98 B. Clark et al. an amplitude-dependent linear model. Measurements from a representative nonlinear structure are then used to predict the consequences on the environment testing. 13.2 Setting of Problem We construct our model to match the behavior of the S4 benchmark beam structure [2]. For this study we assume a single degree-of-freedom modal model with the generalized coordinate being the displacement of Mode 2. This mode of vibration exhibits noticeable nonlinearity based on the data from [3], and is shown as the blue dots in Fig.13.1. This model was then fit to a single-degree-of-freedom modal model with a mass, linear spring, linear damper, and an Iwan joint in order to be able to extrapolate the behavior to higher amplitudes (green dashed line in Fig.13.1). Approximating the system with a quasi-linear modal model, the equation of motion can be written in terms of the amplitude-dependent natural frequency .ωn(A) and damping ratio .ζ(A) where A is the amplitude of the response q, i.e., .q =Asin(ωt −φ). We assume a broad-band base excitation model for the force .F(t) so that the resonance frequency will be excited even if the natural frequency shifts at higher amplitude. Our modal equation of motion is . ¨q +2ζ(A)ωn(A)˙q +ω 2 n(A)q =F(t) (13.1) The response of the system to a harmonic force is then given by the well-known equation, which gives the peak velocity in modal coordinates. .q(t) =Re −Fp (ω2 n −ω2) +i(2ζωnω) eiωt → Vp = Fp 2ζωn . (13.2) Now suppose that we know the response .Vp1 due to a force .Fp1 and we wish to find the response .Vp2 to a different force .Fp2 =αFp1. Using the prior equation, while allowing the damping and frequency to differ at each point because they are amplitude dependent, we obtain the following: .Vp2 = αFp1 2ζ2ωn2 =αVp1 ζ1 ζ2 ωn1 ωn2 =VLinPred ζ1 ζ2 ωn1 ωn2 (13.3) This equation must be solved iteratively because . ζ2 is a function of the amplitude .Vp2. In doing so, one can discover how the response of the system changes as the force (i.e., the environment) increases or decreases. This can then be compared 10-2 10-1 100 101 102 103 104 105 106 Amplitude (in./s) 10-4 10-3 10-2 10-1 100 Damping Ratio Damping vs. Velocity Amplitude Measured Fit Fig. 13.1 Damping vs. velocity amplitude measurements from [3] and fit to an SDOF system with an Iwan element
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