4 C. Yoong and M. Legrand f|[0;2L]= ⎧ ⎪⎪⎨ ⎪⎪⎩ u0(ξ) 2 + 1 2c ξ 0 v0(s)ds ξ ∈ [0;L], g0 − u0(2L−ξ) 2 + 1 2c L 0 v0(s)ds − 1 2c ξ L v0(2L−s)ds ξ ∈]L;2L], (1.13) Consequently, f can be completely defined over Rfor every phase. To summarize, the displacement of the elastic bar with Robin BC on the left end is obtained through Eq. (1.9) provided that f is defined everywhere on the real axis. The successive switches in boundary conditions at x =L, reflecting the unilateral contact constraint, are incorporated through appropriate extensions: Eq. (1.10) for the free phase or Eq. (1.11) for contact phase. The following section is concerned with analytical derivations for the computation of periodic solutions by employing the expressions (1.9)–(1.11). 1.3 Periodic Solutions and Nonsmooth Modal Analysis Nonsmooth modes of vibration of the spring–bar system depicted in Fig. 1.1 are defined as continuous families of periodic solutions satisfying the formulation (1.1)–(1.4) together with periodicity conditions in displacement and velocity: ∃T >0 such that u(x, t +T) =u(x, t) and v(x, t +T) =v(x, t), ∀x ∈ [0;L] and ∀t >0. Finding such solutions translates into finding corresponding initial conditions u0 andv0 andperiodT which generate periodic motions. Without loss of generality, it is assumed that within one period, over the interval t ∈ [0;T], the initial time segment is a free phase that initiates at t =0+ and the final time segment is a contact phase that ends at t =T− and switches back to the initial free phase state at t =T+. In general, various successions of free and contact phases might arise within one period. Knowing that the motion of the bar can be uniquely defined by a single function f, the targeted trajectory is then an unknown periodic sequence of functions uin the form (1.9) where f switches between (1.10) and (1.11). Let us consider the simplest combination of one free phase and one contact phase of duration tf and tc = T −tf, respectively. When the gap is open, displacement and velocity satisfy the following equalities ∀x ∈ [0;L] and ∀t ∈ [0;tf]: u1(x, t) =f0(ct +x) +f0(ct −x) −2αe α(x−ct) ct−x 0 eαsf0(s)ds, (1.14a) 1 c v1(x, t) =∂t f0(ct +x) +∂t f0(ct −x) −2αf0(ct −x) +2α 2eα(x−ct) ct−x 0 eαsf0(s)ds, (1.14b) where f0|[0;2L], calculated via Eq. (1.12) with the initial conditions u0 and v0, is then expanded throughout the real axis R via (1.10). During gap closure, the motion is described by the composite function f1(f0(·)) corresponding to a boundary condition switch. This function arises by defining the graphf1|[0;2L] withu1(·, tf) andv1(·, tf) as the “initial conditions” in Eq. (1.13) and then expanding it onRvia Eq. (1.11). During a contact phase, the displacement and velocity read∀x ∈ [0;L] and ∀t ∈]tf ;T] u2(x, t) =f1(c(t −tf) +x) +f1(c(t −tf) −x) −2αe α(x−c(t−tf)) c(t−tf)−x 0 eαsf1(s)ds, (1.15a) 1 c v2(x, t) =∂t f1(c(t −tf)+x) +∂t f1(c(t −tf)−x)−2αf1(c(t −tf)−x)+2α 2eα(x−c(t−tf)) c(t−tf)−x 0 eαsf1(s)ds. (1.15b) Accordingly, admissibleT-periodic motions involving one lasting contact phase per period are described by functions f1◦f0 and f0 that satisfy the following periodicity condition, arising fromu0(x) =u2(x,T) and v0(x) =v2(x,T), ∀x ∈ [0;L] u0(x) =f1(ctc +x) +f1(ctc −x) −2αe α(x−ctc) ctc−x 0 eαsf1(s)ds, (1.16a) 1 c v0(x) =∂t f1(ctc +x) +∂t f1(ctc −x) −2αf1(ctc −x) +2α 2eα(x−ctc) ctc−x 0 eαsf1(s)ds. (1.16b) together with the admissibility conditions reflecting Signorini’s conditions at x =Lin Eq. (1.3):
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