Dynamics of rotationally periodic structures

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INTRODUCTION
Many engineering structures have a more or less cylindrical form. In some cases, such as certain pipes, chimneys, or pressure vessels, the structures are axisymmetric, and there is now a considerable range of computer programs available for analysing these, taking full advantage of their axisymmetric nature. However, if a structure is not exactly axisymmetric, and cannot be represented approximately by an axisymmetric idealization, it is at present necessary to analyse the whole structure.
The class of structure considered in this paper consists of those which have rotational periodicity. By this is meant that, if the geometry for any radial and axial position is defined at some angle 8, it is identical at (8 + n&), where do is 2 7~/ N , and n and N are integers. N is structure-dependent and n is any integer less than N. It follows that once the geometry has been defined over a sector or substructure from 6' to 8 +do, the remainder of the structure can be obtained by repeated rotations through do. (Note that the sector can start at different 8 periodicity can be exploited to enable a series of analyses on one sector to be used to obtain all the information available from an analysis of the complete assembly.
A cooling tower with legs is an example of a rotationally periodic structure. Figure 1 shows a tower from Didcot Power Station. This has 40 pairs of legs. If the geometry of the tower is completely defined for an angular segment of 2~r / 4 0 rad, such as the substructure boundaries  Figure 1, the rest of the tower can be generated by repeated rc E ions of the segmen through 2~/ 1 4 0 rad. It is not possible to consider the tower to be axisymmetric unless the legs are approximated as a uniform ring. Nevertheless the axisymmetric assumption has been made in almost all analyses of cooling towers performed to date,'.' as it is impracticable to perform calculations in sufficient detail to include all the legs simultaneously. However, an analysis of the sector containing 1 pair of legs shown in Figure 1 is perfectly feasible. The methods described in this paper show how a series of analyses of this one sector could be used to obtain an analysis of the complete tower, without introducing any additional approximations.
In this paper the free vibration of rotationally periodic structures is considered initially. These are frequently analysed using the finite element method, and this paper shows how such an analysis can be very considerably simplified for such structures (alternatively, how considerably more complicated structures than hitherto can be treated). However, the method proposed in this paper is not confined to use with finite elements, but can be used in any matrix analysis of the structure, provided it is linearly elastic. Furthermore it is not confined to vibration, but can be used in other structural eigenvalue problems, such as buckling.
The finite element analysis of infinite linear periodic structures has been described by Orris and Petyt.3 A very brief description of the work on finite rotationally periodic structures described in this paper, in which the theory is considered as an extension of Orris and Petyt's work, has already been given. 4 The normal modes of the structure were considered to be standing waves, for which only certain values of the complex propagation constant are allowed, In this paper an alternative description of the modes is given, and the analysis is developed to cover a wider range of problems.
In a finite element analysis of a structure, real stiffness and mass matrices, [ K ] and [MI are set up, and the eigenvalue equation is then solved to give the natural frequencies w and eigenvectors { u } of the system. 'There are many computer programs available for carrying out this process. However, for complicated rotationally periodic structures it may be impracticable to solve equation (1) directly because of limitations on the number of degrees-of-freedom which can be handled by eigenvalue subroutines.
In section 'Rotating mode shapes' of this paper it is shown how the eigenvectors of equation (1) can be expressed as rotating eigenvectors, for rotationally periodic structures. Section 'Use of complex constraints' shows how the phase relationship between adjacent substructures undergoing a rotating vibration, can be exploited, by the use of complex constraints, so that a modified form of equation (1) is solved for a single substructure. No additional approximations are introduced using this technique. Some examples of the use of the method are given in section 'Examples'.
After the discussion of the free vibration problem, this paper considers the direct solution of the forced damped vibration problem. Static problems are of course a special case of this. For steady state conditions, the equation of motion becomes: where [ K ] , [C] and [MI are the real stiffness, damping and mass matrices, and { F } and { u } are complex force and displacement vectors. It is assumed that the excitation is at an angular frequency w . There are many programs available for assembling the stiffness, mass and damping matrices of complete structures, and solving equation (2) for { u } .
S7ie tbrced-vibration method is based on a consideration of the response of a structure to a rotating force distribution.
It is possible to decompose any arbitrary force, rotating or not, into rotating components.
In this paper we first show how any rotating force acting on the complete structure can be described in terms of a series of components acting on only one substructure. This is then generalized to include stationary forces or any arbitrary force distribution. We then show how it is only necessary to solve a modified form of equation (2) for one substructure to obtain the response to each component of force. Just as in the free vibration case, it is thus possible to deal with considerably more complex structures than would be the case if the whole structure had to be considered at one time.
MacNeal et al' have described a facility similar to that discussed in this paper, which they have incorporated into the program NASTRAN. Their description of the method uses real arithmetic throughout, whereas in the present paper complex arithmetic is used. This enables the components into which a general applied force is split to be interpreted physically as members of a series of rotating forces, which the present author feels increases the understanding of the nature of these components. Such an interpretation emphasises that, for some structures, many of the force components are completely missing, which considerably reduces the computational effort required. The equations describing the method are considerably simplified when complex arithmetic is used, as it is not necessary to consider sine and cosine components of force separately.
MacNeal et alS consider rotationally periodic structures for which each substructure is connected only to the substructures on its immediate right and left. However in this paper, the treatment is generalized so that each substructure can be mechanically connected to any number of the other substructures, so long as the connections are rotationally periodic.

Classification of mode shapes
Orris and Petyt's paper on infinite linear periodic structures lists references to other work in that field. The vibrations of such structures are normally described in terms of the propagation of free waves. These vibrations can occur at any frequency in a number of allowable bands. The free vibrations of rotationally periodic structures, however, occur only at certain discrete frequencies. It is possible to obtain these by consideration of the standing waves in the structure using a modification of the infinite linear periodic t h e~r y .~ However, in this paper, the conventional normal modes of a finite structure are used to describe its motion.
A rotationally periodic structure consists of N identical substructures. Equation (1) is the eigenvalue equation for the complete structure. The total number of degrees-of-freedom in the structure is NJ. The degrees-of-freedom are ordered so that the J degrees-of-freedom of the first substructure are followed by J degrees-of-freedom on the second substructure, and so on. An eigenvector of the whole structure, { u } , can be written With axisymmetric structures, it is found that most modes of vibration occur in degenerate orthogonal pairs. This is because, if a mode has a maximum deflection at some point on the structure, it is clearly possible, because of the axisymmetric nature of the structure, to rotate the mode shape through any angle and not change the frequency of vibration. A similar effect is found with rotationally periodic structures. The possible mode shapes, { u } , fall into three classes, depending on the relationship between the shapes for individual substructures. These are: (a) Each substructure has the same mode, shape as its neighbours, i.e.
(b) Each substructure has the same mode shape as its neighbours, but is vibrating in antiphase with them, i.e.
(all i) (4) u ( i ) ----u(i+l) (c) All other possible mode shapes. Modes of class (a) or (b) do not exibit degeneracy (except for any 'accidental' degeneracy which may occur if an unrelated mode shape is associated with the same natural frequency).
For class (a) modes, the eigenvector { u } can be written and it is obvious that rotating the mode shape through any integral number of substructures leaves it unchanged, so that no other mode shape is needed to describe the rotation. Class (b) modes can exist only if N is even. { u } has the form: Clearly rotation of the mode shape through an even number of substructures leaves it unchanged. When it is rotated through an odd number of substructures, it becomes -{u}. This does not, however, represent a new mode shape, but a change of phase of T in the vibrations. All other mode shapes fall into class (c), and exhibit two-fold degeneracy, and must be considered in greater detail.

Orthogonal pairs of eigenvectors
A class (c) eigenvector { u } has u(')# u('+') and u ( ' ) # -u('+') . It will be assumed that it is normalized so that {u}'{u} = 1. Since all the substructures are identical, the deflected shape { u ' } obtained by rotating { u } round one substructure, is also an eigenvector, distinguishable from { u } , and with the same eigenvalue. It is not, however, orthogonal to { u } in general. There must be a normalized eigenvector, {U}, orthogonal to { u } , with the same eigenvalue, to enable any eigenvectors other than { u } to exist with this eigenvalue. { u ' } has the form . . . -9=

{U'}T{U'} = ( C { U } T + S { U } T ) ( -S { u } + C { U } )
It is now possible to write the deflections on any one substructure in terms of the deflections and, for the jth substructure Repeated application of equation (14) allows the components of { u } and {ii} acting on any substructure to be expressed in terms of those acting on substructure 1.

Complex eigenvectors
The where n is an integer. Similarly, the relationship between the components of { z } acting on neighbouring substructures, given in terms of { u } and {a} in equation (14), can be written Equation (18) allows the complex eigenvector acting on any of the substructures to be expressed in terms of the vector acting on only one substructure.
In the method of calculating the eigenvalues and eigenvectors described in the following  (19) gives the deflected shape as { u } , but at wt = +, equation (19) gives and, from equation (1 l), this is { u ' } . At t = $ / w the deflected shape of the structure is therefore the same as at t = 0, except that it has rotated round one substructure. The complex eigenvectors { z } therefore describe rotating modes of vibration of the structure, in which the same instantaneous deflected shape reappears after successive intervals of time t = $ / w , rotated round an additional substructure each time.

Values of n
The value of n used to define $ in equation (17) will now be discussed. n depends on the particular rotating mode shape { z } which is being considered, and is a measure of the periodicity of the mode. Clearly, from equation (17) U For N = 6, n = 3 , p = 6, and there is no such relationship between the u(j)'s. However, if N / n is not an integer there is not a relationship as simple as equation (22). If N = 7 and n = 2, N / n = 3 3 . This implies that the deflected shape repeats after every 3 3 substructures, which in general is not possible. However, if the deflected shape is { u } at t = 0 , then after a time t = $ 1 2~ (that is, half the time it takes for the deflected shape to move round one substructure) { u } will appear to have moved round four substructures. { u } may thus be thought of as having a latent periodicity of 3-5 substructures, which manifests itself when it is rotated through half a substructure.
The discussion of complex eigenvectors { z } has been concerned with class (c) modes of the structure. However, class (a) and (b) modes may be expressed in a similar way. As there is no orthogonal mode in either of these cases,

Summary
In this section we have seen how most possible modal shapes of a rotationally periodic structure exist in orthogonal pairs. By combining these into complex eigenvectors { z } , they may be considered as rotating modal shapes. The displacement of any substructure can be related to the displacement of one particular substructure by a phase angle $. The non-rotating class (a) and (b) modes can be considered as special cases of the rotating modes, with $ = 0 and n-. There are N possible values of $, but only (N/2)+ 1 (even N ) or ( N + 1 ) / 2 (odd N ) of these represent different mode shapes, the remainder corresponding to similar shapes rotating in the opposite sense.
Any complex mode shape of the structure can be expressed in terms of the deflected shape of just one substructure, by use of the appropriate value of $. As the number of values of $ is limited it is possible to examine each in turn. In the following section, it is shown how, for a given value of 4, it is possible to calculate all the corresponding rotating mode shapes of one substructure, and thence of the whole structure, from the mass and stiffness matrices of one substructure without any approximation. By repeating this for every possible value of 4, it is possible to obtain all the mode shapes of the complete structure.

Equation of free vibration
The rotationally periodic structure is divided into N identical substructures, each of which contains J degrees-of-freedom. However, each substructure is connected to a number of other substructures. This number is at least 2 and may include all the other substructures. The stiffness matrix for one substructure can be expressed as a number of J x J submatrices. For the ith substructure these submatrices are K$i), which represents' the interaction between the J degrees-of-freedom in the substructure, and up to N -1 triads of submatrices K:;), Kjj' and K:;). These triads represent the interaction between the ith and jth substructure. The equation of motion of the complete structure, equation (l), can therefore be expanded to give equation (23).

Solution with complex constraints
The first row of equation ( The conventional eigenvalue problem encountered in structural mechanics involves two real matrices, as in equation (1). The eigenvalue problem in equation (28), however, involves complex mass and stiffness matrices. However, it may be seen by inspection of equation (28) that both matrices are Hermitian, that is, that the real parts are symmetric and the imaginary parts are skew-symmetric. It is a property of Hermitian matrices that the eigenvalues are reai, but the eigenvectors are complex. Solution of equation (28)

The constrained stiffness matrix [K,] is obtained by applying a transformation to [ K O ] :
where and [TT*] is its conjugate transpose. It may readily be verified that [K,] evaluated from equations (29) and (31) is the stiffness matrix used in equation (28). The mass matrix may be obtained in the same manner.
For most practical problems, each substructure will only be connected to a small number of other substructures. Many of the submatrices in equation (29) may therefore have all zero terms. If only a few degrees-of-freedom are joined on those substructures which are connected, many of the rows of the submatrices which remain in equation (29) will also be zero. The size of [KO] may not therefore be particularly large, and the computational effort required to evaluate [K,] from equation (3 1) can thus be considerably reduced.

Computer time and storage
It is interesting to compare the computer time and storage required in a solution of a structural eigenvalue problem directly, solving equation (l), and using complex constraints on a single substructure, solving equation (27 (1) the time is therefore proportional to N 3 J 3 . To use complex constraints, only J degrees-of-freedom are used, but each multiplication is complex, and thus equivalent to about four real multiplications. The time to extract the eigenvalues for one value of 9 is thus -4J3 For an even value of N (which is the worse case) there are N/2 + 1 values of (ignoring negative values). The CPU time to extract all the eigenvalues of the structure is therefore proportional to (2N +4)J3. The ratio of CPU times for analysing the whole structure, and for an analysis using complex constraints, is N3/(2N+4). Even for quite small values of N this ratio is substantial, e.g. for N = 6 it is 13.5.
To store the matrices for the whole structure, core storage proportional to N 2 J 2 is required. When complex constraints are used, the number of array elements to be stored is J 2 , but as these are complex the storage required is proportional to 25'. The ratio of core storage for the mass and stiffness matrices required for the whole structure, and for one substructure using complex constraints, is thus N2/2. Again, this ratio is large even for small values of N, e.g. for N = 6 it is 18.
With many in-core eigenvalue subroutines core storage limitations put an upper limit of about 200 degrees-of-freedom on the size of matrices that can be solved directly (using equation (1)). This means that a rotationally periodic structure with more than a very few substructures cannot be analysed at all, except by gross over-simplification, if the complete structure is included. With the use of complex constraints the storage limit is about 200/J2, or 140, degrees-of-freedom, after application of the constraints. As only one substructure is represented, however, a fairly detailed substructure can be considered.

EXAMPLES
To illustrate the use of the method in a practical problem, its application to the calculation of the normal modes of a fully supported alternator end-winding is described in this section. The end-winding consists of 48 identical pairs of conductors. Each conductor leaves the cylindrical core of the machine (see the schematic idealization in Figure 2), describes an involute of a circle on a conical surface to the nose region, at which point it is connected to another conductor, lying on a different conical surface, which returns to the core. Each conductor has interconnexions along the whole of its length with its nearest neighbours, and also with the It has been found that one conductor can be adequately represented by about 18 beam finite elements. The nodes are shown schematically in Figure 2 . There are 6 degrees-of-freedom for each node, giving 102 unconstrained degrees-of-freedom per conductor. In order to simplify the structure, it is assumed that each bracket can be replaced by two brackets, each of half the correct stiffness, so that the number of brackets and the number of conductors are equal. The end-winding then consists of 48 identical substructures, each including one conductor, a support bracket, and one set each of inter-coil and inter-layer connexions. One such substructure is shown in Figure 2.

NODE ELIMINATED WITH COMPLEX CONSTRAINTS (THE NUMBER REFERS TO THE NUMBER OF SUBSTRUCTURES BETWEEN THE NODE AND THE RETAINED SUBSTRUCTURE)
The nodes of the first boundary of the substructure, and internal nodes, are shown as solid circles. The second boundary nodes, which form part of other substructures, are shown as open circles. By each of these there is an integer n. This represents the number of substructures through which it is necessary to rotate the original substructure in order that the open circle node lies on the original substructure.
If calculations of the normal modes of the complete structure are performed for a particular value of $, as defined in previous sections, it is necessary to consider only the substructure shown in Figure 2. Each open circle node has a constraint applied, of the form relating it to the corresponding solid circle node. It is therefore possible to perform calculations for the complete machine, using 120 complex degrees-of-freedom. To analyse the complete machine, without using complex constraints, would take 120 x 48, or 5760, real degrees-of-freedom, which is impracticable for a free vibration problem. Table I shows some of the natural frequencies of an end winding which has been analysed using complex constraints, for various values of n. For comparison, the frequencies of a coil, without any supports of connexions to other coils, are shown. It can be seen that the influence of the supports on the behaviour of this machine is considerable. Another example of a structure which has been analysed using the methods described in this paper is the cooling tower shown in Figure 1. The substructure was idealized with Five of DebNath's shell elements7 together with two beam elements representing the pair of legs. All the degrees-of-freedom on one boundary were related to those on the other using complex constraints. Full details of the calculations,8 and of measurements on both a model and a full scale structure,' are given elsewhere, but the results are summarized in Table II. It can be seen that the agreement between model and calculated frequencies is excellent. The lower modes of the full scale structure have slightly lower frequencies than predicted. This is because of the effects of foundation flexibility.'" After application of the constraints, the substructure contains 72 degrees-of-freedom, so that the extraction of the eigenvalues is a simple and inexpensive procedure. However, to analyse the full structure, using an equivalent idealization, would require 40 x 72, or 2880 degrees-offreedom, which would be a very complex and expensive problem. Another structure which has been analysed using this method is a 151-bladed turbine wheel. The mesh of 20-noded isoparametric elements used for one substructure, which represents a single blade rigidly clamped at the disk and coupled to its neighbours at the tip, is shown in Figure 3. The mesh contains 77 elements and 605 nodes. To reduce the number of degrees-offreedom to an acceptable level, the nodal condenstation technique6 was used. It is essential that ail degrees-of-freedom common to more than one substructure are retained as masters, and that a sufficient number are retained on the remainder of the structure to ensure sufficient accuracy. 123 master degrees-of-freedom were retained in the analysis. The node condensation was performed first, followed by the application of complex constraints for each value of n, and extraction of the eigenvalues. This means that the expensive node condensation stage need only be performed once. The results of the analysis are shown in Figure 3, in the form of the frequencies plotted as a function of modal periodicity n. It can be seen that the existence of coupling between the blades has a major influence on the frequencies, demonstrated by their variability with n.

Rotating forces
The previous section have shown how the free vibration problem can be solved for rotationally periodic structures using the methods of complex constraints. Forced vibration problems can either be solved using modal synthesis, in which case the methods described above are of use, or by direct solution. In this section we show how the direct solution can be simplified to a series of solutions on a single substructure, by decomposing the force into a series of rotating components. We first consider the decomposition of a general rotating force, and then generalize the treatment to include any arbitrary force distribution.
We first consider a rotating force acting on the complete structure. This consists 0f.N identical substructures. These are divided into M groups, containing equal numbers of substructures. The force system acting on the complete structure may be called a rotating force system if the force acting on any degree-of-freedom on one substructure has the same amplitude as the force acting on the same degree-of-freedom in the N/Mth following substructure and there is a fixed difference between them. Thus, if Fk,j eiwt is the force acting on any degree-of-freedom k, in the jth substructure: where p is an integer. After a time t = 4 / w , the instantaneous force distribution is identical to that at t = 0, except that it has moved round N/M substructures. If the complete structure can be divided into L groups, each subject to a force distribution identical in both amplitude and phase, then: so that: where p = 1 , 2 , . . . . Equation (40)shows how a system of rotating forces acting on a rotationally periodic structure can be expressed in terms of a number of components, Ak,p, each of which describes a rotating force, which has the same amplitude on all substructures, and a constant phase relationship between adjacent substructures (see equation (36)). It is shown below how this property can be used to enable the deflections on the whole structure to be obtained from computations on a single substructure.
The decomposition of the forces described above assumed that the initial force distribution Fk,j eiw' is rotating (i.e., amplitude repeats after every N / M substructures) and periodic (amplitude and phase repeat after every NIL substructures). However, a non-periodic rotating force is obtained with L = 1, and a completely general force can therefore still be expressed in terms of N rotating components by setting M = L = 1, so that the analysis puts no restrictions whatever on the spatial distribution of the force. If a number of different frequencies is present, the analysis above can be applied to each frequency component separately.

Response calculations
The response of the structure to one of the rotating force components is now considered. The vector {A,}, which is the pth component of the force on the first substructure, is the pth column of [A], found from equation (40). From equation (36) the force on the jth substructure is: Since the force on the jth substructure (for any j ) has the same amplitude, but a phase lag of 4 = 27r(j -1)(L + M ( pl))/N behind that on the first substructure, it follows that after a time t = 4 / w the instantaneous force distribution over the whole structure is the same as at t = 0, except that it has moved round ( j -1) substructures. Furthermore, at t = 4 / w + T, where T has any value, it is the same as it is at t = r, except for being displaced by ( j -1) substructures. The response of the entire structure at t = 4 / w + T is therefore identical to that at t = T , except for a displacement round ( j -1) substructures. The displacement vector for the first substructure, The ordering of the degrees-of-freedom in the substructure analysed is immaterial. If the order differs from that assumed above, appropriate rows and columns in [YC,] and [TI should be interchanged.
To illustrate the use of complex constraints on a periodic structure, the example shown in Figure 4 has been analysed. This consists of a highly idealized alternator end-winding. The structuie lies in a plane, and only in-plane motions are considered. There are therefore three degrees-of-freedom at each unclamped node. There are six phase groups each containing 1 coil which runs from the core of the machine, to the nose, and back to the core. Each coil is connected to its neighbours by a ring beam at the nose. Oscillating forces of unit magnitude, and phases of 0, -120, -240 degrees are applied at the points shown.
Initially the complete structure was analysed. A single substructure (see Figure 4) was then analysed, and the complex constraint facility described in the previous section was used on all three degrees-of-freedom on node 8, relating them to the corresponding degrees-of-freedom of node 3, with JI=-120 degrees. For a damped forced motion, the results for substructure 1 were identical to those obtained when the whole structure was analysed. However, the analysis of the complete structure required 1-9 times as much core storage, and 4.5 times as much computer time, as the analysis using complex constraints. For more realistic, larger structures, these ratios become very much greater, rendering a complete analysis impracticable.
The method has been used to calculate the forced vibrations of the alternator end-winding shown schematically in Figure 2. It was not possible to analyse the full structure without using the complex constraint method. However, to test its correctness when applied to a large structure, a simplified structure, containing 48 coils and the connexions between adjacent coils, but no brackets or connexions between layers, was analysed using the complex constraint method, and using a full idealization. Excellent agreement between the two calculations was obtained.

CONCLUSIONS
A method for solving eigenvalue problems for rotationally periodic structures without introducing any additional approximations has been described. The working of the method is performed on a single substructure throughout.
The method enables an economical analysis to be performed on structures too complex to be analysed at all hitherto. The method can also be used for the direct solution of forced vibration problems, by reducing the force to a set of rotating force components. Each element in the series defined in equation (52) is a unit vector in the complex plane, rotated through an angle -2~( j -l)(kl ) M / N with respect to the x-axis. The angle between successive vectors in the series is -27r(j--l)M/N. After N / M terms, the total angle turned through is therefore -27r(j -1). This is an integral multiple of 2~. The vectors in the series form a closed polygon (see Figure 5 for example) and the vector sum is therefore zero. Thus