A method for solving dynamic problems for cylindrical domains

Transcription

1 Tansactions of NAS of Azebaijan, Issue Mechanics, 35 (7), (016). Seies of Physical-Technical and Mathematical Sciences. A method fo solving dynamic poblems fo cylindical domains N.B. Rassoulova G.R. Mizoyeva Received: / Accepted: Abstact. This aticle exploes the tansient dynamic poblems of the theoy of elasticity fo bodies with cylindical symmety. Poblems of hamonic vibations daw moe attention than non-stationay poblems in dynamic theoy of elasticity. If we conside that in most cases, the souces of excitation of wave popagation in elastic media ae shock o explosive natue, it becomes clea that in pactical applications of dynamic elasticity theoy one has to deal, as a ule, with non stationay poblems. The existing analytical solutions cylindical poblems of dynamic elasticity theoy ae athe few, among these, the most popula is the wok [6], whee an asymptotic solution of the poblem of a cylindes stoke at a fixed igid baie is obtained. The pupose of the pape is to give methods fo constucting the solutions of an elastic poblem of non-stationay dynamics in the geneal fom fo cylindical wave notions. It was stated, fo the sake of simplicity, in the example of a semi-infinite, cicula cylinde, and on the basis of this method, an exact solution fo the initial stages of the pocess Was constucted. It is appopiate to notice that the poposed method makes possible to obtain analytical solutions fo any finite peiod of time and fo any body with an axially symmetic cylindical configuation (in the special case fo layeed cylindical bodies as well). Keywods. wave motion, cicula cylinde, Lame equations, longitudinal impact and integal tansfoms. Mathematics Subject Classification (010): 74B05 N.B. Rassoulova Institute of Mathematics and Mechanics, Baku, Azebaijan asulova@gmail.com G.R. Mizoyeva Institute of Mathematics and Mechanics, Baku, Azebaijan gulna.mizayeva@gmail.com

2 N.B. Rassoulova and G.R. Mizoyeva 69 1 Fomulation of the poblem Impact to the end of a cicula cylinde. The poblem of wave popagation in semi-infinite cicula cylinde of adiusa, unde the action of axial shock foces applied to the end aea of the cylinde. In the cylindical coodinate system z, φ, z elated with the end of the cylinde, the solution of the axially symmetic poblem is elated to the solution of the following system of Lame equations: σ + σ z z + σ σ θθ = ρ u t σ z + σzz z + σz z = ρ u z t (1.1) σ = µε + λe; σ θθ = µε θθ + λe; σ zz = µε zz + λe. σ z = µε z ; e = ε + ε zz + ε θθ ; ε = u ; ε θθ = u ; ε zz = u z ; (1.) ε z = 1 ( u z + u ) z ; ε zθ = ε θ = 0. fo z > 0; 0 a; t 0 Hee u and u z ae adial and axial displacement of paticles, ρ is density. This system joins the initial and bounday conditions; they ae: u = u z = 0 u t = u z t = 0 σ zz = σ 0 ()f(t) u = 0 } } in t = 0. (1.3) in z = 0 (1.4) σ = σ z = 0 in = a; 0 < z < (1.5) (1.1) and (1.) leads to an axially symmetic system of Lame equations: (µ + λ) u + (λ + µ) u z z + µ u (λ + µ) u (λ + µ) + z u = ρ u t (µ + λ) u z + (λ + µ) u z + µ u z (λ + µ) + u z + µ u z = ρ u z t. By applying sin and cos Fouie in the coodinate z, to the functions u and u z espectively and then the Laplace tansfom in t, system (1.1) - (1.5) can be educed to the fom: (λ + µ) u (s) (λ + µ) q u(s) + (λ + µ) 1 + (λ + µ) q u(s) ū (s) + µ 1 ( µq + ρ p ) ū (s) u (c) z (λ + µ) q ū(c) z [ ρ p + (λ + µ) q ] ū (c) z (λ + µ) ūs = 0 = σ 0 f(p) Hee p and q - ae the paametes of Laplace and Fouie tansfoms espectively. Select two new functions by the fomulas: (1.6)

3 70 A method fo solving dynamic poblems fo cylindical domains ū (s) ū (s) z = φ q ψ = qφ 1 ( ψ The system (1.6) afte applying the substitution (1.7) takes the simplest fom: (λ + µ) (B 1 φ) = q(b ψ) q (B ψ) 1 d d (B ψ) d (B ψ) d = σ 0τ(p) µ Hee B 1 and B ae modified Bessel opeatos (of zeo index). B k = d d + 1 ( ) d p d c + q k ) k = 1,. (1.7) (1.8) The second equation of system (1.8) is also a modified Bessel equation with espect to the function (B ψ): B 0 B ψ = σ 0f(p) ; µ hee B 0 = d d + 1 d d q. A bounded solution of the system in the aea 0 a ae the functions: φ = σ 0f(p) qν1 (λ + µ) + A 0I 0 (ν 1 ) p ψ = σ 0f(p) q ν µ + B 0I 0 (ν ) hee ν k = c + q, k = 1, and I 0 ae modified Bessel functions. k The unknown coefficients ae detemined fom the lateal conditions on the suface = a; } σ = 0 σ z = 0 in = a; z > 0. With egad to (1.7) and (1.9), the last elations tun into a system of two linea algebaic equations fo detemining A 0 and B 0 : [ A 0 (λ + µ) I 0 (v 1 a) + λ ] a I 0(v 1 a) λq I 0 (v 1 a) B 0 [ (λ + µ) qi 0 (v a) + λ a qi 0(v a)] λq[ 1 a I 0(v a) + I 0 (v a) ] = λqσ 0f(p) v1 (λ + µ) (1.9) [ A 0 qi 0(v, a) B 0 q I 0(v a) + 1 a I 0(v a) + 1 ] a I 0 (v a) + I 0 (v a) = 0. (1.10) The deteminant of this system is the function {[ ( ) ] c F = µ v 1 c v1 c 1 c q (q + v )I 1 (av )I 0 (v 1 a) + p v 1 c a I 1 (v 1 a)i 1 (v a) + 4q v 1 v I 1 (v 1 a)i 0 (v a) } (1.11) System (1.10) has the following solution: λσ A 0 = 0 q ( q (λ + µ)v1 F + v ) v I 1 (v a) B 0 = λσ 0q v 1 (λ + µ)f I 1(v 1 a) (1.1)

4 N.B. Rassoulova and G.R. Mizoyeva 71 Substituting these expessions in fomulas of common solutions (1.9) the solutions in tansfomations ae completely detemined. We must now etun to the eal coodinates, theefoe, invese tansfomation of the function of the fom (1.9), with taken (1.14) into account should be epoduced. Judging fom expessions (1.1) and (1.11), this opeation is faught with difficulties and to this day only the asymptotic solution of the stated poblem as t is known. It can be shown that the functionf, defined by (1.11), and a pat of the denominato in (1.1), at the values of ealq, has no zeos in the complex plane p, outside of an imaginay axis. On the imaginay axis Rep = 0 by eplacement p = ik the equation F = 0 (1.13) is educed to the well-known Pochhamme equation fo the fequencies of vibations of infinitely long cicula cylinde, when the longitudinal waves popagate in it. This equation was fist obtained and studied in [4], futhe analytical and numeical study of this equation was conducted by a numbe of authos [3, 5]. Fom these studies it follows that equation (1.13) fo evey eal q has on the axis Rep = 0 an infinite numbe of ootsp m = if m (q) (m = ±1, ±...), all of them ae simple and paiwise complex conjugate (p m = p m). But hee we will not deal with finding these oots, we use the method developed in [6] fo the solution of tansient dynamics of a ectangula beam, and in the special case, of a laye. Fo simplicity, we show this method on an example of finding the velocity of paticles of the cental axis, that in tansfomations is expessed by the fomula: u z = σ 0 c v1 c + λσ 0c 1 c 1 [ q ( q + v ) v v 1 F I 1 (v 1 ) I 0 (v 1 ) q v v 1 F I 1 (v 1 a) I 0 (v ) We modify it to the fom: σ u z = 0 v1 (λ + µ) + λσ 0q [ q + v I 1 (av ) (λ + µ) v 3 I 0 (v a) 1 v1 1 I 1 (v 1 a) v 1 I 0 (v 1 a) 1 ] I 0 (v ) v I 0 (v a) I 0 (v 1 ) I 0 (v 1 a) ] ; (1.14) v4 F (1.15) hee {[ ( ) F c = v 1 v1 c 1 + p v 1 c a ] (q c c q + v ) I 1 (av ) I 0 (av ) + } I 1 (v 1 a) I 1 (v a) I 0 (v 1 a) I 0 (v a) + I 4q v 1 v 1 (v 1 a) I 0 (v 1 a) Following [], the function v4 expands in seies: F v 4 F = n=1 a n 1 v n (1.16) Each membe of this seies is the Laplace and Fouie tansfoms. If we ae satisfied with the solutions fo small values of t, it is sufficient to etain the fist few tems: v 4 F = 1 [ v 1 + c ( ) ] [q c av v 1 + c c + c c a. The expessions in squae backets of Fomula (1.15) as shown is a meomophic function in the half plane Rep > 0, disappeaing at infinity as p = R n and also having simple poles at the points of the axis Rep = 0. Thus, we can apply to it the second expansion theoem [1], accoding to which f(t) = p k es pk f (p)e pt

5 7 A method fo solving dynamic poblems fo cylindical domains hee f (p) = L(f(t) and it possesses above popeties. Fist note that fo small values t, in (1.15) by the fist tem dominates: σ 0 1 λ + µ v1 LF 0 = 0 π ) σ 0 c 1 (t λ + µ H zc1 Since the anothe membe, by the cude estimate, in time t by two degees is a less quantity, 1) q + v v 3 I 1 (v a) L c a sin(qc t) c I 0 (av ) = k=0 a u z t O(t) + O(t) fo small values of t. Now we give the invese functions of images that ae the pats of the solutions (1.15): ( ) ) q α k a sin (c q + α k a t ) 1 v i I 0 (v i ) L I 0 (v i a) = 3) 1 I 1 (v 1 a) L v 1 I 0 (v 1 a) = αk ( ) sin (c c i q sin(qc it) J 0 α i k a k=0 J 1 (α k ) c 1 a 4) 1 c J 0 (c qt) v = k=0 sin (c 1 q + α k a q + α k a 5) 1 πc t v 3 = qγ (3/) I 1(c qt) hee α k ae the zeos of the function J 0 (x) and v i = ) p c i t q + α k a α k q + α k a + q ; i = 1,. q + α k ) a t ; (1.17) Fig.1. Distibution of longitudinal speed u z along cental axis at moment t 1 = a c 1.

6 N.B. Rassoulova and G.R. Mizoyeva 73 The solution (1.6) is a plane longitudinal wave that stats fom the end aea and popagates at a ate of c 1. To the moment t 0 = a c 1 the solution on the axis oz will be pesented by this plane wave, since the diffaction waves stat fom the side suface = a with speeds c 1 and c, each the cental axis with the moments t 0 = a c 1 and t 1 = a c espectively. The calculation of u z the following values of paamete and subsequent time intevals: t 1 = a c 1 ; t = 1, 5a ; t 3 = a ; t 4 = c 1 c 1 c 1 = c = m/c λ = µ = 1, kγ/m σ 0 = 10 6 kγ/m, 5a ; t 5 = 3a and etc. Fig.1 Fig. 5. c 1 c 1 Fig.. Distibution of longitudinal speed u z along cental axis at moment t 1 = 1.5a c 1. Fig.3. Distibution of longitudinal speed u z along cental axis at moment t 1 = a c 1

7 74 A method fo solving dynamic poblems fo cylindical domains Fig.4. Distibution of longitudinal speed u z along cental axis at moment t 1 =.5 a c 1. Fig.5. Distibution of longitudinal speed u z along cental axis at moment t 1 = 3 a c 1. These esults ae nealy identical with the esults obtained in the [6], and these solutions also satisfy theoetical expectations, and agee well with the expeimental esults. Conclusions 1. Fo the integation of an axially symmetic system of Lame equations, a new method has been applied as a esult of which, fistly, this system takes the simple fom, and secondly, the bounday defined function at the end z = 0, appeas in the ight pats of the equations of motion as a fee membe. Following fact allows you to quickly build solutions acoss the aea of the cylinde, fo a sufficiently boad class of functions, appeaing in the bounday conditions at the end z = 0.. Solutions to these poblems moe complex configuation cylindical solids (hollow, layeed cylinde, etc.), appaently, it can be easily built in tansfomations. Then, fo the epesentation of solutions in

8 N.B. Rassoulova and G.R. Mizoyeva 75 eal domains (as above the case shown with ound cylinde) tansfomations found necessay to be epesented as a poduct of two membes; the fist of them gives analytic tansfomation to the eal vaiables, and the second expands into seies of types (1.6), which can be conveted easily tem wise. It should be noted that with inceasing complexity of the aea in the pocess of building a complete solution only inceases the difficulty of algebaic natue. Refeences 1. Ditkin V.A., Pudnikov A.P. Integal tansfoms and the opeational calculus. Moscow, fizmatgiz, 54 (1961).. Doetsch G. Handbuch de Laplace-Tansfomation. Bikhause Basel,, 64 (1950). 3. Kolsky H. The popagation of longitudinal elastic waves along cylindical bas. Philos. Mag, 45 (36), (1954). 4. Poshhamme L. Ube Die., Fotflanzungsgeschwindigkeiten kleine schwingungen in einem unbegenzten izotopen Keiszylinde.- J. Reine und angew. Math., 81 (4), (1876). 5. Rassoulova. N. B. On dynamics of ba of ectangula coss section. Tansactions of the ASME,68, (001). 6. Skalak R. Longitudinal impact of a semi-infinite cicula elastic ba. J. Appl. Mech., 1957, 4 (1), (1957).