Non-stationary oscillation of a string on the Winkler foundation subjected to a discrete mass–spring system non-uniformly moving at a sub-critical speed

Гаврилов, С. Н.; Shishkina, Ekaterina V.; Poroshin, Ilya O.

doi:10.1016/j.jsv.2021.116673

Cited by 4 publications

(38 citation statements)

References 39 publications

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“…If the instantaneous value of the vector of the zeroth order system parameters (2.30) at ξ = 0 leaves the localization domain, then we expect very fast vanishing of the displacements U(t). In a particular case, this was demonstrated numerically in [19]. Later, we have shown that at least for ǫ = 0 the displacement U(t) in the composite system vanishes asymptotically faster than in the corresponding pure continuum system.…”

Section: Remark 23mentioning

confidence: 54%

“…This formula cannot be obtained as a particular case of general formulae (5.6), (5.7). The results given by the correct formula derived later in [19], see also Sect. 6.5, are very close to ones given by (6.6) and differ considerably only for large enough mass M (see Sect.…”

Section: A Non-uniformly Moving Inertial Loadmentioning

confidence: 71%

“…6.5, are very close to ones given by (6.6) and differ considerably only for large enough mass M (see Sect. 5.2.2 of [19]).…”

Section: A Non-uniformly Moving Inertial Loadmentioning

confidence: 96%

“…For ξ > 0 and ξ < 0 we look for the solution in the form, which we called in [19] "the single-frequency ansatz", or better to say, "the single mode ansatz" corresponding to the trapped mode frequency Ω 0 (T ):…”

Section: Derivation Of the First Approximation Equationmentioning

confidence: 99%

“…In [18,19,[53][54][55], the described above approach was applied to various particular cases of the general system considered in this paper, as well as to a more complicated problem where the continuum system was a Bernoulli-Euler beam [56]. The resonant case, i.e., passage through a resonance, was considered in [57].…”

Section: Introductionmentioning

confidence: 99%

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Universal formal asymptotics for localized oscillation of a discrete mass-spring-damper system of time-varying properties, embedded into a one-dimensional medium described by the telegraph equation with variable coefficients

Gavrilov,

Poroshin,

Shishkina

et al. 2024

Preprint

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We consider a quite general problem concerning a linear free oscillation of a discrete mass-spring-damper system. This discrete subsystem is embedded into a one-dimensional continuum medium described by the linear telegraph equation. In a particular case, the discrete subsystem can move along the continuum one at a sub-critical speed. Provided that the dissipation in both discrete and continuum subsystems is absent, if parameters of the subsystems are constants, under certain conditions (the localization conditions), a non-vanishing oscillation localized near the discrete subsystem can be possible. In the paper we assume that the dissipation in the damper and the medium is small, and all discrete-continuum system parameters are slowly varying functions in time and in space (when applicable), such that the localization condition is fulfilled for the instantaneous values of the parameters in a certain neighbourhood of the discrete subsystem position. This general statement can describe a number of mechanical systems of various nature. We derive the expression for the leading-order term of a universal asymptotics, which describes a localized oscillation of the discrete subsystem. In the non-dissipative case, the leading-order term of the expansion for the amplitude is found in the form of an algebraic expression, which involves the instantaneous values of the system parameters. In the dissipative case, the leading-order term for the amplitude, generally, is found in quadratures in the form of a functional, which depends on the history of the system parameters, though in some exceptional cases the result can be obtained as a function of time and the instantaneous limiting values of the system parameters. In previous studies, several non-dissipative particular cases of the problem under consideration are investigated using a similar approach, provided that only one parameter of the discrete-continuum system is a slowly time-varying non-constant quantity, whereas all other parameters are constants. We show that asymptotics obtained in previous studies are particular cases of the universal asymptotics. The existence of a universal asymptotics in the form of a function is a non-trivial fact, which does not follow from the summarization of the results obtained for particular cases. Finally, we have justified the universal asymptotics by numerical calculations.

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Section: Remark 23mentioning

confidence: 54%

Section: A Non-uniformly Moving Inertial Loadmentioning

confidence: 71%

“…6.5, are very close to ones given by (6.6) and differ considerably only for large enough mass M (see Sect. 5.2.2 of [19]).…”

Section: A Non-uniformly Moving Inertial Loadmentioning

confidence: 96%

Section: Derivation Of the First Approximation Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Universal formal asymptotics for localized oscillation of a discrete mass-spring-damper system of time-varying properties, embedded into a one-dimensional medium described by the telegraph equation with variable coefficients

Gavrilov,

Poroshin,

Shishkina

et al. 2024

Preprint

Self Cite

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Generalized multiple scale approach to the problem of a taut string traveled by a single force

2023

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The strongly nonlinear dynamics of taut strings, traveled by a force moving with uniform velocity, is analyzed. A change of variable is performed, which recasts the equations of motion in terms of a linearized dynamic displacement, measured from the nonlinear quasi-static response. Under the hypothesis the load velocity is far enough from the celerity of the string, the system appears in the form of linear PDEs whose coefficients are slowly variable in time. Since the classic perturbation methods are not applicable to such kind of equations, the Generalized Method of Multiple Scales is developed, by directly attacking the PDEs, to derive asymptotic solutions. The validity of the analytical predictions is assessed by comparisons with numerical simulations, aimed to prove the accuracy of (1) linearization, and (2) the asymptotic approach.

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Formal asymptotics for oscillation of a discrete mass-spring-damper system of time-varying properties, embedded into a one-dimensional medium described by the telegraph equation with variable coefficients

Gavrilov,

Poroshin,

Shishkina

et al. 2024

Nonlinear Dyn

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Non-stationary oscillation of a string on the Winkler foundation subjected to a discrete mass–spring system non-uniformly moving at a sub-critical speed

Cited by 4 publications

References 39 publications

Universal formal asymptotics for localized oscillation of a discrete mass-spring-damper system of time-varying properties, embedded into a one-dimensional medium described by the telegraph equation with variable coefficients

Universal formal asymptotics for localized oscillation of a discrete mass-spring-damper system of time-varying properties, embedded into a one-dimensional medium described by the telegraph equation with variable coefficients

Generalized multiple scale approach to the problem of a taut string traveled by a single force

Formal asymptotics for oscillation of a discrete mass-spring-damper system of time-varying properties, embedded into a one-dimensional medium described by the telegraph equation with variable coefficients

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