## 2 Theoretical Modelling
### General Sandwich Beam Specimens
The complex coupling parameter \(Z^{*}\) was also introduced in reference [12], which satisfies the following expressions for constrained-layer sandwich structures  
\[\left(EI\right)^{*}=\left(EI\right)_{\text{O}}(1+Z^{*}Y), \tag{5}\]  
\[\eta=\frac{\text{Im}(Z^{*})Y}{1+\text{Re}(Z^{*})Y}, \tag{6}\]  
where  
\[Z^{*}=\frac{X^{*}}{1+X^{*}}, \tag{7}\]  
\[Z^{*}=\frac{X(X+1)+X^{2}\beta^{2}}{\left(X+1\right)^{2}+X^{2}\beta^{2}}+\text {i}\frac{X^{2}\beta^{2}}{\left(X+1\right)^{2}+X^{2}\beta^{2}}. \tag{8}\]  
The real part and imagine part of the complex coupling parameter are respectively expressed as  
\[\text{Re}(Z^{*})=\frac{X(X+1)+X^{2}\beta^{2}}{\left(X+1\right)^{2}+X^{2}\beta ^{2}}, \tag{9}\]  
\[(EI)^{*}=(EI)_{\text{re}}(1+\mathrm{i}\eta), \tag{2}\]0  
Here we introduce the bending stiffness ratio \(S\) as the ratio of the effective bending stiffness and the uncoupled bending stiffness of sandwich structures, which is expressed as  
\[(EI)^{*}=(EI)_{\text{re}}(1+\mathrm{i}\eta), \tag{2}\]1  
According to Equations (5) and (6), the relationship of the stiffness ratio and the complex coupling parameter can be expressed as  
\[(EI)^{*}=(EI)_{\text{re}}(1+\mathrm{i}\eta), \tag{2}\]2  
\[(EI)^{*}=(EI)_{\text{re}}(1+\mathrm{i}\eta), \tag{2}\]3  
Combining Equations (9), (10) and Equations (12), (13), the loss factor and the effective shear parameter can be obtained as follows  
\[(EI)^{*}=(EI)_{\text{re}}(1+\mathrm{i}\eta), \tag{2}\]4  
\[(EI)^{*}=(EI)_{\text{re}}(1+\mathrm{i}\eta), \tag{2}\]5  
As for cantilever beam specimens, the modal flexural wavenumbers can be achieved by [2]  
\[(EI)^{*}=(EI)_{\text{re}}(1+\mathrm{i}\eta), \tag{2}\]6  
where \(l\) is the length of the beam and \(C_{n}\) is the coefficient for mode \(n\) of clamped-free uniform beam, \(C_{1}=0.55959\), \(C_{2}=3.5069\), \(C_{3}=9.8194\), \(C_{4}=19.242\), \(C_{5}=31.809\), and \(C_{n}=(\pi/2)(n-0.5)^{2}\), for \(n>3\). Equation (18) implies that beams of identical lengths undergo the same modal wavenumbers. Thus when the VCBTM is used to determine the dynamic mechanical properties of viscoelastic materials, the wavenumbers of the base beam specimens and sandwich beam specimens are approximately equal. According to the definition expression of the flexural wavenumber, we arrive at \(EI=m\big{(}2\pi f/k^{2}\big{)}^{2}\). Thus the bending stiffness ratio of the tested sandwich beam specimen can be obtained by  
\[(EI)^{*}=(EI)_{\text{re}}(1+\mathrm{i}\eta), \tag{2}\]7  
where \(m_{i}\) and \(f_{i}\) represents the areal density and the resonant frequency of the \(i\)th layer respectively, \(f_{s}\) represents the resonant frequency of the sandwich specimen. For sandwich test beam specimens, the bending stiffness of the damping layer is far smaller than the bending stiffness of the elastic base layers. Thus Equation (17) can be rewritten as  
\[(EI)^{*}=(EI)_{\text{re}}(1+\mathrm{i}\eta), \tag{2}\]8  
where \(\rho_{i}\) represents the density of the \(i\)th layer. In Equation (18), the densities, thicknesses of each layer, and the resonant frequencies of the base beams and sandwich specimens are the parameters to be tested during the vibration cantilever beam test. According to the definition formulas Equation (4), the geometrical parameter of a sandwich beam can be expressed as  
\[(EI)^{*}=(EI)_{\text{re}}(1+\mathrm{i}\eta), \tag{2}\]9  
According to the definition expression of the effective shear parameter, the shear modulus of the cantilever sandwich beam can be obtained by  
\[X^{*}=X(1+\mathrm{i}\beta), \tag{3}\]0  
Substituting Equation (20) into Equation (15), we arrive at  
\[X^{*}=X(1+\mathrm{i}\beta), \tag{3}\]1  
\[X^{*}=X(1+\mathrm{i}\beta), \tag{3}\]2  
where \(\eta_{s}\) is the modal system loss factor of the sandwich beam. Equation (21) and Equation (22) are calculation formulas of damping loss factor and shear modulus respectively for general sandwich beam specimens.