Akışkan yüklü plakların bağlaşık titreşimi

Abstract

ÜZET Bu doktora çalışmasında üç boyutlu bir yapı-akışkan dinamik etkileşim sorunu ele alınmıştır. Dikdörtgen sıvı haznelerinin elastik varsayılan akışkan yüklü yan duvarlarının bağlaşık titreşimi incelenmektedir. Sıvı haznesinin duvarı olan yapı, enine yerdeğiştirmeleri küçük, ince, elastik bir plâk olarak ideali eştirilmiş, haznenin diğer duvarlarına ve tabanına basit olarak mesnetlendirildiği, üst kenarının ise boşta olduğu varsayılmıştır. Akışkan ise, lineer olarak sıkıştırılabilen ideal (sürtünmesiz) bir akışkan olarak ele alınmıştır. Haznenin diğer duvarları ve tabanı rijit kabul edildiği için zemin-akışkan ve yapı-zemin dinamik etkileşimi öngörülmemiştir. Yapı ve akışkanın hareket denklemleri ile yapı ve akışkan alanlarındaki sınır koşulları elde edilmiştir. Akışkanın hareket denklemi elde edilirken, akışkan hızlarının küçük olduğu varsayıla rak hareket denkleminin 1 ineerleştirilmesi yoluna gidilmiştir. Bu şekilde matematik olarak lineer, homogen bir sınır değer problemi elde edilmiştir. Titreşim hareketinin h armonik olduğu varsayılmıştır. Akışkana ait hareket denklemi kullanılarak p(jc,y,z,t) hidrodinamik basınç çözümü, bu çözüm yardımı ile plak hareket denklemi kullanılarak w(*,y,t) yanal yerdeğiştirme çözümü elde edilmiştir. Bu çözümler, yapı-akışkan ortak yüzeyinde yapı ve akışkan yerdeğiştirme! erin in uygunluğunu ifade eden sınır koşulunda kullanılarak ve trivial olmayan bir çözüm elde etme düşüncesi yürütülerek Frekans Denklemi'ne varılmıştır. Frekans Denklemi, elemanları a boyutsuz bağlaşık frekansına bağlı, sonsuz satır ve kolonlu bir matrisin determinantının sıfıra eşit olmasını ifade eder. Sözkonusu matris, sonlu sayıda satır ve kolonda kesilerek, yaklaşık ilk iki bağlaşık frekans, simetrik ve antimetrik titreşim modi arı için, ayrı ayrı, bir bilgisayar programı geliştirilerek elde edilmiştir. En uygun kesme mertebesi araştırıl mıştır. Bağlaşık frekanslara, akışkan serbest yüzeyinin dalgalanması nın etkisinin küçük bağlaşık frekanslar için, akışkan sıkışabilirli- ğinin etkisinin ise büyük bağlaşık frekanslar ve büyük v boyutsuz değişken değerleri için önemli olduğu gösterilmiştir. Hazne doluluk oranı büyüdükçe dinamik etkileşimin kuvvetlendiği görülmüştür. Bağlaşık frekansları hazne boyutlarının, y, relatif kütlesinin ve XD boyutsuz değişkeninin önemli ölçüde etkilediği gösterilmiştir. Simetrik ve antimetrik ilk iki bağlaşık titreşim biçimi ve dört ayrı hazne doluluk oranı için yapı-akışkan ortak yüzeyindeki hidro dinamik basınç ve plak enine yerdeçiştirme modları, bir bilgisayar programı geliştirilerek, elde edilciştir.

SUMMARY COUPLED VIBRATIONS OF LIQUID LOADED PLATES In this thesis, a three dimensional fluid-structure dynamic interaction problem is investigated. Coupled vibrations of a liquid loaded side wall of a rectangular liquid container are examined. The structure, the liquid loaded side wall of the container, is idealized as a thin, elastic plate. Linear plate theory is used because of small lateral displacements assumption. The plate is assumed to be simply supported to the other side walls and to the bottom of the container, the top edge is free. The liquid is considered as linearly compressible nonviscid fluid. Furthermore it is assumed that the other side wall and the bottom of the container is rigid, therefore, soil -structure and soil-fluid dynamic interaction are not considered. The governing equation of the structure field is the equation of motion of the plate 2 2 Vç 3 wı i 4«i+- - t- = ~ tt P (-x»y>o,t) (ky<b (la) h<y<H (lb) where w, (x,y,t) and W£ (x,y,t) are the lateral displacements of the plate on wet and dry surfaces, respectively, /i$ is mass per unit area of the plate, D is flexural rigidity of the plate. Fluid field is governed by Euler's Equations of motion and continuity equation. If low motion of fluid is assumed the convective component of the accelaration, which is nonlinear term in terms of velocity, can be neglected. In this way the problem is linearized. When velocity vector is eliminated in Euler equations and rontinuity equation, we obtainV-7 İ <2> where p(x,y,z,t) is hydrodynamic fluid pressure in excess hydrostatic, c=/ K/p^ is the speed of the pressure wave in the fluid medium, Pf is mass density and K is bulk modulus of the fluid. In this way the problem is linearized. Thus, a linear, homogeneous boundary value problem is obtained. The most important boundary condition of the problem appears along the plate-flu^d 4nterface is the one which is given. 3p an*, 3z z-o r 3t6 This boundary condition states that the structure and the fluid are in full contact along the interface during the motion, that is, separation is not permitted. It can easily be seen in the first equation of motion of the plate and in the above mentioned boundary condition that two different field variables, lateral displacements of the plate on wet surface w-i(x,y,t) and hydrodynamic fluid pressure p(x,y,z,t), influence each other. This case is known as Structure-Fluid Dynamic Interaction. It means that there are Coupling Effects between the structure and the fluid, and they are formed a Coupled Dynamical System. There are two extreme cases of the problem where no coupling effects can be mentioned. In the first case, the flexural rigidity of the plate is infinitely large and the motion of the fluid in a rigid container is considered. This motion is known as Sloshing Vibrations of the fluid. In the second case, there is no fluid in the container and the motion of the plate in vacuum is considered. This motion is known as Bending Vibrations of the plate. In between of the above mentioned extreme cases, the coupling effects appear. In this general case the mathematical analysis gives two types of vibrations. One of these types of vibrations resembles sloshing vibrations of the fluid and is called as Coupled Fluid Vibrations, The other type of vibrations resembles bending vibrations of the structure and is called as Coupled Structural Vibrations. In this thesis coupled structural vibrations are examined only. i*Harmonic motion is assumed, therefore time dependence can be removed as usual. The equation of fluid motion together with boundary conditions in fluid field is solved by using Separation of Variables Method. On the free fluid surface two different types of boundary condition can be stipulated and it depends on whether surface waves are admitted «or not. If surface waves are admitted, on the free fluid surface ft * 2 [^ =0 (4a) 3y y=h g St y=h is valid. Otherwise [p] = 0 (4b) y=h is used, where h is fluid depth, g is acceleration of gravity. Hydrodynatiric pressure solution is obtained as an infinite series. PU,y,z,t)-ew [^ ^ Plte coso^cosB^siny^Lsiny^* ?cosYlk£LcosYl£)* + k=k^l J +1 P2k£COSakxcos^y(-thY2kilLshY2kilz* '0 ` *=£0 +chY2k£z)] 0<x<a, Û*y«h, 0<z<L (5) where kır » if surface waves are neglectedf, 21,-1 7T if- x 6JT-2- 17 (6a) otherwise for a definite frequency u>, S. 's are infinite number of roots of.the trancendental equation ü)2cosph + g3sin$h = 0 (6b) and k, z are the largest integers which satisfy inequality c and wo2 2 02»l/2 2,2ifl2 w.1/2 for a definite frequency w, In structure field for the lateral displacement solution w(x,y,t)= e1ut l Ym(y) sino* (7) A V*>sinv m=i is proposed. This solution satisfies the boundary conditions given in x direction, on the vertical edge of the plate. Y Ay)* y dependent part of the solution, is determined so as to satisfy the equation of motion and the boundary conditions in y direction, on both wet and dry surface of the plate separately. XTThus, on wet surface of the plate. kî. nı i»t o o PlkicosYlk],L o 1 D k=o £=1 X4-(c£+Bjr m=l km <AniAmsheıray+Bm^cheımy+cm«ınsine3ray+ 00 +DniiUncos63my>sinV+ * 4l akm<A12lMsh*liBy* o +cosakxcos$.y ] + P2k£ r1`0 +^12tochBlray*hl2iUnsinB3my+D112itmwsB3iny)sinVt 00 + «JLi akiD(A122£msh6lmy+B122£n,chBlniytC122jln,sh6my+ o +D122£mche2my^sinamx4coSakJtcosB£y^ Ooc^a, (tey^h (8a) iwt ko £o Plk«cosYlkcL mo *B21lM,ch6lm>,<C21lMSİnB3m>wD211toCOS63l/)SİnV* xn+ z item +1 W^2Ums^ln/+B22UmchBlm>` ^zzumshh^hzMâhh^simıfi P2k£ - ° /J^ JL-i T^SST [ i 1^2fc'W1^ *hmmchh^hmmsin^+^zmC05h^)s`mmM + «,r*i akm(A222£msh0lmy+B222)lmchBlmyt m=ro +1 +C222tosh&2my+D222tochB2nı y)sinV3 } (8b> 0«x<a, h^y^H where m is the largest integer which satisfies inequality a,,, < X = (-5- u ) and *2nf (£*2),/2 xmfor a definite frequency w. Here a,L,H are the linear dimensions of the container. Integration constants A,, -j &), Bi-j iPi Am » c-hi2i3 ^ and D^j2^ £`, are determined in sucK away that the boundary conditions in y direction are satisfied. Finally by using the boundary condition given on the plate- fluid interface, in the compact.form, k=o £=1 K£ KSm is obtained. This is a system of linear, homogeneous, algebraic equations for unknown vector {PkJJc. The presence of a non-trivial solution of this system of equations requires that the determinant of coefficient matrix should be zero. Ak£jnn (ü))= 0 k= 0,1,2,.., Jl,m,n*l,2,3,., (10) This equation is called Frequency Equation. The roots of the frequency equation which are infinite in number and `they corresponds to frequencies of trie system. The frequency equation is the determinant of the coefficient matrix having infinite number of rows and columns. Truncating this coefficient matrix up to a finite number of rows and columns, approximate coupled frequencies are obtained. The more accurate coupled frequencies can be obtained by increasing the truncating order of this coefficient matrix. If the surface waves are neglected, for b£s (6a) is used in the frequency equation. Otherwise the 6^s are the roots of the trancendental equation (6b). In this case it is necessary to apply a successive approximation procedure. To begin with the successive approximation procedure the coupled frequency corresponding to no surface waves assumption is used as an initial value. The incompressible fluid solution may be obtained by substituting c= ~ in the above formulation. To accomplish the above mentioned calculations, a computer program is written, For saving in memory capacity and time and ob tarring more precise results, symmetrical and antimetncal components of the coupled vibrations are analized separately. It is found out that the effects of the surface waves and compressibility of the liquid on coupled frequency are in negligible order. xivFor the first and second symmetrical and antimetrical vibrational modes, the variations of the coupled frequencies of the system with respect to the filling ratio of the container are examined. For the small values of the filling ratio the interaction seems to be weak. As filling ratio increases the interaction effects become pronounced and for large values of the filling ratio strong interaction effects are observed. For the same vibrational modes» the effects of linear dimensions of the container on coupled frequencies is examined. The variations of the same coupled frequencies with respect todimensionless parameters ur, X$ and Xç are obtained. pr is relative mass of the system. The case of y^ 0 corresponds to the bending vibrations of the plate. With increasing values of pr the interaction effects appear and the coupled frequencies of the system decrease in a fast manner. Since two different types of the field variables are considered, two types of vibrational modes for a coupled frequency of the system can be found. W(x,y) is the coupled mode of the lateral displacement in the structure field, and p (x»y,z) is the coupled mode of the hydrodynamic pressure in the fluid field. In the analysis k=o A=l Pk*Ak»(u)'° m,n= 1,2,3,,., (9) has been obtained. This equation represents a system of infinite number of linear, homogeneous equations for {Pjt£> « The coefficient determinant of this system yields the frequency equation for estimating the coupled frequencies of the system as follows. iAk£ron Ml = ° k= 0,1,2£,m,n»l,2,3,., (10) Since equation (10) is satisfied for a coupled frequency, {P^}» in the system of equations (9), can be solved in such away that one of the {Pill's is defined, say P-m= 1. Having determined the vector {PkJ Tn this manner, the vibrational modes of lateral displacement ana^hydrodynami c pressure can be determined by using equations (8a, b) and (5) respectively. The first and the second symmetrical and antimetr-ical vibrational modes of lateral displacement and hydrodynamic pressure for four different filling ratio of the container and empty container are given. xv