, to which he achieved inceases, while emaining less than one. Meaning α, which peaks ae shown in Table and Table. In the zeos hamonic fo any α thee is zeo eigenvalue and α smalle units exist puely imaginay eigenvalues, which tends to infinity when appoaching the unit on the left. Zeo eigenvalue coesponds to the motion of the cylinde as a i- gid body. In the case of n =, fo the fist mode, thee ae citical values α =.9, α =.97 espectively, stating fom which complex eigenvalues becomes puely imaginay, i.e. oscillatoy pocess is eplaced by a peiodic (Figue 4 and Figue 4). At α big data citical values of the imaginay pat of the bifucated, Figue 3. The dependence of the eal pat of thei own values of α (n = 4); Dependence of the imaginay pats of the eigenvalues values of α (n = 4). 637

10 Figue 4. (а) The dependence of the eal pat of thei own values of α (n =, ); Dependence of the imaginay pats of the eigenvalues values of α. Table. The values of the attenuation coefficient depending on the numbe of hamonics. Model numbe The ooms of hamonics Table. The values of the maximum oscillation damping depending on the numbe of hamonics. The ooms of hamonics Mode numbe

11 with one banch tend to zeo α tends to infinity, and the second inceases indefinitely. (See Figue 4, the numbes maked on the chats numbeed hamonics.) Fo n = 3 also exists a citical value α 3 =.9, banch fom which the fist and second modes ae meged into one, and when minimum Eigen values of becomes ed. Nea the citical point at α < α3 the imaginay pat of the fist two events thee is a maximum in excess of the maximum of the othe modes consideed. In addition, the segment α [.9864;.99] discoveed puely imaginay oots, depending on which α shown in Figue 5. A simila case is obtained fo n = 4. At the citical value α 4 =.95 banches of the second and thid modes ae meged, while α > α4 second eigenvalues becomes a multiple. Nea the citical point at α < α4 the imaginay pat of the second mode has a maximum exceeding the maximum of the est of the consideed modes. On the inteval α [.99;.995] also found puely imaginay oots, whose dependence on α. Thus fo all n thee is consideed imaginay banch of the natual fequencies of the paamete α, which in the vicinity of the unit is boken continuity. Figue 4. (а) The dependence of the eal pat of thei own values of α (n =, ); Dependence of the imaginay pats of the eigenvalues values of α. 639

12 Figue 5. The dependence of the imaginay pat of its own values of α (n = 4). 5. Numeical Solution of the Poblem on Its Own Plane Oscillations of a Cylinde Solution of the esulting task was caied out by sepaation of vaiables and note to the solution of the tanscendental equation. All esults of the calculations ae given in dimensionless system of units in which the value of the shea modulusµ, density cylinde ρ, adius of the cylinde Rconsideed to be equal to unity. Poisson s atio ν is assumed to be.5. In case α = и α = the oots of the natual vibation fequencies of the fee elastic cylinde. At α =, α o α =, α oots, obtained by numeical method tend to own oscillating elastic fixed to the oute suface of the cylinde. In all of these cases, the solution of the spectal poblem, unlike composite semi-infinite od, exists fo any of the paametes of fiction, as in the case ant plana cylinde oscillation. Figue 6 and Figue 7 show the dependence of the eal and imaginay pats of the eigen values of the spectal poblem of the paamete α ( α = ). Figue 8 and Figue 9 α ( α = ) n = to 3, espectively. The numbes in the gaphs denote the eigenvalues in ascending ode of thei eal pats. Fo n = the poblem is divided into two independent tasks. In this case, as in the case ant plana cylinde oscillation, depending on the eal pats of the eigen values have the fom of smooth deceasing steps with a maximum angle of inclination of the tangent in the case of adial oscillations in the inteval α [.6;.8] (Figue 6), tosions in the inteval α [.9;.] (Figue 8). In these segments coesponding imaginay pats have a chaacteistic maximum Imω max at α = αm ( α = ) (Figue 6), α = αm ( α = ) (Figue 8). As in the case anticline, with inceasing numbes of hamonics inceases the maximum value, except fo the fist adial oot (see Table 3). It diffes fom the est of the eigen values of the fact that the eal pat vanishes that coesponds to the motion of the cylinde as a igid body. Fo all the above cases, fo n >, thee is a clea sepaation of the oots into two types. The diffeences between these types of oots appea as a chaacte of the dependence of the eigen values of the paametes of the extenal fiction, and in the value of the fom. Fo example, fo the fist oot of the fist hamonic- twist, the maximum value of the eal pat of the adial component of the voltage wavefom is about thee times smalle than the value of the eal pat component foms tosion stesses. Fo the second oot of the fist hamonic- adial the maximum value of the eal pat of the adial component of its own fom of stess thee times moe than the maximum tosion component (see Figue, the fist oot; Figue, the second). Actual own foms pat of the defomation, on the edge of the cylinde, chaacteized by two times (see Figue ). The imaginay pats of thei own foms of valid ode of magnitude smalle and do not have such a ponounced diffeence. If you change depending on the eal pats of the eigen values of the fist type- adial the paamete of fiction have the fom of smooth deceasing steps with a maximum angle of inclination of the tangent to the ange α [.6;.8] except fo the fist mode of the second, thid, fouth hamonics. Depending on the eal pats of the eigenvalues of the second type tosion have in this case the fom is close to the line (see Figue 7). In case of change α. Depending on the eal pats tensional eigenvalues have the fom of deceasing levels of smoothed with a maximum angle of inclination of the tangent to the ange α [.9;.], with the exception of the second mode of the second, thid, fouth hamonics; a dependence of the eal pats of the adial eigen val- 64

13 Figue 6. The dependence of the eal pat of the eigenvalues of α; The dependence of the imaginay pat of the eigenvalues of α. 64

14 Figue 7. The dependence of the eal pat of the eigenvalues of α; The dependence of the imaginay pat of the eigenvalues of α. Figue 8. The dependence of the imaginay pat of the eigenvalues of α; The dependence of the imaginay pat of the eigenvalues of α. 64

15 Figue 9. The dependence of the eal pat of the eigenvalues of α (n = 3); The dependence of the imaginay pat of the eigenvalues of α. 643

16 Figue. The dependence of the eal pat of the eigenvalues of α; The dependence of the imaginay pat of the eigenvalues of α. Figue. The dependence of the eal pat of the eigenvalues of α. Table 3. The values of the maximum oscillation damping depending on the mode numbe. Mode numbe Imωmax α Imω α M max M ues-lose to a linea fom (see Figue 8 and Figue 9). Actual pat allocated in both cases the eigenvalues coesponding to the above defined anges, eaches cetain values and then deceases to zeo. When changing α imaginay pat of the adial eigenvalues have a chaacteistic maximum in the inteval α [.6;.8], the value of which is ten times the value of the imaginay pats of the tensional eigenvalues on the whole inteval unde consideation (i.e., α [ ; + ], α = ) (see Figue 7). When changing α behave similaly imaginay pat of the tensional eigenvalues, eaching a maximum in the inteval α [.9;.] (see Figue 8 and Figue 9). The imaginay pats of the selected eigenvalues at the vanishing eal pat of the bifucated, one ushes to the oot + (see Figue 8 and Figue 9). The imaginay pats of the selected eigen values at the vanishing eal pat of the bifucated, one ushes to the oot ( Imω M ) calculated as n inceases and shifts to the ight along the axis α, split in two when α = α (see Table 4). Unlike the case of Ante plane p 644

17 Table 4. The values of the maximum oscillation damping depending on the numbe of hamonics. Values Im M Consideing the fist mode Consideing the second mode ω α α = Im p M ω α ( α = ) p vibations of the cylinde thee is no expession of the gowth of the maximum values with inceasing mode and shift it to the axis α. 6. Conclusions Fo all the cases consideed flat fluctuations atn >, thee is a clea sepaation of the oots into two types. The diffeences between these types of oots appea as a chaacte of the dependence of the eigen values of the paametes of the extenal fiction, and in the value of the fom. Fo example, fo the fist oot of the fist hamonic- twist, the maximum value of the eal pat of the adial component of the voltage wavefom is about thee times smalle than the value of the eal pat component foms tensional stesses. The imaginay pats of thei own foms of valid ode of magnitude smalle and do not have such a ponounced diffeence. When changing α depending on the eal pats of the eigenvalues of the fist type adial the paamete of fiction have the fom of smooth deceasing steps with a maximum angle of inclination of the tangent to the ange α [.6;.8] except fo the fist mode of the second, thid, fouth hamonics. Depending on the eal pats of the eigen values of the second type tosion have in this case the fom is close to linea. Fo all cases consideed anti plane oscillations n thee imaginay banch of the natual fequencies of the paamete α, which in the vicinity of the unit is boken continuity. Refeences [] Rashidov, T.R. (973) Dynamic Theoy of Seismic Stability of Complex Systems of Undegound Stuctues. Tashkent, 8 p. [] Rashidov, T.R., Doman, I.J. and Ishankhodjaev, A.A. (975) Seismic Stability of Tunnel Constuction of Subways. Moscow, p. [3] Guz, A.N. and Golovchan, V.T. (97) Diffaction of Elastic Waves in Multiply Bodies. Kiev, 54 p. [4] Kubenko, V.D. (979) Unsteady Inteaction of Stuctual Elements with the Envionment. Kiev, 83 p. [5] Safaov, I.I. (99) Oscillations and Waves in Dissipative Media and Nedoodnyhkonstuktsiyah. Tashkent, 5 with. [6] Bazaov, M.B., Safaov, I.I. and Shokin, Y.M. (996) Numeical Modeling of Vibations Dissipative Inhomogeneous and Homogeneous Mechanical Systems. Sibeian Banch, Russian Academy of Sciences, Novosibisk, 89 p. [7] Safaov, I.I., Chat, Z.I., Rakhmonov, K.K. and Umaov, A.O. (3) Foced to Steady-State Oscillations of Cylindical Bodies with an Extenal Bounday Fiction [8] Neumak, M.A. (968) Linea Diffeential Opeatoy. Moscow, 58, p. [9] Safaov, I.I., Teshaev, M.Kh. and Boltaev, Z.I. () Wave Pocesses in the Mechanical Waveguide. LAP LAMBERT Academic Publishing, Gemany, 7 p. 645