116 A. M. Puhwein and M. J. Hochrainer appears rather complicated, it has proven effective in laboratory operation because the nonlinear absorbtion investigated has a high parameter sensitivity. Studies in the laboratory confirm, that the iterations indicated in Fig 3 typically must be carried several times before the performance of the test-setup turns out to be optimal. Fig. 3 Flowchart for successful implementation of RTHT; modified from [22]. Analytical Model and Coupled Simulation When analyzing the motion of complex structures, it is common to decompose the response of complex structures into modal components, thereby identifying the dominant degrees of freedom in a frequency range of interest. In many situations it is even possible to approximate the system dynamics using only a single degree of freedom. If this simplification is acceptable for host structure and absorber, their coupled dynamics can be described by Duffing type oscillators with cubic and quintic stiffness terms [21] mS¨xS +cS ˙xS +kSxS +kS3xS 3 +kS5xS 5 −c A( ˙xA− ˙xS) −kA(xA−xS) −kA3(xA−xS) 3 −kA5(xA−xS) 5 =FE cos(ωt) mA¨xA+cA( ˙xA− ˙xS)+kA(xA−xS)+kA3(xA−xS) 3 +kA5(xA−xS) 5 =0 (1) mS, cS, kS, kS3, kS5,FE describe the mass, damping coefficient, linear, cubic and quintic stiffness as well as the excitation force of the host structure and mA, cA, kA, kA3, kA5 denote the mass, damping coefficient, linear, cubic and quintic stiffness of the absorber. Introducing the the equivalent static deformation xS,stat = FE/kS, structure and absorber displacements can be expressed dimensionless as qS =xS/xS,stat andqA =(xS−xA)/xS,stat, the external forcing amplitude is commonly described by the equivalent acceleration amplitude aF =FE/mS. The natural frequencies ωnS=pkS/mS, ωnA=pkA/mA, and the damping ratios ζS=cS/(2mSωnS), ζA=cA/(2mAωnA) of host structure and absorber can be used to describe the underlying linear two degree of freedom dynamics. The normalized excitation frequency is given by γ = ω/ωnS and the mass ratio is denoted µ = mA/mS. Although the equations of motion, eq. (1), can be solved numerically by time integration, analytical solutions can be given by the harmonic balance method (HBM) for steady state conditions. Comparisons with exact numerical results have revealed, that in the current work single term approximations for both, structure and absorber, are sufficiently accurate. The HBM approach yields a set of nonlinear coupled equations, which must be solved numerically to obtain the desired response amplitudes. In [21] a systematic method is described to discover and identify Isolas by evaluating a turning point condition to generate and follow the turning point curve (TP-curve), see Fig 4. Numerically, it is also possible to determine the closed curve of an isolated branch using a path following algorithm for constant excitation force (FRF-Isola). However, these results cannot be reproduced experimentally because instable branches of the coupled system cannot be followed. Consequently, it is also not possible to follow the turning point curve experimentally. An Isola can only be reached by keeping the system states on the stable regions of the generally complex shaped response surfaces. This often requires a simultaneous adjustment of amplitude aF and frequency γ of the structural excitation. Typically, a desired isolated state can be reached experimentally by starting from an initial stable configuration with a higher excitation force, followed by force reductions and frequency corrections until the desired state is reached. In the experiment, however, the actual time to drive the system to an isolated state can take up to serval minutes, because the dynamic system must remain in quasi periodic conditions during the entire process. This time can be reduced significantly by the proposed variable system configuration.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==