2. Cyclic repeatability

2.1. Definition

We say that a structure has a cyclic repeatability of axis \(\mathrm{Oz}\), if there is an angle \(0<\alpha <\pi\) such that the structure is geometrically and mechanically invariant by rotation around \(\mathrm{Oz}\) of this angle. If \(\alpha\) is the smallest angle satisfying this property, then any angular portion of angle \(\alpha\) of the structure is called the « base sector » (or « irreducible sector »).

The global structure is then composed of \(N\) sectors:

(2.1)\[ N=\ frac {2\ Pi} {\ alpha}\]

2.2. Wave propagation

Note \(\theta\) the rotation of axis \(\mathrm{Oz}\) and angle \(\alpha\) defined in \({R}^{3}\).

Consider a base sector of a repeatable structure with axis \(\mathrm{Oz}\), and two similar points of two contiguous sectors \(G\) and \(D\):

We have the relationship between the points \(G\) and \(D\):

(2.2)\[ G=\ theta (D)\]

Note that the structure is left invariant by any \({\theta }^{m}\) rotation (with \(m\) being an integer).

It can be noted that all the rotations leaving the structure invariant (geometrically and mechanically) are in a finite number:

(2.3)\[ {\ theta} ^ {m} m\ in\ text {{}\ mathrm {0.1},\ mathrm {...} N-1\ text {}}\]

Consider a scalar state variable from the mechanical system studied \(U\), and \(Z\) the associated complex:

\[\]

: label: eq-4

U=text {Re} (Z) =text {Re} (U+mathrm {jV})

It is possible to demonstrate, using finite group theory, the following relationship for points \(D\) and \(G\) [bib5]:

\[\]

: label: eq-5

exists mintext {{}mathrm {0.1},mathrm {…} ,frac {N} {2}text {}}text {}}mathrm {tel}mathrm {that} Z (G) = {e} ^ {mathrm {mm}alpha} Z (D)

Notes:

• quantities are expressed in the cylindrical coordinate system \((r,\theta ,z)\) ,

• for an axisymmetric structure (special case of cyclic repeatability), \(m\) is called the index of FOURIER,

• in the case of an undamped plane wave, \({e}^{\mathrm{jm}\alpha }\) is the complex phase shift between two contiguous sectors; the equation means that this phase shift can only take a finite number of known values,

• it is possible to limit the number of values of \(m\) to values between 0 and \(N/2\) ; in fact, it is shown that the wave associated with the phase \(N-m\) is identical to that associated with the phase \(m\), but progresses in the opposite direction [bib5].

If \(N\) is even: \(m=0\) and \(m=N/2\) correspond to real modes:

(2.4)\[\begin{split} \ begin {array} {} m=0\ Rightarrow\ forall DU (\ theta (D)) =U (D)\\\ m=n/2\ Rightarrow\ forall DU (\ theta (D)) =-U (D)\ end {array}\end{split}\]

All the other values of \(m\) correspond to modes appearing in orthogonal pairs at a given frequency (we then speak of degenerate modes):

(2.5)\[ U=\ text {Re} (Z)\ mathrm {and} V=\ text {Im} (Z)\]

Si

is odd: \(m=0\) corresponds to a real non-degenerate mode:

(2.6)\[ m=0\ Rightarrow\ forall DU (\ theta (D)) =U (D)\]

All other values in \(m\) correspond to degenerate modes that appear in orthogonal pairs:

(2.7)\[ U=\ text {Re} (Z)\ mathrm {and} V=\ text {Im} (Z)\]

2.3. Notion of diameters and nodal circles

The property of cyclic repeatability, translated by equation (), makes it possible to know a priory the appearance of the natural modes of the structure, which is very similar to what can be observed for axisymmetric structures. If one considers an eigenmode of a cyclic symmetric structure, all the sectors have the same deformation but with an amplitude that is a function of their angular position, which can be translated by a phase shift between substructures. This mode can be classified based on the number of diameters and nodal circles that characterize it. A nodal diameter (which is only confused with a diameter if the structure is axisymmetric) is a line of points of zero movement passing through the axis of repetitiveness; a nodal circle (which has a circular shape only for axisymmetric structures) is a line of points of zero movement, itself with cyclic repetitiveness. It can be seen that it is the deformation of the mode of the substructure on which the mode of the complete structure is based that determines the number of nodal circle (s). On the other hand, the number of node diameter (s) is defined by the phase difference between two consecutive sectors.

Deformed from sector	Phase between sector	Deformed overall	Family
Flexion 1	N sectors in phase		0 circle 0 diameter
Flexion 1	N/2 sectors in phase		0 circle 1 diameter
Flexion 2	N sectors in phase		1 circle 0 diameter
Flexion 2	N/sectors in phase		1 circle 1 diameter

2.4. Boundary conditions

In general, the implementation of cyclical substructuring methods with a reduction technique should not require any particular treatment in the case where the sector has nodes located on the axis of rotation. We would simply find ourselves in a case where some nodes belong simultaneously to the right and left interfaces. However, taking into account boundary conditions in Code_Aster requires treating the two cases separately:

• • on the one hand, the relationships of continuity between the facing faces, excluding the axis

  • on the other hand the relationships associated with the nodes carried by the axis of rotation, if any exist.

2.4.1. Linking equations between the right and left faces — axis excluded

Let us consider a structure with cyclical repeatability, and two successive base sectors of it:

Since the links between sectors are considered to be perfect, we have the conditions between the sectors:

(2.8)\[ {q} _ {g} ^ {k} = {q} _ {d} ^ {k\ text {+} 1}\]

(2.9)\[ {f} _ {{L} _ {g}}} ^ {k}} ^ {k} =- {f} _ {d}} ^ {k\ text {+}} 1} ^ {k\ text {+} 1}\]

The exponent indicates the number of the sector in question. The preceding link conditions are expressed in the global frame of reference.

By the relationship () (wave propagation in the structure) and by setting: \(\beta =m\alpha\), we have:

(2.10)\[\begin{split} \ begin {array} {c} {({q} ^ {k+1})} _ {k+1})} _ {k+1} = {e} ^ {k} {k}}} _ {k}\\ {\ {({f} ^ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} _ {k+1})} end {array}\end{split}\]

The index \(k\) means that the quantity is expressed in the coordinate system linked to the sector \(k\): \({R}^{k}\).

The link equations (), written in the coordinate system linked to sector \(k\) therefore involve the transition matrix from sector \(k\) to sector \(k+1\). This matrix is none other than the rotation matrix of the degrees of freedom from right to those of the left, i.e. the rotation matrix with axis \(\mathrm{Oz}\) and angle \(\alpha\), noted \(\theta\).

So we get the following system:

(2.11)\[ \ begin {array} {c} {({q} _ {g} _ {g} ^ {k})} _ {k})} _ {k} = {e} ^ {beta}\ theta {({q} _ {d} ^ {k})}} _ {q} _ {d} ^ {k})} _ {d} ^ {k})} _ {k})} _ {k})} _ {d} ^ {k})} _ {k})} _ {e} ^ {k})} _ {k})} _ {e} ^ {k})} _ {k})} _ {k})} _ {k})} _ {e} ^ {k})} _ {k})} _ {e} ^ {k})} _ {k}) beta}\ theta {({f} _ {{L} _ {d}}} ^ {k})} _ {k}\ end {array}\]

The boundary conditions () make it possible to calculate the natural modes of the whole structure from a single base sector.

2.4.2. Equations verified by the degrees of freedom carried by the axis

This formalization can be extended to the case of axis nodes. We then obtain, for a given sector:

(2.12)\[ \ begin {array} {c} {q} _ {a} = {e} = {e} = {e} ^ {e} ^ {j\ beta} {\ theta} _ {\ {f} _ {L} _ {a}} =- {e}}} =- {e}}} =- {e}}} =- {e}} =- {e}} ^ {e} ^ {j\ beta} {\ theta} _ {f} _ {a}} _ {a}}\ end {array}}\]

Since the complex exponential has module 1, the continuity of the movements of the axis can also be put in the more classical form.

(2.13)\[ {\ theta} _ {a} {q} _ {a} _ {a} = {e} ^ {-j\ beta} {q} _ {a}\]

This is an eigenvalue problem, and the axis can only admit displacements if pairs \(({q}_{a},{e}^{-j\beta })\) correspond to the eigenvectors and eigenvalues of the rotation matrix \({\theta }_{a}\). The eigenvalues of \({\theta }_{a}\) are \((\mathrm{1,}{e}^{j\alpha },{e}^{-j\alpha })\), associated respectively with the eigenvectors that are the axis of rotation, and with two axes orthogonal to each other and orthogonal to the axis of rotation. The only values of \(\beta\) that allow axis movements to be obtained are therefore:

• \(\beta =0\), which is m=0. The movements take place only in the direction of the axis of rotation.

• \(\beta =\alpha\), i.e. m=1. The movements take place in a direction normal to the axis of rotation.

On the other hand, equilibrium relationships can also take the form of an eigenvalue problem.

(2.14)\[ {\ theta} _ {a} {f} _ {{L} _ {a}} = {e} ^ {-j (\ beta +\ pi)} {f} _ {L} _ {a}} = {e}}} = {e}} ^ {a}} = {e} ^ {-j (\ beta -\ pi)} {f} _ {a}}\]

As before, this system only admits a non-identically zero solution if \({e}^{-j(\beta -\pi )}\) is the eigenvalue of \({\theta }_{a}\), i.e. \(\beta \in (\pi ,\alpha +\pi ,\pi -\alpha )\). For the cases m=0 and m=1, this is equivalent to having \(\alpha =\pi /2\) or \(\alpha =\pi\). If the angular opening of the sector is different from \(\pi\) or \(\pi /2\), i.e. a problem with two or four sectors, then we will necessarily have \({f}_{{L}_{a}}=0\) for cases with 0 and 1 diameter.