Simetriden sapmaların ve ilave elastik yayların makina temellerinin davranışına etkisi

Abstract

Makina temellerine statik yüklerin yanında dinamik yükler de etkimekte ve sonuçta atalet kuvvetleri oluşmak tadır. Sistemin dayanımının yanında işletme esnasında oluşacak titreşim genliklerinin sınırlandırılması, makina temellerinin boyutlandırılmasının esasını oluşturur. Bir makina-temel-mesnet sistemi, makina ve temel (katı cisim, sistem A) ile temele bağlanan elastik kısım dan (sistem B) ve de mesnetlerden oluşmaktadır. Makina temelleri ile ilgili mevcut yayınlarda, sis temin simetrik olduğu durum gözönüne alınmıştır. Çeşitli nedenlerle sistemin simetrisinde bozulma olabilmektedir. Bu durumun detaylı bir şekilde incelenmesi gerekmektedir. Bu çalışmada, planda birbirine dik iki düzleme göre simet rik olmayan makina-temel-mesnetler sisteminin serbest ve zorlanmış titreşimleri incelenmiştir. Ayrıca bu tit reşimlerin frekans ile genliklerinin hesabı ve sınırlan dırılması da incelenmiştir. Amaca uygun olarak önce sistem tanıtılmış ve sis temin simetriden uzaklaşma nedenleri ayrıntılı olarak verilmiştir. Sonra, D'Alembert prensibi kullanılarak genel halde sistemin hareket denklemleri çıkarılmış, serbest ve zorlanmış titreşimleri incelenmiştir. Simet rinin olmadığı genel durumdan tamamen simetrik olma duru muna kadar tüm özel durumlar genel amaç doğrultusunda birer birer incelenmiştir. Simetrik olma ve olmama durum larında, sisteme dışarıdan eklenen elastik bir B sistemi nin, tüm sistemin titreşimlerine etkisi araştırılmıştır. Bu sayede, genlikleri çok büyük olan makina temellerinin basit B sistemi şeklinde ilavelerle genliklerinin küçül tülüp kabul edilebilir sınırlar içine çekilebileceği gös terilmiştir. Konunun kolay anlaşılabilmesi için sayısal uygula malar da yapılmıştır.

The loads which are applied on a machine foundation are composed of static and dynamic loads. Inertial forces and loads produced by machine are dynamic character. The specification of the dynamic loads necessitates the investigation of the free and forced vibrations of the machine - foundation - supports system. The basic principle of the design of a machine foundation is to assure that the amplitudes of the foundation which come into being during the operation of the machine do not pass over the permissible limits. On the other hand the foundation has to be design in a way that it should have an adequate strength to resist to the static^and dynamic loads of the system. In the literature related to that subject one can find studies dealing with the design of the machine foundations. However, most of the authors assume a machine- foundation-supports system having symmetry with respect to two vertical planes which are perpendicular to each other. In this case, the total weight of the system W passes through the centroid of base plane and through the elastic center of elements of the support. However, in such cases when the machine can not be placed on foundation symmetrically, when the machine foundation itself is not symmetric and when the settling of supports can not be assured to be completely symmetric, deviations from the symmetry mentioned above can appear. Furthermore, similar deviations may come into being when the existing machine foundations which do not work satisfactorly have to be improved or to be strengthened. During the strengthening, the supplementary elastic parts connected with the machine foundation may not be placed symmetrically because of the form of the foundation base. This case may also cause a deviation from the symmetry of the system. xiv Due to reasons mentioned above, it is of practical importance to investigate the machine foundations which do not have any symmetry in detail. The subject of this study is the investigation of the free and forced vibrations of the machine-foundation- supports system which is not symmetric with respect to the two planes that are perpendicular each other in the plan. However, the aim of the study is to calculate the frequencies and the amplitudes of those vibrations and, to show how to assure the amplitudes of the forced vibrations' within acceptable limits. In the present study, a non-symmetric case is considered and the general equations of motion of the system are written and, free and forced vibrations are examined. The system has, in general, six degrees of freedom and, therefore, six natural frequencies (one corresponding to each mode of vibrations). Three of them are translations along the three principal axes (x, y and z) and the other three are rotations about the three axes. The geometrical and mechanical specifications of a non-symmetric machine-foundation-supports system are shown in Fig. 3.1 schematically. The following assumptions related to the system used in the derivation of the governing equations of the problem. 1) The machine-foundation-supports system is not symmetric with respect to the two planes which are perpendicular each other in the plan. 2) The centroid of the base plane and the elastic center of elements of the support coincides. 3) At the system, there are only deviations from the symmetry in the horizontal plane and the deviation, ö and 6, are small. 4) The supporting elements of the system are on the same horizontal plane and they are of a distance s from the gravitational center. 5) The effect of the damping is not considered. Sometimes the complete of the foundation mass and sometimes only any part of it may be assumed a rigid body together with the machine. In this second case, the foundation system is composed of the two subsystems. a) Rigid body, system A, b) Elastic parts, system B. The system B may contain the distributed or lumped masses. In general, the effect of them is small and it may be neglected (Fig. 3.1). xv It is assumed that the elements of supports of the system A (rigid body) are independent from each other and without mass and are linear elastic and/ they are on the same plane (x y plane). Since they are independent from each other ana without mass, it is known that only the stiffness influence coefficients of the support elements will appear in the equations of motion. However, the system B is composed of beams or frames along axis x and y and generally they may be assumed independent from each other. In the case that there is a slab between them, this part may be assumed a rigid body for the motions in the horizontal plane. In the case that the system B is composed of the separate frames or beams as described above, it is obvious that the effect of this part to the rigid body motion will be obtained by summing the effects of separate parts which are independent from each other. In the analysis of the frame the effects of a) the axial deformations, b) the flexural stiffness of beams about the z-axis, c) the torsional stiffness of beams around its own axes, are neglected, the effect of this frame to the rigid body at the connecting points may be explained as follows : The axial forces in the xy-plane are given in Eq. (3.1), the moment about the principal axes (x, y, z) and the shear force along the axis z are given in Eq. (3.2)., The.parameter k,k,k,k,h,h,d,,b,,b^... xn'. yn znx/ Z.nyl.c nx'. nyi nx d and d in the equations are. the stiffness influence coefficients at the points connecting the beams of the system B to the system A and, their subscripts indicate to the directions of rotation. Sig x and sig y show the signs of absis and ordinate of Çhe connecting point of the beam N. When the masses on the system B are not neglected, it is known that the coefficients in Eqs. (3.1) and (3.2) depend on the circular frequency q of the complete system and also the masses depends on system B. Here, the parameter q shows the circular natural frequencies A. of the system for the free vibrations and the circular frequency a of the machine for the forced vibrations. When the number of mass is large, it is recommended to reduce the dynamic degree of freedom of the system B. xvi In the connecting point N of the beams which form the the system B to the system A, if it is desired the bending moments and the shear forces around the z-axis can be considered. In Pig. 3.1 and equations, the following notations are used 5 : the center of gravity of the rigid body, O : the elastic center of supports of the system A in the horizontal plane, the centroid of base plane, W, M_. : the weight and the mass of the rigid body, k, k, k ;d,d,d : the stiffness influence coeffxcxents of the support elements (translations along the three principal axes x, y, z, respectively, and rotations about them) u, v, w : translations of any point N of the system along the principal axes x, y, z, respectively (They are assumed to be positive in the directions of the axes), , , : rotations of any point N ^ of ine principal axes x, y, z, respectively (for the positive directions for rotations the right hand rule is applied), 6, S, s : distances between the x v ¦* gravity center of the rigxd body S and the elastic center 0 along of the x, y and z-axes, respectively (6 and' <5 have the negative sign in the positive area of the corresponding axes and have the positive sign in the negative area of them), It is assumed that the translations and the rotations are very small, namely the quadratic terms such as u, u.w,...,(j),... may be neglected in comparison to the terms of u, v, w,..., $,... and accordingly one can write sin = tg 0 - , cos = 1. Using the rule of superposition and Eq. (3.7), the translations and the rotations of the point N of the rigid body are given by Eqs. (3.8) and (3.10). During the motion, the forces effecting to the system may be classified as follows : 1) Inertial forces : Inertial forces which are opposite to the direction of motion and are assumed to be positive along this direction and they are given in Eqs. (3.17) and (3.18). xvix 2) Exciting forces and exciting moments : These effects are positive in the positive directions of the principal axes and in the positive rotations. They are given in Eqs. (3.19) and (3.20). 3) The reactions at the supports : These reactions are given in Eqs. (3.12), (3.13) and (3.14). Their positive directions are shown in Fig. 3.7. 4) The effects of the system B : These effects are given in Eqs. (3.15) and (3.16). Their positive directions are the same as that described for the reactions at the supports. The general equations of motion are written by using d'Alembert's Principle. The coefficient at the equations of motion given in Eq. (3.47) and the abbreviations are stated. In these expressions, the terms having 6 and 6 show the effects of the deviations from the symmetry. However, the terms having the superscript (b) show the effects of system B. By using the sinusoidal expressions for the translations, rotations and exciting forces and moments, free and forced vibrations of the system are examined. The complete special cases are derived by considering the non-symmetric general case step by step. When the system has the symmetry with respect to the xz-plane, the points S and O are on the same plane. The expressions of (6.1) and (6.2) are valid and Eqs. (3.47) are uncoupled into the two systems having three equations. They are given in Eqs. (6.3) and (6.4). The complete symmetric case is presented in Chapter 9. In this case the expression of (9.1) is valid and the equations of motion are expressed in Eqs. (9.2), (9.3), (9.4) and (9.5). The natural frequencies for free vibrations are given in Eqs. (9.11), (9.12), (9.30) and (9.31). The amplitudes w, i >, u, , v, of the forced vibrations are given in Eqs. (9i?2), (9.S5), (9.47), (9.48), (9.49) and (9.50), respectively. The required difference between a> and the natural frequencies X, X.,A, p for the free vibrations, and the permissible amplitudes are given by the related codes of various countries. The translations of the system are explained for the symmetric case when an earthquake-excitation is present along one of the coordinate axis and the differences in the equations of motion are noted. When the system is symmetric and when there are only the trans lational springs between the machine and foundation in one direction, the equations of motion for the machine-foundation-supports system are obtained. xviii Finally, a special case is examined and the system is assumed to be symmetric and the system B is not present. The equations of motion of the system are written under the assumption that there are an earthquake-excitation along the direction of the x-axis and the springs between the machine and foundation and machine permit only the translation at the direction of the x-axis. For the clear presentation of the subject, the numerical solutions are given and their results are discussed., The important results obtained in the present study can be summarized as follows : In the present literature related to the design of the machine foundations, the machine-foundation-supports system is assumed to be symmetric with respect to two planes which are perpendicular each other in plan. Due the reasons discussed above, the deviations from the symmetry of the system can appear. This facts make it necessary to investigate the non-symmetric systems in detail. In the general case the equations of motion of the system are written arid, free and forced vibrations are examined and the equation of natural frequencies are found and, the amplitudes for the forced vibrations are obtained. The effects of the deviations from the symmetry are denoted by 6x and 8. The special case are examined according to the aim of this study, separately. Both for symmetric and non symmetric system, the effects of the system B which is joined to the main system from the outside are investigated. It is assumed that the system B composes of the elastic springs. It is shown how to reduce the amplitudes of the machine foundations when they are large and above the permissible limits by joining simple supplements such as the system B. The symmetric system is considered and it is assumed to be subjected to an earthquake ground-motion excitation along the x-axis and the translations of the system are investigated and the differences at the equations of motion are explained. For a symmetric system, the motion equations of the system are written when there are the springs which permit the translation only along the direction of the x-axis between the machine and foundation. Considering the two cases mentioned above, the motion equations are written for the symmetric system without the system B. xix At the numerical solutions, the effects of the deviations from the symmetry and the earthquake ground- motion excitation and the springs between the machine and foundation, on the vibrations of the system are shown.