Theoretical elements =================== In this section, the theoretical foundations of the approach are presented. Introductory ------------ General expression of the problem ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We start by recalling the formulation of a linear dynamics problem in the form of minimizing the Hamiltonian: .. math:: :label: eq-1 \ underset {\ mathit {0}}} {\ mathit {min}}}\ underset {{\ mathit {enc}} _ {1}} {\ overset {{\ mathit {enc}} _ {2}} {2}}} {\ int}}} {\ int}}} {\ int}}}\ left ({\ mathit {0}}}\ left ({\ mathit {0}}} ^ {\ mathit {halt}}} {\ mathit {1}}} {\ mathit}} {\ mathit}} {\ mathit}} {\ mathit}} {\ mathit}}\ mathit {}} - {\ dot {\ mathit}} - {\ dot {\ mathit}} √}}} ^ {\ mathit {halt}}}\ mathit {residue}}\ dot {\ mathit {0}}\ right)\ mathit {}}\ right)\ mathit {}} With a few algebraic manipulations, we can obtain the modal calculation expression that is important to us: .. math:: :label: eq-2 \ mathit {thanx} - {\ mathit {8,5}}} ^ {2}\ mathit {forx} =0 Generalized modes and unknowns ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We also recall that we denote the rectangular matrix of the :math:`{n}_{m}` modes as follows: .. math:: :label: eq-3 \ mathrm {\ Phi} =\ left [\ begin {array} {ccc} {\ mathrm {\ Phi}} _ {1} &\ text {...} & {\ mathrm {\ Phi}}} _ {{n}} _ {m}}\ end {array}\ right] Since each vector :math:`{\mathrm{\Phi }}_{i}` is of length :math:`{n}_{\text{eq}}`, the matrix :math:`\mathrm{\Phi }` has dimensions :math:`{n}_{\text{eq}}\times {n}_{m}`. Finally, we recall the relationship between a vector of unknowns in physical space :math:`x` and a vector of generalized unknowns :math:`\mathrm{\eta }`: .. math:: : label: eq-4 x=\ mathrm {\ Phi}\ mathrm {\ eta} = {\ mathrm {\ eta}} _ {1} {\ mathrm {\ Phi}} _ {1} + {\ mathrm {\ eta}}} + {\ mathrm {\ eta}}} _ {2}} +\ mathrm {\ Phi}} _ {1} +\ mathrm {\ eta}} + {\ mathrm {\ eta}} + {\ mathrm {\ eta}} + {\ mathrm {\ eta}} + {\ mathrm {\ eta}} + {\ mathrm {\ eta}} + {\ mathrm {\ eta}} _ {{n}} _ {n} _ {m}} {\ mathrm {\ Phi}} _ {{n} _ {m}} .. _RefHeading___Toc 20369_2140609791: Problem under constraints ~~~~~~~~~~~~~~~~~~~~~~~~~~~ The last point presented concerns the resolution of a dynamic problem under constraints. The latter is expressed in the general form: .. math:: : label: eq-5 \ mathit {Bx} =0 In the present case, the matrix :math:`B` is a rectangular matrix whose number of rows is equal to the number of constraints applied to the unknowns. The problem under constraints is expressed as follows: .. math:: :label: eq-6 \ underset {\ mathit {740}} =0} {\ mathit {min}}}\ underset {{\ mathit {enc}} _ {1}} {\ overset {{\ mathit {enc}}} _ {2}}} {2}}} {\ int}}} {\ int}}}\ left ({\ mathit {8}}} ^ {\ mathit {10}}} {\ mathit {10}}} {\ mathit {1}}} {\ mathit {10}}} {\ mathit {2}}} _ {\ mathit {mathit {0}}}} ^ {\ mathit {‡ {‡}}\ mathit {10}}\ dot {\ mathit {0}}\ right)\ mathit {]} We assume that we know how to build a :math:`{n}_{\mathit{ker}}` dimensional base of the core of the constraint matrix. It is noted: .. math:: :label: eq-7 \ mathrm {\ Psi} =\ left [{\ mathrm {\ Psi}} _ {1}\ mathrm {...} {\ mathrm {\ Psi}} _ {{n} _ {\ mathit {ker}}}\ right] As the solutions sought verify the :math:`\mathit{Bx}=0` constraints, they belong to the core of :math:`B`. So we can write them on this basis: .. math:: :label: eq-8 x=\ mathrm {\ Psi}\ mathrm {\ eta} By injecting this expression into the minimization problem, we see that we can then relax the constraint and write the problem in the form: .. math:: :label: eq-9 \ underset {\ mathrm {\ eta}}} {\ mathit {min}}} {\ mathit {min}}}\ underset {{\ mathit {enc}}} {\ mathit {enc}} _ {2}} {\ int}}} {\ int}}}\ left ({\ mathrm {\ int}}}\ left [{\ mathrm {\ Psi}}\ left [{\ mathrm {\ psi}}} ^ {\ mathit {REGULO}}\ mathit {332}\ mathrm {\ Psi}\ right]\ mathrm {\ eta} - {\ dot {\ mathrm {\ eta}}}}}} ^ {\ mathit {scrap}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}}} ^ {\ mathit {\ eta}}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}} ^ {\ right]\ dot {\ mathrm {\ eta}}}\ right)\ mathit { The problem is therefore solvable using conventional techniques (minimization without constraints). Implementation ------------- We consider a structure divided into 2 substructures. For each, the following were determined: * the matrices of mass :math:`{M}_{1}` :math:`{M}_{2}` and stiffness :math:`{K}_{1}` :math:`{K}_{2}`, of respective dimensions :math:`{n}_{{\text{eq}}_{1}}\times {n}_{{\text{eq}}_{1}}` and :math:`{n}_{{\text{eq}}_{2}}\times {n}_{{\text{eq}}_{2}}` * the matrices of the modes with blocked interface :math:`{\mathrm{\Phi }}_{1}` and :math:`{\mathrm{\Phi }}_{2}` of respective dimensions :math:`{n}_{{\mathit{eq}}_{1}}\times {n}_{{m}_{1}}` and :math:`{n}_{{\mathit{eq}}_{2}}\times {n}_{{m}_{2}}`. They are related to generalized unknowns :math:`{\mathrm{\eta }}_{1}` and :math:`{\mathrm{\eta }}_{2}` respectively. * the matrices of the interface modes :math:`\overline{{\mathrm{\Phi }}_{1}}` and :math:`\overline{{\mathrm{\Phi }}_{2}}` of respective dimensions :math:`{n}_{{\mathit{eq}}_{1}}\times {n}_{i}` and :math:`{n}_{{\mathit{eq}}_{2}}\times {n}_{i}`. They are related to generalized unknowns :math:`\overline{{\mathrm{\eta }}_{1}}` and :math:`\overline{{\mathrm{\eta }}_{2}}` respectively. Based on these definitions, we can write the link between the physical and generalized degrees of freedom of each of the substructures: .. math:: :label: eq-10 {\ mathit {x}} _ {1} =\ left [{\ mathrm {\ Phi}}} _ {1} {\ overline {\ mathrm {\ Phi}}}} _ {1}\ right]\ right]\ left\ left\ {\ begin {array} {\ left\ {\ begin {array} {c}} {c}} {\ mathrm {\ eta}}} _ {1}\ {\ overline {\ mathrm {\ Phi}}} _ {1}\ {\ overline {\ mathrm {\ eta}}} _ {1}\ end {array}\ right\} By noting :math:`\stackrel{~}{{\mathrm{\Phi }}_{i}}=\left[{\mathrm{\Phi }}_{i}{\overline{\mathrm{\Phi }}}_{i}\right]` and :math:`\stackrel{~}{{\mathrm{\eta }}_{i}}=\left\{\begin{array}{c}{\mathrm{\eta }}_{i}\\ {\overline{\mathrm{\eta }}}_{i}\end{array}\right\}`, we can write the previous relationship as: .. math:: :label: eq-11 {\ mathit {x}} _ {1} = {\ stackrel {~} {\ stackrel {~}} {\ mathrm {\ Phi}}} _ {1} {\ stackrel {~}} {\ stackrel {~}} {\ mathrm {\ eta}}}} _ {1} For each substructure, it is then possible to calculate the mass and stiffness matrices projected onto the blocked interface modes and the interface modes: .. math:: :label: eq-12 {\ stackrel {~} {\ mathit {492}}}} _ {1}} _ {1}} = {\ stackrel {~} {\ mathrm {\ Phi}}} _ {1} ^ {T} {\ mathit {390}} {\ mathit {390}}} {\ mathit {}}} {\ mathit {}}} {\ mathit {}}} {\ mathit {}}}} {\ mathit {390}}} _ {1}} We then build the global matrices that relate to the 2 substructures: .. math:: :label: eq-13 {\ mathit {492}} _ {\ mathit {zern}} =\ left [\ begin {array} {cc} {\ stackrel {~} {\ mathit {390}}}} _ {1}} & 0\\ 0& 0\\ 0& {\ stackrel {\ 0& {\ stackrel {~}} {\ mathit {902}}} _ {2}\ end {array}}} _ {1}} & 0\\ 0& 0\\ 0& {\ stackrel {~}} {\ mathit {902}}} _ {2}\ end {array}\ right] The generalized degrees of freedom are constructed in the same way: .. math:: :label: eq-14 {\ mathrm {\ eta}} _ {g} =\ left\ {\ begin {array} {c} {\ stackrel {~} {\ mathrm {\ eta}}}} _ {1}\\ {\ stackrel {\ stackrel {~} =\\ {\ stackrel {~}} =\\ {\ {\ stackrel {~}} {\ stackrel {~}}} _ {1}\\ {\ {\ stackrel {~}}} _ {1}\\ {\\ stackrel {~}} =\\ {\ {\ stackrel {~}} {\ stackrel {~}}} _ {1}\\ {\ {\ stackrel {~}} =\ left\ {\\ stackrel {~}} c}\ left\ {\ begin {array} {c} {c} {\ mathrm {\ eta}}} _ {1}\\ {\ overline {\ eta}}}} _ {1}\ end {array}\\ array}\ right\\ array}\ right\ right\}\ right\}\ right\}\ right\}}\\ {\ overline}}\\ {\ overline {\ mathrm {\ eta}}} _ {2}\\ {\ array}\ right\ right\}\ right\}}\ right\}\ right\}}\ right\}\ right\}}\ right\}}\ right\}\ right\}}\ right\}\ right\}}\ right\}\ right\}}\ right\}\ right\}}\ right\ thrm {\ eta}}} _ {2}\ end {array}\ right\}\ end {array}\ right\} We recall that :math:`{\mathrm{\eta }}_{i}` are the generalized unknowns associated with blocked interface modes and :math:`\overline{{\mathrm{\eta }}_{i}}` those associated with interface modes. So to ensure the continuity of movements in physical space when crossing the interface, we must impose: .. math:: :label: eq-15 \ overline {{\ mathrm {\ eta}}} _ {1}} =\ overline {{\ mathrm {\ eta}}} _ {2}} This can be expressed simply using a constraint matrix in the form: .. math:: :label: eq-16 B {\ mathrm {\ eta}} _ {g} =0 with: +----------------------------------------------------------------------------------------------------------------------------++ | || + .. image:: images/100002000000015C0000009BADCF883F8884610B.png ++ | :width: 1.622in || + :height: 0.7217in ++ | || + ++ | || +----------------------------------------------------------------------------------------------------------------------------++ The global problem can then be expressed in the form of: .. math:: :label: eq-18 \ underset {\ mathit {bola} {\ mathrm {\ eta}}} _ {g} =0} {\ mathit {min}}\ underset {{t} _ {1}} {\ overset {{t} _ {t} _ {2} _ {2}}} {2}}} {\ int}}} {\ int}} {\ int}}\ left ({\ mathrm {\ eta}}} _ {g} ^ {\ mathit {‡}} {\ mathit {‡}} {332}} _ {g} {\ mathrm {\ eta}}} _ {eta}} _ {\ eta}} - {\ dot {\ mathrm {\ eta}}} {\ mathit {halt}}} {\ mathit {exactly}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {147}}}} {\ mathit {146}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {exactly}}} By calculating the core of :math:`B`, the problem above is solvable as presented in section :ref:`5.1.3 `. Implementation examples can be found in the sdlv102 and sdlv103 tests.