Nonsmooth modal analysis via the boundary element method for one-dimensional bar systems

Nonsmooth modal analysis of a varying cross-sectional area bar in unilateral contact

“…As evident from literature on modal analysis of the Signorini problem, NSM can be described by via a continuum of solutions .q 0 .T /; T / on a closed interval of periods, referred to as a backbone curve [1,11,12,31]. This continuum of solutions can can be found by applying sequential continuation on the system of equations (59) [12,31].…”

“…As evident from literature on modal analysis of the Signorini problem, NSM can be described by via a continuum of solutions .q 0 .T /; T / on a closed interval of periods, referred to as a backbone curve [1,11,12,31]. This continuum of solutions can can be found by applying sequential continuation on the system of equations (59) [12,31]. Applying sequential continuation to the NBM numerically, it has been noticed that the sequential continuation for curves of N 10 faces difficulties in obtaining solutions on the backbone curve .q 0 .T /; T /.…”

“…To verify the validity of the nonsmooth modes presented here, we will compare each nonsmooth mode with its corresponding forcedresponse diagram since it is expected that the backbone branch will align with the frequency and energy at resonance [7,13]. All results in Section 6.2 are generated for a gap distance g D 0:001 to comply with the models investigated in [13,31]. Furthermore, all NSMs were generated via FE models of 20 elements and quadratic shape functions for which the backbone branches were depicting resonant points sufficiently.…”

“…The internally resonant bar of uniform area A.x/ D 1 is an example that has been investigated both numerically [12,31] and analytically [1,11,20]. The system is known to manifest an intricate modal space consisting of families of iso-periodic periodic solutions in a dense set of periods and families of periodic solutions of the same frequency and energy.…”

“…Similarly to [13,31], the bar with soft support features a uniform cross-sectional area A.x/ D 1 where the homogeneous Dirichlet condition at x D 0 is replaced with the Robin condition ku.0; t/ D u x .0; t /; k D N kL=.EA 0 / (66)…”