ABSTRACTIn this paper some recently

ABSTRACT
In this paper some recently developed techniques to evaluate both analytically and numerically stability features of a non-smooth
dynamical system, are used to investigate in detail the stability boundaries of regions corresponding to given stable periodic responses.
The problem analyzed is that of a rocking block simply supported on a harmonically moving rigid ground; in this case, if the block is
assumed to be a rigid body, the strong discontinuities characterizing the dynamic evolution are due to the impacts occurring each time
the block crosses the initial equilibrium configuration.
Therefore some special tools, specific for non-smooth functions, must be introduced to perform the stability analysis. In the present
study, the theory due to Müller [19] is used to handle the evaluation of Lyapunov’s exponents upon discontinuities, by introducing the
treatment, both analytical and numerical, of “saltation matrices”. Such a general theoretical method on one hand has been adapted to
the numerical algorithms needed for the solution of the complete, non-linearized, problem and on the other hand, it allowed the
development of the closed-form analytical reference solutions, obtained by linearizing assumptions less restrictive than those used by
Hogan [8, 9].
The approximated stability boundaries obtained by the linearized closed-form solutions have been the starting point to guide the
choice of the system parameters values to locate the responses in regions where bifurcations can arise.
Inside these ranges several examples can be presented to illustrate the trends exhibited by the numerically evaluated Lyapunov’s
exponents when the values of the forcing amplitude increase over the stability boundary of the symmetric responses while the value of
the forcing frequency is fixed.
Among these, investigations on sequences of responses composing period doubling cascades toward chaos, can provide a good and
interesting test to appreciate the indications offered by the numerically derived Lyapunov’s exponents.

ผลลัพธ์ (อังกฤษ) 2:

ABSTRACT
In this Paper Some recently developed Techniques to evaluate both analytically and numerically stability features of a non-Smooth
dynamical System, are used to Investigate in detail the stability boundaries of Regions corresponding to Given Stable periodic Responses.
The Problem analyzed is that of a rocking. block simply supported on a harmonically moving rigid ground; in this Case, if the Block is
assumed to be a rigid Body, the strong discontinuities characterizing the Dynamic Evolution are Due to the impacts occurring each time
the Block crosses the Initial equilibrium Configuration.
Therefore Some special Tools, specific for non-Smooth functions,. must be introduced to perform the stability analysis. In the present
Study, the Theory Due to Müller [19] is used to Handle the evaluation of Lyapunov's exponents upon discontinuities, by introducing the
Treatment, both analytical and Numerical, of "Saltation matrices". Such a general theoretical method on one Hand has been adapted to
the Numerical algorithms needed for the Solution of the Complete, non-linearized, Problem and on the Other Hand, it allowed the
Development of the Closed-form analytical reference Solutions, obtained by Linearizing. assumptions less restrictive than those used by
Hogan [8, 9].
The approximated stability boundaries obtained by the linearized Closed-form Solutions have been the Starting Point to Guide the
Choice of the System Parameters values ​​to Locate the Responses in Regions where bifurcations Can ARISE. .
Inside these Ranges several examples Can be Presented to illustrate the Trends exhibited by the numerically evaluated Lyapunov's
exponents when the values ​​of the forcing amplitude increase over the stability Boundary of the symmetric Responses while the Value of
the forcing frequency is fixed.
Among these, Investigations. Responses on composing sequences of period Doubling Cascades toward Chaos, Can provide a good and
Interesting Test to appreciate the indications offered by the numerically derived Lyapunov's exponents.

ผลลัพธ์ (อังกฤษ) 3:

ABSTRACT
In this paper some recently developed techniques to evaluate both analytically and numerically stability features. Of a non-smooth
dynamical system are used, to investigate in detail the stability boundaries of regions corresponding to. Given stable periodic responses.
The problem analyzed is that of a rocking block simply supported on a harmonically moving. Rigid ground; in, this caseIf the block is
assumed to be a, rigid body the strong discontinuities characterizing the dynamic evolution are due to. The impacts occurring each time
the block crosses the initial equilibrium configuration.
Therefore some, special tools specific. For, non-smooth functions must be introduced to perform the stability analysis. In the, present
studyThe theory due to M V ller [] is 19 used to handle the evaluation of Lyapunov 's exponents upon discontinuities by introducing,, The
treatment both analytical, and numerical of ", saltation matrices". Such a general theoretical method on one hand has. Been adapted to
the numerical algorithms needed for the solution of, the complete non-linearized problem and, on the other. Hand it allowed, the
.Development of the closed-form analytical, reference solutions obtained by linearizing assumptions less restrictive than. Those used by
Hogan [,]. 8 9
The approximated stability boundaries obtained by the linearized closed-form solutions have. Been the starting point to guide the
choice of the system parameters values to locate the responses in regions where bifurcations. Can arise.
.Inside these ranges several examples can be presented to illustrate the trends exhibited by the numerically evaluated Lyapunov s
exponents. ' When the values of the forcing amplitude increase over the stability boundary of the symmetric responses while the value. Of
the forcing frequency is fixed.
Among these investigations on, sequences of responses composing period doubling cascades. Toward, chaosCan provide a good and Sb interesting test to appreciate the indications offered by the numerically derived Lyapunov ’ s exponents sb EOS.