11.2.1 Step 1 – Setting up System Configurations

The normal modes of the plate are taken from equation 5.46. The modes are sorted in frequency order. The first mode shape with $n=(1,1)$ is at $f_1=54.2$ Hz; until $f_{\text{max}}=4780.1$ Hz there are 140 modes. The SEA properties of the cavities are taken from Chapter 10.

The direct field radiation stiffness matrices are calculated using a mesh of $N_x=40$ and $N_y=25$ nodes. This leads to a mesh of element length $\mathrm{\Delta }x\approx \mathrm{\Delta }y\approx 0.02$ m, which is sufficient for this frequency. When the structural plate modes shall be calculated using a FE-mesh, this must be fine enough for the bending wavelength and can thus be much finer than the mesh that is required for a precise radiation stiffness calculation in the desired frequency range. However, in this example it is not necessary, because we map the solution onto our mesh directly from analytical results.

When using fine FE meshes for the structure, it is better to use two meshes, each appropriate for structure and fluid, for the better use of (8.42).

11.2.2 Step 2 – Setting up System Matrices and Coupling Loss Factors

The total system matrix is created according to (11.3). All degrees of freedom are shared by the connected systems and no specific treatment of connecting region is required.

The SEA matrix is set up with the same principles as in section 7.3.2.1 and using equation (7.50). In the SEA matrix the plate does not occur as subsystem. Hence, for the energy degrees of freedom, we have a two system SEA matrix.

The result for the hybrid transmission loss of the plate is shown in Figure 11.3. In addition the SEA transmission loss is also included. We recognize the typical shape of the transmission loss as described, for example, in Bies et al. (2018). At low frequencies below the first resonances, the transmission is stiffness controlled and very high. The transmission loss drops down at the first panel resonances, increases, and reaches mass law slope above 500 Hz. At this frequency the wavelength in the fluid becomes similar to the panel dimensions. There is a good agreement between the SEA and the hybrid result in the mass law regime. In the coincidence area both curves also coincide well, despite small variations around the ensemble average of the SEA curve. Thus, the reverberant field model of the plate and area coupling loss factors are in good accordance to the hybrid method.

In addition to the inert damping of the SEA systems, there is the damping due to the connected FEM subsystems given in modal form by (7.51). The damping loss for both cavities is shown in Figure 11.4, revealing that this dissipation is relatively low. The reason for the low dissipation is the small area of the panel in relation to the total surface of the room. When all walls of the room would be FEM subsystems, the situation would be different. The highest dissipation occurs at the resonances of the plate.

11.2.3 Step 3 – External Loads

First, the SEA cavity 1 is excited by a power load of ${\mathrm{\Pi }}_{1,\mathrm{ext}}=1$ W. Second, the plate is excited by a point force in the $z$-direction of the plate with ${\hat{F}}_{z,\mathrm{rms}}=10$ N that is located at $x=0.31$ m and $y=0.11$ m in plate coordinates assuming one corner as origin. Both cases are calculated separately.

11.2.4 Step 4 – Solving System Matrices

11.2.4.1 Step 4.1 – Power Input due to FEM System Excitation

The cross spectral density of the point force in modal coordinates must be calculated with

  (11.4)

and the radiated power with

  (11.5)

Even in modal form the triple matrix multiplication of (11.4) is computationally expensive. As the point force is fully coherent (to itself), it is faster to calculate the cross spectral density of the plate from the deterministic response, hence solving

  (11.6)

calculating the cross spectral density directly from the deterministic response, and determining the power input into the SEA subsystems from external excitation that is not related to SEA. The rms-value $F_{z,\mathrm{rms}}=10$ N of the point force corresponds to an amplitude of ${\hat{F}}_z=\sqrt{2}10$ N. From the above equation we calculate a power radiation from the plate into the SEA systems as given in Figure 11.5. Compared to the source in cavity 1, the power can be neglected. However, at the odd resonances and near coincidence the panel becomes an efficient radiator, leading to high power radiation into the cavities. As both cavities are directly connected to the vibrating plate, the power radiation is the same for both rooms.

The velocity rms-value is reconstructed from the modal coordinates by

  (11.7)

  (11.8)

The resulting velocity due to the force excitation is shown in Figure 11.7 in combination with the velocity caused by excitation from the reverberant fields.

11.2.4.2 Setp 4.2 – Solve the SEA Equations with SEA and FEM Power Input from Step 4.1

The result for each single case can be seen in Figure 11.6. In comparison the power load in cavity 1 leads to higher pressure levels at low frequencies, but at high frequencies the radiation from the plate becomes more efficient and radiates more energy into cavity 2 than is transmitted from cavity 1 into cavity 2 through the plate. The SEA results of the corresponding model show that the random modelling gives reliable results above 600 Hz for the power excitation in cavity 1 and above 2 kHz for the force excitation. In case of force excitation, the point force does not lead to equal excitation of all modes, and the conditions for SEA are met later in frequency. In addition we must consider that the mass law provides a correct value for the coupling of both cavities, and therefore the results agree earlier in frequency with the hybrid result.

11.2.4.3 Step 4.3 – Calculate FE Response due to Energies in SEA Subsystems from Step 4.2

As the external forces were treated by step 4.1, we finally calculate the response of the FEM systems resulting from the reverberant field excitation in each subsystem using (7.43), omitting the deterministic external loads

  (11.9)

As the response is calculated in the modal base, the cross spectral density matrix must be converted into the mesh space by

  (11.10)

and the rms-velocity results from the diagonal of the above matrix

  (11.11)

11.2.5 Step 5 – Adding the Results

In the final step all results are added together for the deterministic systems. The cross spectral density of the plate results from the external excitation and the reverberant load of the connected SEA systems. In Figure 11.7 the rms-velocity of the plate is shown in combination with the SEA result. For the power load case there is no external excitation of the plate, so the reverberant load is the exclusive excitation. The SEA simulation covers only the resonant energy in the plate. This is the reason why both curves don’t agree even at high frequencies, and the resonant SEA solution underestimates the velocity level compared to the FEM system response. When we add the nonresonant motion to the engineering result of the plate using (10.19), both curves agree well above 500 Hz.

In case of the force excitation, the external load response is dominant compared to the reverberant response $S_{qq,\mathrm{rev}}$. This is obvious, because the reverberant load from the cavities is the indirect feedback of the force exciting the plate. Globally, the results from section 6.4.2.1 are still valid. A point force requires a large modal overlap to fulfil the conditions of a smooth ensemble average. However, above 1kHz the SEA solution is valid.

11.3.1 The Trim Stiffness Matrix

In order to cope with the hybrid theory, the acoustic treatment must also be modelled by a a stiffness matrix. This is rather unusual, as, for example, FE implementations are using displacement degrees of freedom for the structure side and pressure degrees of freedom for the fluid side (Doutres et al., 2007) due to the fact that the fluid is represented by pressure degrees of freedom in most solvers.

The trim dynamic stiffness matrix has different sets of degrees of freedom for the left (1) and right (2) side. This suggests writing the matrix in blocked form

  (11.12)

The calculation of the above stiffness matrix is done in two steps: First, the transfer matrix of the infinite layer from section 9.3 is transformed into the stiffness matrix in wavenumber domain.

  (11.13)

Second, this stiffness matrix must be converted into the discrete space. For the approximation of this discrete stiffness matrix, we assume a regular mesh with constant element lengths $\mathrm{\Delta }x$,$\mathrm{\Delta }y$. The transfer matrix of infinite layers (9.96)

  (9.96)

is to be converted into the stiffness matrix (11.13). The internal pressure ${\mathit{p}}_2$ must be replaced by the external pressure ${\mathit{p}}_2^{\prime }=-{\mathit{p}}_2$

  (11.14)

and the coefficients of the stiffness matrix in wavenumber domain are hence

  (11.15)

  (11.16)

  (11.17)

In Figure 11.8 such a treatment connected to a structure is shown. Tournour et al. (2007) proposed a local approximation that assumes that only the opposite nodes are connected. This assumption is valid for very thin layers ($h\ll \lambda$), and the block matrices in equation (11.12) are diagonal in that case. It is shown by Peiffer (2018) that this assumption is too strict and a nonlocal approach is required for most treatments.

This matrix must be converted into the space domain by inverse Fourier transform. Using the jinc-function approach from Langley (2007) and assuming isotropy in each layer, the transformation becomes simple and implies a finite wavenumber integration.

  (11.18)

The transformation is applied to each coefficient of the stiffness matrix with wavenumber arguments

  (11.19)

being the projected distance $x_{ij}$ between nodes, and the maximum wavenumber $k_s$ that can be represented by the spacial sampling of the mesh, respectively. The integral in (11.18) is solved numerically, and the stiffness matrix of the trim is available for further considerations.

11.3.2 Hybrid Modal Formulation of Trim and Plate

The trim constitutes an additional deterministic layer that must be considered in the total stiffness matrix. When a layer of noise control treatment is applied to the right surface, the plate degrees of freedom are shared with those from left side of the trim. The right side of the noise control treatment is connected to the fluid as shown in figure 11.8.

This configuration gives rise to the following block matrix configuration

  (11.20)

adding up to the total stiffness matrix with trim

  (11.21)

The coupling loss factor can be calculated when we write (7.27) in block matrix form

  (11.22)

The above equation can be used as is but leads to double size matrices in this case. The idea is to stay with the plate displacement coordinates and condense the above equation to pure structural coordinates, here ${\mathit{w}}_1$. Writing equation (11.20) using the block matrix gives

  (11.23)

  (11.24)

We eliminate ${\mathit{w}}_2$ in order to express the total stiffness in structural coordinates. Doing this with equations (11.23) and (11.24) leads to

  (11.25)

There is a modified version of the total stiffness matrix

  (11.26)

which is valid for coordinates $\left\{\begin{array}{c}{\mathit{w}}_1\end{array}\right\}$ Due to the condensation $\left\{\begin{array}{c}{\mathit{F}}_2\end{array}\right\}$ is transformed to $\left\{\begin{array}{c}{\mathit{w}}_1\end{array}\right\}$ degrees of freedom by

  (11.27)

using the transformation matrix

  (11.28)

Equation (11.27) provides the deterministic equation of motion for displacement degrees of freedom $\left\{\begin{array}{c}{\mathit{w}}_1\end{array}\right\}$ of the plate. For use in hybrid coupling loss factor equation (7.27), we replace the direct radiation stiffness by an expression that includes the trim. Let us first consider that the trimmed side is the radiating side. The power radiated by the degrees of freedom from side two $\left\{\begin{array}{c}{\mathit{w}}_2\end{array}\right\}$ is given by

  (11.29)

Reordering of (11.24) and assuming no external forces ${\mathit{F}}_2=0$ allows transforming $\left\{\begin{array}{c}{\mathit{w}}_1\end{array}\right\}$ to $\left\{\begin{array}{c}{\mathit{w}}_2\end{array}\right\}$

  (11.30)

and we get for the radiated power

  (11.31)

The term between both $\left\{\begin{array}{c}{\mathit{w}}_1\end{array}\right\}$ is called the reduced radiation stiffness

  (11.32)

Thus, the radiated power can be calculated with

  (11.33)

and (11.32) replaces $\left[\begin{array}{c}{\mathit{D}}_{\text{dir}}^{\left(2\right)}\end{array}\right]$ in equation (7.27).

  (11.34)

From reciprocity (6.112) it follows that the reduced radiation stiffness can also be used for the reverse direction.

  (11.35)

In addition to the reciprocity argument, the above expression can also be derived using the fact that in the condensed equation of motion (11.27), forces $\left\{\begin{array}{c}{\mathit{F}}_2\end{array}\right\}$ are converted to $\left\{\begin{array}{c}{\mathit{F}}_1\end{array}\right\}$ by the transformation matrix $\left[\begin{array}{c}{\mathit{D}}_{\text{trans}}^{SP}\end{array}\right]$.

The diffuse field excitation from side two is then given by

  (11.36)

Entering this expression in (7.21) leads also to (11.35). It can be shown that a trimmed surface can be considered in general by using the total stiffness matrix given by

  (11.37)

using the different expression for trimmed sufaces

According to equations (11.34) and (11.35), the radiation stiffness must be replaced by the reduced radiation stiffness if there is trim on the surface. The same approach is used for all other expressions used in the hybrid formulation, for example (7.52).

11.3.3 Modal Space

The stiffness matrix must be transformed into the modal space for both sets of degrees of freedom. The transformation reads:

  (11.38)

Or, in matrix form when we write all connecting region degrees of freedom in one column vector

  (11.39)

The transformation matrix is a block matrix with twice the modal transformation matrix. All system matrices are transformed into modal space by this matrix meaning to apply the modal transformation matrix to every block matrix. All above given matrices are converted to modal space with the usual transformation.

The same is true for the block matrices of the trim leading to:

  (11.40)

From the conversion to modal space follows a useful property for evaluating the radiation: the modal radiation efficiency given by

  (11.41)

For configurations with trim the reduced radiation stiffness from equation (11.32) shall be considered.

11.3.4 Plate Example with Trim

We apply this theory to the plate from section 11.2. The panel surface connected to cavity 2 is treated with the 5 cm fiber-mass system from section 10.5.3. With the given theory the transmission loss is calculated and shown in Figure 11.8 in comparison with the SEA result. The agreement between hybrid and SEA is even better than for the untrimmed case. Both approaches start to coincide above 250 Hz in contrast to figure 11.3 where the agreement started around 1 kHz. This is the reason why SEA is considered often as valid even at low frequencies for airborne problems. The dissipation by the noise treatment introduces damping into the system that results in an earlier fulfillment in frequency of the requirements of SEA.