nonlinear stiffness and damping effects are usually inseparable, it is generally considered to be an advantage to also consider the energy dissipation in nonlinear systems rather than the damping alone [7]. In the proposed damping model, the interfacial forces from the stiffness model are used in conjunction with the known dynamic friction coefficients to arrive at an equivalent viscous damping. Each node, in each layer, is tracked to determine the relative motion while the FN is modeled as a linear motion sliding tangential joint much like that in Figure 1. One method that can be used to evaluate the effectiveness of the Ffr, with respect to the steady state excitation, is through energy dissipation. Consider the debonded layered composite being excited at the tip by a force of Fsinωt. As the beams pass from zero amplitude to maxima and minima amplitudes, and assuming that they are into the macro-slip region, that is Fsinωt≥Ffr, then Coulombic damping will occur. Since the exciting force is sinusoidal, it is reasonable to assume that ( ) ω φ − = t x Xsin which leads to = − − = − 2 2 1 sin ω ω ω ω e fr F t F kx m x kx The energy dissipated per cycle is then represented as − − = 2 1 4 ω ωe fr N k F F Ffr E It is quite easy to see that the ratio of F/Ffr is important for maximizing the energy (damping) dissipated. For debonded layered composites, the damping within each layer is non-uniform for two reasons. Firstly, the interfacial forces from bending are nonlinear along the length of each layer. The highest interfacial forces, due to bending, are located at the free end of the layer and represented by a low order polynomial. This becomes less significant if there is a presence of an externally applied compression. Secondly, the highest rotations are also located at the free end of each layer. This results in the highest relative motion between layers being at the tip. To determine the damping in each of the layers, the average nodal FN and the average relative motion can be determined by Equations 20 and 21, respectively. L F dx F N N ∫ = L rdx r ∫ = It is important to understand that the relative motion between layers will be greater than for just a single layer at the interface. As a result, both layers of the interface need to be summed together. As one face of the layer is moving in a positive direction, the adjacent face, of the adjacent layer, is moving in the opposing direction due to tensile and compressive stresses. The amount of relative motion is dependent on the thickness of the layer and where the neutral axis of that layer is located. For layers of the same thickness, the motion for the interface is simply twice that of the individual beam surface at the interface. The proposed model is slightly flawed by not accounting for when the layers overlap each other at the tips and some of the tip of the interface is no longer in contact. However, this will result in a very small overall percentage of the damping in the system and will decrease as the length of the layers increase. Once FN and r are solved for each individual layer, the summation of FN and r for all layers can be used, with a single interface, in a Coulombic free vibration decay for a SDOF model. This takes the form of (17) (18) (19) (20) (21) 294
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