Taşıma Matris Metodu

Abstract

Yatay eğilme ve burulma zorlanması gibi problemlerin çözümünde ortaya çıkan hiperstatik sistemler için, taşıma matris metodu uygun bir çözüm yolu olmaktadır. Bir çubuğun veya bir sistemin, iki ucu arasındaki kuvvet ve deformasyon büyüklükleri arasındaki lineer bağıntının belirlenmesi, taşıma matris metodunun anafikrini oluşturmaktadır. Mukavemet ve mekanik problemlerinde sınır değerler çok fazla olursa, bu sınır değerleri başlangıç değerlerine dönüştürmek ve bu suretle ara şartlardan dolayı girebilecek yeni sabitlerin önüne geçebilmek, taşıma matris metodu ile mümkündür. Bunun için de, problemi matris şeklinde formüle etmek gerekir. Bu çalışmada ilk olarak, giriş bölümünde taşıma matris metodunun genel tarifi yapılmıştır. Bir sistemde, meydana gelebilecek deformasyon büyüklükleri ve sol uçtaki büyüklüklere göre sağ uçtaki büyüdüklerinin nasıl bulunabileceği ifade edilmiştir. Ayrıca, eğilmeye zorlanan bir kirişte oluşan elastik eğrinin diferansiyel denklemi verilmiştir. Bölüm 2'de, taşıma matris metodunun, şasinin yatay düzlemde eğilmesine uygulanması anlatılmıştır. Kirişlerin eğilmesinde geçerli olan diferansiyel denklemi kullanarak, sol uçtaki büyüklüklerden sağ uçtaki büyüklükleri bulmak için alan matrisi teşkil edilmiştir. Daha sonra, bir bölgeden diğerine geçişi sağlayan nokta matrisi oluşturulmuştur. Böylece, sol baştaki büyüklüklerden sağ baştaki büyüklüklere kadar, her bölge ve noktadaki büyüklüklerin bulunabileceği ifade edilmiştir. Bölüm 3'de, yatay düzlemde eğilme probleminde olduğu gibi, alan ve nokta matrisleri oluşturularak, sol uçtaki büyüklüklerden sağ uçtaki büyüklükleri tespit ederek, burulma probleminin de çözümü yaplmıştır. Bölüm 4'de, bölüm 2 ve bölüm 3'de olduğu gibi, titreşim probleminin çözümü, farklı olan alan ve nokta matrislerinin teşkil edilmesiyle, her noktadaki büyüklüklerin bulunmasıyla yapılmıştır. Bölüm 5'de iki ve beş gözlü çerçeve modellerinin eğilme ve burulma yönünden, aynı beş gözlü çerçeve modelinin titreşim yönünden incelenmesine ait bilgisayar programıyla yapılan çalışmalara yer verilmiştir. Son olarak, bölüm 6'da sonuçlar ve öneriler sunulmuştur.

The matrix methods of structural analysis described in the two previous sections are suitable for the analysis of frameworks consisting of beam elements. For some framework problems procedures using transfer matrices have advantages. The theory is presented very briefly here, concentrating on its use in the analysis of commercial vehicle chassis frames. As is well known, the basis of the theory was developed by Falk, but a practical presentation for use in civil engineering is given by Kersten. In-plane bending of two-dimensional frames : As an example, Figure 2.1 shows the lateral horizontal loads on a chassis frame of a commercial vehicle and the resulting bending moment diagram. The analysis uses the load-displacement relations, as well as the equilibrium and compatibility conditions of the beam elements to find the loads at one end of the frame from those at the other. The state variables are the displacements and the internal loads at the ends of the beams. These are compiled in a column matrix, sometimes called the state vector, which, for a beam in bending, has the form (1.1) where v is the lateral displacement at the ends, p is the slope, M the bending moment, and Q the lateral or normal load (shear force) acting on the beam. The element "1" is added to include the external load, as will be shown later. Given the loads and displacements in the front cross member the state variables can be calculated for the next cross member by the use of transfer matrices. In fact, only some of the state variables at the front are known and the remainder depend on the boundary conditions at the rear end of the frame. Since the state variables at the rear are expressed in terms of those at the front, the structure can be analysed. Figure 2.3 shows the loads and displacements at the ends of a beam element in the kth bay of the frame. Since the loads at a "cut" are equal and opposite only those at the right hand side of the cut are define. A clockwise moment at the right hand end of the beam therefore appears as an anticlockwise moment at the left hand end of the adjoining structure. The left hand end of the element is indicated by the suffixes kk and the other end by kk+1. The relation between the state variables at the ends of the beam is given by (1.2) with the transfer matrix (figure 2.4) the elements of this matrix are the load-deflection characteristics of a beam in bending. Figure 2.2 shows, as an example, a 4 bay grillage acting as a simply supported beam with lateral loads at each cross member. While the support shown at IX the ends may be fictitious, displacements can be measured from the line joining them. The right hand support is free to move in the longitudinal direction. In this analysis the change in length of the beam elements is ignored, bending is therefore the only significant mode on deflection in the system. With this assumption the state variables at the end kk of the kth bay are (2.3). Following equation (1.2) the state variables at the end kk+1 are given by (2.9). The transfer matrix F/d linking the state variables at the two ends of the bay and including the effects of the two side members is shown overleaf. The elements in this matrix can be obtained from the transfer matrix for the single bay, remembering that there are two parallel beams (side members) transferring the load. Only the sum of the lateral loads at each cross member are known. They are only equal at each end of the cross member when the side members are the same, but unequal side members do occur, as in the subframes of buses, so that to keep the calculation general the lateral loads are eliminated from the variables by using the lateral displacement vhch- This displacement is found from the state variables at end kk from the first row of equation (2.9). In the transfer matrix F/d the rows for the slope and bending moment contain the terms to be multiplied by Vhch. Along the boundary k+1 the load in the cross member QtH introduces internal loads, which together with the external load at this section, combine to give the state variables at the front end of the next bay k+1 k+1 as in the equation (2.22) where the matrix Uk« is defined here as the nodal transfer matrix. The displacements at the rear of the kth bay are equal to those at the front on the k+lth bay, so that (2.13a), (2.13b), (2.13c) and (2.13d) similarly the load are transferred across the boundary and the main diagonal of (2.22) consists of ones,as shown in figure 2.9. The cross member Q^-i is deformed by the displacements uhcHj, Phcmj, Pwcfi.r. The moments at the ends of the cross member result from the load- displacement relations of the cross member considered as a beam element in bending. These relations are combined in the coupling matrix cm which can be written in figure 2.8. This matrix is inserted in the nodal transfer matrix Uk+i in the position indicated in figure 2.9. The last column of u^i contains the external loads, which are usually lateral loads, but, as will be shown, can also be moments acting in the plane of the frame. The state variables at the front end of bay k+1 can be written in terms of those at the front of bay k by using equations (2.9) and (2.22) in the (2.23). With this equation the state variables at the rear end of the frame in figure 2.2 can be calculated from those at the front. In general, the rear end of the frame will be denoted by the section rm. Examination of figure 2.2 shows that negligible elongation is assumed for the side members at the rear end of the frame. This assumption has no effect on the variables at the interior of the frame but gives the boundary conditions at the rear end shown later in equations (2.24a), (2.24b) and (2.24c). Out of the state variables in equations (2.3) at the front end of the frame the known variables are vn=0, Qn=Fi. The unknows are unjr, Pnj, piu and the bending moments Mn,i and Mn;. These are connected to the three unknown displacements by the coupling matrix which is in figure 2.8. At the rear end there are three known conditions (2.24a), (2.24b) and (2.24c). For practical calculations it is convenient to split the column matrix znyd into the sum of column matrices with the unknown variables as coefficients, and a separate column matrix of the external loads at the front end of the frame, as in figure 2.11. This relation can be summarized as (2.26), where the superscripts in brackets refer to the row number in the matrices of the state variable. Using equation (2.23) repeatedly on each term until the end nn is reached, the last equation will be (2.27). From the conditions at the rear end summarized in equations (2.24a), (2.24b) and (2.24c), the 1st, 5th, and 6th terms of matrix z^3"1 are zero, so that (2.28a), (2.28b) and (2.28c), where vm(2:i is the first term in the column matrix zj® etc. The unknows unjr, Pıı,ı, Pn, follow from equations (2.28a), (2.28b) and (2.28c). The bending moments in the front cross member can now be found from these displacements. Therefore, all the variables are known for the front cross member, and, using equation (2.23) they can be found for all the bay ends and thus for all the ends of the beam elements. The chassis frame type of structure shown in figure 2.2 has, as external loads, both lateral loads and in-plane moments at the nodes. These moments hardly ever occur in practice but they do, however, make it possible to calculate approximately the effect of longitudinal loads on one side of the frame. Since the changes in length of the side members can be neglected in the analysis, the longitudinal external loads can be introduced at any section along the length of the frame. Therefore, they can be introduced as a load system at the rear cross member as illustrated. Torsion in a symmetrical chassis frame : Torsion is caused by the action of unsymrnetrical loads acting normal to the plane of the frame. The analysis is restricted to symmetrical frames which are almost universally used as vehicle chassis frames. Figure 3.1 shows a frame with 4 cross members with both the general and particular notation used in the analysis. It will be noted that, due to symmetry, it is only necessary to analyse half the frame. Also from symmetry, the bending moments are zero at the centre of the cross members, which, however, carry a constant torque so that the conditions at these points can be represented by longitudinal hinges. Transfer matrices are readily applicable to the side members where the cross members are attached. The additional displacements (rotations) and loads, shown as the angle of twist, ui, and the torque, Mo, in figure 3.2 can be added to the state variables at the front end of the kth bay to form the new column matrix (3.1). The transfer matrix connecting the state variables at the two ends of the bay element is (3.2). This transfer matrix is obtained from the transfer matrix given in figure 2.4 by the addition of the torsional flexibility of the beam in the second row (with a negative sign), and a unit in the fifth row corresponding to the torque applied at the end kk. The matrices in equation (3.2) are given in full as figure 3.3 and the nodal transfer matrix (3.3) follows from the conditions at the k+1 fh cross member which is in bending due to XI the vertical displacement vkk+1 and the angle of twist u/hc« of the side member at its end. It is also being tvvisted at its outer end through the angle pkjcfi. Equation (3.3) is fully written out in figure 3.9. Equations (3.2) and (3.3) can be combined to give an equation similar to (2.23). Through the continued use of this equation the state variables at the rear cross member can be expressed in terms of those at the front cross member. Assuming, as shown in figure 3. 1, that there is a support at the rear corner of the frame, i.e. the rear cross member is horizontal, then the state variables at the front are unknown and can be written as vn, \j/n, Pn, Mn, Mon, Qn. These are incorporated in the u matris as before. At the section immediately aft of the rear cross member, section nn in figure 3.1, the zero state variables are (3.18a), (3.18b) and (3.18c) from which the state variables at the front cross member can be found. To do this the column matrix Znb is split up into separate terms with the state variables as coefficients, as in figure 2.11. The above procedure is suitable for the analysis of torsionally stiff frames. In the case of torsionally flexible frames, the torsion constant, and the second moment of area have such different magnitudes that, after successive multiplication of the transfer matrices, numerical errors can occur. For an approximate analysis of flexible frames this limitation is not essential and can be ignored since the ERZ method. Neither this method nor the transfer matrix method so far described involves the analysis of the joint properties of the frame. Therefore both methods give results to the same level of approximation and involve little effort. For frames with open section members the approximation is poor because the effect of the inhibition of warping is not taken into account. For the analysis of stiff frames with closed section members, especially doubly symmetrical warp free section, the above method is sufficiently accurate. Bending vibration of a chassis frame : The transfer matrix procedure makes it possible to find the bending natural frequencies and mode shapes of a chassis frame in a simple way. The frame shown in figure 4. 1 is assumed to be massless with its own weight, as well as the remainder of the vehicle weight and the payload, replaced by equivalent lumped masses at nodal points along the frame. The bending stiffness of the load-carrying body is added to the bending stiffness of the chassis over the appropriate length. The internal loads resulting from these assumptions are brought into the analysis through the nodal transfer matrices. Since the cab is normally mounted on flexible mounts it is represented by two lumped masses, each mounted on a separate spring, thus neglecting the mass coupling within the two-degress of freedom sub-system. The frame is supported on mass-spring systems, with the road springs, axle masses, and tyre springs represented separately. Although, in the case of trucks, leaf springs with widely spaced spring hangers are almost universally used, it has been found that it is sufficiently accurate to assume that they act at a single spring support, as shown infigure4.1. x« It is an advantage when using transfer matrices to keep the number of effective bays as low as possible. Closer/ spaced masses or springs can be combined in single nodes having equivalent properties. For instance, in figure 4.1 the mass itifa, representing the rear of the cab, mounted on the spring with a stiffness cFA is considered to act at the same nodal cross section as the front lumped mass of the load platform, m3. Comparative calculations have shown that the body cross members in front of and behind the rear axle position can be combined at node 6 without serious error for the first two natural frequencies. It can also be shown that an idealization with approximately 7 nodes gives sufficiently accurate results. In order to include the effects of the lumped masses and springs in the analysis the dynamic displacements, v, have to be specified as a variable at each node. The load increment at each node is therefore the product of the effective dynamic stiffness and the displacement, which for the rear of bay k-1, k (node k) can be expressed as (4.2). For oscillations with frequency eo at node k the effective stiffness, ojc, is found as shown in figure 4.5. When there are other dynamic effects at the same node they can be added to q^. For example, at node 6 in figure 4.1, because of the effects of tiie mass of the body and payload, as shown above, and of the axle mass, road, and tyre spring, q,, becomes (4.3). The variables can now be written as (4. 1). The transfer matrix given in figure 2.4 is used, but, as there are no static external loads, the fifth row and column are eliminated. The u matrix is now in figure 4.4. At the front of the frame the displacements vn and Pn are unknown. The loads are (4.4) and (4.5). Equation (4.1) can be written as in figure 2.11 with the state variables as coefficients in figure 4.6. At the rear of the frame the load variables beyond the last node n are (4.7) and (4.8) so that the third and fourth row terms of Zj» are zero and the equivalent rows of figure 4.6 become (4.9) and (4. 10). The unknows Vı i and Pi ı can be found. For non-trivial solutions of this set of homogenous equations the determinant of the coefficients must be zero. By successive multiplication of the transfer matrices, using the elements containing eo2 in the nodal transfer matrices, the higher exponents of ©2 and consequently the higher orders of natural frequencies can be obtained. The recommended method of solution is to calculate the left hand side for values of co2 close to the expected value and interpolate to find the zero value of the left hand side. Now that m2, MtJ-1^, MU(2), QpJ-1^ and QpJ® are known. For a given deflection and associated slope at the front of the frame the displacements at the other nodes can be found from the transfer matrices when the frequencies obtained. In this way the mode shapes of the vibrating frame can be found at the natural frequencies calculated.