28 Brian Evan Saunders what extent an isola motion can eat up the fatigue life of a structure, and how to better qualify a structure designed to operate in an environment that may be at risk of experiencing nonlinear behaviors such as isolas. Methodology and System Model Three simple and well-known methods are combined for this fatigue analysis: the log-linear (i.e., Basquin) model of the stress-cycles curve N=cS−β, Palmgren-Miner linear damage D=Pk i=1 ni Ni , and rainflow cycle counting [7]. An experimental apparatus of the dynamical system, along with parameter values, can be found in [3] and is modeled as follows: ¨x+2ωnζ ˙x+ω2 nx+ α mx 3 +Fc(x) m = p mcos(ωt) , (1a) Fc = Kc (x+j1), 0, Kc (x−j2), x<−j1 −j1 ≤x≤j2 x>j2 (1b) σ(t)= 12Eb L2 x(t) (2) Euler-Bernoulli beam theory, for a fixed-fixed beam with centered load, is assumed in order to approximate the maximum stress in the system in Equation (2) [8]. Figure 1 shows the nonlinear forced response curves for the system model, including superharmonic, subharmonic, and isola solutions. 0 5 10 15 20 25 30 Frequency (Hz) 10 -4 10 -3 10 -2 Amplitude (m) N H = 12, = 1 N H = 12, = 2 N H = 12, = 2 N H = 12, = 3 N H = 12, = 5 N H = 12, = 9 N H = 12, = 4 N H = 12, = 4 N H = 12, = 6 N H = 12, = 6 N H = 12, = 1 N H = 36, = 3 N H = 12, = 2 N H = 36, = 3 N H = 12, = 1 N H = 36, = 3 N H = 12, = 2 N H = 12, = 2 N H = 36, = 5 Fig. 1 Nonlinear forced response curves for the dynamical system described by Eq. (1). NH is a solver convergence parameter, andν is the subharmonic order of each solution curve Fatigue Damage Analysis Figure 2 shows the calculated fatigue damage for each of the solution curves in Figure 1. A fatigue damage coefficient of β = 4.22 was used in the following calculations. Complete analysis details and parameter values can be found in the final
RkJQdWJsaXNoZXIy MTMzNzEzMQ==