k F t x t A t A N d µ ω ω ± + = sin cos ( ) 2 1 where the sign convention alternates between positive and negative directions. For the first half-cycle (0≤ωt≤π), the displacement is less than the previous displacement amplitude by k Fd N µ 2 This trend continues until the displacement reaches the “dead zone”, which is equal to k Fd N µ To find the damping of the system we can view it as being equivalent viscous damping. The energy dissipated per cycle by the viscous damping force in a single degree of freedom vibrating system is approximated by If it is assumed that x=Xsinωt occurs for each complete cycle. The energy dissipated is therefore ∫ = 2 0 2 2 2 2 cos 4 π ω ω π ω tdt c X cX Since it is known that the energy dissipated per cycle by Coulomb damping is 4FNµX, approximately, an equivalent viscous damping coefficient for Coulomb damping can be solved for and can be expressed as X F c N eq ωπ µ 4 = 3. FINITE ELEMENT MODEL Multiple Finite Element Models (FEM’s) have been used to investigate and validate the analytical models. Each of the models are full 3-D models using hexahedron 20 node higher-order elements. For this work, three primary different FEM’s, with multiple simulations, were used for the validation process. The first FEM was a nonlinear static model, with frictional contact elements and without nonlinear materials or geometry, was used to represent the layered beams for the stiffness model. The contact elements employed a Coulomb friction model without the friction decay. The second FEM was the same as the static model but was used for flexible dynamic analyses in the form of free vibrations. Solving these types of problems numerically is computationally costly, especially when there are contact nonlinearities. These contact nonlinearities are riddled with complexities that can often lead to unconverged results for implicit analyses. Various simulations were performed with and without the application of external compression loads. For transient models, the time steps are extremely important in order to capture the true dynamics of the system. The time steps in this case were based on the first natural frequency (bending mode) when stiction is present between the layers. This produces a higher natural frequency and in turn will create a smaller time step for the following criteria. For the maximum time step 1/30th of the period of the natural frequency was used, while for the minimum time step 1/300th of the period was used. The results seemed to indicate reasonable capture of the physics of the layers when they were used as having frictional interfaces. For representation of the ESL model, for the stiffness and damping models, a third FEM was used. This consisted of a single geometric layer using an elastic-plastic isotropic hardening material model initially. When using this material model for free vibrations, hysteretic damping from the Bauschinger Effect [8] was present since the material was used to go into the plastic region of the material model. This is an artificial hysteresis and does not really occur as the material model is for an elastic-plastic material and in this work it is being exploited for the use of accounting for macro-slip. An (22) (24) (23) (25) (26) (27) ∫x cxdx 0 4 & 295
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