Journal of Physcs: Conference Seres PAPER OPEN ACCESS Forced vbratons of a two-layered shell n the case of vscous resstance To cte ths artcle: L A Aghalovyan and L G Ghulghazaryan 08 J. Phys.: Conf. Ser. 99 000 Related content - STARS HAVING SHELL SPECTRA Paul W. Merrll - STARS HAVING SHELL SPECTRA Paul W. Merrll - FDTD Investgaton on Electromagnetc Scatterng from Two-Layered Rough Surfaces under UPML Absorbng Condton L Juan Guo L-Xn and Zeng Hao Vew the artcle onlne for updates and enhancements. Ths content was downloaded from IP address 48.5.3.83 on 3/07/08 at 8:59
TPCM-07 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres 34567890 99 (08 000 do :0.088/74-6596/99//000 Forced vbratons of a two-layered shell n the case of vscous resstance L A Aghalovyan and L G Ghulghazaryan Insttute of Mechancs of the Natonal Academy of Scences of Armena Yerevan Armena Kh. Abovyan Armenan State Pedagogcal Unversty Yerevan Armena E-mal: lagal@sc.am lusna@mal.ru Abstract. Forced vbratons of a two-layered orthotropc shell are studed n the case of vscous resstance n the lower layer of the shell. Two versons of spatal boundary condtons on the upper surface of the shell are posed and the dsplacement vector s gven on the lower surface. An asymptotc method s used to solve the correspondng dynamc equatons and relatons of the three-dmensonal problem of elastcty. The ampltudes of the forced vbratons are determned and the resonance condtons are establshed.. Basc equatons and statement of the boundary-value problems Forced vbratons of the two-layered orthotropc shell Ω = {α β γ; α β Ω 0 h γ h } are consdered n the case of vscous resstance n the lower layer. Here Ω 0 s the surface of contact between the layers α and β are the curvature lnes of the surface of contact and γ s the rectlnear axs drected perpendcularly to the surface of contact between the layers. In the chosen three-dmensonal orthogonal coordnate system t s requred to fnd nonzero solutons of dynamc equatons of elastcty whch satsfy the boundary condtons of the second or mxed boundary-value problems of elastcty on the surfaces of the shell. To smplfy the computatons the asymmetrc stress tensor components τ j [ are denoted by γ = +γ/r ( = where R and R are the basc rad of curvature of the surface of contact between the layers. We have: the equatons of moton (I (Bτ α αα k β τ (I ββ + (II (Bτ αα k β τ (II α ββ + k γ γ U (II t (I (Aτ β βα + k ατ (I αβ + γ τ (I αγ γ (II (Aτ β βα + k ατ (II αβ + γ (I τ + αγ τ (II αγ γ R (II τ + αγ = ρ II γ γ U (II t (A B; α β; R R ; U V = ρ I γ γ U (I t R Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3.0 lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s and the ttle of the work journal ctaton and DOI. Publshed under lcence by IOP Publshng Ltd
TPCM-07 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres 34567890 99 (08 000 do :0.088/74-6596/99//000 τ (I γγ γ τ (II γγ γ ( (I τ αα (I ββ R + τ R ( (II τ αα + τ R + k β τ (II αγ + A (II ββ R + k α τ (II βγ + A τ (I αγ α τ (II αγ α + B τ (I βγ β + B k W (II γ γ t τ (II βγ β + k βτ (I αγ + k α τ (I βγ = ρi γ γ W (I t = ρ II γ γ W (II t where γ τ (j αβ = γ τ (j βα j = I II (symmetry condtons; the equatons of state (relatons of elastcty ( U (j γ A α + k αv (j + W (j = γ a (j R τ αα (j + γ a (j τ (j ββ + a(j 3 τ γγ (j (A B; α β; R R ; U V ; a (j a(j ; a(j 3 a(j 3 W (j γ γ γ U (j β γ ( B = γ a (j 3 τ αα (j + γ a (j 3 τ (j ββ + a(j 33 τ γγ (j k βv (j U (j γ γ γ γ U (j + R A γ W (j α ( V (j + γ A α k αu (j = γ a (j τ (j αγ (A B; α β; R R ; U V ; a (j a(j 44 = γ a (j 66 τ (j αβ j = I II where k α k β are geodesc curvatures A B are coeffcents of the frst quadratc form ρ (j are the layer denstes a (j k are elastcty constants and j s the layer number. On the face surface γ = h the boundary condtons are or and on the surface γ = h the condtons are τ I αγ(h = 0 τ I βγ (h = 0 τ I γγ(h = 0 ( U I (h = 0 V I (h = 0 W I (h = 0 (3 U II ( h =u (α β sn(ωt V II ( h =v (α β sn(ωt W II ( h =w (α β sn(ωt. (4 On the surface of the contact between the layers the condtons of complete contact must be fulflled: ταγ(γ I = 0 = ταγ(γ II = 0 τβγ I II (γ = 0 = τβγ (γ = 0 τ γγ(γ I = 0 = τγγ(γ II = 0 (5 U I (γ = 0 = U II (γ = 0 V I (γ = 0 = V II (γ = 0 W I (γ = 0 = W II (γ = 0. (6 The condtons on the lateral surface are not specfed and n the problems of ths class they mply orgnaton of a boundary layer [ 3.. Soluton of the outer problem To solve above-formulated boundary-value problems ( (6 we pass to dmensonless coordnates and dsplacements n equatons ( by the formulas α = Rξ β = Rη γ = εrζ = hζ U = Ru V = Rv W = Rw h = max{h h } h R (
TPCM-07 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres 34567890 99 (08 000 do :0.088/74-6596/99//000 where R s the characterstc dmenson of the shell (the smallest value of the rad of curvature and the lnear dmensons of the surface of contact between the layers and ε = h/r s a small parameter. The soluton of the transformed equatons [4 5 s sought n the form Q (j αβ (x y z t = Q(j (ξ η ζ sn(ωt + Q(j (ξ η ζ cos(ωt (α β γ j = I II (7 where Q (j αβ are any values of stresses and dsplacements and Ω s the frequency of the external forcng effect. As a result a system of equatons for Q (j Q(j sngularly perturbed by the small parameter ε s obtaned and ts soluton s the sum of solutons of the outer problem and boundary layers: I = Q out + R b [ 3 6. The solutons of the outer problem are sought as the asymptotc expanson [7 τ out(j mk ( u out(j (ξ η ζ = ε +s τ (js mk (ξ η ζ m k = 3 s = 0 N j = I II = (ξ η ζ v out(j (ξ η ζ w out(j (ξ η ζ =ε s( u (js (ξ η ζ v (js (ξ η ζ w (js (ξ η ζ (8. Here and further s = 0 N means that by the mute (repeated summaton ndex s ranges between 0 and N. The soluton of the problem must satsfy condtons ( (6. If the above-mentoned structure of the general soluton s known then to determne τ (js mk u(js and (u v w n the outer problem from condtons ( (4 we must satsfy the followng boundary condtons at ζ = ζ (ζ = h /h: or and the condtons at ζ = ζ (ζ = h /h where τ (Is 3 (ζ = τ (Is 3b (ζ = ζ (3 3 33 = (9 u (Is (ζ = ū (Is b (ζ = ζ (u v w = (0 u (IIs = u (IIs (ξ η (u v w ( u (II0 = u R u (II0 = 0 τ (I0 m3b = 0 s 0 (u v w m = 3 =. ū(i0 b = 0 u (IIs = ū (IIs b (ζ = ζ ( The values τ (Is 3b ū(is b ū (IIs b and (u v w are determned after constructng the boundary layer soluton. Standardly substtutng expressons (8 nto the obtaned sngularly perturbed system of 3
TPCM-07 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres 34567890 99 (08 000 do :0.088/74-6596/99//000 equatons we obtan τ (Is 33 τ (IIs 33 τ (IIs 33 A (Is (Bτ k β Rτ (Is + (Is (Aτ + k α Rτ (Is + τ (Is 3 + r τ (Is 3 + ρ I Ω u (Is + (r + r ζρ I Ω u (Is + r r ζ ρ I Ω u (Is = 0 (A B; α β; r r ; ξ η; u v; τ τ ; τ τ ; τ 3 τ 3 r τ (Is r τ (Is + A τ (Is 3 + B τ (Is 3 + k β Rτ (Is 3 (Is τ 3 + r ζ + k α Rτ (Is 3 + ρ I Ω w (Is + (r + r ζρ I Ω w (Is + r r ζ ρ I Ω w (Is = 0 (IIs (Bτ k β Rτ (IIs + (IIs (Aτ + k α Rτ (IIs + τ (IIs 3 (IIs τ 3 + r ζ + r τ (IIs 3 + KΩ u (IIs + ρ II Ω u (IIs = (r + r ζkω u (IIs r r ζ KΩ u (IIs (r + r ζρ II Ω u (IIs r r ζ ρ II Ω u (IIs (A B; α β; r r ; ξ η; u v; τ τ ; τ τ ; τ 3 τ 3 r τ (IIs r τ (IIs + A τ (IIs 3 + B τ (IIs 3 + k β Rτ (IIs 3 + k α Rτ (IIs 3 + KΩ w (IIs + ρ II Ω w (IIs = (r + r ζkω w (IIs r r ζ KΩ w (IIs (r + r ζρ II Ω w (IIs r r ζ ρ II Ω w (IIs (IIs (Bτ k β Rτ (IIs + (IIs (Aτ + k α Rτ (IIs + τ (IIs 3 (IIs τ 3 + r ζ + r τ (IIs 3 KΩ u (IIs + ρ II Ω u (IIs = (r + r ζkω u (IIs + r r ζ KΩ u (IIs (r + r ζρ II Ω u (IIs r r ζ ρ II Ω u (IIs (A B; α β; r r ; ξ η; u v; τ τ ; τ τ ; τ 3 τ 3 u (js r τ (IIs r τ (IIs + A τ (IIs 3 + B τ (IIs 3 + k β Rτ (IIs 3 + k α Rτ (IIs 3 KΩ w (IIs + ρ II Ω w (IIs = (r + r ζkω w (IIs + r r ζ KΩ w (IIs (r + r ζρ II Ω w (IIs + k α Rv (js + r w (js + r ζ ( A (js = r ζa j τ (js + r ζa j τ (js + r r ζ ρ II Ω w (IIs u (js ( A B; α β; r r ; ξ η; u v; τ τ ; a a ; + k α Rv (js (js (js + r w (js (3 4
TPCM-07 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres 34567890 99 (08 000 do :0.088/74-6596/99//000 w (js B u (js u (js + ζ(r + r w(js + r ζ k β Rv (js ( A v (js + ζ(r + r u(js w (js + ζ w (js r r ( + r ζ B k α Ru (js u (js + ζ u (js r r w (js (js = 3 k β Rv (js + + r ζ A A (A B; r r ; ξ η; u v; τ 3 τ 3 ; a a 44 τ (js + r ζτ (js = τ (js + r ζτ (js (js k = a k τ (js + a k τ (js +r ζa j 3 τ (js + r ζa j 3 τ (js + A v (js = a j 66 τ (js + r ζa j 66 τ (js r u (js = a j τ (js 3 + r ζa j τ (js 3 ζr r u (js + a k3 τ (js = k = 3 j = I II. 33 k α Ru (js Takng nto account that Q (jm 0 for m < 0 from system (3 we derve the followng equatons for determnng the decomposton coeffcents (8: τ (Is 3 u (Is v (Is w (Is τ (IIs 3 τ (IIs 3 + ρ I Ω u (Is a I τ (Is 3 = P (Is u a I 44τ (Is 3 = P (Is v (Is 3 = P (Is w = P (Is 6τ (3 3 33; u v w; 6τ 5τ 4τ + KΩ u (IIs + ρ II Ω u (IIs = P (IIs 6τ (3 3 33; u v w; 6τ 5τ 4τ KΩ u (IIs + ρ II Ω u (IIs = P (IIs 6τ (3 3 33; u v w; 6τ 5τ 4τ τ (js = P (js τ τ (js = P (js τ r ζτ (js + r ζτ (js (js u (js = P (js τ (js a j τ (js 3 = P (js u = P (js 3τ j = I II = v (js a j 44 τ (js 3 = P (js v w (js (js 3 = P (js w (4 5
TPCM-07 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres 34567890 99 (08 000 do :0.088/74-6596/99//000 where P (Is 4τ = r τ (Is + r τ (Is A τ (Is 3 B τ (Is 3 k β Rτ (Is 3 k α Rτ (Is 3 (r + r ζρ I Ω w (Is r r ζ ρ I Ω w (Is P (Is 5τ = P (IIs 4τ (Is τ 3 r ζ (Is (Aτ + k α Rτ (Is (Is (Bτ k β Rτ (Is r τ (Is 3 (r + r ζρ I Ω v (Is r r ζ ρ I Ω v (Is (5τ 6τ; A B; u v; α β; r r ; ξ η; τ τ ; τ τ ; τ 3 τ 3 = r τ (IIs + r τ (IIs A τ (IIs 3 B τ (IIs 3 k β Rτ (IIs 3 k α Rτ (IIs 3 (r + r ζkω w (IIs r r ζ KΩ w (IIs (r + r ζρ II Ω w (IIs r r ζ ρ II Ω w (IIs P (IIs 5τ = P (IIs 4τ (IIs (Aτ + k α Rτ (IIs (IIs (Bτ (IIs k β Rτ (IIs τ 3 r ζ r τ (IIs 3 (r + r ζkω v (IIs r r ζ KΩ v (IIs (r + r ζρ II Ω v (IIs r r ζ ρ II Ω v (IIs (5τ 6τ; A B; v u; α β; r r ; ξ η; τ τ ; τ τ ; τ 3 τ 3 = r τ (IIs + r τ (IIs A τ (IIs 3 B τ (IIs 3 k β Rτ (IIs 3 k α Rτ (IIs 3 + (r + r ζkω w (IIs + r r ζ KΩ w (IIs (r + r ζρ II Ω w (IIs r r ζ ρ II Ω w (IIs P (IIs 5τ = (IIs (Aτ + k α Rτ (IIs (IIs (Bτ (IIs k β Rτ (IIs τ 3 r ζ r τ (IIs 3 + (r + r ζkω v (IIs + r r ζ KΩ v (IIs (r + r ζρ II Ω v (IIs r r ζ ρ II Ω v (IIs (5τ 6τ; A B; v u; α β; r r ; ξ η; τ τ ; τ τ ; τ 3 τ 3 [ u (js ( k β Rv (js u (js + r ζ k β Rv (js B B P (js τ = a j 66 P (js τ + A v (js u (js j = A ( + r ζ A u (js k α Ru (js + k α Rv (js + k α Rv (js ( + r ζ A + r w (js v (js + r w (js k α Ru (js (τ 3τ; A B; α β; r r ; ξ η; u v; τ τ ; a a r ζa j 66 τ (js r ζa j τ (js r ζa j τ (js (5 6
TPCM-07 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres 34567890 99 (08 000 do :0.088/74-6596/99//000 P (js u P (js w = ζ(r + r u(js w (js ζ u (js r r w (js + r u (js + ζr r u (js r ζ + r ζa j A A 3 (u v; A B; r r ; ξ η; τ 3 τ 3 ; a a 44 j = I II = ; = ζ(r + r w(js ζ w (js r r + r ζa j 3 τ (js + r ζa j 3 τ (js. We use relatons (4 to express the stress tensor components n terms of u (js where τ (js 3 = [ (js u a (j τ (js = P (js [ τ (js = (j P (js u τ (js τ τ (js = P (js τ (j w (js 3 = a (j 44 [ v (js P (js v r ζτ (js + r ζτ (js + (j 3 P (js τ + (j P (js 3τ (j P (js w ( 33; 3 ; 3 ; 3 3 j = I II = (j = a j 3 aj 3 aj 33 aj (j = a j aj 3 aj aj 3 (j 3 = a j 3 aj aj aj 3 v (js and w (js (j = a j (j 3 + aj 3 (j + a j (j (j nk = aj nna j kk (aj nk n k = 3. For determnng the dsplacement vector components n the frst layer we obtan the equatons : (6 u (Is w (Is + a I ρ I Ω u (Is + (I (I F (Is w = (I ρ I Ω w (Is [ (I P (Is 4τ = a I P (Is 6τ + = F (Is w = (I P (Is τ (Is P u (u v; a I a I 44; 6τ 5τ (I 3 P (Is 3τ + (I P (Is w (7 whose solutons have the form u (Is v (Is w (Is = C (Is = C (Is 3 = C (Is 5 sn(χ (Iu ζ + C (Is sn(χ (Iv ζ + C (Is 4 cos(χ (Iu ζ + ū (Is (ξ η ζ cos(χ (Iv ζ + v (Is (ξ η ζ sn(χ (Iw ζ + C (Is 6 cos(χ (Iw ζ + w (Is (ξ η ζ = (8 where χ (Iu = a I ρi Ω χ (Iv = a I 44 ρi Ω χ (Iw = (I ρ I Ω and ū (Is v (Is w (Is are partcular solutons of equatons (7. For determnng the (I 7
TPCM-07 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres 34567890 99 (08 000 do :0.088/74-6596/99//000 dsplacement vector components n the second layer we obtan the equatons u (IIs u (IIs w (IIs w (IIs + a II (ρ II Ω u (IIs + KΩ u (IIs = a II P (IIs 6τ + + a II (ρ II Ω u (IIs KΩ u (IIs = a II P (IIs 6τ + + (II (II + (II (II F (IIs w = (II From (9 u (IIs = w (IIs = we derve the equatons (ρ II Ω w (IIs + KΩ w (IIs = F (IIs w (ρ II Ω w (IIs KΩ w (IIs = F (IIs w [ (II P (IIs 4τ (II P (IIs τ [ KΩ a II a II P (IIs 6τ + (u v; a a 44 ; 6τ 5τ; (II (II KΩ [ F (IIs w (II 3 P (IIs u w (IIs (IIs P u (u v; a a 44 ; 6τ 5τ (IIs P u (u v; a a 44 ; 6τ 5τ P (IIs 3τ u (IIs (II (II + (II (9 P (IIs w =. a II ρ II Ω u (IIs ρ II Ω w (IIs (0 4 u (IIs 4 Q (IIs u Q (IIs w + a II ρ II Ω u (IIs (u v w; a a 44 / = Ω a II + a II Ω (ρ II Ω + 4K u (IIs = Q (IIs u ( a II KP (IIs 6τ K + 3 P (IIs u Ω a II 3 (u v; a a 44 ; 6τ 5τ = (II Ω (II ( (II The solutons of equatons ( are F (IIs w (II Ω P (IIs u + ρ II Ω a II P (IIs 6τ + Ω ρ + Ω ρ II F (IIs w + P (IIs 6τ Ω (IIs II P u KF (IIs w ( u (IIs = u (IIs 0 (ξ η ζ + u (IIs p (ξ η ζ (u v w ( where the ndex 0 denotes the general soluton of the correspondng homogeneous equaton and the ndex p denotes a partcular soluton of the nhomogeneous equaton. The solutons of the correspondng homogeneous equatons are u (IIs 0 (ξ η ζ = C (uiis (ξ ηϕ u (ζ + C (uiis (ξ ηϕ u (ζ + C (uiis 3 (ξ ηϕ 3u (ζ + C (uiis 4 (ξ ηϕ 4u (ζ (u v w (3 8
TPCM-07 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres 34567890 99 (08 000 do :0.088/74-6596/99//000 where ϕ u (ζ = cosh(γ u ζ cos(δ u ζ ϕ u (ζ = snh(γ u ζ sn(δ u ζ ϕ 3u (ζ = cosh(γ u ζ sn(δ u ζ ϕ 4u (ζ = snh(γ u ζ cos(δ u ζ a II γ u = Ω ( a II ρ II Ω + 4K Ω δu = Ω ( ρ II Ω + 4K + Ω (u v w; a a 44 /. For the dsplacements wth ndex II we have u (IIs = u (IIs 0 (ξ η ζ + u (IIs p (ξ η ζ (u v w u (IIs 0 (ξ η ζ = C (uiis (ξ ηϕ u + C (uiis (ξ ηϕ u + C (uiis 3 (ξ ηϕ 4u C (uiis u (IIs p (ξ η ζ = a II KΩ 4 (ξ ηϕ 3u (u v w ( a II P (IIs 6τ + P (IIs u u (IIs p a II ρ II Ω u (IIs p. Satsfyng condtons (9 ( (5 (6 and (0 ( (5 (6 we correspondngly obtan three algebrac systems of equatons for C (Is k (ξ η C m (uiis (k = 6 m = 4 = ; u v w. The obtaned systems have solutons f ther determnants are dfferent from zero. So the followng condtons correspondng to boundary condtons (9 ( (5 (6 must be satsfed: uτ = δ u + γ u a II cos(δ u ζ cos (χ (Iu ζ [cosh(γ u ζ + cos(δ u ζ + (χ(iu sn (χ (Iu ζ [cosh(γ u ζ χ(iu a I aii a I (4 sn(χ (Iu ζ [δ u sn(δ u ζ + γ u snh(γ u ζ 0 (u v w (5 together wth the condtons correspondng to boundary condtons (0 ( (5 (6 uu = δ u + γ u a II + (χ(iu a I sn (χ (Iu ζ [cosh(γ u ζ + cos(δ u ζ cos (χ (Iu ζ [cosh(γ u ζ cos(δ u ζ + χ(iu a I sn(χ (Iu ζ [δ u sn(δ u ζ + γ u snh(γ u ζ 0 (u v w. aii In the case of one-layered shell [ wthout vscous resstance resonance (nfnte ampltude always occurs and n the presence of vscous resstance [4 at resonance n the shell the ampltude of forced vbratons s fnte. In the case consdered here the presence of vscous resstance only n the lower layer of the shell results n resonance wth an unlmted ampltude of forced vbratons as the equatons uτ = 0 uu = 0 (u v w have real roots whch are the basc frequences of natural vbratons and resonance occurs when the mparted frequency of the forced effect concurs wth the frequency of natural vbratons. Concluson Dynamcal three-dmensonal problems for a two-layered orthotropc shell are solved asymptotcally n the case of vscous resstance n one of the layers for two versons of the boundary condtons on the face surfaces. The ampltudes of forced vbratons are determned. It s shown that for such a confguraton resonance wth an nfnte ampltude always occurs n the shell n contrast to the case of a sngle-layered shell wth vscosty when the ampltude s fnte. (6 9
TPCM-07 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres 34567890 99 (08 000 do :0.088/74-6596/99//000 References [ Aghalovyan L A 05 Asymptotc Theory of Ansotropc Plates and Shells (Sngapore - London: World Scentfc Publshng p 375 [ Goldenveser A L 976 Theory of Elastc Thn Shells (Moscow: Nauka p 5 [n Russan [3 Aghalovyan L A 0 On the classes of problems for deformable one-layer and multlayer thn bodes solvable by the asymptotc method Mekh. Kompoz. Mater. 47 ( 85 0 [Mech. Compos. Mater. (Engl. Transl. 47 ( 59 7 [4 Ghulghazaryan L G 05 Forced vbratons of orthotropc shells when there s vscous resstance J. Appl. Math. Mech. 79 (3 8 9 [5 Azatyan G L 007 Forced vbratons of an orthotropc plate when there s vscous resstance Izv. Nats. Akad. Nauk Armen. Mekh. 60 ( 9 40 [6 Aghalovyan L A 000 To the asymptotc method of soluton of dynamcally mxed problems of ansotropc strps and plates Izv. Vuzov. Severo-Kavkaz. Regon. Est. Nauk: Natural scences No 3 8 [7 Aghalovyan L A Gevorgyan R S and Ghulghazaryan L G 00 The asymptotc solutons of 3D dynamc problems of orthotropc cylndrcal and torod shells Izv. Nats. Akad. Nauk Armen. Mekh. 63 ( 6 [8 Aghalovyan L A 00 Asymptotcs of solutons of classcal and non-classcal boundary-value problems of statcs and dynamcs of thn bodes Appl. Mech. 38 (7 3 4 [9 Aghalovyan L A and Ghulghazaryan L G 006 Asymptotc solutons of non-classcal boundary-value problems of natural vbratons of orthotropc shells J. Appl. Math. Mech. 70 ( 0 5 [0 Ghulghazaryan L G 008 The own vbratons of two-layered orthotropc shells under the complete contact between the layers In Proc. 6th Int. Conf. The Problems of Dynamcs of Interacton of Deformable Meda September 6 Gors - Stepanakert 008 (Gors - Stepanakert: Inst. Mekh. NAN RA 0 [n Russan [ Agalovyan L A and Gulgazaryan L G 009 Forced vbratons of orthotropc shells: nonclasscal boundaryvalue problems Int. Appl. Mech 5 (8 888 903 [ Ghulghazaryan L G 03 Own vbratons of two-layered orthotropc shells under an ncomplete contact between the layers Vestnk Fonda Fund. Issled. 65 (3 5 69 0