and structures as a result of the multiple parameters that are available to change their damping characteristics. Another benefit to this type of damping is that it does not exhibit the level of temperature dependence as compared to viscoelastic materials, which are commonly used for vibration damping applications due to their high loss factors. With metallic debonded layered systems, as the materials begin to increase in temperature from the frictional and thermoelastic heating, the material properties remain nearly unchanged. This is an extreme benefit as these materials will behave in a consistent way over a wide range of temperatures. In addition, these dampers can be used as primary structural elements without the undesired added mass of typical dampers and could be used along the primary load paths to provide system damping. 2. ANALYTICAL MODEL DEVELOPMENT For this work, the analytical models are developed on the basis of using simplified geometries. A slender cantilever beam of rectangular cross-section and uniform thickness with linear elastic material properties will be used to model each layer of the naturally debonded composite. It is assumed that the cantilever beams are inextensible and axial displacements are negligible since these will be relatively small with respect to the bending displacements. A further assumption will be that the stiction of all nodes at a single interface are overcome simultaneously since the beams are assumed to be inextensible. It is also assumed that the cross-sections of the beams remain constant along their length, ignoring the effect of Poisson’s ratio and any geometric nonlinearities. The beams are based on Bernoulli-Euler beam theories and assume that the curvature of the beams is proportional to the bending moment. During dynamic motion, since friction is present in the system, there are oscillating intervals of time in which stiction occurs and intervals of time in which relative sliding at the interfaces occurs. This can be represented by the following two equations of motion in Equations 1 (stiction) and 2 (sliding) and in Figure 1. Fig. 1. Single degree of freedom (SDOF) system with (a) hysteretic damping and (b) Coulombic damping When stiction occurs, the damping is solely hysteretic from the material and is denoted as a complex stiffness in the system. On the other hand, when sliding occurs, the damping is a combination of Coulombic damping and hysteretic damping. Damping that occurs in debonded layered systems is primarily Coulombic for metallics and for other materials that have small magnitude loss factors. In this paper, focus will only be on the Coulombic portion of the total damping and the model will neglect any material damping that is present. It must be noted that the frictional force in Equation 2, opposes the motion always. Therefore, as the body is displaced in the opposing direction, the sign convention changes polarity. In the case of the Coulombic damping system in Figure 1, the normal force and dynamic friction occurs at the layer interfaces and the hysteretic damping is ignored. 2.1 STIFFNESS MODEL Since this model is based on only macro-slip friction, there is no decay between the static and dynamic friction coefficients. This would indicate that velocity has no influence on the overall coefficients during interfacial sliding. Much like the Classic Coulomb, Regularized Coulomb, and Karnopp friction models, the presliding is not characterized (1) (2) m k FN µd k* m (a) Hysteretic Damping (b) Coulombic Damping 0 * + = mx k x && fr mx kx F + = && 290
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