8 Application of Quasi-Static Modal Analysis to an Orion Multi-Purpose Crew Vehicle Test 67 ϕT 0rfJ (Φ0q, θ) ≈g(qr) (8.5) However, we recognize that the other modal displacements, qj for j =r are generally not zero. So, we assume that they are slaved to the rth DOF in the same ratios that are observed in the static response. Specifically, we excite each mode in turn with a force applied over the entire structure in the shape of that linear mode by solving the following quasi-static problem, Kx +fJ (x, θ) =Mϕ0rα (8.6) where α is the load amplitude. This produces the quasi-static response x(α) of the structure up to some maximum load amplitude. The response is computed at several intermediate steps yielding the entire load displacement curve. The resulting displacements are then mapped to each of the modes using: qr(α) =ϕ0r TMx(α). The modes that were not directly excited are also retained and used to assess the degree to which the modes are statically coupled at each load amplitude. It is also important to note that we have obtained a single valued relationship between x(α) and α by requiring that the load always ramps from an unloaded state to a maximum load αmax. As a result, it is no longer necessary to track any internal states, θ, e.g. slider states for the Iwan joint. Using the curve of the modal force fr(α) =ϕ0r TMϕ0rα =α versus modal displacement, qr, the full hysteresis curve is constructed using Masing’s rules [15, 16], i.e. ˆf1 (qr) =2fr qr+qr(α) 2 −αand ˆf2 (qr) =α−2fr qr(α)−qr 2 as illustrated in Fig. 8.1. Hysteresis curves such as these can now be used to estimate the instantaneous stiffness and damping for each mode. The secant to the curve is used to estimate the effective natural frequency, as follows. ωr (α) α qr (α) (8.7) The energy dissipated per cycle of vibration is the area enclosed by the full hysteresis curve Dr (α) = qr(α) −qr(α) ˆf1 (qr) − ˆf2 (qr) dqr (8.8) where α is the constant maximum load at the level of interest, and fr is a function of displacement qr. Then, using the relationship between energy dissipated per cycle and the damping ratio for a linear system, the effective damping ratio can then be calculated from the following. ζr (α) = D(α) 2π(qr (α)ωr (α)) 2 (8.9) It is important to note that these effective “modal properties” are not a linearization of the structure about some state. Indeed, we are considering the effective behavior of the structure over a typical cycle of vibration. Such a model is typically called a quasi-linearization or a “describing function” model [17, 18]. A comprehensive verification of QSMA is presented by Lacayo & Allen [5] and is not repeated here for brevity. Fig. 8.1 Sample loading curve obtained by solving Eq. (8.6). (blue) Hysteresis curve (forward ˆf1 (qr) and reverse ˆf2 (qr)) computed using Masing’s rules r f rq Quasi-Stat ic Response 2 ( ) r ( ) r D ( ) rq 1 ˆf 2 ˆf w a a a a
RkJQdWJsaXNoZXIy MTMzNzEzMQ==