Nonlinear Modal Analysis of a One-Dimensional Bar Undergoing Unilateral Contact via the Time-Domain Boundary Element Method

Abstract

Finite elements in space with time-stepping numerical schemes, even though versatile, face theoretical and numerical difficulties when dealing with unilateral contact conditions. In most cases, an impact law has to be introduced to ensure the uniqueness of the solution: total energy is either not preserved or spurious high-frequency oscillations arise. In this work, the Time Domain Boundary Element Method (TD-BEM) is shown to overcome these issues on a one-dimensional system undergoing a unilateral Signorini contact condition. Unilateral contact is implemented by switching between free boundary conditions (open gap) and fixed boundary conditions (closed gap). The solution method does not numerically dissipate energy unlike the Finite Element Method and properly captures wave fronts, allowing for the search of periodic solutions. Indeed, TD-BEM relies on fundamental solutions which are travelling Heaviside functions in the considered one-dimensional setting. The proposed formulation is capable of capturing main, subharmonic as well as internal resonance backbone curves useful to the vibration analyst. For the system of interest, the nonlinear modeshapes are piecewise-linear unseparated functions of space and time, as opposed to the linear modeshapes that are separated half sine waves in space and full sine waves in time.

Nonsmooth Modal Analysis With Boundary Element Method

“…The corresponding backbone curves are shown in figure 2. The backbone curve shows the frequency-dependence of the total energy, and compares well with existing solutions [11], except for backbone curve near linear mode. Since contact period length decreases as frequency !…”

“…For a periodic motion, the equality q.q 0 ; T / D q 0 holds. In this work, v 0 D 0 is assumed [11], and the problem to be solved thus reduces to: Find vector u 0 and strictly positive integer m, where T D mt, such that…”

“…Through (11), Q u at frequency ! can be solved for a combination of Dirichlet and/or Neumann boundary conditions.…”

“…Since initial velocity is assumed to be zero [11], all sin terms vanishes. Only constant and cos terms remain in equation ( 13) and equation ( 14).…”

“…Nonsmooth Modal Analysis (NSM) is a version of modal analysis dedicated to such systems. A few numerical schemes exist to perform NSM, with inherent limitations [11,12]. In the present work, it is proposed to explore the numerical capabilities of two numerical schemes grounded on the Boundary Element Method (BEM) to perform NSM.…”