Sensitivity analysis of viscoelastic structures

In the context of control of sound and vibration of mechanical systems, the use of viscoelastic materials has been regarded as a convenient strategy in many types of industrial applications. Numerical models based on finite element discretization have been frequently used in the analysis and design of complex structural systems incorporating viscoelastic materials. Such models must account for the typical dependence of the viscoelastic characteristics on operational and environmental parameters, such as frequency and temperature. In many applications, including optimal design and model updating, sensitivity analysis based on numerical models is a very usefull tool. In this paper, the formulation of first-order sensitivity analysis of complex frequency response functions is developed for plates treated with passive constraining damping layers, considering geometrical characteristics, such as the thicknesses of the multi-layer components, as design variables. Also, the sensitivity of the frequency response functions with respect to temperature is introduced. As an example, response derivatives are calculated for a three-layer sandwich plate and the results obtained are compared with first-order finite-difference approximations.


Introduction.
Passive damping approaches to the problem of vibration attenuation constitute an important subject in modern Mechanical Engineering. The use of viscoelastic materials to reduce noise and vibration levels in various types of engineering systems such as robots, automobiles, airplanes, communication satellites, buildings and space structures has been intensively investigated lately. Much of the knowledge available to date is compiled in the books by Nashif et al. (1985) and Mead (1998) and in some review papers such as those by Rao (2001) and Samali and Kwok (1995).
In the last two decades, a great deal of effort has been devoted to the development of mathematical models for the dynamic behavior of viscoelastic materials, accounting for its typical dependence on operational and environmental effects, such as the excitation frequency, static preloads and temperature (Christensen, 1982). Some of those models are considered to be particularly suitable to be used in combination with finite element discretization, which makes them very attractive for the modeling of complex engineering systems. Among those models, it should be mentioned the socalled Fractional Derivative Model (FDM) (Bagley and Torvik, 1979, 1983, the Golla-Hughes-McTavish Model (GHM) (Golla and Hughes, 1985;McTavish and Hughes, 1993) and the Anelastic Displacement Fields Model (ADF) suggested by Lesieutre and co-workers (Lesieutre, 1992;Lesieutre and Bianchini, 1995;Lesieutre and Lee, 1996). These models can represent the viscoelastic behavior in the frequency domain. Moreover, the GHM and ADF models can also provide convenient time domain representations. For themorheologically-simple viscoelastic materials, the influence of temperature can be accounted for based on the concept of reduced frequency, according to the relation proposed by Drake and Soovere (1984).
In the context of analysis and design of structural systems, an important topic to be addressed is the so-called sensitivity analysis, which enables to evaluate the effect of variations of physical and/or geometrical parameters on the mechanical behavior. Moreover, sensitivity analysis constitutes an important step in various types of problems such as model updating, optimal design, system identification and control.
The sensitivity analysis is based on the evaluation of the derivatives (most frequently limited to the first order) of the system response with respect to a set of parameters of interest. It can be associated to different kinds of mechanical responses: static displacements, eigenvalues and eigenvectors, frequency response functions and time responses (Haug et al., 1986). According to Murthy and Haftka (1988), the optimal design structural systems has a narrow connection with sensitivity analysis, since a significant part of typical optimization algorithms generally perform a large number of evaluations of the system response for different values of the design variables. Derivatives can be used to approximate the response of modified systems, thus reducing the cost of re-analysis, specially for high-order systems. Several approaches are available for performing sensitivity analysis of dynamic responses, such as the Modal Method (Fox andKapoor, 1968), Nelson's Method (Nelson, 1976) and the Improved Modal Method (Lim et al., 1987). However, applications to the case of structural systems containing viscoelastic components are not numerous.
In this remainder, sensitivity analysis is applied to plates treated with surface constraining damping layer treatments, considering, as dynamic responses, complex frequency response functions. As design parameters, goemetrical characteristics, such as the thicknesses of the multi-layer components are considered. Also, an approach for evaluating the sensitivity of the frequency responses functions with respect to temperature is introduced. To illustrate the use of the formulation developed, response derivatives are calculated for a three-layer sandwich plate and the results obtained are compared with first-order finite-difference approximations.

The complex modulus approach.
According to the linear theory of viscoelasticity (Christensen, 1982), the one-dimensional stress-strain relation can be expressed, in Laplace domain, as follows: where: In the equation above, G is the static modulus, representing the elastic behavior and is the relaxation function, associated to the dissipation effects. When evaluated along the imaginary axes of the s-plane ( , the complex modulus and the capacity of dissipation of the viscoelastic materials, are expressed in the following forms: where , and are defined, respectively, as storage modulus, loss modulus and loss factor of the viscoelastic materials.

Influence of temperature on the viscoelastic behavior.
According to Nashif et al. (1985), the temperature is usually considered to be the single most important environmental factor affecting the properties of viscoelastic materials. This makes it important to account for this factor in the modeling of structural systems containing viscoelastic elements. This is possible by making use of the so-called Frequency-Temperature Superposition Principle -FTSP, which establishes an relation between the effects of the excitation frequency and temperature on the proprieties of the thermorheologically simple viscoelastic materials. This relation leads to a function of the temperature known as the shift factor, which implies that the viscoelastic behavior at different temperatures can be related to each other by changes (or shifts) in the actual values of the excitation frequency. This leads to the concept of reduced frequency (Christensen, 1982). Symbolically, the FTSP can be expressed under the following forms: where is the reduced frequency, is the actual frequency, is the shift function that must be identified from experimental tests for a specific type of viscoelastic material, and T is a reference value of temperature.  Nashif et al. (1985)).

Viscoelastic behavior incorporated into finite element models.
The finite element equations of motion of viscoelastic structure containing degrees of freedom can be expressed as follows: where are the mass (symmetric, positive-definite), damping (symmetric, nonnegative-definite), are the vectors of displacement and external loads, respectively.
It is assumed that the structure contains both elastic and viscoelastic elements, so that the stiffness matrix can be decomposed as follows: is the stiffness matrix corresponding to the purely elastic substructure and is the stiffness matrix associated with the viscoelastic substructure. The inclusion of the frequency-dependent behavior of the viscoelastic material can be made by using the so-called elastic-viscoelastic correspondence principle (Christensen, 1982), according to which, for a given temperature, is first generated for specific types of elements (rods, beam, plates, etc.) assuming that the longitudinal modulus ( ) T ω, E and/or the shear modulus ( ) T ω, G (according to the stressstate) are constant, independent on frequency. After the finite element matrix is constructed, such moduli are made frequency-dependent according to a particular viscoelastic constitutive model adopted. Moreover, by assuming a constant Poisson ratio for the viscoelastic material, ( ) Then, of the two moduli can be factored-out of the stiffness matrix of the viscoelastic substructures. By writing: equations (7), (8) and (9) are combined to give: The difficulty in dealing with Eq. (10) for predicting the vibration response in the time domain and performing eigenvalue analysis commes from the fact that the stiffness matrix is not constant and depends on frequency. Some procedures for overcoming such difficulty have been suggested, based on the adoption of particular representations for the frequency-dependent behavior of the viscoelastic materials. Such an approach is used in the Fractional Derivative, Golla-Hughes-McTavish and Anelastic Displacement Field models (see references in Section 1), which enable to transform Eq. (10) into state-space equations of motion with constant (frequency-independent) state matrices. Equation (10) can be directly used for calculating the steady-state harmonic responses in the frequency domain, by assuming: Upon introduction of Eqs. (11) into Eq. (10), the following relation is obtained between the amplitudes of the excitation forces and the amplitudes of the harmonic responses: where the so-named frequency response function (FRF) matrix is expressed as: In this paper, based on Eq. (13), we consider the sensitivity of FRFs with respect to physical and/or geometrical parameters featuring in the finite element matrices and also with respect to the temperature.

A three-layer sandwich plate finite element.
In this section the formulation of a three-layer sandwich plate finite element is summarized based on the original development made by Kathua and Cheung (1973). Figure 2 depicts a rectangular element formed by an elastic baseplate, a viscoelastic core and an elastic constraining layer, whose dimensions in directions x and are denoted by and , respectively. In the remainder, superscripts (1), (2) and (3) are used to identify the quantities pertaining the baseplate, the viscoelastic core and the constraining layer, respectively. x and , denoted by and θ , respectively. The vectors of nodal and element degrees-of-freedom are then expressed as follows: (3) constraining layer (1) base plate (2)  The following assumptions are adopted: • normal stresses and strains in direction z are neglected for all the three layers.
• the elastic layers (base-plate and constraining layer) are modeled according to Kirchhoff's theory which neglects the effects associated to transverse (thickness-wise) shear; • for the the viscoelastic core, Mindlin's theory is adopted, which includes transverse shear; • the transverse displacement w is the same for all the three layers and cross-section rotations and are assumed to be the same for both the elastic layers.
x θ y θ By imposing kinematic relations that enforce the continuity of displacements along the interfaces between the layers, the following expressions are found for the longitudinal displacements of the viscoelastic core middle plane in directions and in terms of previously defined nodal coordinates: x y  Based on the hypotheses of the stress-states assumed for each layer, the following stress-strain relations apply: • For the base-plate: • For the viscoelastic core: • For the constraining layer: where matrices are formed by derivatives of the shape functions according to the differential operators appearing in the strain-displacement relations for each layer.
Based on the stress-strain relations above, neglecting viscous damping, the strain and kinetic energies of the composite plate element are formulated. Lagrange equations are then used, considering the nodal displacements and rotations as generalized coordinates, to obtain the element stiffness and mass matrices. Such procedure has been fully developed by Stoppa (2003) using symbolic computation.
According to Eqs. (8) and (9), the element stiffness matrix can be expressed as follows: and are, respectively, the element stiffness matrices of the base-plate, constraining layer and viscoelastic core.
The assembling of element matrices into global matrices follows the standard procedure based on the enforcement of connectivity expressing rotation and displacement continuity between neighborg elements through common nodes (Maia et al., 1997).

Sensitivity analysis of structural responses.
The global finite element matrices, , and M C ( ) T ω, K appearing in Eq. (7), establish the dependency of the response of the system with respect to a set of design parameters, which include physical and geometrical characteristics and also the temperature. Such functional dependency can be expressed in a general manner as follows: where and designate vectors of structural responses and design parameters, respectively. r p The sensitivity of the structural responses with respect to a given parameter , evaluated for a prescribed set of design parameter is defined as the following partial derivative: where is a variation applied to the initial value parameter , while all other parameters remain unchanged.  As the result of a comprehensive experimental work, Drake and Soovere (1984) proposed the following analytical expressions to represent the complex modulus and the shift factor as functions of the reduced frequency for temperatures in the interval [210K -360K] and frequencies in the interval [1Hz -1x10 6 Hz], for the 3M™ISD112 viscoelastic material: The derivatives of Eqs. (28) and (29) For the purpose of illustration, Fig. 4 depicts the curves representing the variations of the storage modulus, loss modulus and loss factor as functions of the reduced frequency, as obtained from Eq. (28), and the plot of the shift factor as a function of the temperature, as given by Eq. (29).

Sensitivity of FRF with respect to structural parameters.
In this example, the thicknesses of the constraining and viscoelastic layers were considered as design variables in the computation of the sensitivities of the FRF . The results obtained by using the first-order derivatives according to Eq. (24), as compared to the corresponding values calculated by finite differences (having been adopted a variation from 2.5% of the nominal values of the parameters given in Table 1 Figure 6 shows the real and imaginary parts of the sensitivities of the FRF with respect to temperature, for two different values of the nominal temperature, as compared to their counterparts calculated by finite differences, using variations of 2.0% of the nominal temperature values. From these figures, it can be concluded that the first-order derivatives compare fairly well with finite differences and enable to evaluate the degree of influence of temperature changes within the frequency band of interest. Figure 6. Sensitivities of the FRF with respect to temperatures for nominal temperature values 298K and 308K. As a complementary demonstration of the utility of first-order derivatives, Figure 7 enables to compare the FRF amplitudes computed by two different forms: in the first, from a nominal value of the design variable (viscoelastic layer thickness, constraining layer thickness and temperature), variations are voluntarily given to such values (1.5% for both layer thicknesses and for temperature). Then, the exact FRFs of the "perturbed" system are computed. In the second form, the FRFs of the "perturbed" system are estimated from the FRFs of the nominal system by using the first order derivatives through first-order Taylor series expansions. As can be seen, the exact FRFs, represented by red lines, are fairly accurately approximated by the series expansions, represented by blue lines. The agreement is less satisfactory for the temperature variations, leading to conclude that first-order derivatives might not be accurate enough to predict variations of the dynamic behavior associated to large variations of the temperature. In this paper, sensitivity analysis based on finite element models of structural systems containing viscoelastic materials has been addressed. A formulation has been developed for the computation of first-order derivatives of frequency response functions with respect to two different kinds of parameters, namely: a) physical and/or geometrical structural parameters which appear explicitly in the finite element matrices; b) the environmental parameter temperature. Applications have been done to rectangular plates fully treated with constraining layers.
It can be concluded that the use of the complex modulus approach, combined with the concepts of shift factor and reduced frequency -justified by the principle of superposition frequency-temperature -has shown to be an adequate strategy to account for the typical dependency of the viscoelastic characteristics with respect to frequency and temperature in the finite element models of complex structural systems.
As illustrated in the numerical applications presented, the sensitivities of complex frequency response functions convey valuable information about the influence of the design parameters on the dynamic behavior of the system, being also a very useful tool for the design, performance analysis and optimization of viscoelastically damped systems.