Basık dairesel silindirik kabukların aeroelastik analizi
Başlık çevirisi mevcut değil.
- Tez No: 75331
- Danışmanlar: PROF. DR. ZAHİT MECİTOĞLU
- Tez Türü: Yüksek Lisans
- Konular: Uçak Mühendisliği, Aeronautical Engineering
- Anahtar Kelimeler: Aeroelastik analiz, Silindirik kabuklar, Aeroelastic analysis, Cylindrical shells
- Yıl: 1998
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Mekanik Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Bu çalışmada hava akımına maruz ve çeşitli sınır şartlarına sahip plak ve basık dairesel silindirik kabukların aeroelastik davranışı incelenmiş ve flater sınırlan belirlenmeye çalışılmıştır. Basık kabuk denklemleri homojen malzeme, küçük yer değiştirmeler için, Donnel-Mushtari ince kabuk teorisi kullanılarak elde edilmiştir. Plaklar kabukların bir özel hali olduğundan yarıçap çok büyük alınarak plak çözümlerine gidilmiştir. Aerodinamik kuvvet ise dinamik basınç ve kabuk üzerindeki noktaların dönmeleri cinsinden lineerleştirilmiş sesüstü akım teorisiyle ifade edilmiştir. Bu teoride sınır tabakanın hemen üzerindeki vizkoz olmayan bölge ele alınmakta, hava ideal gaz olarak kabul edilmekte ve sistemin izantropik olduğu varsayılmaktadır. Ayrıca kabuk yüzeyini şekil değişimlerinin küçük olduğu varsayılmaktadır. Çözüm aşamasında sonlu elemanlar metodu ve Kitz metodu seçilmiştir. Sonlu elemanlar çözümünde eleman olarak dört düğüm noktalı ve her düğümde beş serbestlik derecesi bulunan, toplam yirmi serbestlik dereceli bir silindirik kabuk eleman seçilmiştir. Bu elemana ait kütle, katılık ve aerodinamik matrisler için bir sonlu elemanlar formülasyonu geliştirilmiş daha sonra çeşitli sınır şartlarını inceleyen bir bilgisayar programı Fortran dilinde yazılmıştır. Ritz metodu ile yapılan çözümde ise bir ucu ankastre olarak mesnetlenmiş bir silindirik kabuk için yaklaşım fonksiyonları uygun polinomlar olarak alınmış, daha sonra çözüme gidilmiştir. Ritz metodu içinde bir bilgisayar programı Matlab dilinde yapılmıştır. Elde edilen sonuçların literatürdeki sonuçlarla uyuştuğu gözlenmiştir.
Özet (Çeviri)
AEROELASTIC ANALYSIS OF SHALLOW CYLINDRICAL SHELLS SUMMARY Aeroelasticity is a science that deals with interaction between the aerodynamic, inertia and elastic forces which appear on a solid body which is exposed to air flow. This interaction can cause an unstable motion on the structure. Flutter is one of these unstable motions. This study is to try to determine the flutter boundary of plates and shallow cylindrical shells which have various boundary conditions. For a given two dimensional control or lifting surface, the flutter mechanism happens as follows: the elastic body deforms due to outer effects, deformation of the body changes the angle of attack. The aerodynamic forces and moments which depend on angle of attack shift also. The changing of aerodynamic forces and moments change the angle of attack again. This cycle continues until the aerodynamic forces and moments become less than bending and torsion stiffness of the body. After this point, the motion turns back. On a certain critical dynamic pressure, the aerodynamic forces which depend on angle of attack become greater than elastic forces which depend on angle of attack. This point is flutter boundary. At the flutter boundary, if the dynamic pressure goes on to increase, the motion of structure becomes unstable. A harmonic motion happens and it's amplitude gets greater and greater. This phenomena causes the structure failure. On the derivation of the equations of motion, the theory used is linear. The air flow is parallel to the straight edge of the surface i.e. x axis of plates and shells as shown following. Figure- 1 Air flow direction versus to shell To obtain the differential equation of the shell, an aerodynamic force has been considered which is normal to the shell surface. Aerodynamic drag has been omitted. This lifting force that is normal to the shell surface has been accounted for by using XIpiston theory. In this theory it is assumed that the flow is inviscid, the considered region is just above the boundary layer, the deformation of the points on the solid surface, in other words the angle of attack of the points due to aerodynamic forces is small, air is assumed as ideal gas, the system is isentropic, the body forces can be neglected, the flow is steady stead. For the supersonic flow the piston theory is obtained by the solution of a linear partial differential equation. This differential equation is a combination of Navier-Stokes equations, continuity equation, energy equation, ideal gas relationship and state equation. The combining of these equations is arrived at by defining a potential function for flow velocities. The Resulting differential equation is nonlinear. To get a linear differential equation, the small perturbation theory is used under the acceptation of small deflection of the body. This linear differential equation is called linearized supersonic potential flow theory equation. The solution of this equation is substituted at the pressure coefficient formula. At the supersonic flows the aerodynamic pressure is obtained as: Pz = 2nqto dw VMW2-1 dx (1) In the above equation, if only one surface is exposed to air flow; n=l and if both upper and lower surface are exposed to air flow; n=2. The valid assumptions during the derivation of equation of motion owing to shallow cylindrical shell are: the thickness is small compered to the shell's edge length (approximately h/L<l/10), displacements and deformations are small enough, the normal stress can be neglected among other stress, Bernoulli-Navier hypothesis is valid, structural damping is omitted. The shallow cylindrical shell assumption for a cylindrical shell depends on the ratio of b/R as shown following. shallow cylindrical shell b/R=0.7 Hfc=0. 009036 2p =40.975 deep cylindrical shell b/R=1.6 B7b=0.25 2^=106.26 Figure-2 Comparison of shallow shell and deep shell The differential equation of the cylindrical shallow shell which is exposed to air flow has been obtained by using Newton's second law, stress-strain relationship, geometric xuproperties come from Beraoulli-Navier hypothesis, the shallow shell approximation (R + z = R) and piston theory for the aerodynamic force. The result equation is“. E.h d4w 2nq0 R2 ^ VmT^I ^phv4^?) ox dt2 (2) Here V ”5x8+ öX6öy2+6ÖX45y4+4ac2ay6+öX8 and V4 = a4. + 2 S4 + ? ox4 3x2Sy2 5y4 are the defined statements. In the above equation; w denotes the middle surface displacement in z direction. Analytical solution of this differential equation cannot be found unless convenient boundary conditions exist. For this reason in this study numerical methods have been preferred for the solution. Selected numerical solutions are the Ritz method and finite element method. In the Ritz method, an approximate solution is obtained for the varitional statement from which the differential equation has been derived instead of the given differential equation. Thus there needs to be a varitional statement for the differential equation. But in solid mechanics there is always a variational statement for the equation of motion. It is the Hamilton principle which is given as t2 J[8(T-V)+8W]dt = 0 (3) where T is the kinetic energy, V is the potential energy, SW is the virtual work that the outer force does. For the shallow shells the kinetic energy can be found as T = -phjj(w2 +û2 + v2)dxdy (4) The potential energy is «-İJI da dv w] ^ + ?^- + ^?1 -2(l-v) du dv daw 1 (dv du dx' dy öx R 4 vox dyj + D a2w _ a V ox2 dy1) -2(l-v) 52w ö2w f S2w^2 Sx ' oy voxdy. dxdy (5) xmAnd the aerodynamic force can be found by integrating the pressure which was formulated before by the piston theory over surface area: The virtual work which the aerodynamic force does is 5W = Q5w = ff ^f-^8wdxdy (7) JAJ Vm^T dx y KJ In Ritz method, calculations have been made on a cantilevered plate. The approximate function is selected to_ satisfy the essential boundary conditions and it is assumed that the dependent variables has the form of M u(x,y,t) = ][]<Pm(x,y).qm(t) m=l M v(x,y,t) = 2>m(x,y).qm(t) m=l M w(x,y,t)=2>m(x,y).qm(t) (8) m=l Here cpm,<j>m and v|/m are the approximate functions which satisfy the essential boundary conditions. For cantilevered plates following function family supplyies the essential boundary conditions. Hence the aerodynamic virtual work could be obtained by substituting defined approximate function into (7): In flutter analysis for the time dependent variable qm, the following equation is assumed: XIVqm = e 1m (10) Here co has two companents, i.e: © = CD r + İ© j (11) In the above expression © r denotes the real root which means stable motion and © j denotes the imaginer root which means unstable motion. Thus, if there is any imaginer root in the solution, it means that the system is unstable i.e in the flutter. If (8) and (9) expressions are substituted into (3) finally this is obtained: 20 JJ l(qJphcû2(cpm9nHnA,+M>mVn) + C A m=l 3<Pm. dch. + ? Vr dx dy R K dx dy ^n, ^n, Vn“ R (1-v) dy dx dx dy R dx dx R 2\ dx dy J\dx c$m dcpa + g(pa ctym + \|/m dy J + D d\Lm d\ V dx2 + ? dy2 J ax a2VlN v dx2 +”dy2 j (1-v) a2\|/m d2ya a2\|/m a2\|/0 ay2 ax2 ax2 ay2 2a2M/ma2v|/n axay axay, _2n<h,dwm Vm.2-i ax -\ kixdy=0 (10) In the above equation n, m denotes row and column of the eigenvalue matrice respectively For finite elements formulation Hamilton principle can be written as so: |[5(T-V) + 8W]dt = i;(|[8(Te-Ve)+5We]dt 1=0 (11) So elements matrice can be obtained by using t2 Le = J[8(Te-Ve) + 8We]dt (12) It can be easily seen, the equation (12) is the same with the equation (3), but the right hand side of the equation is not zero this time. Thus (10) formulation can be used for the finite element formulation. The selected element for the finite element solution has four nodes at four corners, each has five degrees of freedom so totally twenty degrees of freedom as shown below. XVFigure-3 The shell element and it's degrees of freedom The shape function has been obtained from polynomials by utilising compatibility and displacement conditions on nodes. The calculated shape functions are: 1.. x... y^ 8X~ a'v“ W 2 2 x y x y v,(x,y) = 7(i-7Xi-rX2---£-^-pr) 1 x -- y^, Yn2 v1(«.y)-jbO--Xi + t»-t) V3(t,y) = -j>(i+-Xi-fxi-7)2 ö a b a 1 x v xv x v v.O-.y^lO+lW-fxz^-J-^-fcr) XVIv,(x,y)4(i + f)(i+^)(2+f+i-^-^) v,(x,y) = -ib(i+^)(i-i)(i+^)2 ',(x,y) = |a(l-%l+^)(l+^)2 Saba 1 x y x y x2 y2 «0-7)a+i:X2--+£--r-r5 8 a b a b a b V”(x,y) = -^b(l-^)(l-^)(l + ^)2 V,2(x,y) = -^a(l + -)(l + ^)(l-^)2 o a b a cp13(x,y) = <j>17(x,y) = ^(i-^)(i-^) q>Mtey) = <h.tey) = 7(1+7X1-7) cp15(x,y) = Mx,y) = ^(i+f)(i+7) 9,«(x.y) = <M*.y) = 7(1-7X1+7) (13) For the following approximate functions, the shape functions which are defined above is valid. Thus: 16 20 u(x,y,t)=£q>m(x,y).qn(t), v(x,y,t)= £<|>m(x,y).qm(t) m=13 m=17 12 w(x,y,t)=][>m(x,y).qm(t) (14) m=l For the finite element solution, the element matrices (mass, stiffness and aerodynamic matrices) have been calculated by a soft ware called as DERIVE. Then an finite element program has been written by FORTRAN language which can consider various boundary conditions for shallow cylindrical shells. For the Ritz method a computer program has been written in MATLAB which consider flutter analysis of cantilevered plates. xvu2500 2000 1500 ? 1000 ? Figure 4 The flutter character of simply supported cylindrical shell Here: X = 2q0 dVmT^T K =©' pha4 D (15) In this study, the results obtained by finite elements and Ritz method are very close to each other and to literature. The Figure 4 is the typical flutter behaviour of a lifting or control surface. As shown above figures two roots try to come near and flutter happens when two mode coalescence. This phenomena can be explained as the aerodynamic forces which depend on angle of attack increase more than elastic forces which depend on angle of attack. Beyond the flutter boundary, the aerodynamic forces are greater than the elastic forces which cause a stable harmonic motion. If the dynamic pressure goes on to increase, the motion of structure becomes unstable. A harmonic motion happens and it's amplitude gets greater and greater. This phenomena causes the structure failure. XVUl
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