Transverse Vibrations of Viscoelastic Sandwich Beams via Galerkin-Based State-Space Approach

Beams and panels with viscoelastic (VE) damping are widely used in state-of-the-art engineering structures, allowing to achieve improved safety, increased durability, and noise and vibration control, as well as reduced costs during the through-life manufacturing, operation, and maintenance of the structural systems (e.g., Lockett 1972; Soong and Dargush 1997; Zhang and Soong 1992; Lee 1997; Rao 2003). This is the case with sandwich beams, where stiff (elastic) external layers are combined with a soft (VE) internal core that has been specifically designed to offer significant damping capabilities in a variety of engineering situations (e.g., Palmeri 2010; Hohe and Librescu 2004; Arikoglu and Ozkol 2010). This in turn has driven the continuous development of new VE materials and laminate configurations and improved and cost-effective manufacturing processes, as well as the formulation of more efficient modeling and design methods. Composite steel-concrete beams (e.g., Adam et al. 1997; Berczyński and Wróblewski 2005; Shen et al. 2011) are another well-known example of multilayered structural members, in which each layer is optimized to achieve an overall better performance and the mechanical properties of the interface (typically represented by shear studs) play a fundamental role in their dynamics (e.g., Kasinos et al. 2015). Different model assumptions for the free undamped vibration of sandwich beams were recently studied by Lenci and Clementi (2012), showing how shear deformations, axial and rotatory inertia, and interface stiffness affect modal frequencies and modal shapes, while Lenci and Warminski (2012) used the multiple-scale method to determine the effects of a nonlinear interface in combination with a pure viscous damping.

The dynamics of three-layered sandwich beams has been addressed in several studies involving the finite element (FE) method (e.g., Soni 1980; Sainsbury and Zhang 1999; Zhang and Sainsbury 2000; Daya and Potier-Ferry 2001; Galucio et al. 2004; Hu et al. 2008; Abdoun et al. 2009; Amichi and Atalla 2009; Won et al. 2013), implementing both linear and nonlinear VE models. For the linear case, the dependance of stiffness and dissipation of the structural elements on the vibration frequency is usually accounted for via complex-valued, frequency-dependent material coefficients. This approach generally involves specialized numerical methods for the solution of nonlinear eigenvalue problems (e.g., Daya and Potier-Ferry 2001), with an increased computational cost for the frequency-domain response functions, as the complex-valued dynamic stiffness matrix has to be formed and decomposed at many different frequencies. A different computational strategy consists of defining the dynamic stiffness matrix for the composite beam (Howson and Zare 2005), but this requires the solution of a transcendental eigenvalue problem with the so-called Wittrick-Williams algorithm (Williams and Wittrick 1983).

In addition to FEs, some researchers (e.g., Vu et al. 2000; Oniszczuk 2000; Oniszczuk 2003; and further works cited in Palmeri and Adhikari 2011) have derived analytical solutions for this class of dynamic problems by modeling the sandwich beam as two parallel Euler-Bernoulli beams continuously connected by a layer of distributed springs. This approach, although very appealing from a theoretical point of view and useful to obtain benchmark solutions, can hardly be applied to cases with different boundary conditions (BCs), functionally graded materials, or both. Such difficulties have been tackled by Palmeri and Adhikari (2011), who proposed a Galerkin-type state-space formulation for the vibration analysis of double-beam systems, in which the computational cost is reduced by (1) adopting simple harmonic functions to discretize the equations of motion and (2) using modal relaxation functions (Palmeri et al. 2004; Muscolino and Palmeri 2007) to represent the VE behavior of the inner layer. In their formulation, however, the inner springs are assumed to provide stiffness in the transverse direction only, so that their contribution is associated with extension or compression of the inner layer, not to any shear strain. On the contrary, such a mode of deformation may play a fundamental role in the dynamics of VE sandwich beams.

Motivated by these considerations, this paper presents a novel Galerkin-type approximation for the fields of both axial and transverse displacements in the outer beams, which allows to represent their relative shear slip. The proposed technique also allows considering (1) any BCs, which in general are different for the two outer layers; and (2) rate-dependent constitutive laws for the inner layer through an enlarged state-space model conveniently built in a reduced modal space. Numerical applications to composite beams with different configurations demonstrate the accuracy and versatility of the proposed approach.

Fig. 1(a) shows the composite system under investigation, which consists of two parallel elastic beams of the same length $L$, continuously connected by an inner layer of shear-type VE springs. In the right-handed Cartesian reference system, $z$ is the abscissa along the beam (which is pointing to the right) and $\phi$ is the in-plane rotation (which is positive anticlockwise).

The two elastic beams are prismatic and slender enough for the classical Euler–Bernoulli theory to be valid (i.e., shear deformations and rotatory inertia are negligible). Different material and sectional properties are allowed for the beams, and each of them is fully characterized by modulus of elasticity $E_r$, mass density ${\rho }_r$, frequency-independent viscous damping ratios ${\zeta }_r$, cross-sectional area $A_r$, and second moment $I_r$, where the subscript $r$ denotes the top ($r=1$) and bottom ($r=2$) beam, respectively.

The inner layer is homogeneous and, within the limits of the linear viscoelasticity theory, its behavior in shearing is fully characterized by the complex-valued stiffness per unit length, $k_{\mathrm{inn}}(\omega )$, which depends on the vibration frequency $\omega$; $m_{\mathrm{inn}}={\rho }_{\mathrm{inn}}{A}_{\mathrm{inn}}$ is the mass density per unit length; and $d_0$ is the distance between their centroids.

Without loss of generality, the Standard Linear Solid (SLS) model is adopted as constitutive law for the inner layer. As illustrated within Fig. 1(b), the model consists of a primary elastic spring (called equilibrium modulus), $K_0\equiv k_{\mathrm{inn}}(0)$, acting in parallel with a Maxwell element, which in turn is given by a secondary elastic spring, $K_1$, in series with a viscous dashpot, $C_1=K_1{\tau }_1$, where ${\tau }_1$ is the single relaxation time of the VE material.

For the SLS model, the function $k_{\mathrm{inn}}(\omega )$ can be expressed aswhere $i=\sqrt{-1}$ = the imaginary unit. The shear reaction force, $S(t)$, experienced by the VE inner layer can be related to the associated displacement, $s(t)$ (i.e., the internal slippage between top and bottom beam), aswhere ${\mathcal{F}}^{-1}$ = the inverse Fourier transform operator; ∗ stands for the convolution operator; and the over-dot means derivative with respect to time $t$, so that $\dot{s}(t)$ is the pertinent velocity (i.e., the rate of variation of the shear strain within the inner layer). The reaction force $S(t)$ can be expressed as (Palmeri et al. 2003)where $K_{0}s(t)$ = the elastic part in the VE constitutive law; $K_1{\lambda }_1(t)$ = the contribution of the Maxwell element; ${\lambda }_1(t)$ = the additional internal variable, which measures the elongation of the spring, $K_1$, and is ruled by

The extension to more sophisticated linear VE models such as the generalized Maxwell (GM) model is straightforward (Palmeri et al. 2003) and requires considering more internal variables ${\lambda }_k(t)$ on the right side of Eq. (3). The associated parameters $K_i$ and ${\tau }_i$ can be determined from the storage modulus and loss modulus of the VE material, which are measured experimentally for different frequencies of vibration.

It is worth mentioning here that Eqs. (1)–(4) are formally similar to those used for double-beam systems by Palmeri and Adhikari (2011), with the key difference that shear forces and shear deformations are considered in the current work instead of extensions and compressions of the core.

By resorting to a Galerkin-type approximation, the in-plane vibration of the sandwich beam of Fig. 1(a) can be fully characterized by three independent time-varying fields of displacement: (1) two axial displacements, $u_1(z,t)$ and $u_2(z,t)$, for the top and bottom beams, respectively; and (2) a single transverse displacement, $v(z,t)$, as long as the inner VE layer can be assumed to be incompressible in the transverse direction. It follows that the top and bottom beams will have the same transverse displacements, rotations $\phi (z,t)=-v^{\prime }(z,t)$, and curvatures $\chi (z,t)=-v^{\prime \prime }(z,t)$, in which the prime means a derivative with respect to the spatial coordinate $z$.

Eq. (17) in the Lagrangian space and Eq. (23) in the modal subspace do not include energy dissipation. Following a similar approach to Palmeri and Adhikari (2011), both viscous and VE damping can be introduced. Assuming that the former is proportional to the mass density ${\rho }_r$ and modulus of elasticity $E_r$ of the $r$th outer layer ($r=1$, 2), and adopting the Rayleigh model (Clough and Penzien 2010; Chopra 2007), the following expression can be devised for the $N\times N$ viscous damping matrix in the Lagrangian space:in which the elementary $n\times n$ blocks are given bywhere ${\alpha }_K={({\tilde{\omega }}_1+{\tilde{\omega }}_m)}^{-1}$ and ${\alpha }_M={\tilde{\omega }}_1{\tilde{\omega }}_m{\alpha }_K$ are the two proportionality coefficients for stiffness and mass, respectively.

Since the VE damping is provided by the shear deformations within the core, their effects on the sandwich beam are proportional to the shear stiffness of the inner layer, whose $N\times N$ influence matrix in the Lagrangian space is

Now, exploiting the same modal transformation of variables as in the previous section [i.e., $\mathbf{u}(t)=\mathbf{X}·\mathit{\theta }(t)$], the matrices of Eqs. (30) and (31) can be projected onto the reduced modal subspace, which then gives the $m\times m$ matrices of modal viscous damping, $\mathbf{\Xi }={\mathbf{X}}^{\top }·\mathbf{C}·\mathbf{X}$, and rigidity influence of the inner layer on the modal subspace, ${\mathbf{B}}_{\mathrm{inn}}={\mathbf{X}}^{\top }·{\mathbf{L}}_{\mathrm{inn}}·\mathbf{X}$. These new matrices allow including viscous and VE damping directly into the modal equations of motion [Eq. (23) for the undamped case], which now take the formwhere the $m$-dimensional array ${\mathit{\lambda }}_1(t)={[{\lambda }_{1,1}(t)\cdots {\lambda }_{1,m}(t)]}^{\top }$ has an additional time-varying internal variable for each modal coordinate. Since the spectral matrix $\mathbf{\Omega }$ is diagonal, it appears from Eq. (32) that the modal coordinates collected by the array $\mathit{\theta }(t)$ are only coupled by the modal matrices $\mathbf{\Xi }$ and ${\mathbf{B}}_{\mathrm{inn}}$. If these matrices are diagonal (or if their out-of-diagonal terms are negligible), the dynamic system becomes classically damped in the sense that the modes of vibration are decoupled (Palmeri et al. 2004; Adhikari 2006).

Eq. (32) can then be arranged in a more convenient state-space form:where ${\mathbf{y}}_1(t)={\{\begin{array}{ccc}\mathit{\theta }{(t)}^{\top }& \dot{\mathit{\theta }}{(t)}^{\top }& {\mathit{\lambda }}_1{(t)}^{\top }\end{array}\}}^{\top }$ = enlarged state array (which includes modal displacements, modal velocities, and additional internal variables in the modal subspace), while dynamic matrix ${\mathbf{D}}_1$ and load influence matrix ${\mathbf{G}}_1$ areand the subscript 1 means that only one relaxation time (${\tau }_1$), and therefore only one additional state variable [${\lambda }_1(t)$], have been used for the VE behavior of the core. If more than one relaxation time is required, the GM model can be used instead. In this case, the shear force $S(t)$ within the inner layer becomeswhere a viscous dashpot of damping constant $C_0$ and further ($\ell -1$) Maxwell elements are ideally appended in parallel to the spring-dashpot system of Fig. 1(b). The additional internal variable ${\lambda }_i(t)$ then describes the elongation of the elastic spring $K_i$ within the $i$th Maxwell element, whose evolution in time is ruled bywith ${\tau }_i$ as the $i$th relaxation time of the VE core.

The introduction of the GM model has the effect of enlarging the size of the dynamic problem. One can prove that if $\ell$ Maxwell elements are considered in the analysis, the state-space equations of motion can be posed in the formwhere the expanded array ${\mathbf{y}}_{\ell }(t)={\{\begin{array}{ccc}\mathit{\theta }{(t)}^{\top }& \dot{\mathit{\theta }}{(t)}^{\top }& {\mathit{\lambda }}_1{(t)}^{\top }\cdots {\mathit{\lambda }}_{\ell }{(t)}^{\top }\end{array}\}}^{\top }$ collects the state variables of the system; and dynamic matrix and load influence matrix take the generalized expressions

The time-domain solution of Eq. (33) or (37) can be sought either with standard techniques [e.g., the classical RK4 Runge-Kutta method (2012)] or specifically tailored numerical schemes (Palmeri et al. 2003; Palmeri and Muscolino 2011).

Alternatively, the dynamic response of the system can be evaluated in the frequency domain. In this case, the solution of Eq. (25) can still be used, provided that the FRF matrix of Eq. (27) is generalized to take into account the effect of viscous and VE damping, as modeled previously. If $m$ modes of vibration are retained in the analysis, and $\ell$ is the number of relaxation times needed to describe the VE behavior of the core, the $N\times N$ complex-valued FRF matrix can be evaluated aswhich at given frequency $\omega$ requires the inversion of a square matrix of size $(2+\ell )m$; i.e., the size of the problem increases linearly with both $m$ and $\ell$.

A new state-space formulation has been developed and numerically validated for the dynamic analysis of sandwich beams consisting of two parallel Euler-Bernoulli elastic beams continuously connected by a shear-type VE layer. The proposed approach can be used with any boundary conditions of the outer layers and any number $\ell \ge 0$ of Maxwell’s elements to represent the constitutive law of the core. The derivation of the state-space model requires the following three steps: