Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 46, 1, pp. 21-40, Warsaw 2008 DYNAMICS OF A THICK-WALLED SPHERICAL CASING LOADED WITH A TIME DEPENDING INTERNAL PRESSURE Edward Włodarczyk Military University of Technology, Faculty of Mechatronics, Warsaw, Poland e-mail: edward.wlodarczyk@wat.edu.pl Mariusz Zielenkiewicz Military Institute of Armament Technology, Zielonka, Poland e-mail: m.zielenkiewicz@chello.pl Theproblemof radial vibrationof a thick-walled spherical casing, loaded with an internal pressure, which is a time function, is studied.We assu- med that the material of the casing is incompressible. Furthermore, the linear elasticity theory is applied in the considerations.Bymeans of the- se simplifications, an analytical solution of dynamics of the thick-walled spherical casing loaded with the internal pressure has been obtained. This solution may be used for estimation of strength of spherical balli- stic casings which are applied to explosive driven specimen in ring tests. Moreover, the results of this paper contribute to the theory of vibration of continuous engineering systems. Key words: dynamics, thick-walled spherical casing, forced vibrations, internal surge-pressure, incompressible material 1. Introduction One of experimental methods on investigating mechanical properties of dyna- mically loadedmaterials is the ring testmethod (Włodarczyk andJaniszewski, 2004;Włodarczyk et al., 2005b). In thismethod, a thin-walled ringmadeof the testedmaterial is driven by detonation products of HighExplosive (HE) or by a high energy impulse of an electromagnetic field, and the radial displacement of the ring or velocity of its expansion is recorded. 22 E. Włodarczyk, M. Zielenkiewicz Thick-walled metal casings of cylindrical or spherical symmetry are used to eliminate the influence of perturbations caused byHEdetonation products. The thin-walled ring is placed on the external surface of the casing that is fully or partially filledwithHE.After detonation, the compression shockwave propagates in the casing wall, which reflecting from the ring contact surface, drives it up to high velocity. Reaction of high explosive detonation products with various geometry casings takes place also in artillery shells, grenades and bombs as well as in barrels of projectiling systems (Walters and Zukas, 1989; Włodarczyk et al., 2004). Theseproblemsare the subject of various theoretical and experimental researches. Radial vibrations of a thick-walled pipe forced by internal surge-pressure uniformly distributed along its length, was studied in Włodarczyk et al. (2005a). The material of the pipe was approximated by an incompressible linearly-elastic medium. For this kind ofmaterial, a closed-form analytical so- lution to the problem of dynamics was obtained. It was found that the pipe made of an incompressible material, loaded by an internal pressure, responds like a system with one degree of freedom. For example, a pipe loaded with a rapidly applied and constant internal pressure vibrates with a constant angu- lar frequency ω0 around the state determined through the static solution to Lame’s problem. In this paper, an analogous problem of a thick-walled spherical casing has been considered. The comparison of analytical results determining dynamical states of both kinds of objects has significant practical meaning and is an additional contribution of knowledge to the theory of vibrations of continuous media. 2. Formulation of the problem Dynamic states of the radial displacement and of the stress as well as strain in a spherical casing loaded by an internal pressure p(t) have been determined. Let a and b denote the internal and external radii of the casing. The problem has been solved in the spherical system of Lagrangian coordinates r,φ,θ. Taking into account spherical symmetry, the problem can be assumed as a spatially one-dimensional boundary value problem. Therefore, the states of stress and strain in the material of the casing can be represented by the following components: σr – radial stress σϕ =σθ – tangential stresses Dynamics of a thick-walled spherical casing... 23 εr – radial strain εϕ = εθ – tangential strains. The rest of components of the stress and strain tensors equal to zero in this coordinate system. The problem has been solved according to the linear elasticity theory. Therefore, the following relations can be written (Nowacki, 1970) εr(r,t)= ∂u(r,t) ∂r εϕ(r,t) = εθ(r,t) = u(r,t) r (2.1) σr(r,t)−σϕ(r,t)= 2µ(εr −εϕ) where u and r denote the radial displacement of an infinitesimal element of the casing and its Lagrangian coordinate, while µ denote Lame’s constant µ= E 2(1+ν) (2.2) The symbols E and ν denote Young’s modulus and Poisson’s ratio. Using theLagrangian coordinate r for the infinitesimal element and taking into account the spherical symmetry and themass conservation law, it can be written (r+u)2 ( 1+ ∂u ∂r ) = ρ0 ρ r2 (2.3) where symbols ρ0 and ρ denote the initial and current density of thematerial of the casing. For metals exposed to pressures of the order of a few thousands MPa, it can be assumed that ρ≈ ρ0 = const. This simplification has been discussed in Włodarczyk and Janiszewski (2004). The error caused by this is less than a few percents. Taking into account this assumption, for small strains εϕεr = u r ∂u ∂r ≈ 0 ε2ϕ = (u r )2 ≈ 0 equation (2.3) can be reduced to the following form ∂u ∂r +2 u r =0 (2.4) Using the Lagrangian coordinate r to formulate the condition of dynamic equilibrium, the equation of motion of the infinitesimal element, after trans- formation, can be written as ρ0 ∂2u ∂t2 = ( 1+ u r )2∂σr ∂r +2 ( 1+ u r )( 1+ ∂u ∂r )σr−σϕ r (2.5) 24 E. Włodarczyk, M. Zielenkiewicz For small strains, and neglecting small quantities of higher orders, equation (2.5) can be reduced to the following form ∂σr ∂r +2 σr −σϕ r = ρ0 ∂2u ∂t2 (2.6) The linearized system of equations (2.4) and (2.6) has been solved for the following boundary conditions σr(a,t)=−p(t) for r= a σr(b,t)≡ 0 for r= b u(r,0)= 0 and υ(r,0)= ∂u ∂t ∣ ∣ ∣ ∣ t=0 =0 (2.7) The structure of the analytical solution to the above formulated problem has been presented below. 3. Analytical solution to the problem The general solution of equation (2.4) has the following form u(r,t) = C(t) r2 (3.1) where C(t) denotes a continuous and twice differentiable time function. From expressions (2.1) and (3.1), we have εr(r,t)=−2 C(t) r3 εϕ(r,t) = εθ(r,t) = C(t) r3 (3.2) σr(r,t)−σϕ(r,t) =−6µ C(t) r3 Upon substitution of expressions (3.1) and (3.2)3 into equation of motion (2.6) and integration in respect to r, the following expression has been obta- ined σr(r,t) =−4µ C(t) r3 −ρ0 C̈(t) r +A(t) (3.3) where C̈(t)= d2C(t)/dt2, and A(t) denotes an arbitrary time function. From boundary condition (2.7)2 and solution (3.3), we obtain A(t)= 4µ C(t) b3 +ρ0 C̈(t) b (3.4) Dynamics of a thick-walled spherical casing... 25 Finally, after substitution of expression (3.4) into equation (3.3) and simple transformations, the radial stress σr(r,t) can be expressed by the following formula σr(r,t)= ρ0 r− b br C̈(t)+4µ r3− b3 b3r3 C(t) (3.5) In turn, after substitution of expression (3.5) into boundary condition (2.7)1 and transformation, we have C̈(t)+4 µ ρ0 b2+ab+a2 a2b2 C(t)= ab ρ0(b−a) p(t) (3.6) Let us introduce the following symbols c= √ µ ρ0 β= b a (3.7) ω20 =4 µ ρ0 b2+ab+a2 a2b2 =4 (c a )2β2+β+1 β2 where c denotes the propagation velocity of a transverse wave in thematerial of the casing, and ω0 is the angular frequency of its free vibrations. It has been found that, similarly to the incompressible pipe (Włodarczyk et al., 2005a), the thick-walled spherical casing has only one angular frequency (pulsatance) and responds like a systemwith one degree of freedom. Taking into account expressions (3.7) and (2.7)3, the function C(t) can be determined by the following equation C̈(t)+ω20C(t)= a ρ0 β β−1 p(t) (3.8) with the homogeneous initial conditions C(0)=0 Ċ(t)= dC(t) dt ∣ ∣ ∣ ∣ t=0 =0 (3.9) The solution to equation (3.8) with initial conditions (3.9) has the following form C(t)= a ρ0ω0 β β−1 t ∫ 0 p(τ)sin[ω0(t− τ)] dτ (3.10) In turn, from equation (3.8), we obtain C̈(t)= a ρ0 β β−1 p(t)−ω20C(t) (3.11) 26 E. Włodarczyk, M. Zielenkiewicz As it can be seen, the function C(t) is determined by means of formula (3.10). In turn, the function C(t) fully determines all unknown quantities of the problem, namely u(r,t) = C(t) r2 υ(r,t) = ∂u ∂t = Ċ(t) r2 Ċ(t)= dC(t) dt εr(r,t) = ∂u ∂r =−2 C(t) r3 εϕ(r,t)= εθ(r,t)= u r = C(t) r3 (3.12) σr(r,t)= 1 β−1 ( 1− b r ) p(t)+4 µ b3 [ 1− (b r )3 − b2+ab+a2 a2 ( 1− b r )] C(t) σϕ(r,t) = 1 β−1 ( 1− b r ) p(t)+2 µ b3 [ 2+ (b r )3 −2 b2+ab+a2 a2 ( 1− b r )] C(t) If the pressure inside of the casing is generated statically, i.e.: p(t)= ps = = const, then C(t)=Cs = const, C̈(t)= 0, and taking into account expres- sion (3.5), we have σr(r,t) =σrs(r)=−4 µ b3 b3−r3 b3+r3 Cs (3.13) Subsequently, from the boundary condition σrs(a) = −ps and expression (3.13), we obtain Cs = 1 4 ps µ a3b3 b3−a3 = 1 4 ps µ β3 β3−1 a3 (3.14) Through replacing the function C(t) with the symbol Cs in expressions (3.12)1,3−5, the unknown quantities of the problem can be determined for the static load. 4. Dynamic state of parameters of the spherical casing loaded with an internal surge pressure p= const In order to simplify quantitative analysis of the particular parameters: displa- cement, velocity, strains and stresses, the following dimensionless quantities have been introduced Dynamics of a thick-walled spherical casing... 27 ξ= r a η= t T0 β= b a U = u a V = υ c C = C a3 F = µ p Sr = σr p Sϕ = σϕ p Us = us a Cs = Cs a3 Srs = σrs ps Sϕs = σϕs ps ω0 = ω0 (c/a) (4.1) where T0 denotes the natural vibration period of the casing T0 = 2π ω0 (4.2) After substitution p(τ) = p = const into expression (3.10), integration within the range 0−t and taking into account dimensionless quantities (4.1), we obtain C(t)= 1 4F β3 β3−1 (1− cos2πη)=Cs(1− cos2πη) (4.3) Cs = 1 4F β3 β3−1 The remaining quantities determining the mechanical state of the casing, ac- cording to expressions (3.12), (3.13) and (4.3), can be written as follows: — for the dynamic load U(ξ,η)= Cs ξ2 (1−cos2πη) (4.4) V (ξ,η) = 1 2F β2 √ β2+β+1 β3−1 1 ξ2 sin2πη (4.5) εϕ(ξ,η) =− 1 2 εr(ξ,η) = Cs ξ3 (1− cos2πη) (4.6) Sr(ξ,η) =− 1 β3−1 [(β ξ )3 −1 ] +Ar(ξ)cos2πη (4.7) Ar(ξ)= 1 β3−1 [(β ξ )3 − (β2+β+1) (β ξ −1 ) −1 ] (4.8) Sϕ(ξ,η)= 1 2(β3−1) [ 2+ (β ξ )3] −Aϕ(ξ)cos2πη (4.9) Aϕ(ξ)= 1 2(β3−1) [(β ξ )3 +2(β2+β+1) (β ξ −1 ) +2 ] (4.10) 28 E. Włodarczyk, M. Zielenkiewicz ω0 =2 √ β2+β+1 β (4.11) — for the static load Us(ξ)= 1 4F β3 β3−1 1 ξ2 = Cs ξ2 (4.12) εϕs(ξ)=− 1 2 εrs(ξ)= Cs ξ3 (4.13) Srs(ξ)=− 1 β3−1 [(β ξ )3 −1 ] (4.14) Sϕs(ξ)= 1 2(β3−1) [ 2+ (β ξ )3] (4.15) For comparison of the results, analogous relations obtained for the cylin- drical casing (Włodarczyk et al., 2005a) can be quoted: — for the dynamic load Cs = a2b2 b2−a2 p 2µ Cs = C a2 = 1 2F β2 β2−1 (4.16) U(ξ,η) = 1 2F β2 β2−1 1 ξ (1− cos2πη)= Cs ξ (1− cos2πη) (4.17) V (ξ,η) = 1 F β2 β2−1 √ β2−1 2lnβ 1 ξ sin2πη (4.18) εϕ(ξ,η)=−εr(ξ,η) = Cs ξ2 (1− cos2πη) (4.19) Sr(ξ,η) =− 1 β2−1 [(β ξ )2 −1 ] +Ar(ξ)cos2πη (4.20) Ar(ξ)= 1 β2−1 [(β ξ )2 −1 ) − ln(β/ξ) lnβ (4.21) Sϕ(ξ,η) = 1 β2−1 [(β ξ )2 +1 ] −Aϕ(ξ)cos2πη (4.22) Aϕ(ξ)= 1 β2−1 [(β ξ )2 +1 ] + ln(β/ξ) lnβ (4.23) ω0 = √ 2(β2−1) β2 lnβ (4.24) Dynamics of a thick-walled spherical casing... 29 — for the static load Us(ξ)= 1 2F β2 β2−1 1 ξ = Cs ξ (4.25) εϕs(ξ)=−εrs(ξ)= Cs ξ2 (4.26) Srs(ξ)=− 1 β2−1 [(β ξ )2 −1 ] (4.27) Sϕs(ξ)= 1 β2−1 [(β ξ )2 +1 ] (4.28) From the analysis of the relations presented above, it results that the con- sidered casings loaded with the internal surge pressure p = const respond like mechanical systems with one degree of freedom and vibrate radially with the angular frequencydetermined through formulae (4.11) and (4.24). Thedy- namic state of mechanical parameters of the loaded casings oscillates around their static values obtained for the statically generated pressure of the same value, i.e.: p= ps = const. The quantitative analysis of the mechanical parameters of studied casings is presented below. 5. Quantitative analysis of mechanical parameters of casings We assume that the considered casings are made of steel which mechanical characteristics are: density in normal conditions ρ0 =7800kg/m 3, shear mo- dulus µ = 75GPa. The value of pressure has been set to p = 400MPa. For these values we have: F = µ/p = 187.5, c = √ µ/ρ0 = 3100m/s. The rema- ining values of appropriate quantities are to be presented during analysis of particular parameters. According to formulae (4.11) and (4.24), the relative angular frequencies of free vibration of the considered casings are functions of their walls thicknesses, which are characterized by the parameter β. Graphs of these functions are presented in Fig.1. It is seen that the angular frequencies decrease exponentially with the increase of wall thicknesses. This fact is caused by the increase of mass of the casings. It is significant that the angular frequency of the spherical casing is nearly twice greater than theangular frequencyof a cylindrical one. It is caused 30 E. Włodarczyk, M. Zielenkiewicz Fig. 1. Relative angular frequencies of free vibration of casings as functions of the parameter β Fig. 2. Relative radial displacement of selected concentric surfaces of the casings within the range 0¬ η¬ 1 for selected values of the parameter ξ and β=2 by greater stiffness of the spherical casing in comparison with the cylindrical casing of the same wall thickness. The graphs of relative radial displacements of selected concentric surfaces of the casings are presented in Fig.2. They are plotted according to formulae (4.4) and (4.17) for selected values of ξ and β=2. The range 0¬ η¬ 1 con- tains the full free vibration period 0¬ t¬T0. It is significant that the radial displacement of the internal (ξ = 1) and remaining surfaces of casings has a positive value in the whole range. This means that during elastic vibrations forcedwith an internal pressurepulse, their internal diameters donot decrease below the initial values. The graphs depicted in Fig.2 also show that the ra- dial vibration amplitudes of particular concentric surfaces of spherical casings Dynamics of a thick-walled spherical casing... 31 are several times smaller than corresponding amplitudes of cylindrical casings. This indicates that the spherical casing is more rigid than the cylindrical one. Like in the one-degree-of-freedom mechanical system, the dynamic coeffi- cient of loading for both kinds of casings, according to formulae (4.4), (4.12), (4.17) and (4.25), is determined by the following expression Ψ = U(ξ,η) Us(ξ) = 1− cos2πη (5.1) and its maximum value is 2. Fig. 3. Oscillation amplitudes Ar(ξ) of relative radial stresses Sr(ξ,η) in walls of the casings for selected values of their thickness (β=1.2, 1.5, 2, 3 and 4) In turn, the graphs of amplitudes Ar(ξ) along the casing wall thickness (1 ¬ ξ ¬ β) for selected values of the parameter β are depicted in Fig.3. It shows that the functions Ar(ξ) have negative values in the whole range 1¬ ξ¬β, and reach theminimum for ξ= ξe, where: — for the spherical casing ξe =β √ 3 β2+β+1 (5.2) — for the cylindrical casing ξe =β √ 2ln β β2−1 (5.3) 32 E. Włodarczyk, M. Zielenkiewicz Approaching the internal surfaces of casings (ξ → 1), the amplitudes Ar(ξ) decrease and reach zero for ξ=1, according to the boundary condition (Sr(1,η) ≡ −1). The values of functions Ar(ξ) for both kinds of casings are comparable for β < 2. For thicker walls, the differences between functions Ar(ξ) for the cylindrical and spherical casing increase. Analogous graphs of the oscillation amplitudes Aϕ(ξ) of the relative tan- gential stress Sϕ in casingwalls are presented inFig.4. The amplitudes Aϕ(ξ) are decreasing positive functions in the range 1 ¬ ξ ¬ β. They reach their maxima at the internal surfaces of casings (ξ =1), and the maximum values are determined by formulae: — for the spherical casing Aϕmax = 3β3 2(β3−1) (5.4) — for the cylindrical casing Aϕmax = 2β2 β2−1 (5.5) Fig. 4. Oscillation amplitudes of relative radial stresses Sϕ(ξ,η) in walls of the casings for selected values of their thickness (β=1.2, 1.5, 2, 3 and 4) From analysis of formulae (4.7), (4.8), (4.20) and (4.21), it results that the function Sr(ξ,η) can change its sign from negative to positive (tension) on the inside casing wall. For η = 0.5 (the middle of the vibration period), this Dynamics of a thick-walled spherical casing... 33 happens in the spherical casing, when the parameter β characterizing thewall thickness satisfies the following inequality β4+2β3−5β2−6β+5> 0 (5.6) It is satisfied when β > 1.79. This means that in the wall of thickness cha- racterized with β= b/a< 1.79, the radial tensile stress does not appear. The maximum positive value of the function Sr(ξ,0.5) for the spherical casing is reached on the surface determined as follows ξ= ξe(β)=β √ 6 β2+β+1 β> 1.79 (5.7) and amounts Srmax(ξe,0.5)= 1 β−1 (β ξe −1 ) − 2 β3−1 [(β ξe )2 −1 ] (5.8) Analogous relations for the cylindrical casing have the following form ξ= ξe(β)= exp (2(β2+1)lnβ−β2+1 2(β2−1) ) β > 1.87 (5.9) Srmax(ξe,0.5)= ln(β/ξe) lnβ − 2 β2−1 [(β ξe )2 −1 ] Graphs of the function Sr(ξ,η) are presented in Figs.5-7. The graphs presented in Fig.5 for η = 0.25 correspond to the state of static load (cos2π · 0.25 = 0). Values of the function Sr(ξ,η) for η = 0 and η = 0.5 are symmetrically placed relative to the static plot. Graphs of the function Sr(ξ,η) presented in Figs.5 and 6 for both types of casings are qualitatively similar, but slightly differ quantitatively. Absolute values of the function Sr(ξ,η) for the cylindrical casing are mostly greater than for the spherical one. This results from the fact that the displacement and stra- ins for the same load and wall thickness are greater in the cylindrical casing (Fig.2). Analogous graphs of the function Sϕ(ξ,η) are presented in Figs.8-10. It is significant that contrary to the function Sr(ξ,η), the function Sϕ(ξ,η) changes its sign during vibrations for any value of thewall thickness. At the beginning 34 E. Włodarczyk, M. Zielenkiewicz Fig. 5. Relative radial stress Sr(ξ,η) as a function of the parameter ξ for η=0, 0.25 and 0.5 and for β=2 and β=4 of the vibration period, the inertia of the casing wall causes a negative tan- gential stress (compression). Subsequently, the casing wall displaces radially and the sign of tangential stress changes to positive. The value of this stress reaches the maximum on the casing internal surface (ξ =1) in the middle of the period (η = 0.5 – Fig.8). The maximum values of the function Sϕ(ξ,η) for spherical and cylindrical casings are determined by the formulae: Dynamics of a thick-walled spherical casing... 35 Fig. 6. Relative radial stress Sr(ξ,η) as a function of the parameter η for selected concentric surfaces of the wall and for β=2 and β=4 — for the spherical casing Sϕmax = 2β3+1 β3−1 (5.10) — for the cylindrical casing Sϕmax = 3β2+1 β2−1 (5.11) 36 E. Włodarczyk, M. Zielenkiewicz Fig. 7. Spatial graph of the function Sr(ξ,η) for the spherical casing From the analysis of graphs presented in Figs.8-10, it results that, like in the case of the function Sr(ξ,η), plots of the function Sϕ(ξ,η) for spherical and for cylindrical casings are qualitatively similar, but the quantitative diffe- rences are significant. It is seen that for the same diameters, wall thicknesses and internal pressure pulses, the tangential stress is greater in the cylindrical casing. Dynamics of a thick-walled spherical casing... 37 Fig. 8. Relative tangential stress Sϕ(ξ,η) as a function of the parameter ξ for η=0, 0.25 and 0.5 and for β=2 and β=4 6. Final conclusions From the analysis of the considered problem, the following conclusions can be drawn: • Thick-walled casings with cylindrical or spherical symmetry loadedwith an internal pressure pulse vibrate radially with an angular frequency ω0 and behave like one-degree-of-freedom systems. For equal diameters and 38 E. Włodarczyk, M. Zielenkiewicz Fig. 9. Relative tangential stress Sϕ(ξ,η) as a function of the parameter η for selected concentric surfaces of the wall and for β=2 and β=4 wall thicknesses, the pulsatance of free vibrations of the spherical casing is nearly twice greater than that in the cylindrical one. It is caused by greater stiffness of the spherical casing. • Themaximum value of the dynamic coefficient of loading is Ψ =2. • From the analytical solution to the problem, it directly results that the dynamic states of radial displacement u, strains εr, εϕ and stres- ses σr, σϕ generated in the casing walls by the internal pressure p(t) Dynamics of a thick-walled spherical casing... 39 Fig. 10. Spatial graphs of the function Sϕ(ξ,η) for the spherical casing are determined through their material physical properties ρ0, µ, casings geometry a, b and the character of changes of the pressure described by a time function p(t). References 1. Dżygadło Z., Kaliski S., Solarz L., Włodarczyk E., 1966, Vibration and Waves in Solids [in Polish],Warszawa, PWN 40 E. Włodarczyk, M. Zielenkiewicz 2. Nowacki W., 1970,Theory of Elasticity [in Polish],Warszawa, PWN 3. Walters W.P., Zukas J.A., 1989, Fundamentals of Shaped Charges, New York: A.Wiley – Interscience Publication 4. WłodarczykE., Janiszewski J., 2004, Static and dynamic ductility of cop- per and its sinters, J. Tech. Phys., 45, 4, 263-274 5. WłodarczykE., Janiszewski J.,MagierM., 2004,Analysis of axial stress concentration in core of elongated sabot shell duringfire [inPolish],Biul.WAT, LIII, 2/3, 109-132 6. Włodarczyk E., Głodowski Z., Paszkowski R., 2005a, A thick-walled pipe radial vibrations forced by pulsed internal pressure [inPolish],Biul.WAT, LIV, 10, 31-48 7. Włodarczyk E., Janiszewski J., Głodowski Z., 2005b, Dynamical state of stress and strain in thin-walled explosively expanded metal ring [in Polish], Biul. WAT,LIV, 2/3, 109-119 Dynamika grubościennej kulistej osłony obciążonej wewnętrznym ciśnieniem zmiennym w czasie Streszczenie Zbadano problem radialnych drgań grubościennej osłony kulistej obciążonej uda- rowowewnętrznymciśnieniem impulsowym. Założono, żemateriał osłony jest spręży- ście nieściśliwy. Przy takimuproszczeniuuzyskano zamknięte analityczne rozwiązanie zagadnienia dynamiki osłony kulistej w ramach liniowej teorii sprężystości. Okazuje się, że osłonakulistawykonanazmateriałunieściśliwego,podobnie jak rura,obciążona wewnętrznie udarowo, zachowuje się jak układ o jednym stopniu swobody. Częstotli- wośćkołowadrgańwłasnychosłonykulistej jest kilkakrotniewiększa od częstotliwości rury o tej samej średnicy wewnętrznej i grubości ścianki. Przedstawione rozwiązanie możnawykorzystaćdo szacowaniawytrzymałości kulistych osłonbalistycznych stoso- wanychprzywybuchowymnapędzaniu cienkościennychpierścieni używanychwbada- niach dynamicznych właściwości materiałów. Poza tym, przedstawione wyniki badań dają dodatkowywkład wiedzy do teorii drgań technicznych układów ciągłych. Manuscript received March 9, 2007; accepted for print October 3, 2007