## 3 Modelling of Lightweight Structures with Ccld Treatment  
For the estimation of the CCLD potential, a parametric model based on analytical relations of a classical CLD treatment was developed. The considered beam with CCLD is schematically shown in Figure 3. The mechanical and physical properties of the layers are denoted by subscripts \(s\), \(v\), \(c\) for beam structure, viscoelastic and constraining layer, respectively. The width of all layers is denoted as \(b\). To analyse the CCLD principle, a CLD model presented by [PERSON] (1997) has been extended by the possibility of the compression-driven modification of the viscoelastic layer's geometrical and material properties. The initial Zheng model was elaborated following the assumptions:  
1. All three layers have the same transverse deflection when bended;
2. Shear strains in the base layer and constraining layer and also the rotary inertia of all layers are negligible;
3. The viscoelastic layer carries transverse shear but no normal stresses. A frequency-dependent, complex shear modulus, \(G_{v}\left(\omega\right)^{*}=G_{v}\left(\omega\right)\left[1+i\eta_{v}\left( \omega\right)\right]\) where \(\eta_{v}\left(\omega\right)\) is the determined loss factor, is used for the description of the dynamic behaviour of the layer;
4. All displacements are small compared to the structural dimensions and hence linear theories of elasticity and viscoelasticity can be used;
5. The interfaces of the layers are characterised by the lack of slipping;
6. The plane transverse to the middle plane remains plane under bending. The deformation pattern of the CCLD-beam at a given compression level is presented in Figure 4. The extension of the classical Zheng model of CLD for the analysis of structures with CCLD consists in the implementation of the possibility of compression-driven change of viscoelastic layer \(\epsilon\) properties. The impact of the compression level on the following features is introduced:  
* Thickness of the viscoelastic layer \(h_{v}\)
* Density of the viscoelastic layer \(\rho_{v}\)
* Storage shear modulus of the viscoelastic layer \(G_{v}\)
* Loss factor of the viscoelastic layer \(\eta_{v}\)  
Figure 4: Deformation pattern of the considered model of the beam with CCLD treatment at a given compression level (drawing after Reference [PERSON] (1997)). Figure 3: Configuration of the simply supported beam with CCLD analysed within the study. The volume of the damping layer during the CCLD actuation in thickness direction is changing according to the compression ratio. The mass of the whole system was assumed as constant by neglecting the variable oscillating air mass. In short, the equation of motion for generalized displacements yields:  
\[\left[M\right]\left\{\tilde{q}\left(t\right)\right\}+\left[K\right]\left\{q \left(t\right)\right\}=\left\{P\left(t\right)\right\} \tag{1}\]  
with: \(\left[M\right]\) and \(\left[K\right]\) generalized mass and stiffness matrices of the whole system, which \(\left[K\right]\) is a complex matrix including the viscoelastic properties of the damping layer; \(\left\{P\left(t\right)\right\}\) is the vector of generalized forces; and \(\left\{q\left(t\right)\right\}\) is the displacement vector. The system's dynamic response is determined for steady-state harmonic excitation. After introduction of the time dependence, \(e^{i\omega t}\) in Equation (1):  
\[\left\{q\left(t\right)\right\}=\left\{\tilde{q}\right\}e^{i\omega t};\ \ \ \ \ \ \ \ \left\{P\left(t\right)\right\}=\left\{\tilde{P}\right\}e^{i\omega t} \tag{2}\]  
the complex differential equation simplifies to linear system expressed as:  
\[\left[-\omega^{2}\,M+K\right]\left\{\tilde{q}\right\}=\left\{\tilde{P}\right\} \tag{3}\]  
The full derivation for modelling the simply supported beam with fluidically actuated CCLD can be found in Reference [38]. The linear system (Equation (3)) is solved numerically for \(\left\{\tilde{q}\right\}\) at each frequency of interest resulting in frequency response curves covering a desired frequency range.