Basik Dairesel Silindirik Kabuklarin Aeroelastik Analizi

Abstract

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.

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 XI piston 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/Lm(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. xvu 2500 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.