The method of strained coordinates for vibrations with weak unilateral springs

Tong

2014

Nonlinear Dyn

The NNM (Nonlinear normal modes) have recently been investigated with the method of strained coordinates for spring mass models with some springs with piecewise linear stiffness. The N d.o.f. case and the mathematical validity of the method for large time were rigorously proved. The time validity is related to the nature of contact. For grazing contact, this method and also the multiscale expansion lose nonlinear features. For a small piecewise linear stiffness, we show that this method is less precise for a weak unilateral grazing contact. Thus, the validity of the asymptotic expansion for large time can not be improved and the method has to be modified.

“…in [3] and independently in [8]. Numerical experiments have been investigated in [3] and also in [4]. The mathematical validity is obtained in [4].…”

Section: Introductionmentioning

confidence: 86%

“…Numerical experiments have been investigated in [3] and also in [4]. The mathematical validity is obtained in [4]. But the accuracy is less precise for the grazing contact case.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Limitation on the method of strained coordinates for vibrations with weak grazing unilateral contact

Tong

2014

Nonlinear Dyn

“…A similar approach with a simplified piecewise linear model is studied in Ref. 20. More sophisticated smooth models, involving finite non-null crack compressibility, have been developed for applications to engineering 2 and geomechanical problems.…”

Section: Physical Motivationmentioning

confidence: 99%

Interaction Between Periodic Elastic Waves and Two Contact Nonlinearities

Math. Models Methods Appl. Sci.

Lombard

2012

Propagation of elastic waves is studied in a 1D medium containing two cracks. The latter are modeled by smooth nonlinear jump conditions accounting for the finite, non-null compressibility of real cracks. The evolution equations are written in the form of a system of two nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. Perturbation analysis indicates that, under periodic excitation, the periodic solutions oscillate around positive mean values, which increase with the forcing level. This typically nonlinear phenomenon offers non-destructive means to evaluate the cracks. Existence, uniqueness and attractivity of periodic solutions is then examined. At some particular values of the ratio between the wave travel time and the period of the source, results are obtained whatever the forcing level. With a much larger set of ratios but at small forcing levels, results are obtained under a Diophantine condition. Lastly, numerical experiments are proposed to illustrate the behavior of the periodic diffracted waves.

“…The optimal α is more difficult to get and it is obtained in the following subsections, see also [19]. The evaluation of exponent α also implies new asymptotic expansions involving piecewise smooth functions in [20].…”

Section: About Non Degeneracy Conditionsmentioning

confidence: 99%

Averaging lemmas with a force term in the transport equation

Berthelin

Journal de Mathématiques Pures et Appliquées

2010

Self Cite

We obtain several averaging lemmas for transport operator with a force term. These lemmas improve the regularity yet known by not considering the force term as part of an arbitrary right-hand side. Two methods are used: local variable changes or stationary phase. These new results are subjected to two non degeneracy assumptions. We characterize the optimal conditions of these assumptions to compare the obtained regularities according to the space and velocity variables. Our results are mainly in L 2 , and for constant force, in L p for 1 < p ≤ 2. RésuméNous obtenons plusieurs lemmes de moyenne pour deséquations de transport avec un terme de force. Ces résultats améliorent la régularité connue en ne considérant pas le terme de force comme un terme source arbitraire. Deux techniques sont utilisées : des changements de variables locaux ou des phases stationnaires. Ces résultats sont quantifiées par deux hypothèses de non dégénérescence. Nous caractérisons les conditions optimales de ces hypothèses pour comparer les régularités obtenues, par rapport aux variables d'espace et de vitesse. Les résultats sont principalement dans L 2 , et pour le cas constant, dans L p pour 1 < p ≤ 2.