From Equation 13 the breakaway force and displacement can be calculated. The breakaway force is the ratio of the tip force that was used to get convergence of the artificial shear stress and the sum of the shear stress of the interfaces of the layers. The corresponding deflection comes from the tip displacement using the effective modulus of elasticity and the effective second area moment of inertia using a Volume Fraction (VF) approach to arrive at an ESL. To calculate the secant modulus, the coordinate system for the elastic modulus must be replaced with the secant modulus (F=0, δ=0). Once τxy is equal to ψ, the layers will slip relative to one another. The secant modulus is simply the remaining balance of the load while accounting for the opposing friction once slipping begins. To calculate the secant modulus deflection, the remaining tip force must be calculated. This is done by Once the remaining force has been calculated, the displacement for the secant modulus of the bilinear curve can be resolved. Since the layers are now at the point of sliding, and µd has been removed from the remaining force as an opposing force, the layers will behave as though they are frictionless. Therefore the solution for solving this is relatively straightforward and is described as It should be noted that although the secant modulus should be a factor of 2n, as indicated in Equation 16, it is not, as the shift in the opposing frictional force and has been removed. If this is added back into F” then the secant modulus does follow this behaviour. As the ratio between static and dynamic friction coefficients changes, so does the secant modulus. The comparison between the analytical model and the numerical model can be seen in Figure 3. Fig. 3. Numerical and analytical bilinear behavioural comparison 2.2 DAMPING MODEL In most nonlinear systems, only the frequency response functions (FRFs) can be linearized, at best, for frequencies close to a single resonance of a system [6]. The parameters that are linearized are usually for short frequency ranges. This often leads to difficulties when assessing the energy dissipation and the damping in nonlinear systems. Because snl nl G =Σ ψ ( ) ( ) ( ) ( ) F F At y I F s F F F nl d eff − − − = − − 2 2 8 " µ µ ( )n eff eff E I F L y 2 3 " 3 = δ Ffr = 42 Ffr = 30 Ffr = 9 Ffr = 18 (14) (15) (16) 293
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