2. Harmonic equation

We establish the dynamic equation in the case of harmonic stress for three types of mechanical systems:

• pure structures (without fluid),

• pure fluids (without structure) with linear acoustic behavior,

• mixed structures and fluids systems in fluid-structure interaction.

2.1. Harmonic equation of structures

The vibratory behavior of a pure structure results from the external forces applied to it. The quantity to be calculated is the displacement at any point \(P\) of the model.

2.1.1. Direct calculation of elastic structures

In the case of direct calculation on the finite element model we can write:

(2.1)\[ \ mathrm {M}\ ddot {\ mathrm {u}}} +\ mathrm {u}} +\ mathrm {u}} +\ mathrm {K}\ phantom {\ rule {0.5em} {0.5em} {0em}} {0ex}}}\ mathrm {u}}\ mathrm {u}} = {\ mathrm {f}}} _ {\ text {ext}}\ phantom {\ rule {0.5em}} {0em} {0ex}}}\ mathrm {u}} = {\ mathrm {f}}} _ {\ text {ext}}} (P, t)\]

where \(\mathrm{M}\) is the (real) mass matrix, \(C\) is the (real) damping matrix, is the (real) damping matrix, \(K\) is the (real) stiffness matrix, \({\mathrm{f}}_{\text{ext}}(P,t)\) is the (complex) field vector of the external forces applied to \(S\). Finally \(\mathrm{u}\), \(\dot{\mathrm{u}}\) and \(\ddot{\mathrm{u}}\) are the (complex) vectors of displacement, speed, and acceleration, functions of \(P\) and \(t\).

In a harmonic problem, a loading is imposed that is dynamic, spatially random, but sinusoidal in time. We are then interested in the stabilized response of the system, without taking into account the transitory part. The field of external forces is written as:

(2.2)\[ {\ mathrm {f}} _ {\ text {ext}} (P, t) =\ {{\ mathrm {f}}} _ {\ text {ext}} (P)\} {e}} {e}} ^ {j\ omega t}\]

The travel field is written as:

(2.3)\[ \ mathrm {u} (P, t) =\ {\ mathrm {u} (P)\} {e} ^ {j\ omega t}\]

The speed and acceleration fields are written by time derivation of the displacements:

\[\]

: label: eq-4

begin {array} {c}dot {mathrm {u}} (P, t) =jomega{mathrm {u} (P)} {e} ^ {jomega t}\ omega t}\ ddot {ddot {mathrm {u}}} (P, t)text {=}text {-} {omega} ^ {2}\ ddot {mathrm {u}}} (P, t)text {=}text {-} {omega} ^ {2}{mathrm}} {u} (P)} {e} ^ {jomega t}end {array}

Finally, structure \(S\) verifies the following equation:

\[\]

: label: eq-5

(mathrm {K} +jomegamathrm {C} - {omega} ^ {2}mathrm {M}){u} ={{mathrm {f}}} _ {text {ext}} (P)} ={{mathrm {f}}}

2.1.2. Modal-based elastic structure calculation

The calculation of the harmonic response by the modal synthesis method consists in looking for the unknown field of displacement, resulting from finite element modeling, on an appropriate space, of reduced dimension (Ritz transformation). Reference will be made to documents [R4.06.02] and [R4.06.03].

If we use this method instead, equation () is projected onto the modal basis of \(S\) and we end up with the following harmonic equation:

(2.4)\[ (\ overline {\ mathrm {K}}} +j\ omega\ overline {C} - {\ omega} ^ {2}\ overline {\ mathrm {M}})\ {\ eta\}} =\ {{\ overline {\ overline {\ overline {\ mathrm {f}}}}} _ {\ text {ext}}}\}\]

where \(\overline{\mathrm{M}}={\mathrm{\Phi }}^{\mathrm{T}}\mathrm{M}\mathrm{\Phi }\) is the (real) generalized mass matrix, \(\overline{\mathrm{C}}={\mathrm{\Phi }}^{\mathrm{T}}\mathrm{C}\mathrm{\Phi }\) is the (real) generalized damping matrix, is the (real) generalized mass matrix, is the (real) generalized mass matrix, and is the (complex) vector of the generalized harmonic external forces applied. \(\overline{\mathrm{K}}={\mathrm{\Phi }}^{\mathrm{T}}\mathrm{K}\mathrm{\Phi }\) \(\{{\overline{\mathrm{f}}}_{\text{ext}}\}={\mathrm{\Phi }}^{\mathrm{T}}\{{\mathrm{f}}_{\text{ext}}\}\) We have noted \(\mathrm{\Phi }\) the (real) matrix of the modal vectors of the Ritz base, \(\{\mathrm{\eta }(P)\}\) is the (complex) vector of generalized harmonic displacements.

Once \(\{\mathrm{\eta }(P)\}\) is determined by () we make a return on a physical basis (cf. [R4.06.02]).

2.1.3. Direct calculation of viscoelastic structures

The law of behavior of a linear viscoelastic material is of the type \(L\sigma =Mϵ\) where \(L\mathit{et}M\) are linear differential operators of the time variable (see [R5.05.04]). In the harmonic domain, this law of behavior becomes of the \(\sigma ={H}_{c}(\omega )ϵ\) type where \({H}_{c}(\omega )\) is a complex tensor depending on the pulsation. The discretized equation of the structure is then written in harmonic:

(2.5)\[\begin{split} ({\ mathrm {K}}} _ {\ mathrm {r}}} (\ omega) +j {\ mathrm {K}}} _ {\ mathrm {i}} (\ omega) - {\ omega} ^ {2} ^ {2}\ mathrm {2}}\ {2}\\ mathrm {2}}\\ mathrm {2}}\\ mathrm {M}})\ {u\}} =\ {{\ mathrm {f}}} (\ omega) - {\ omega} ^ {2} ^ {2}\ mathrm {2}}\\ mathrm {M}})\ {u\}} =\ {{\ mathrm {f}}} (\ omega) - {\ omega} ^ {2}}\end{split}\]

Note \({\mathrm{K}}_{\mathrm{c}}={\mathrm{K}}_{\mathrm{r}}(\omega )+j{\mathrm{K}}_{\mathrm{r}}(\omega )\) the complex stiffness matrix depending on the frequency.

Special case: if the damping is of the hysteretic « global » type, the law of behavior is written \(\sigma =(1+j\eta )Hϵ\) where \(H\) is the real Hooke tensor and \(\eta\) is a global loss coefficient (cf. [R5.05.04]). The equation () is then written as:

(2.6)\[ (1+j\ eta)\ mathrm {K} - {\ omega} ^ {2}\ mathrm {M})\ {u\} =\ {{\ mathrm {f}}} _ {\ text {ext}} (P)\}\]

Note that \({\mathrm{K}}_{\mathrm{c}}=\mathrm{K}+j\eta \mathrm{K}\) is a complex stiffness matrix. which no longer depends on frequency.

2.2. Harmonic equation of acoustic fluids

The document [R4.02.01] describes the finite element modeling of a fluid system (without transport) with linear acoustic behavior.

Fluid system \(F\) is subject to a harmonic acoustic velocity stress on part of its border. The harmonic response is described by the following equation, where the quantity to be calculated is the sound pressure at any point \(P\) of the model.

(2.7)\[ (\ mathrm {K} +j\ omega\ mathrm {C} - {\ omega} ^ {2}\ mathrm {M})\ {\ mathrm {p} (P)\}} =-j\ omega\ {\ omega\ {{\ mathrm {v}}} _ {\ mathrm {n}} (P)\} =-j\ omega\ omega\ {{\ mathrm {v}}} _ {\ mathrm {n}} (P)\}\]

where \(\mathrm{M}\) is the (complex) acoustic « mass » matrix, \(\mathrm{C}\) is the (complex) acoustic « damping » matrix (and in this case the edge \({\partial }_{z}F\) where acoustic impedance is applied) and \(\mathrm{K}\) is the (complex) acoustic « stiffness » matrix.

We note \({\mathrm{v}}_{\mathrm{n}}(P,t)=\{{\mathrm{v}}_{\mathrm{n}}(P)\}{e}^{j\omega t}\) the speed, with \(\text{{}{V}_{n}(P)\text{}}\) the (complex) field vector of normal acoustic velocities applied to the border \({\partial }_{v}F\) of \(F\) where acoustic velocities are applied and \(\mathrm{p}(P,t)=\{\mathrm{p}(P)\}{e}^{j\omega t}\) the pressure, with \(\text{{}p(P)\text{}}\) the (complex) vector of acoustic pressures, with the (complex) vector of acoustic pressures.

2.3. Harmonic equation of fluid-structure systems

The document [R4.02.02] describes the finite element modeling of a system \(F+S\) consisting of a fluid part (without transport) \(F\) interacting with a structure part \(S\) (interaction in \(F\cup S\)). Fluid and structure have a linear behavior.

Fluid system \(F\) is subject to harmonic stress at normal acoustic speed over part of its border. The harmonic response is described by the following equation where the quantities to be calculated are:

• the sound pressure at any point \(P\) of the fluid \(F\),

• the movement at any point \(P\) of the structure \(S\),

• auxiliarily the potential \(\varphi\) for movement at any point \(P\) of the fluid \(F\) or other auxiliary variables described in [R4.02.02].

(2.8)\[\begin{split} (K- {\ omega} ^ {2} m-J {\ omega} ^ {3} I)\ left\ {\ begin {array} {c} u (P)\\ p (P)\\ varphi (P)\\ varphi (P)\ end {array}\\\ varphi (P)\\\\\\\\ varphi (P)\\\\\\\ P (P)\\\\\\end{split}\]

where \(\mathrm{M}\) is the (real) fluid-structure « mass » matrix from domains \(F\) and \(S\), \(I\) is the (real) fluid « impedance » matrix coming from the edge \({\partial }_{z}F\) of the domain \(F\) where an impedance is applied and \(K\) is the (real) fluid-structure « stiffness » matrix from domains \(F\) and \(S\).

Harmonic solution fields. \({v}_{n}(P,t)=\{{v}_{n}(P)\}{e}^{j\omegat }\) is the (real) vector of the field of normal acoustic velocities applied to the border \({\partial }_{v}F\) of \(F\), \(u(P,t)=\{u(P)\}{e}^{j\omegat }\) is the (complex) vector of the field of displacement in the structure \(S\), \(p(P,t)=\{p(P)\}{e}^{j\omegat }\) is the (complex) vector of the field of sound pressure in the fluid \(F\) and \(\varphi (P,t)=\{\varphi (P)\}{e}^{j\omegat }\) is the vector ( complex) of the displacement potential field in fluid \(F\).

2.4. General harmonic equation

In order to take into account all the cases of harmonic equations, the operator DYNA_VIBRA solves the following general harmonic equation:

(2.9)\[ (-j {\ omega} ^ {3} I- {\ omega} {\ omega} ^ {2} M+j\ omegac +K)\ {q\} =\ left\ {\ sum _ {i=1} ^ {k} {h} {h}} _ {h}} _ {h}} _ {h}} _ {h}} _ {h}} _ {h} _ {h}} _ {h} _ {h} _ {h}} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h}} _ {h} _ {h} _ {h} _ {h} _ {h}} _ {h} _ {h} _ {h}} _ {h} _ {\ varphi} _ {i}} {\ text {180}}}}\ cdot {g} _ {i} (P)\ right\}\]

where \(\mathrm{I}\) is the possible fluid « impedance » matrix, \(M\) is the « mass » matrix, \(C\) is the « damping » matrix and \(K\) is the « stiffness » matrix.

The solution is \(\left\{q(P)\right\}\). The second member contains \(\left\{{g}_{i}(P)\right\}\) which is a field vector at the nodes corresponding to one or more loads of force or acoustic speed or potential or imposed movement, \({h}_{i}(f)\) a real or complex function of the frequency, \(f\), \(\omega =2\pi f\) is the pulsation, \({n}_{i}\) is the power of the pulsation, is the power of the pulsation, where the load is a function of the pulsation, and \({\varphi }_{i}\) is the phase in degrees. of each component of the excitation with respect to a phase reference.

As an example, if we take the case of a fluid system modeled acoustically, with no imposed degrees of freedom, simply solicited on part of its border by a normal speed field \({\mathrm{v}}_{\mathrm{n}}(P,t)=\{{\mathrm{v}}_{\mathrm{n}}(P)\}{e}^{j\omega t}\), the terms of equation () become:

(2.10)\[ (- {\ omega} ^ {2}\ mathrm {M} +j\ omega\ mathrm {C} +\ mathrm {K})\ {\ mathrm {p}\} =- {\ omega} ^ {} ^ {} ^ {}\ left\ {}\ left\ {{\ mathrm {v}}} _ {\ mathrm {n}} (P)\ right\}\]

• no \(\mathrm{I}\) impedance component,

• acoustic mass matrix \(\mathrm{M}\),

• Optionally, a damping matrix \(\mathrm{C}\) resulting from acoustic finite element modeling if impedance on a border,

• acoustic stiffness matrix \(\mathrm{K}\),

• the vector of unknowns \(\left\{\mathrm{q}(P)\right\}\) is reduced to \(\left\{\mathrm{p}(P)\right\}\), vector of pressures at the nodes

• load \(\left\{{\mathrm{g}}_{\mathrm{i}}(P)\right\}\) is \(\left\{{\mathrm{v}}_{\mathrm{n}}(P)\right\}\), vector field with normal speed to the faces

• \({h}_{i}(f)\) is constant and is equal to \(-1.\)

• the power \({n}_{i}\) is equal to \(1\) and the phase \({\varphi }_{i}\) is zero