Eksenel kuvvete maruz hareketli bandların enine serbest titreşimleri

Abstract

ÖZET Transport bandlarında artan hızlarla birlikte doğan dinamik etkilerin bandın çalışması Üzerinde nasıl sonuçlar doğuracağı merak konusu olmuştur. Bu çalışma bu nedenle ele alınmış, yatay doğrultuda hareketli, eksenel kuvvete maruz, eğilme rijitliği olan bir bandın enine serbest titreşimlerinin teorik araştırması yapılmıştır. Birinci bölümde, konuyla ilgili çalışmalar tanıtılmış kısaca içerikleri ve elde edilen sonuçlar hakkında bilgi verilmiştir. İkinci bölüm, titreşim hareketinin diferansiyel denkleminin kurulmasına ve çözümünde uygulanan yöntemin tanıtılmasına ayrılmıştır. Titreşim hareketinin denklemi, 4`üncü mertebeden kısmi türevli lineer bir diferansiyel denklem olarak elde edilmiştir. üçüncü böl ümde, hareketin dördüncü mertebe diferansiyel denklemi, sonlu fark eşitlikleri cinsinden yazılmıştır. Bunun sonucunda, hareket denklemi, bir lineer denklem sistemine dönüşmüştür. Homojen olan bu denklemler, sınır şartlarının da uygulanmasıyla bir özdeğer problemi şekline dönüşmüştür. Dördüncü bölümde, frekans katsayılarının hesaplanma sında kullanılan bilgisayar programı tanıtılmıştır. Beşinci böl ümde, katsayılar matrisi, bilgisayar programıyla çözülerek doğal frekanslar elde edilmiştir. 60, 80, İOO adet düğüm noktası için çözülen katsayılar matrisinden elde edilen doğal frekanslar arasında ekstrapolasyon uygulanarak, kesin çözümlere çok yakın frekans değerleri bulunmuştur. Elde edilen frekans değerleri, ana programda yerine konularak bulunan lineer denklem sistemine, Gauss-Jordan metodu uygulanarak, o frekansa karşı gelen mod değerleri elde edilmiştir. Altıncı böl Umde, kritik hızlarının araştırılmasına yer verilmiştir. Kritik hızlar doğal frekansların kaybolduğu band hızları olarak tanımlanmış, bu hızlarda kararsız titreşimlerin ortaya çıktığı saptanmıştır. Kritik hızlar değişik gergi kuvvetlerinde ve muhtelif harmonikler için ayrı ayrı hesaplanmıştır. Sonuç bölümünde bazı önemli noktalar ve elde edilen önemli sonuçlar özetlenmiştir. EK Al de, frekans tablolarının elde edilmesini sağlayan Fortran dilinde hazırlanmış ana ve alt programlar verilmiştir.

SUMMARY THE FREE TRANSVERSE VIBRATIONS OF THE MOVING BANDS SUBJECTED TO AXIAL FORCE In this study, the natural frequencie and modes of the moving bands and the critical velocities are exami ned by using the finite differences method and solving the equations of motion. The fixed conditions at both ends are applied and their effects to the frequencies and mode shapes are investigated. In the first chapter, an introduction to the vibra tion of continuous systems are given and then the types and effects of vibration on these systems are explained and also the importance of vibration is displayed. Also, up-to-date references about the moving bands and similar structures are given according to the historical prog ress. As a result of research, the vibration characte ristics and the equations of motion of the moving band are similar to those of bandsaw, pipelines containing flowing fluids. In the second chapter, the basic equations of a mov ing band subjected to axial force are given. For a mov ing band which moves along the x coordinate with velo city V, choosing the coordinate axes C X, w3 also moving with the band, the differential equation of the free transverse vibration is, EJwIV - Sw*1 pFw** = O where, EJ : Bending rigidity of the beamC elastic modulus times moment of inertia of the cross-section!) S : Axial force pF mass per unit length to the band X : horizontal coordinate moving together with the band VIIn this equation, shearing force and rotatory iner tia effects on the vibration are neglected. And the Z. being the constant distance between the two carrying rolls, we accept the values E, J and pF to be constant along the band. If we want to express the w function according to the fixed coordinates Cx,w), we must apply the variable transformations, x » f CX.T) and t = f (X.T) 1 2 So, the partial diferential equation of a moving band according to the fixed (x,w) coordinates is, EJ^L + CpFV2 - Si^L + 2VpF_^L + pF^L = O IV 2 2 öx ax axot at In order to make the diferential equation non-dime nsional, the following quantities are introduced, As a result, if the dimensionless quantities shown above and the coefficients VI a = C1'2 and l` 2 ft = - CS - pFV > EJ are used, the differential equation of motion will be as follows: VII77 - (*n + 2c*77 * + 77* ` »O Thus t, he problem reduces to solving the above equa tion subject to prescribed boundary conditions. In order to solve this differential equation, we firstly accept TîCÇ.T) = yCÇ)e`T Then, separating its variables the main equation reduces to the following form, y ~ fty + 2auy + a) y = O This is a linear and homogen ordinary differential equation of forth degree with constant coefficients This differential equation has no exact solutions and for this reason, the numerical solutions are obtained by using the finite differences method. In the third chapter, this finite differences method is widely eluciadeted. If the main equation is rewritten in terms of the parameters a and ft, one obtains, i 2 a J dlf d? dÇ where, F - l.O i Fz - -ft F = 2a« a Then, we divide the x-axie of the beam io N-l equal parts and evaluate the central finite differences for each pivotal points. These differences for the first, second and the fourth derivatives are, y = --4 y - y ) 2H n+1 n_1 y` ` `^ yn+1 ` 2yn + yn-S H2 vitiiv 1 y = - C y - 4y + 6y - 4y + y 3 H* where» N is the number of pivotal points on the x-axls and H is the number of equal spacings. Finally, we obtain a matrix equation in the type of CA3CY3 = X [Y3 where [A3 is a reel matrix with the <N*N+d> order, X's are the eigenvalues and CY3 are the mod functions. After introducing the boundary conditions which are for x=0 y s y J 2 'o y = y the matrix CA3 transforms into the matrix [B] which is in the order of <N*N> and takes the form of CB3CY3 = X CY3 In the fifth chapter, the set of N linear algebraic, homogenous equations has the trivial solutions Yi = O, corresponding to the straight configuration of equilib rium of the beam, but may have a non-zero solutions if and only if the determinant a of its coefficients which is a function of X is identically zero. The determinan- tal equation. a(/) ¦ O is an algebraic equation whose roots approximates to the N characteristic values. This determinantal equation was solved by using 60, 80 and lOO pivotal points and the frequency constants CKO were obtained for the first IX2 4 eight modes. Then, the C H - H ) extrapolations were applied. The modal functions corresponding to the frequen cies were obtained by normalizing them according to the displacements of t<he/ fixed ends. Then, N-l simultaneous algebraic equations were solved by a computer program using Gauss-Jordan method. The graphs of eigenf unction for the combination of a and 5 parameters were plotted in the fifth chapter. In the sixth chapter, the critical velocities are investigated. When the frequency curves are examined, we Bee that the frequency decreases while the band velocity increases. Some a values make the to eigenvalues zero. These band velocities are called as ` critical veloci ties`. The differential equation of the motion for u> = O becomes, J* -I2 ±_y + (&JL = o dÇ4 dÇ2 where, A l2S 2 /9 «a - a EJ and the general solution is, ixf -ixf y = C +CÇ+Ce^+Ce ^ ' i 2 a 4 If we apply the boundary conditions of the two types to the equations above, we obtain two homogen equa tion systems which each one has the order of d**4. We equate their determinants to zero and determine the a, val ues. We substitute the a values and obtain the criti- kr cal velocities of the band. This is,,1/2 Vkr C The main results of this study can be summarized as follows: 1- In the free transverse vibrations of the moving bands, while the band velocity increases, the circular natural frequency decreases. K2- When the decrease in frequency is approaching zero, the velocity is called the critical velocity of the band. This means that it is not advisory to run the band at these critical velocities. 3- It has been determined that the axial force has some 'effects on the amplitudes of modes, natural frequ encies and the critical velocites. When the axial force increases, it has been observed that the amplitudes dec reases, natural frequencies and the critical velocities increases. XI