7 A Modal Model to Simulate Typical Structural Dynamic Nonlinearity 65 to find a passband for each mode that successfully eliminated unwanted frequency content without distorting the damping. For this structure the authors have selected to use a 50 % passband for all modes which was robust for every mode when using the SMAC modal filter. Note that the passband used here is not universal and might need modification for a different system. 7.4 Nonlinear Models This work compared the capabilities of three different models/methods to capture the nonlinear dynamics of the test object: Iwan, FREEVIB (FV), and Restoring Force Surface (RFS). A brief description of each model is provided in the sub sections below. As mentioned in Sect. 7.2.3, only those modes that had a damping change greater than 30 % between the low and high-level input tests were modeled as nonlinear. In order to have a fair comparison of their capabilities, each of the three aforementioned models were parameterized with six parameters to capture the dynamics. This quantity was selected as it matches the number of parameters used in the Iwan model. 7.4.1 Modal Iwan Model As discussed in [5, 7] each mode can be modeled with a single degree of freedom system as a modal coordinate. Each modal degree of freedom will be linked to ground with a linear spring and damper. In order to capture the nonlinearity in each mode we then add a four parameter Iwan element in parallel with the linear spring and damper. This element can be described as a joint force as shown in Fig. 7.8. The system is very similar to a standard modal coordinate set-up but with the nonlinear joint force adding complexity due to the nonlinearity of each mode. The equation of motion for the system now takes the form of Rq.t/ CC . q.t/ CK1q.t/ Dˆ TFext CFj (7.14) where the nonlinear force in the joint, Fj, is a function of four parameters, [FS, KT, ¦, “]. FS is the slip force or the force required to begin macro-slip. KT is the stiffness in the joint related to the nonlinear frequency shift from linear conditions to macro-slip. ¦is related to the exponent in a power-law relationship between damping and amplitude in the macro-slip regime. Finally, “ defines the shape of the dissipation curve near the transition from micro to macro-slip. These four parameters can be obtained from experimental measurements as outlined in [5]. In this work data were obtained solely in the linear and micro-slip regimes of response. Thus some of the parameters became more difficult to estimate. The stiffness in the joint, KT, is defined as the change in stiffness as shown in (7.15) KT D! 2 n .!n !n/ 2 (7.15) Fig. 7.8 Schematic of SDOF for Iwan model modal coordinate
RkJQdWJsaXNoZXIy MTMzNzEzMQ==