Measurement and identification of the nonlinear dynamics of a jointed structure using full-field data; Part II - Nonlinear system identification

Data-driven nonlinear model reduction to spectral submanifolds in mechanical systems

“…For adequate accuracy, the normal form indeed needs terms up to to capture the dynamics The model can be used to approximate the beam kinetic energy as where is the number of DIC measurement locations and is the beam mass. As discussed in [ 8 ], the kinetic energy amplitude is a good proxy for the instantaneous decay properties, i.e. the instantaneous damping ratio and frequency, shown in figure 4 d , e , respectively.…”

“…lim||bold-italicρ||false→0[−αjfalse(bold-italicρ,bold-italicθfalse)+iωjfalse(bold-italicρ,bold-italicθfalse)]=λj.Hence, αj and ωj are the nonlinear continuations of these linear quantities, characterizing how dissipation and frequency change with respect to the amplitudes (and phases for internally resonant systems). For a two-dimensional SSM, the parametrized curves αfalse(ρfalse) and ωfalse(ρfalse) are the backbones of transient oscillations [1,8,54,57], representing the instantaneous damping and frequency as nonlinear functions of the normal form amplitude ρ. Normal form amplitudes do not, however, have any direct physical meaning.…”

“…The Brake–Reuss beam (BRB) is a benchmark system in the study of jointed structures [ 7 , 8 , 83 ]. In our study, it consists of two 304 stainless steel beams assembled with a lap joint, as shown in figure 4 a .…”

“…While full models for these structures may not be smooth, we find that trajectory data can be fitted well to smooth models, thereby justifying an SSM-based approach. The data considered here arise from a single resonance decay test, available from [ 8 ], targeting the slowest structural mode. One observable is the measurement from an accelerometer mounted, as shown in figure 4 a , on the shaker with time history illustrated in figure 4 c .…”

“…Low-dimensional models are expected to reduce computational cost and capture the essential physics of a high-dimensional system from data. Given the growing interest, for example, in light mechanical structures and MEMS devices, there is the need for truly nonlinear models, capturing amplitude-dependent properties and competing steady-states solutions, which are increasingly important to identify, as highlighted in experiments of nonlinear mechanical vibrations [1][2][3][4][5][6][7][8]. Predicting coexisting stable and unstable forced responses for a broad range of forcing amplitudes and frequencies is paramount in structural dynamics.…”