Eğri eksenli kirişlerin titreşimi

Abstract

Günümüzde transport tekniğinde kullanılan vinç çengellerinden, köprülere kadar birçok sistem eğri eksenli kiriş özelliğine sahiptir. Gelişen şartlar karşısında da bu tür kirişlerin titreşimleri önem kazanmaktadır. Çok çeşitli boyutlarda olabilen bu kirişlerin bilhassa büyük gerilmelere maruz olanlarının iyi etüd edilmesi gerekmektedir. Bunun için de ilk önce kirişin tabii frekansının belirlenmesi problemi karşımıza çıkmaktadır. Bu çalışmada, enine kesiti dairesel olan sikloid, katenari, parabol ve daire formunda eğriliğe sahip olan kirişler incelenmiştir. 1. bölümdeki kısa girişten sonra 2. bölümde eğri eksenli kirişlerdeki eğilme gerilmelerine temas edilmiş, bağıntılar çıkarılmıştır. Böyle bir sistemin kesin çözümü incelenirken çıkan diferansiyel denklemlerin 6. mertebe gibi yüksek bir mertebeye sahip olmasından dolayı Rayleigh-Ritz Metodu kullanılmıştır. Dolayısıyla tabii frekans değerleri yaklaşık değerlerdir. 3. bölümde ise bu metoda ait prensipler ele alınmıştır. 4. ve 5. bölümlerde bu metod yardımıyla ilgili denklemler elde edilmiş ve her bir form için tabii frekansı verecek 0*ye bağlı ifadeler çıkartılmıştır. 4. bölümde sadece ankastre mesnetli hal için kendi düzleminde oluşan titreşimler incelenmiştir. 5. bölümde ise sabit mesnet olması durumunda kendi düzleminin dışındaki titreşimler ele alınmıştır. 6. bölümde de ' ye bağlı ifadeler için sayısal değerler, tablo ve grafikler halinde verilmiştir.

Several systems used in current technology are in the form of elastic arcs such as crane hooks, bridges etc. With the improvement in technology the vibration of these arcs becomes more significant. Being in several different sizes especially the one's with high strength, should be examined more carefully. To handle the problem first the natural frequency of the arc should be determined. Some scientists have studied on this subject. They have found some formulas about determining the lowest natural frequency of elastic arcs. One of them, Prof. J. P. Den HARTOG has dealt with the lowest natural frequency of circular arcs with clamped ends and of vibrations occuring in the plane of initial curvature of the arcs. Another scientist, Prof. E. VOLTERRA has extended the results of Den Har tog' s studies. He has derived these expressions to elastic arcs having the center lines in the form of a cycloid, of a catenary, or of a parabola. The problem of finding the natural frequencies of vibration of a circular ring has been attacked by LAMB who derived the differential equation and solved it for the case of small curvatures. The exact integration for the general case of a central ex, having any value between 0° and 360 becomes extremely complicated. In this study the approximate method of Rayleigh-Ri tz has been applied for calculating the lowest natural frequency of circular arcs subtending any angle, for hinged as well as for clamped ends. In this study, fistly we have examined the natural frequency of circular arcs with clamped ends, secondly we have found the lowest natural frequency of elastic hinged arcs. In the first situation, vibrations occur ed in the plane of the initial curvature of the arcs were taken into consideration, in addition to this in the second situation vibrations occur ed inside and outside the plane of the initial curvature of the arcs were taken into consideration also. In the clamped ends case, initial radius of curvature center lines is assumed R, radial displacement is assumed v, axial displacement is also assumed w. These values are put into Kinetic and Potential Energy equations. Hence U and T can be showed like the following equations: U » EI JQL a2u as R aw 1 2 OS EO ds + rLr- OL dw ds u R i2 ds T = fO. ds where L, p, EI, O are the total length of the arc, the density of the arc, flexural rigidity of the arc and the cross-sectional area of the curved bar which is supposed to be circular, respectively. are: For extensional vibrations the boundary conditions u = w du âs = O for s = r o L Sui t abl e anal yt i c al expr es s i ons displacements satisfying equations are: for the u = A(l - cos <2rcs/L>) sinovt w = B sin<2?Ts/L> si neat Since, sine and cosine So finding U,T.We max max these sine and cosine terms. terms at most equal 1, can put 1 instead of When u and w are put into the energy equations, we can obtain the following expressions: U EOL 1 4 CA2k + B2m + 2ABn) (OOL.fci2(3A2 + B2) where k = S2<2n>4 + JO v R ; ds + ds - cL r, %2 m = 5"<2n>' n = S2<2rc>3 a L Jo JO R 1 f ?' Jo fans'). R, fZns cos ds cos r t.n N.. v L. fi2ns cos ds + (2ti)' ds - C2") ds + (2rr>- ds U = T max max is written by means of Rayleigh-Ritz's principle, so 02(3A2 + B2) = CA2k + B2m + 2ABn) is obtained, such that 5 =. 4> p" L r r2 = j_ E \ fi being the radius L E l O of gyration of the cross-section of the curved bar We find, 3 By substituting the bending and twisting moments, :[< M^ = EI JC/5/R)-Câ2v)/Cös2)l M a EI y y [ca*U VCös"). + cı/R)CöwVCds) Mz = GI CÖ/?)/Cös) + (l/FOC^vVCds) (where components of the elastic displacement u, v, w, angle of twist ft, flexural and torsional rigidities EI, EI and GI, initial radius of curvature R of the x y p center line and the lengths of the arc) into total potential energy associated with elastic deformation is gi ven by, ri ll = M M M 2 -i EI EI GI ds PJ in view of the total kinetic energy associated with the particle motion which is given by» pCi T = n aft l öt J J 2-i ds According to Rayleigh's principle we can obtain, U = T lmox max U * T 2 max 2max with» U = ı T = i ds ds (ix> U s 2 EI pCl rLn OL d/3 ds 1 R &v âs ds From above equations, the frequecy equations for the vibrations occur i ng in the plane of initial curvature of the bar and for the vibrations occurring outside the plane of initial curvature of the bar will be derived. In the case of 1 nextensi onal vibration occur i ng in the plane of initial curvature of the bar the following boundary conditions must be verified: u = w = M=0 p - + ex y and, also the condition, dw/dtp = u must be satisfied. Suitable analytical expressions displacements satisfying above equations are, u = A sinÇ^TT/oO si not, w = -(Aof/rr) I cos^rr/a) + 1 I sinwt If we substitude displacements into, U » T lrnox lrnax oi can be obtained as, in the formula of, for the u> CEI ypCiRy F x = rCX sin R* Tip CX dp ?ex R JO sin Tip ex CX n cos np ex + 1 dp In case of vibrations occuring outside the plane of initial curvature, the boundary conditions are, v = ft = M = O x for s c Suit-able analytical expressions for the displacements satisfying the boundary conditions are, v ? = A si nCnrs/L) si ncot ft - B sinÇnrs/L) si nwt When these displacements are inserted to energy equations, the following equations, can be obtained: '[< <*>2|(A/L)2 + aS2B2l = 52f< CA/L)2k + B2m + 2BCA/L)nl where 4> - pw L / E By expressing that. = O where A = T - U as defined before ÖA ÖA 3A 3B the following equation is obtained, 204 ~ 02(2<52k + m) + 52(km-n2) = O where k - n + l+v an' 4 L rL ds m 1+v O*" R sin' ÎTS ds n = n 2 L sin 0 R ITS* ds + 1+v 4 L rL cos R rrs ds v is Poisson's ratio. In this section, by inserting the values of k, m and n that are obtained before, to the frequency equations, the value of