Analysis of Piecewise Linear Systems With Displacement and Time Regions by Vector Fields

1999 ◽  
Author(s):  
Mikio Nakai ◽  
Shinji Murata ◽  
Seiji Hagio
Keyword(s):  
Linear Systems ◽  
Vector Fields ◽  
Matrix Representation ◽  
Piecewise Linear ◽  
Generalized Solution ◽  
State Variables ◽  
Piecewise Linear Systems ◽  
Restoring Forces ◽  
Pass Through ◽  
Solution Methodology

Abstract A generalized solution methodology based on piecewise linear vector fields is proposed for piecewise linear systems with singular regions or asymmetric restoring forces which vary spatially and temporally. In matrix representation for these systems, state variables in each region can be explicitly expressed as a function of the time the orbit spends between two boundaries or the time the orbit takes to pass through the boundary. The time can be determined by the Brent method, and periodic solutions can then be obtained. Analytical solutions are validated on a system with 3-regions of displacement and 2-regions of time, a circumferential vibration of gear meshing system, by using the newly developed numerical method.

Analysis of Piecewise Linear Systems With Boundaries in Displacement and Time

2002 ◽  
Vol 124 (4) ◽  
pp. 527-536 ◽  
Author(s):  
Mikio Nakai ◽  
Shinji Murata ◽  
Seiji Hagio
Keyword(s):  
Linear Systems ◽  
Analytical Solutions ◽  
Vector Fields ◽  
Matrix Representation ◽  
Piecewise Linear ◽  
Generalized Solution ◽  
State Variables ◽  
Piecewise Linear Systems ◽  
Restoring Forces ◽  
Solution Methodology

A generalized solution methodology based on piecewise linear vector fields is proposed for piecewise linear systems with singular regions or asymmetric restoring forces which vary spatially and temporally. In matrix representation for these systems, state variables in each region can be explicitly expressed as a function of the time the orbit spends between two boundaries or the time when the orbit hits the boundary. The time can be determined by the Brent method, and periodic solutions can then be obtained. Analytical solutions are validated on a system with 3-regions of displacement and 2-regions of time, a circumferential vibration of gear meshing system, by using the newly developed numerical method.

Analysis of Piecewise Linear Systems With Boundaries in Displacement and Velocity Using Vector Fields

2001 ◽  
Author(s):  
Mikio Nakai ◽  
Seiji Hagio ◽  
Kazuki Mizutani ◽  
Hossain Md. Zahid
Keyword(s):  
Linear Systems ◽  
Present Method ◽  
Vector Fields ◽  
Piecewise Linear ◽  
Generalized Solution ◽  
Dynamic Responses ◽  
Spring Stiffness ◽  
Piecewise Linear Systems ◽  
Solution Methodology ◽  
Machinery System

Abstract A generalized solution methodology based on piecewise linear vector fields has been proposed for piecewise linear systems with boundaries for both displacement and time. In this paper, this methodology is applied in analyzing dynamic responses of torsional vibration through bilinear spring stiffness with changing hysteretic torque in the rotating machinery system, which has boundaries for both displacement and velocity. For asymmetric torque with hysteresis, two cases exist when restoring torque changes smoothly and discontinuously. Period-one and period-two solutions are sought in two cases and compared with ones obtained by the numerical method. Our present method can be applied to dynamic responses of piecewise linear systems having velocity and displacement boundaries.

Analysis of Vibro-Impact Dynamical System Using Piecewise Linear Vector Fields

2009 ◽  
Author(s):  
Mikio Nakai
Keyword(s):  
Vector Fields ◽  
Matrix Representation ◽  
Piecewise Linear ◽  
Generalized Solution ◽  
Dynamic Responses ◽  
State Variables ◽  
Single Degree Of Freedom ◽  
Linear Vector ◽  
Solution Methodology ◽  
Machinery System

An analysis of the dynamic responses has been developed for all piecewise linear single-degree-of-freedom systems with boundaries for only displacement by applying analyses based on piecewise linear vector fields. Furthermore, a generalized solution methodology has been proposed for piecewise linear systems with boundaries for both displacement and time such as the system with parameters which give rise to separations due to clearances or backlash and for rotating machinery system with boundaries for both displacement and velocity. In this paper, this methodology was developed for analyzing dynamic responses of two vibro-impact systems: one with stops at both sides and another with a stop only at one side. In matrix representation for these systems, state variables can explicitly be expressed at each region as a function of the time when an orbit hits the stop. The time can be determined by the Brent method, and periodic solutions can then be obtained. Analytical solutions are validated on vibro-impact systems by using conventional method.

BIMODAL PIECEWISE LINEAR DYNAMICAL SYSTEMS: REDUCED FORMS

2010 ◽  
Vol 20 (09) ◽  
pp. 2795-2808 ◽  
Author(s):  
JOSEP FERRER ◽  
M. DOLORS MAGRET ◽  
MARTA PEÑA
Keyword(s):  
Dynamical Systems ◽  
Linear Systems ◽  
Piecewise Linear ◽  
Equivalence Classes ◽  
State Variables ◽  
Linear Dynamical Systems ◽  
Piecewise Linear Systems ◽  
Wide Range ◽  
General Systems ◽  
Reduced Forms

Piecewise linear systems constitute a class of nonlinear systems which have recently attracted the interest of researchers because of their interesting properties and the wide range of applications from which they arise. Different authors have used reduced forms when studying these systems, mostly in the case where they are observable. In this work, we focus on bimodal continuous dynamical systems (those consisting of two linear systems on each side of a given hyperplane, having continuous dynamics along that hyperplane) depending on two or three state variables, which are the most common piecewise linear systems found in practice. Reduced forms are obtained for general systems, not necessarily observable. As an application, we calculate the dimension of the equivalence classes.

BIFURCATION SETS OF SYMMETRICAL CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH THREE ZONES

2002 ◽  
Vol 12 (08) ◽  
pp. 1675-1702 ◽  
Author(s):  
EMILIO FREIRE ◽  
ENRIQUE PONCE ◽  
FRANCISCO RODRIGO ◽  
FRANCISCO TORRES
Keyword(s):  
Linear Systems ◽  
Canonical Form ◽  
Vector Fields ◽  
Piecewise Linear ◽  
Oscillatory Behavior ◽  
Van Der Pol ◽  
Linear Vector ◽  
Piecewise Linear Systems

Planar symmetrical continuous piecewise linear vector fields with three zones are thoroughly analyzed. Special emphasis is placed on their oscillatory behavior. The study is made by using a Van der Pol canonical form which captures the most interesting dynamics and minimizes the number of parameters to be dealt with. This work is a continuation of a previous paper [Freire et al., 1998] and uses the same approach and techniques.

Bifurcation Sets of Continuous Piecewise Linear Systems with Two Zones

1998 ◽  
Vol 08 (11) ◽  
pp. 2073-2097 ◽  
Author(s):  
Emilio Freire ◽  
Enrique Ponce ◽  
Francisco Rodrigo ◽  
Francisco Torres
Keyword(s):  
Linear Systems ◽  
Periodic Orbits ◽  
Canonical Form ◽  
Vector Fields ◽  
Piecewise Linear ◽  
Oscillatory Behavior ◽  
Linear Vector ◽  
Piecewise Linear Systems ◽  
The Creation

Planar continuous piecewise linear vector fields with two zones are considered. A canonical form which captures the most interesting oscillatory behavior is obtained and their bifurcation sets are drawn. Different mechanisms for the creation of periodic orbits are detected, and their main characteristics are emphasized.

scholarly journals Limit cycles from a monodromic infinity in planar piecewise linear systems

2021 ◽  
Vol 496 (2) ◽  
pp. 124818
Author(s):  
Emilio Freire ◽  
Enrique Ponce ◽  
Joan Torregrosa ◽  
Francisco Torres
Keyword(s):  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Piecewise Linear Systems
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