3. Cyclic substructuring methods#

3.1. Craig-Bampton method#

The problem is considered to the eigenvalues of the global structure expressed on the base sector. The latter is therefore subject to the connecting forces applied to it by the contiguous sectors. In addition, the base sector verifies the link equations (). So we have:

(3.1)#\[\begin{split} \ begin {array} {c} (K- {\ omega} ^ {2} M) q= {f} _ {L}\\\ {q} _ {g} = {e} ^ {j\ beta}\ theta {q}\ theta {q} _ {q} _ {q} _ {g}} =- {e} ^ {j\ beta}\ theta {f} _ {g}} =- {e} ^ {j\ beta}\ theta {q}}\ theta {f} _ {{L} _ {d}}\ end {array}}\end{split}\]

We assume that the base is composed of the dynamic eigenmodes of the base sector embedded in its interfaces, noted \(\varphi\), and constrained modes relating to the degrees of freedom of the right and left interfaces, noted \({\Psi }_{d}\) and \({\Psi }_{g}\).

Given that the only contribution to the movements of a degree of interface freedom comes from the corresponding constrained mode, the transformation of RITZ can be written as:

(3.2)#\[\begin{split} q=\ left\ {\ begin {array} {c} {c} {q} _ {i}\\ {q} _ {d}\\ {q} _ {g}\ end {array}\ right\} =\ left [\ begin {array} {c} {ccc} _ {c}}\ varphi & {\ Psi} _ {g}\ end {array}\ right]\ begin {array}\ right]\ begin {array} {\ right] {ccc} {ccc}\ varphi & {\ Psi} _ {g}\ end {array}\ right\ right]\ left\ {\ begin {array} {c} {\ eta} _ {i}\\ {q} _ {d}\\ {q} _ {g}\ end {array}\ right\} _ {g}\ end {array}\ right\}} =\ Phi\ eta\end{split}\]

Therefore, using the transformation of RITZ, the system of equations [éq 3.1-1] becomes:

(3.3)#\[\begin{split} \ begin {array} {c} (\ overline {K} - {\ omega} ^ {2}\ overline {M})\ left\ {\ begin {array} {\ eta} _ {i}\\ {q} _ {i}\\ {q} _ {d}\ {q} _ {d}\\ {q} _ {g}\ {q} _ {g}\ end {array}\ right\} = {\ left [\ begin {array} {\ eta} _ {i}\\ {q} _ {ccc} q} _ {c}\ varphi & {\ Psi} _ {d} & {\ Psi} _ {g} _ {g}\ end {array}\ right]} ^ {T}\ left\ {\ begin {array} {c} 0\\ {f} _ {{L} _ {L} _ {d}}}\\ {d}}\\ {l}}\\\\\ {q} _ {g}} = {e} ^ {j\ beta}\ theta {q} _ {d} _ {d}\\ {f} _ {L} _ {g}} =- {e} ^ {j\ beta}\ theta {f}}\ theta {f} _ {f} _ {f} _ {d}}\ end {array}\ theta {f} _ {f} _ {d}}\end{split}\]

The matrices surmounted by a bar are the projections of the finite element matrices on the modal basis of the base sector (generalized matrices).

It can be shown that the constrained modes are orthogonal to the normal modes with respect to the stiffness matrix [bib5]. So the corresponding products are void.

Let’s adopt the following notations:

\(m\): index relating to the sector’s clean modes,

\(d\): index relating to the constrained modes of the right interface,

\(g\): clue relating to the constrained modes of the left interface.

We can therefore write these matrices in the form:

(3.4)#\[\begin{split} \ overline {K} =\ left [\ begin {array} {ccc} {ccc}\ overline {{K}} _ {\ mathit {mm}}}} & 0&\\ 0&\ overline {{K} _ {\ mathit {k}} _ {\ mathit {dd}}}\\ 0&\ overline {{K}} _ {\ mathit {dd}}}}\\ 0&\ overline {{K} _ {\ mathit {dd}}}}\\ 0&\ overline {{K} _ {\ mathit {dd}}}}\\ 0&\ overline {{K}} _ {\ mathit {dd}}}}\\ 0&\ overline {{K}} _ {\ mathit {dd}}}}\\ 0{ gd}}} &\ overline {{K} _ {gg}}}\ end {array}}}\ end {array}\ right]\ overline {M} =\ left [\ begin {array} {ccc}\ overline {{M}} _ {\ mathit {mm}} _ {\ mathit {mm}}} &\ overline {{M}} _ {\ mathit {mm}}} _ {\ mathit {mm}}} &\ overline {{M}} _ {\ mathit {mm}}} &\ overline {{M}} _ {\ mathit {mm}}} mg}}}\\\ overline {{M} _ {\ mathit {dg}} _ {\ mathit {dm}}} &\ overline {{M}} _ {\ mathit {dg}} _ {\ mathit {dg}}}}}\\ mathit {dg}}}}\\ mathit {dg}}} &\ overline {{M} _ {\ mathit {dg}}}}\\ overline {{M} _ {\ mathit {dg}}}}\\ overline {{M} _ {\ mathit {dg}}}}\\ overline {{M} _ {\ mathit {dg}}}}\\ overline {{M} _ {\ mathit {dg}}}}}} &\ overline {{M} _ {gg}}\ end {array}\ right]\end{split}\]

Given their definition, constrained modes verify:

\[\]

: label: eq-21

{Psi} _ {d} =left [begin {array} {c} {c} {Psi} _ {mathit {di}}}\ {Psi} _ {mathit {dd}}}\ {Psi}}\{Psi} _ {mathit {dg}} _ {mathit {dg}}} _ {mathit {dg}}}}end {array}right] =left [begin {array} {c} {}} {mathit {Psi}} _ {mathit {dg}} _ {mathit {dg}}} _ {mathit {dg}}}}end {array}right] =left [begin {array} {c} {dd}}}\mathit {Id}\ 0end {array}right] {Psi}} _ {g} =left [begin {array} {c} {Psi} _ {mathit {gi}}}\ {pi}}\ {Psi}} _ {mathit {gd}} _ {array} {gg}end {array}right] =left [begin {array}right] =left [begin {array}right] =left [begin {array}right] =left [begin {array}right] =left [begin {array}} {c} {Psi} _ {mathit {gi}}\ 0\mathit {Id}end {array}right]

The second member of the matrix equation () becomes:

(3.5)#\[\begin{split} \ left [\ begin {array} {ccc} {\ varphi} {\ varphi} _ {i}} ^ {T} & 0& 0\\ {\ Psi} _ {\ mathit {di}} ^ {T} &\ mathit {Id}} &\ mathit {Id}} &\ mathit {Id}} &\ mathit {Id}} &\ mathit {Id}} &\ mathit {Id}} &\ mathit {Id}} &\ mathit {Id} &\ mathit {Id}\ &\ mathit {Id} &\ mathit {Id} &\ mathit {Id} &\ mathit {Id} &\ mathit {Id} &\ mathit {Id} &\ mathit {Id}\ {\ begin {array} {c} 0\\ {f} _ {{L} _ {d}}\\ {f}}\\ {L} _ {g}}\ end {array}\ right\}} =\ left\\ left\ {\ begin {f}} _ {\ f}}\\ {f}}\\ {f}\\ {f}\ right\}} =\ left\ {\\ begin {f}} =\ left\ {\ begin {array}}} =\ left\ {\ begin {array}}} =\ left\ {\ begin {array}}} =\ left\ {\ begin {array}}} =\ left\ {\ begin {array}}} =\ left\ {\ begin {array}}} =\ left\ {\ begin {array} {f}} end {array}\ right\}\end{split}\]

Taking these ratings into account, let’s develop the matrix equation verified by the base sector:

(3.6)#\[\begin{split} \ begin {array} {c}\ overline {{K} _ {\ mathit {K}} _ {\ mathit {mm}}}} {\ eta} _ {{M} _ {\ mathit {mm}} _ {\ mathit {mm}}}} {\ mathit {mm}}} {\ mathit {mm}} _ {mm}} _ {mm}} _ {d}}} {\ mathit {mm}}} {\ mathit {mm}} _ {mm}} _ {d}}} {\ mathit {mm}}} {\ mathit {mm}}} {\ mathit {mm}} _ {d}}} {\ mathit {mm}}} {\ mathit {mm}} _ {mm}}} {\ mathit {mm}} _ {mm overline {{M} _ {\ mathit {mg}}}} {q}} {q} _ {g}) =0\\ overline {{K} _ {dd}}} {q} _ {d} +\ overline {{K}} +\ overline {{K}} _ {K} _ {g}}} {q} - {\ omega} ^ {2}} (\ overline {{M}} _ {\ mathit {dm}}} {\ eta} _ {i} +\ overline {{M}} _ {\ mathit {dd}}} {q} _ {d} +\ overline {{M} _ {\ mathit {dg} _ {g}}} {\ mathit {dg}}} {\ mathit {dg}}} {\ mathit {dg}}} {\ mathit {dg}}} {\ mathit {dg}}} {\ mathit {dg}}} {\ mathit {dg}}} {\ mathit {dg}}} {\ mathit {dg}}} {\ mathit {dg}}} {\ mathit {dg}}} {\ mathit {dg}}} {\ mathit {dg}}} {\ mathit {dg}}} it {gd}}} {q} _ {d} +\ overline {{K}} _ {gg}} {q}} {g} - {\ omega} ^ {2} (\ overline {{M}} _ {\ mathit {gm}} _ {gm}}} {\ mathit {gm}}} {\ mathit {gm}} _ {d}}} {q} _ {d}}} {q} _ {d}}} {\ mathit {gm}}} {\ mathit {gm}}} {\ mathit {gm}}} {\ mathit {gm}}} {\ mathit {gm}}} {\ mathit {gm}}} {\ mathit {gm}}} {\ mathit {gm}}} {\ mathit {gm}}} {+\ overline {{M} _ { gg}} {q} _ {g}) = {f} _ {{L} _ {L} _ {g}}\\\\ {q} _ {g} = {e} ^ {j\ beta}\ theta {q} _ {d} _ {d}}\\ {f} _ {f} _ {f} _ {f} _ {f} _ {d} _ {f} _ {f} _ {d}}\ end {array}\end{split}\]

Let’s introduce the last two equations of this system into the first three:

(3.7)#\[\begin{split} \ begin {array} {c} (\ overline {{K}} _ {\ mathit {mm}}}} - {\ omega} ^ {2}\ overline {{M} _ {\ mathit {mm}}}}) {\ eta}}}) {\ eta} _ {i} _ {i} - {\ omega}} - {\ omega} ^ {2}} (\ overline {{M}} _ {\ mathit {md}}}}) {\ mathit {mm}}}}) {\ eta} _ {mm}}}) {\ eta} _ {i} - {i} - {\ omega} ^ {2} (\ overline {M}} _ {\ mathit {md}}}} + {e} ^ {j}}}) {\ eta} _ {\ beta}\ overline {{M} _ {\ mathit {mg}}}}}\ theta) {q} _ {d} =0\\ (\ overline {{K} _ {\ mathit {dd}}}}} + {e}}}} + {e}} ^ {j\ beta}}}\ theta) {\ mathit {dd}}}}} + {e}}}\ mathit {dd}}}} + {e}} ^ {j\ beta}}} + {e} ^ {j\ beta}}\ overline {{K}}}\ theta) {q} _ {d}}} + {e}} ^ {j\ beta}}\ overline {{K}}}\ theta) {q} _ {d}}} + {e}} ^ {2}\ left (\ overline {{M}} _ {\ mathit {M}}} {\ mathit {dm}}}} {\ eta} _ {\ mathit {dd}}}} + {\ mathit {dd}}}}} + {\ mathit {dd}}}} {\ mathit {dd}}}} {\ mathit {dd}}}} {\ mathit {dd}}}} {\ mathit {dd}}}} {\ mathit {dd}}}} {\ mathit {dd}}}} {\ mathit {dd}}}} {\ mathit {dd}}}} {\ mathit {dd}}}} {\ mathit {dd}}}} {\ mathit {dd}}}} {\ mathit {dd} {d}\ right) = {f} _ {{L} _ {d}}}\\ (\ overline {{K} _ {\ mathit {gd}}}} + {e} ^ {j\ beta}}\ overline {{K}}\ overline {K} _ {d}}\ overline {{K}} _ {g}}}\ left (\ overline {{M} _ {\ mathit {gm}}}}} {\ eta}}} {\ eta} _ {i} +\ text {(}\ overline {{M}} _ {\ mathit {gd}}}}} + {e}}} + {e}}} + {e}}} + {e}}} {\ mathit {gd}}}}} + {e}}} + {e}}} + {e}}} + {e}}} + {e}}} {\ mathit {gd}}}}} {\ mathit {gd}}}} {\ mathit {gd}}}}} {\ mathit {gd}}}}} {\ mathit {gd}}}}} {\ mathit {^ {j\ beta}\ theta {f} _ {{L} _ {d}}\ end {array}\end{split}\]

The combination of the last two equations makes it possible to eliminate the terms of the bond forces. We then end up with a final eigenvalue problem that can be put in the form:

(3.8)#\[ \ left (\ stackrel {} {K} {K} (\ beta) - {\ omega} ^ {2}\ stackrel {} {M} (\ beta)\ right)\ stackrel {} {k} (\ beta) (\ beta) (\ beta) (\ beta) (\ beta) (\ beta)\ right)\ stackrel {} {right)\ stackrel {} {right)\ stackrel {} {\]

With: \(\stackrel{̃}{q}=\left\{\begin{array}{c}{\eta }_{i}\\ {q}_{d}\end{array}\right\}\)

(3.9)#\[\begin{split} \ stackrel {459} {K} =\ left [\ begin {array} {cc} {cc}\ overline {{K} _ {\ mathit {mm}}} & 0\\ 0&\ overline {{K}} _ {\ mathit {dd}} _ {\ mathit {dd}} _ {\ mathit {dd}} _ {\ mathit {dg}} _ {\ mathit {dg}} _ {\ mathit {dg}} _ {\ mathit {dg}} _ {\ mathit {dg}} _ {\ mathit {dg}} _ {\ mathit {dg}} _ {\ mathit {dg}} _ {\ mathit {dg}} _ {\ mathit {dg}} _ {\ mathit {dg}} _ {\ mathit {dg}}} ^ {-j\ beta} {\ theta} ^ {T} ^ {T}\ overline {{K}} _ {\ mathit {gd}}}} + {\ theta} ^ {T}\ overline {{K}}\ overline {{K} _ {G} _ {gg}}\ theta\ end {array}\ right]\end{split}\]

\[\]

: label: eq-27

stackrel {459} {M} =left [begin {array} {cc} {cc}overline {{M} _ {mathit {mm}}} &overline {{M} _ {mathit {md}}}}} {mathit {md}}}} {mathit {md}}} {mathit {md}}} {mathit {md}}}} {mathit {md}}}} {mathit {md}}}} {mathit {md}}}} {mathit {md}}}} {mathit {md}}}} {mathit {md}}}} {mathit {md}}}} {mathit {md}}}} {mathit {md}}}} {} _ {mathit {dm}}}} + {e}} ^ {e} ^ {-jbeta} {theta} ^ {T}overline {{M}}}} &overline {{M}} _ {mathit {M}} _ {mathit {M}} _ {mathit {M}}} _ {mathit {dg}} _ {mathit {dg}} _ {mathit {dg}} _ {mathit {dg}} _ {mathit {dg}} _ {mathit {dg}} _ {mathit {dg}} _ {mathit {dg}} _ {mathit {dg}}} _ {mathit {dg}}} _ {mathit {dg}}} eta + {e} ^ {-jbeta} {theta} {theta} ^ {T}overline {{M} _ {mathit {gd}}}} + {theta} ^ {T}}overline {{M}}overline {{M}} _ {gg}}thetaend {array}right]

The mass and stiffness matrices of the final problem are Hermitian. The solution eigenvalues are therefore real. On the other hand, the problem is small.

Solving the problem with complex eigenvalues () makes it possible to determine the complex generalized coordinates of the eigenmodes of the global structure. The complex values of the displacements of the base sector in the global mode are given, from the generalized coordinates, by the following formula:

\[\]

: label: eq-28

qtext {”} =left [begin {array} {cc} {cc}varphi & {Psi} _ {d} + {e} ^ {jbeta}theta {Psi} _ {d} _ {d}d}end {array} {cc}cc}varphi & {Psi}right]stackrel {} {q}

To determine the real values of the displacements, it is necessary to distinguish three cases according to the values of the inter-sector phase difference:

Case 1: \(\beta =0\):

The displacements \(q\text{'}\) given by the formula () then have real values. All the sectors have even deformed and vibrate in phase. We then have only one real mode of its own:

(3.10)#\[ q=\ text {Re} (q\ text {'}) =q\ text {'}\]

Case no2: \(0<\beta <(N+1)/2\):

The displacements provided by the formula () have complex values. To each of these complex modes correspond two orthogonal degenerate real modes:

\[\]

: label: eq-30

{q} _ {1} =text {Re} (qtext {“}) {q} _ {2} =text {Im} (qtext {“})

Case no. 3: \(\beta =N/2\) (=> \(N\) is even):

The displacements provided by () then have complex values. There are \(N/2\) nodal diameters, two contiguous sectors then vibrate in phase opposition. Each complex mode is the source of a single real mode:

\[\]

: label: eq-31

q=text {Re} (qtext {“}) =-text {Im} (qtext {“})

3.2. Mac Neal method#

(3.11)#\[\begin{split} \ begin {array} {c} (K- {\ omega} ^ {2} M) q= {f} _ {L}\\ {q} _ {g} = {e} ^ {j\ beta}\ theta {q}\ q} _ {q} _ {d} _ {d}\\ {f} _ {f} _ {f} _ {L} _ {L}} =- {e} ^ {j\ beta}\ theta {q}}\ theta {q}}\ theta {q}} _ {L}} _ {d}}\ end {array}\end{split}\]

The modal base used to reduce the dimensions of the problem to be solved is a modal base with free interfaces including dynamic modes and attachment modes relating to the degrees of freedom of the right and left interfaces. Assume that the degrees of freedom of the base sector are ordered as follows:

(3.12)#\[\begin{split} q=\ left\ {\ begin {array} {c} {q} _ {i}\\\ {q} _ {d}\\\\ {q} _ {g}\ end {array}\ right\}\ right\}\ begin {array}}\ begin {array} {c} {c}\ mathrm {degrees}\ mathrm {of}\ mathrm {freedom}\ mathrm {freedom}\ mathrm {internal}\\\ mathrm {internal}\\\ mathrm {internal}\\\ mathrm m {degrees}\ mathrm {of}\ mathrm {freedom}\ mathrm {of}\ mathrm {of}\ mathrm {l}\ mathrm {'}}\ mathrm {interface}\ mathrm {right}\ mathrm {right}\ mathrm {right}\ mathrm {right}\ mathrm {of}\ mathrm {of}\ mathrm {of}\ mathrm {of}\ mathrm {of}\ mathrm {of} l}\ mathrm {\ text {'}}}\ mathrm {interface}\ mathrm {left}\ end {array}\end{split}\]

Let \({B}_{d}\) and \({B}_{g}\) be rectangular extraction matrices such as:

(3.13)#\[ {q} _ {d} = {B} _ {d} q\ text {and} {q} _ {g} = {B} _ {g} _ {g} q\]

With these notations, the boundary condition on displacements becomes:

(3.14)#\[ {B} _ {g} q= {e} ^ {j\ beta} {j\ beta}\ theta {B} _ {d} q\ Rightarrow {B}} _ {\ mathit {dg}} q=0 \ textrm {with} {B} _ {\ mathit {dg}} = {e}} = {e} ^ {j\ beta}\ theta {B} _ {d} - {B} _ {g}\]

For forces, the boundary condition becomes:

(3.15)#\[ {f} _ {L} = {B} _ {g} ^ {g} ^ {g} ^ {g}} _ {g}} + {B} _ {d} {f} _ {{L} {f} _ {L} _ {d}}}}\ Rightarrow {d}}}}}\ Rightarrow {f}}}}\ Rightarrow {f}}}\ Rightarrow {f}}}\ Rightarrow {f}}}\ Rightarrow {f}}}\ Rightarrow {f}}}\ Rightarrow {f}}}\ Rightarrow {f}}}\ Rightarrow {f}}}\ Rightarrow {f}}}\ Rightarrow {f}}}\ Rightarrow eta} {B} _ {d} ^ {T} {\ T} {\ theta} {\ theta} ^ {T}) {f} _ {L} _ {B} _ {\ mathit {dg}}}} ^ {T} {g}}} ^ {T} {f}} _ {g}}\]

Let us consider as a basis, for the transformation of RITZ, all the dynamic modes of the basic sector, distinguishing between identified modes and unknown modes:

(3.16)#\[\begin{split} q=\ left [\ begin {array} {cc} {\ varphi} {\ varphi} _ {1} _ {2}\ end {array}\ right]\ left\ {\ begin {array} {c} {array} {c} {\ eta} {\ eta} _ {1}\\ {\ eta} _ {2}\ end {array}\ right\}\end{split}\]

where the index 1 (resp. 2) refers to the known modes (resp. unknown). In the following, we will assume that the natural modes are normalized to the unit modal mass.

By replacing \(q\) with its expression according to the eigenmodes, and by multiplying on the left by the transpose of the mode matrix, the matrix equations () and () become:

(3.17)#\[ \ begin {array} {c} ({\ lambda} _ {1} _ {1} - {\ omega} ^ {2}\ mathit {Id}) {\ eta} _ {1} = {\ varphi} _ {1} _ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {_ {2} = {\ varphi} _ {2} _ {2} ^ {2} ^ {T} ^ {T} {T} {F} _ {\ varphi}} _ {1} {\ eta} _ {1} _ {1} _ {1} + {1}} + {1}} + {1} {1} {1}} + {1}} + {1}} + {1}} + {1} {1} + {1}} + {1} + {1}} + {1} + {B}} + {B} _ {1} + {B} _ {1} + {B} _ {1} + {B} _ {1} + {B} _ {1} + {B} _ {1} + {B} _ {1} + {\]

where \(\lambda\) is the generalized rigidities matrix (the generalized masses are unitary).

We can therefore derive a formulation of \({\eta }_{2}\) from this:

(3.18)#\[ {\ eta} _ {2} = {({\ lambda}} _ {2} - {\ omega} ^ {2}\ mathit {Id})} ^ {-1} {\ varphi} {\ varphi} _ {\ varphi} _ {\ varphi} _ {\ varphi} _ {\ varphi} _ {\ varphi} _ {2} _ {L}\]

Therefore, we can eliminate \({\eta }_{2}\) from the system of equations (). We then get the following eigenvalue problem:

(3.19)#\[ \ begin {array} {c} ({\ lambda} _ {1} _ {1} - {\ omega} ^ {2}\ mathit {Id}) {\ eta} _ {1} + {\ varphi} _ {1} _ {1} ^ {1} ^ {T} {B} ^ {B} {B} ^ {B} {B} {B} ^ {B} {B} {B}} ^ {B} {B} {B}} _ {B} {B}} _ {B}} _ {B} {B}} _ {B} {B}} _ {B} _ {B}} _ {mathit {dg}} _ {mathit {dg}} _ {mathit {dg}}} it {dg}} {\ varphi} _ {1} {\ eta} _ {\ eta} _ {1} _ {B} _ {\ mathit {dg}} {\ varphi} _ {2} {({\ lambda} _ {2}} - {\ omega} _ {2} - {\ omega}} ^ {2}} {\ lambda} _ {2} {2} {2} {2} {T} _ {2} _ {2} ^ {T} {B} _ {2} _ {\ B} _ {2} mathit {dg}} ^ {T} {f} _ {\ mathit {Lg}} =0\ end {array}\]

The final system to be solved can be written as:

(3.20)#\[ (\ stackrel {b>} {K} - {\ omega} ^ {2}\ stackrel {} {M})\ stackrel {} {K} {K} - {\ omega} {q} =0\]

With:

(3.21)#\[\begin{split} \ stackrel {} {q} =\ left\ {\ begin {array} {\ begin {array} {c} {\ eta} _ {1}\\ {f} _ {L} _ {g}}}\ end {array}\ right\}\end{split}\]

The expressions for stiffness and mass matrices are:

(3.22)#\[\begin{split} \ stackrel {459} {K} =\ left [\ begin {array} {cc} {\ lambda} _ {1} & {\ varphi} _ {1} {B} _ {\ mathit {dg}}}} ^ {T}} ^ {T}}\\ {B}} {\ mathit {dg}} _ {\ mathit {dg}} _ {\ mathit {dg}} _ {\ mathit {dg}} R} _ {e} (\ omega) {B} _ {\ mathit {dg}}} ^ {T}\ end {array}\ right]\ stackrel {} {M} =\ left [\ begin {array} {m} =\ left [\ begin {array} {cc} {cc}\ cc}\ mathit {Id}} & 0\\ 0& 0\ end {array} {M} =\ left [\ begin {array} {{m} =\ left [\ begin {array} {cc} {cc}\ cc}\ mathit {Id} & 0\\ 0& 0\ end {array}\ right]\end{split}\]

Matrix \(\left[\begin{array}{c}{R}_{e}(\omega )\end{array}\right]\) is the residual dynamic flexibility matrix of the unidentified modes:

(3.23)#\[ {R} _ {e} (\ omega) = {\ Phi} = {\ Phi} _ {2} {\ Phi} _ {2} - {\ omega} ^ {2}\ mathit {Id})}} ^ {\ text {Id})}}} ^ {Id})} ^ {Id})} ^ {Id})} ^ {Id})} ^ {Id})} ^ {Id})} ^ {Id})} ^ {Id})} ^ {Id})} ^ {Id})} ^ {\]

The residual dynamic flexibility is approximated by its static contribution, taking into account the attachment methods. So, the retrieval formula that makes it possible to calculate the complex values of the displacements from the generalized coordinates of the solution modes of () is as follows:

(3.24)#\[ q\ text {'} =\ left [\ begin {array} {cc} {cc} {\ varphi} _ {1} & - {R} _ {e} (0) {B} _ {\ mathit {dg}}}} ^ {dg}}} ^ {T} {T}\ end {array}\ right]\ stackrel {} {q}\]

As with the Craig‑Bampton method, the real values of the displacements are determined by the relationships (), () and ().

3.3. Taking into account axis nodes - Craig & Bampton method#

From an algorithmic point of view, the interface modes associated with the degrees of freedom carried by the axes nodes are only taken into account for the cases m = 0 and m = 1, which are the only cases that may present non-zero movements of the axis (see Section 2.4).

NB: It is important to note that calculations taking into account axis movements can only be carried out with a Craig & Bampton reduction method. The approach with the Mac Neal method is not implemented.

In this paragraph, it is assumed that the degrees of freedom carried by the axis nodes, in the same way as the right and left interface nodes, have been blocked for the calculation of the dynamic modes of the base sector and have been the subject of constrained mode calculations.

The projection base is therefore composed of the dynamic eigenmodes of the base sector embedded in its interfaces, noted \(\varphi\), and constrained modes relating to the degrees of freedom of the right, left and axis interfaces, noted \({\Psi }_{d}\), left and axis, noted, \({\Psi }_{g}\) and \({\Psi }_{a}\).

As we saw in section 2.4, if m is greater than or equal to 2, the displacement of the axis nodes is zero. Taking into account the nodes of the axis therefore only makes sense if m=0 or m=1. In practice, to limit memory occupancy and the number of operations, the matrices are assembled taking into account DDL of the axis only in these two cases.

The problem with the eigenvalues of the global structure and the link equations, expressed on this basis, then apply to:

(3.25)#\[\begin{split} \ begin {array} {c} (\ overline {K} - {\ omega} ^ {2}\ overline {M})\ left\ {\ begin {array} {\ eta} _ {i}\\ {q} _ {i}\\ {q} _ {d}\ {q} _ {g}\ {q} _ {a}\ end {array} {\ eta} _ {i}\\ eta} _ {i}\\ {q} _ {i}\\ {q} _ {i}\\ {q} _ {i}\\ {q} _ {i}\\ {q} _ {i}\\ {q} _ {i}\\ {q} _ {i}\\ {q} _ {i}\\ {q} _ {i}\\ {q} _ {} {cccc}\ varphi & {\ Psi} _ {d} _ {d} & {\ Psi} _ {g} & {\ Psi} _ {a}\ end {array}\ right]\ left\ {\ begin {array} {c} {c} {c}} 0\\ {f}} _ {f} _ {d}}\\ {f} _ {g}}\\ {f}}\\ {f} _ {L}}\\ {f} _ {L}}\\ {f} _ {L}}} _ {d}}\ end {array}\ right\}\\ {q}}}\\ {q} _ {g} = {e} ^ {j\ beta}\ theta {q} _ {text {and} {q} _ {a} _ {a} _ {a} _ {a} = {e}} _ {q} _ {g}} = {e} ^ {j\ beta} _ {g}} =- {g}} =- {g}} e} ^ {j\ beta}\ theta {f} _ {{L} _ {g}}\ text {and} {f} _ {{L} _ {a}} =- {e}} ^ {j\ beta} _ {{l} _ {a}}} =- {e} ^ {j\ beta}}\ {j\ beta}}\ theta {f} _ {f}}\ end {array}\end{split}\]

We can therefore write matrices in the form:

(3.26)#\[\begin{split} \ overline {K} =\ left [\ begin {array} {cccc} {cccc}\ overline {{K}} _ {\ mathit {mm}}}} & 0& 0\\ 0&\ overline {{K}} _ {\ mathit {K}} _ {\ mathit {dd}}} _ {\ mathit {dd}}}} &\ overline {{K}} _ {\ mathit {dd}}}} &\ overline {{K}} _ {\ mathit {dd}}}} &\ overline {{K}} _ {\ mathit {dd}}}} &\ overline {{K}} _ {\ mathit {dd}}}} da}}}\\ 0&\ overline {{K}} _ {\ mathit {gd}}}} &\ overline {{K} _ {gg}} &\ overline {{K} _ {\ mathit {ga}}} {\ mathit {ga}}}}}\\ mathit {ga}}}}\\ overline {{ga}}}}\\ overline {{k}} _ {\ mathit {ga}}}}}\\ mathit {ga}}}}\\ 0&\ overline {{ga}}}}\\ mathit {ga}}}}\\ 0&\ overline {{ga}}}}\\ mathit {ga}}}}} &\ overline {{K} _ {\ mathit {aa}}}}}\ end {array}\ right]\ overline {M} =\ left [\ begin {array} {cccc}\ overline {{M}} _ {\ mathit {mm}} _ {\ mathit {mm}}}} &\ overline {{M}} _ {\ mathit {mm}}} &\ overline {{M}} _ {\ mathit {mm}}} &\ overline {{M}} _ {\ mathit {mm}}}} &\ overline {{M}} _ {\ mathit {mm}}} mg}}} &\ overline {{M} _ {\ mathit {ma}}}}\\\ overline {{M} _ {\ mathit {dm}}}} &\ overline {{M} _ { \ mathit {dd}}} &\ overline {{M} _ {\ mathit {M}}}} &\ overline {{M} _ {\ mathit {da}}}}\\ overline {{M} _ {\ mathit {M}} _ {\ mathit {gm}} _ {gm}}} &\ overline {{M}} _ {gg}}} &\ overline {{M}}} &\ overline {{M} _ {\ mathit {ga}}}}}\\\ overline {{M}}}\\ overline {{M}}} &\ overline {{M} _ {\ mathit {ad}}}}} &\ mathit {ad}}}} &\ overline {{am}} _ {\ mathit {aa}}}}\ mathit {ad}}}}\ overline {ad}}}} &\ overline {{an}}}\ overline {aa}}}\ overline {ad}}}}\ overline {ad}}}}\ overline {{ad}}}}\ overline {ad}}}}\ overline {ga}}}\ overline {aa}}}}\ right]\end{split}\]

Given their definition, constrained modes verify:

(3.27)#\[\begin{split} {\ Psi} _ {d} =\ left [\ begin {array} {c} {\ Psi} _ {\ mathit {di}}\\ {\ Psi} _ {\ mathit {dd}}\\ {\ Psi}}\\ {\ Psi} _ {\ mathit {dg}} _ {\ mathit {dg}}}\\ mathit {dg}}}\\ mathit {dg}}\\ mathit {dg}}}\\ mathit {dg}}\\ mathit {dg}}\\ mathit {dg}}}\\ mathit {dg}}\\ mathit {dg}}\\ mathit {dg}}}\\ mathit {dg}}\\ mathit {dg}}}\\ mathit {dg}}\\ mathit {{c} {\ Psi} _ {\ mathit {di}}\\ mathit {di}}\\\ mathit {di}}\\\ mathit {di}}}\\ mathit {di}}}\\ mathit {di}}}\\ mathit {di}}}\\\ mathit {di}}\\\ mathit {di}}\\ {\ array}\ {g} =\ left [\ begin {array} {\ array}} {g} =\ left [\ begin {array} {\ array} {g} =\ left [\ begin {array} {{array}} {c} {\ array}} {c} {\ Psi}} _ {\ psi} _ {gg}\\ {\ Psi} _ {\ mathit {ga}}}\ end {array}\ right] =\ left [\ begin {array} {c} {\ Psi} _ {\ mathit {gi}}\\ 0\\ mathit {ga}}}\\ 0\\ mathit {Id}}}\ end {array}}\ right] {\ Psi} _ {a} =\ left [\ begin {array}}}}\\ 0\\ mathit {ga}}}\ 0\\ mathit {Id}}\\ Id}\\ 0\\ mathit {Id}\\ Id}\\ 0\ end {array}\ Id}\\ 0\ end {array} {Id}\\ Psi}} _ {\ mathit {ai}}\\ {\ Psi}} _ {\ mathit {ad}}\\ {\ Psi} _ {\ mathit {ag}}}\\ {\ Psi} _ {\ Psi} _ {\ Psi} _ {\ mathit {aa}}\ end {array}\ right] =\ left [\ begin {array} {c} {\ Psi} _ {\ mathit {ai}}\\ 0\\ 0\\ 0\\ 0\\\ mathit {Id}\ end {array}\ right]\end{split}\]

The second member of the matrix equation () becomes:

(3.28)#\[\begin{split} \ left [\ begin {array} {cccc} {\ varphi} {\ varphi} _ {i}} _ {\ varphi}} _ {\ mathit {di}} &\ mathit {Id} & 0& 0\\ {\ Psi}} _ {\ {\ Psi}} _ {\ {\ Psi}} _ {\ mathit {\ Psi}} _ {\ mathit {ai}} _ {\ mathit {ai}} _ {\ mathit {ai}} _ {\ mathit {ai}} _ {\ mathit {ai}} _ {\ mathit {ai}} _ {\ mathit {ai}} _ {\ mathit {ai}} _ {\ mathit {ai}} _ {\ mathit {ai}} _ {\ mathit {ai}} 0& 0&\ mathit {Id}\ end {array}\ right]\ left\ {\ begin {array} {c} 0\\ {f} _ {d}}\\ {f}}\\ {f} _ {{L} _ {L} _ {g}}}\\ {g}}\\ {f}}\\ {L} _ {a}}\ end {array}\ {d}}}\\ {f}}\\ {f}}\\ {f}}\\ {f}}\\ {f}}\\ {f}}\\ {f}}\\ {f}}\\ {f}}\\ {f}}\\ {f}}\\ {f}}\\ {f}}\\ {f}}\\ {f}} {c} 0\\ {f} _ {{L} _ {a} _ {d}}\\ {f}}\\ {f}}\\ {f} _ {L} _ {a}}\ end {array}\ right\}\end{split}\]

The connection equations are taken into account by projection. We introduce

\[\]

: label: eq-50

stackrel {b>} {q} =left{begin {array} {c}eta\ {q} _ {d}\ {q} _ {a}{a}end {array}end {array}right}

and we consider projector \(\stackrel{̃}{P}\) defined by

(3.29)#\[\begin{split} \ stackrel {} {P} =\ left [\ begin {array} {ccc}\ mathit {Id} & 0& 0\\ 0&\ mathit {Id} & 0\\ 0& {e} ^ {j\ beta} ^ {j\ beta} ^ {j\ beta} ^ {j\ beta} {e}\ 0\ 0& {e} ^ {j\ beta} {e} _ {a}\ 0& {e} ^ {a}\ 0& {e} ^ {a}\ {e} ^ {a}\ {e} ^ {j\ beta}\ e} ^ {a}\ {e} ^ {a}\ {end array}\ right]\end{split}\]

The problem projected on \(\stackrel{̃}{P}\) becomes

\[\]

: label: eq-52

stackrel {} {{P} ^ {H}}} (overline {H}}}} (overline {K}} - {omega} ^ {2}overline {M})stackrel {} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P} {P}} {P} {P} {P} {P} {P} {P}

This problem naturally verifies the kinematic link equations, since we have

(3.30)#\[\begin{split} \ stackrel {} {P}\ stackrel {459} {q} =\ left\ {\ begin {array} {c}\ eta\\ {q} _ {d}\\ {e} ^ {j\ beta} ^ {j\ beta}\\ beta}\\ beta} {e} _ {a} {q}} ^ {a} {q} _ {a} {a}\ end array}\ right\} =\ left\ {\ begin {array} {c} {c}\ eta\\ {q} _ {d}\\ {q} _ {g}\\ {q} _ {q} _ {a}\ end {array}\ right\}\end{split}\]

and liaison efforts verify

(3.31)#\[\begin{split} \ stackrel {} {{P} ^ {H}} f=\ left\ {\ begin {array} {c} 0\\ {f} _ {d}}} + {e} ^ {-j\ beta} {-j\ beta} {\ beta} {\ beta} {\ beta}} {\ theta} {\ theta} ^ {-j\ beta} {\ beta} {\ beta} {\ beta} {\ theta} ^ {\ theta} ^ {-j\ beta} {\ beta} {\ beta} {\ theta} ^ {theta}} {\ theta} ^ {-j\ beta} {\ beta} {\ beta} {\ theta} ^ {theta}} {\ theta} ^ {-j\ beta} {\ beta} {\ beta} {\} _ {a} ^ {H} {f} _ {{L} _ {a}}\ end {array}\ right\} =\ left\ {\ begin {array} {c} 0\\ 0\\\ stackrel {} {\ stackrel {} _ {{f}} _ {f}}\ end {array}\ right\}\end{split}\]

In the case where the angular opening of the sector is less than \(\pi\), and where there are strictly more than two sectors, it has also been shown that \({f}_{{L}_{a}}=0\) (section 2.4.2).

We then end up with a final eigenvalue problem that can be put in the form:

(3.32)#\[ (\ stackrel {b>} {K} - {\ omega} ^ {2}\ stackrel {} {M})\ stackrel {} {K} {K} - {\ omega} {q} =0\]

With:

(3.33)#\[\begin{split} \ stackrel {459} {K} =\ left [\ begin {array} {ccc}}\ overline {{K} _ {\ mathit {mm}}} & 0&\\ 0&\ overline {{K}} _ {\ mathit {K}} _ {\ mathit {dd}}} _ {\ mathit {dg}}}}\ theta {e} ^ {K}} _ {\ mathit {dg}}}\ theta {e} ^ {K}} _ {\ mathit {dg}}}\ theta {e} ^ {K}} _ {\ mathit {dg}}}\ theta {e} ^ {K}} _ {\ mathit {dg}}}\ theta {e} ^ {K}} _ {\ mathit {dg}}} + {e} ^ {-j\ beta} {\ theta} {\ theta} ^ {T}\ overline {{K}} _ {\ mathit {gd}}} + {\ theta} ^ {T}\ overline {{K}}\ overline {{K}}} + {\ theta} ^ {T}\ overline {{K}}}} {\ theta} ^ {T}\ overline {{K}}}\ overline {{K}}} {a} {e} ^ {e} ^ {j\ beta} + {\ theta} ^ {T}\ overline {{K}} _ {\ mathit {ga}}} {\ theta} _ {a}\\ 0& {e} ^ {-j\ beta} {\ beta} {\ theta}} {\ theta}} {\ theta}} {\ beta} {\ beta}} {\ theta}} {\ beta} {\ beta} {\ theta}} {\ beta} {\ theta}} {\ beta} {\ theta}} {\ beta} {\ theta}} {\ beta} {\ theta}} {\ beta} {\ theta}} {\ beta} {\ theta}} {\ beta} {\ theta}} {\ beta} {\ theta}} {\ beta} {overline {{K} _ {\ mathit {ag}}}}}\ theta & {\ theta} _ {a} ^ {T}\ overline {{K} _ {\ mathit {aa}}} {\ mathit {aa}}}}} {\ theta}}} {\ theta} _ {a}\ end {array}\ right]\end{split}\]

(3.34)#\[\begin{split} \ stackrel {459} {M} =\ left [\ begin {array} {ccc}}\ overline {{M} _ {\ mathit {mm}}} &\ overline {{M} _ {\ mathit {md}}}} {\ mathit {md}}}}\ overline {m}}}\ theta {e} ^ {m} _ {j\ beta} _ {\ mathit {md}}}\ overline {MD}}}\ overline {{M}}}\ theta {e} ^ {j\ beta} _ {\ mathit {md}}}\ overline {MD}}}\ overline {{M}}}\ theta {e} ^ {j\ beta} _ {\ mathit {md}}} {\ mathit {md}}}} _ {\ mathit {ma}}} {\ theta}} {\ theta} _ {a} {e} _ {a} {a} {a}} _ {\ mathit {dm}}} + {e} ^ {-j\ beta} {-j\ beta} {\ beta}} {\ beta} {\ beta}} {\ j\ beta}} {\ beta} {\ beta} {\ theta} {\ theta}} ^ {\ beta} {\ theta} {\ theta}} ^ {\ beta} {\ theta}} {\ theta} ^ {theta}} {\ theta} ^ {theta}} {\ theta} ^ {theta}} ^ {theta} {\ theta}} ^ {theta} {\ theta}} ^ {\ mathit {dd}}} +\ overline {{M} _ {\ mathit {dg}}}}\ theta {e} ^ {j\ beta} + {e} ^ {-j\ beta} {\ theta} {\ theta}} ^ {theta}} ^ {theta}} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ theta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {\ beta} {gg}}\ theta &\ overline {{M} _ {\ mathit {da}}} {\ mathit {da}}} {\ theta} _ {a} {e} ^ {beta} ^ {beta} + {\ theta} ^ {T}}\ overline {T}}\ overline {{M}} _ {\ mathit {ga}}} {\ theta}} {\ theta } _ {a}\\ {e} ^ {-j\ beta} {\ theta} {\ theta} _ {a} ^ {T}\ overline {{M} _ {\ mathit {am}}}} & {e} ^ {-j\ beta} {-j\ beta}} {\ beta}} {\ -j\ beta}} {\ beta}} {\ beta} {\ beta} {\ beta} {\ theta}} _ {\ theta} _ {a} ^ {T}\ overline {{M} _ {\ mathit {am}}}} & {e} ^ {-j\ beta}} {-j\ beta}} {\ beta} {\ beta} {\ beta}} {\ theta}} _ {\ theta} _ {a} ^ {T}\ overline {{M} _ {\ mathit {ag}}}}\ theta & {\ theta} _ {a} ^ {T}\ overline {{M} _ {\ mathit {aa}} _ {\ mathit {aa}}}} {\ mathit {aa}}}} {\ theta} _ {a}\ end {array}\ right]\end{split}\]

Complex modal movements are returned by the following formula:

(3.35)#\[ q\ text {'} =\ left [\ begin {array} {ccc} {ccc}\ varphi & {\ Psi} _ {d} + {e} ^ {j\ beta}\ theta {\ Psi} _ {d} _ {d} & {d} & {\ Psi} & {\ Psi} _ {a} _ {a}\ end {array}\ right]\ stackrel {}} {\ mathrm {q}}}\]

NB: Special case of the two- and four-sector problem

The proposed formulation remains legitimate in the particular case of the two-sector problem, taking into account the orthogonality between \({q}_{a}\) and \({f}_{{L}_{a}}\) in the case of cyclic symmetry. Indeed, the link equations () lead to

(3.36)#\[ {q} _ {a} ^ {H} {f} _ {f} _ {L} _ {a}} _ {a}} = {({e} ^ {a} {q} _ {a})}} ^ {a})} ^ {H})} (- {e} (- {e}} (- {e}}) ^ {e} _ {a})} ^ {a})} ^ {a})} ^ {a})} ^ {H} (- {e}} (- e} ^ {e} ^ {e} ^ {e} ^ {e} ^ {j\ beta} {theta} _ {a}}) =- {a}}) =- {a}} q} _ {a} ^ {H} {f} _ {{L} _ {a}} =0\]

Moreover, the solutions of () are also the solution of the quadratic problem

(3.37)#\[ \ stackrel {} {q} =\ underset {\ stackrel {}} {{q} _ {0}}} {\ text {argMin}}\ left (\ stackrel {} {{q} {{q}} _ {0}} _ {0}} _ {0} ^ {H}}} {{Q}}} {\ text {argMin}}}\ left (\ stackrel {}} {{q}} {{q}} _ {{q}} _ {0}}} {\ omega} ^ {H}}} (\ overline {K}} - {\ omega} {{q}} _ {0}} _ {0} ^ {H}}} {\ omega} ^ {2}}\ overline {M})\ stackrel {} {P} {P}\ stackrel {▲} {{q} _ {0}} -\ stackrel {} {{q} _ {0} ^ {H}} {0} ^ {H}}} {0} {H}} _ {0}\ right)\]

However, for the solutions sought, verifying the connection conditions between face and at the level of the axis, we have, according to ()

(3.38)#\[\begin{split} \ stackrel {} {{q} _ {0} _ {0} ^ {H}}}\ stackrel {459} {{P}} {{H}} {f} _ {0} =\ left\ {\ begin {array} {\ array} {ccc} {ccc}} {ccc}} {ccc}} ^ {array} {ccc}} {ccc}} {\ eta}} {\ eta}} _ {\ eta} _ {\ eta} _ {\ eta} _ {\ mathrm {0a}} ^ {H}\ end {array}\ end {array}\ right\}\ left\ {\ begin {array} {c} 0\\\ {f} _ {{L} _ {{L} _ {\ mathrm {0a}}} {\ mathrm {0a}}}}\ end {array}\ right\} =0\end{split}\]

The problem () can then be in the form

\[\]

: label: eq-62

stackrel {459} {q} =underset {stackrel {}} {{q} _ {0}}} {text {argMin}}left (stackrel {} {{q} {{q}} _ {0}} _ {0}} _ {0} ^ {H}}}stackrel {F}} {{P} ^ {H}}} {overline {K}} {{k}} {{q}} _ {{q}} _ {0}} _ {0} ^ {H}}} {omega} ^ {2}} (overline {K}} - {omega} {{q}} _ {0}} {0} ^ {H}}} {omega} ^ {2}}overline {M})stackrel {} {P} {P}stackrel {} {{q} _ {0}}}right)

Given the convexity of the problem (), the solutions of the problem () therefore also verify

(3.39)#\[ \ stackrel {} {{P} ^ {H}}} (\ overline {H}}} (\ overline {K}} - {\ omega} ^ {2}\ overline {M})\ stackrel {} {P} {P} {P}\ stackrel {P}}\ stackrel {F} {P}\ stackrel {F} {P}\ stackrel {} {P}}\ stackrel {F} {P}\ stackrel {F}}\]

Note: the relationship () also indicates that the projector defined in the relation () is not the only eligible one. The term \({e}^{j\beta }{\theta }_{a}\) can be replaced by \(\mathit{Id}\). This choice was not made to maintain a classical form with a reduced problem, similar to that obtained when it is not necessary to separate the terms carried by the axis and those carried by the right and left interfaces. Moreover, since the eigenvalues of the term \({e}^{j\beta }{\theta }_{a}\) are all of module 1, the conditioning of the global problem is not affected by this change.