238 M. Cimpuieru et al. a reduction in beam response with increased damping coefficient and a stiffening effect due to the spring force at impact. Additionally, the highest reduction is obtained when the RVA and the beam move out of phase. 31.2 Methodology To demonstrate the effectiveness of the RVA (Fig. 31.1, green), a computational finite element (FE) model is developed using a beam as host structure as presented in Fig. 31.1. The beam-RVA system is modeled using a number of approximately 741,387 degrees of freedom (DOF), which is reduced using a Craig-Bampton reduced order model with 5 primary nodes and 70 secondary modes. As described in the introduction section, the RVA absorbs energy from the host structure (i.e., the beam) and further dissipates energy through nonlinear impact contacts. Impact contacts between the RVA and the host structure occur at the left and right sides of the “U” shape component present in Fig. 31.1 (red), 2, and 3. The forcing is applied in the proximity of the beam’s root and the response is measured at the free end of the beam. The RVA is tuned to ensure high-energy absorption from the host structure and large nonlinear impact forces for energy dissipation. Thus, the first natural frequency of the RVA is tuned to match the target frequency of the host structure, resulting in two cantilever beam modes: one where the beam and the RVA move in phase with each other (Fig. 31.2), and the other one with the two moving out of phase (Fig. 31.3). The harmonic balance method (HBM) [5] with an alternating frequency-time domain (AFT) procedure is implemented to calculate the beam response. The mass and stiffness matrices of the linear system (i.e., no impact contacts) are obtained through the Craig-Bampton reduction, while the nonlinear forces are calculated within the AFT procedure. The nonlinear forces at impact are calculated in the time domain using a novel microscopic model at each contact node pair. Within the AFT scheme, the forces at impact are converted to the frequency domain using a fast Fourier transformation (FFT) applied over one oscillation cycle. Multiple contact nodes per impact can be added for a more accurate contact model. For each contact node pair, a virtual spring and a virtual damper are used to model the impact contact as exemplified in Fig. 31.4. For the analysis in this chapter, both left and right impacts are modeled as point contacts (i.e., one node pair per contact interface) due to the geometry of the RVA and of the “U” shape component. While the spring models the elastic deformation of the bodies during contact, the damping force dissipates energy during impacts by acting in the opposite direction of the relative motion between the contacting bodies (i.e., contact between the left/right wall and the left/right side of the RVA). The corresponding spring and damping forces are calculated in the time domain as part of the AFT procedure as follows: .fnonlinear j = ki xleft wall,j −xRVA left,j +ci ˙xleft wall,j − ˙xRVA left,j 0 , (31.1 ) . switch condition : ki xleft wall,j −xRVA left,j +ci ˙xleft wall,j − ˙xRVA left,j =0 wher e xRVA left, j is the vector of coordinates (i.e., x,y,z ) of the node on the RVA’s left side corresponding to contact pair j. xleft wall, j is the vector of coordinates of the node placed on the left wall of the “U” component part of contact pair j. Similarly, Eq. 31.1 is used to model the right-side impacts. 31.3 Results and Discussion To analyze the influence of the RVA on the host structure’s response, it is paramount to understand the effect of the stiffness and damping forces during impact on the beam’s response. In Fig. 31.5, the beam tip response when the damper moves freely (i.e., no contact at the left or right walls) shows two peaks corresponding to the two modes that result after tuning. A shift in resonance frequency can be observed for both modes when impacts occur. This phenomenon is attributed to the stiffening effect caused by the virtual spring accounting for the elastic deformation at impact. The damping force causes the amplitude reduction of the beam response through energy dissipation. It can be noticed that for both modes the amplitude reduction is increasing with higher damping value coefficients (ci). However, the response corresponding to the mode at 32.04 Hz when the damper and the host structure are moving in phase shows a smaller amplitude reduction. The reason for this phenomenon is the smaller relative velocity between the damper and the beam that causes lower forces opposing the relative motion of the absorber and the host and therefore less energy dissipation within the system. Finally, Figs. 31.6 and 31.7 show the impact
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