Multiple solutions for the nonlinear algebraic system with the indefinite coefficient matrix

2020 ◽  
Vol 107 ◽  
pp. 106353
Author(s):  
Mengxiang You ◽  
Yu Tian ◽  
Miaomiao Chen ◽  
Yue Yue
Keyword(s):  
Multiple Solutions ◽  
Algebraic System ◽  
Coefficient Matrix ◽  
Nonlinear Algebraic System

scholarly journals Positive solutions for a nonlinear algebraic system with nonnegative coefficient matrix

2017 ◽  
Vol 64 ◽  
pp. 150-155 ◽  
Author(s):  
Yongqiang Du ◽  
Wenying Feng ◽  
Ying Wang ◽  
Guang Zhang
Keyword(s):  
Positive Solutions ◽  
Algebraic System ◽  
Coefficient Matrix ◽  
Nonnegative Coefficient ◽  
Nonlinear Algebraic System

scholarly journals Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yanping Jia ◽  
Ying Gao ◽  
Wenying Feng ◽  
Guang Zhang
Keyword(s):  
Positive Solutions ◽  
Algebraic System ◽  
Periodic Boundary Conditions ◽  
Coefficient Matrix ◽  
Order Difference ◽  
Monotone Iterative ◽  
Existence Of Positive Solutions ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Nonlinear Algebraic System

Abstract Existence of positive solutions for the nonlinear algebraic system $x=\lambda GF ( x ) $ x = λ G F ( x ) has been extensively studied when the $n\times n$ n × n coefficient matrix G is positive or nonnegative. However, to the best of our knowledge, few results have been obtained when the coefficient matrix changes sign. In this case, some commonly applied analysis methods such as the cone theory, the Krein–Rutman theorem, the monotone iterative techniques, and so on cannot be directly applied. In this note, we prove the existence of positive solutions for the above nonlinear algebraic system with sign-changing coefficient matrix taking the advantages of the classical Brouwer fixed point theorem combined with a decomposition condition on the coefficient matrix. We provide an example in solving a second-order difference equation with periodic boundary conditions to illustrate the applications of the results.

scholarly journals Eigenvalue and spectral intervals for a nonlinear algebraic system

2013 ◽  
Vol 439 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Wenying Feng ◽  
Guang Zhang
Keyword(s):  
Algebraic System ◽  
Nonlinear Algebraic System

Paths and Laws of Motion of a Mechanism with Two Successive Conductive Elements and a Triad

2015 ◽  
Vol 772 ◽  
pp. 344-349 ◽  
Author(s):  
Liliana Luca ◽  
Iulian Popescu
Keyword(s):  
Algebraic System ◽  
Contour Method ◽  
Kinematic Scheme ◽  
Structural Scheme ◽  
Successive Elimination ◽  
Ternary Element ◽  
Nonlinear Algebraic System ◽  
Laws Of Motion

It starts from a structural scheme of a mechanism with a triad and two successive conductive elements, and a kinematic scheme with ternary element and another element with void lengths is made. The relations to calculate the positions by contour method are written and the nonlinear algebraic system is solved by the method of successive elimination of the unknowns. There are determined the successive positions, the paths of some points and the variations of lifts, for different correlations between the laws of motion of the two conductive elements. It appears that there result paths and interesting laws.

The Effect of Geometry on Harmonically Varied Helix Milling Tools

2020 ◽  
Vol 142 (7) ◽  
Author(s):  
Markel Sanz ◽  
Alex Iglesias ◽  
Jokin Munoa ◽  
Zoltan Dombovari
Keyword(s):  
Algebraic System ◽  
Point Of View ◽  
Geometrical Parameters ◽  
Stability Properties ◽  
Edge Geometry ◽  
Variable Helix ◽  
Nonlinear Algebraic System ◽  
The Stability ◽  
Milling Tools ◽  
Equivalent Description

Abstract Two different kinds of descriptions for edge geometry of harmonically varied helix tools are studied in this work. The edge geometries of the so-called lag and helix variations are used in this paper, and their equivalency is established from engineering point of view. The equivalent relation is derived analytically and the nonlinear algebraic system is described, with which the numerical equivalency properties can be determined. The equivalent description can be utilized in variable helix tool production to determine an optimized set of geometrical parameters of the edge geometry. The stability properties are shown and compared for a simple one degree-of-freedom case with the nonuniform constant helix tools. The robustness of the results related to the harmonically varied tools is critically discussed in this paper showing advantages compared to the nonuniform constant helix case. The results suggest that the more extreme the edge variation is, the more stable the process performed with the corresponding harmonically varied tool becomes.

scholarly journals A Multimode Approach to Geometrically Nonlinear Free and Forced Vibrations of Multistepped Beams

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Issam El Hantati ◽  
Ahmed Adri ◽  
Hatim Fakhreddine ◽  
Said Rifai ◽  
Rhali Benamar
Keyword(s):  
Approximate Method ◽  
Forced Vibration ◽  
Algebraic System ◽  
Strain Energy ◽  
Dynamic Behaviour ◽  
Forced Vibrations ◽  
Geometrical Parameters ◽  
Geometrically Nonlinear ◽  
Free And Forced Vibrations ◽  
Nonlinear Algebraic System

The scope of this study is to present a contribution to the geometrically nonlinear free and forced vibration of multiple-stepped beams, based on the theories of Euler–Bernoulli and von Karman, in order to calculate their corresponding amplitude-dependent modes and frequencies. Discrete expressions of the strain energy and kinetic energies are derived, and Hamilton’s principle is applied to reduce the problem to a solution of a nonlinear algebraic system and then solved by an approximate method. The forced vibration is then studied based on a multimode approach. The effect of nonlinearity on the dynamic behaviour of multistepped beams in the free and forced vibration is demonstrated and discussed. The effect of varying some geometrical parameters of the stepped beams in the free and forced cases is investigated and illustrated, among which is the variation in the level of excitation.

scholarly journals Geometrically nonlinear free and forced vibrations of Euler-Bernoulli multi-span beams

2018 ◽  
Vol 211 ◽  
pp. 02001 ◽  
Author(s):  
Hatim Fakhreddine ◽  
Ahmed Adri ◽  
Saïd Rifai ◽  
Rhali Benamar
Keyword(s):  
Algebraic System ◽  
Single Mode ◽  
Forced Vibrations ◽  
Nonlinear Frequency ◽  
Bending Vibration ◽  
End Conditions ◽  
Nonlinear Frequency Response ◽  
Simple Support ◽  
Beam Bending ◽  
Nonlinear Algebraic System

The objective of this paper is to establish the formulation of the problem of nonlinear transverse forced vibrations of uniform multi-span beams, with several intermediate simple supports and general end conditions, including use of translational and rotational springs at the ends. The beam bending vibration equation is first written at each span and then the continuity requirements at each simple support are stated, in addition to the beam end conditions. This leads to a homogeneous linear system whose determinant must vanish in order to allow nontrivial solutions to be obtained. The formulation is based on the application of Hamilton’s principle and spectral analysis to the problem of nonlinear forced vibrations occurring at large displacement amplitudes, leading to the solution of a nonlinear algebraic system using numerical or analytical methods. The nonlinear algebraic system has been solved here in the case of a four span beam in the free regime using an approximate method developed previously (second formulation) leading to the amplitude dependent fundamental nonlinear mode of the multi-span beam and to the corresponding backbone curves. Considering the nonlinear regime, under a uniformly distributed excitation harmonic force, the calculation of the corresponding generalised forces has led to the conclusion that the nonlinear response involves predominately the fourth mode. Consequently, an analysis has been performed in the neighbourhood of this mode, based on the single mode approach, to obtain the multi-span beam nonlinear frequency response functions for various excitation levels.

Dynamic Analysis of Flexible Mechanisms With Clearances

1996 ◽  
Author(s):  
Yu (Michael) Wang ◽  
Zaiping Wang
Keyword(s):  
Finite Element Method ◽  
Periodic Solution ◽  
Algebraic System ◽  
Degrees Of Freedom ◽  
Valve Train ◽  
Time Intervals ◽  
Time Period ◽  
Nonlinear Algebraic System ◽  
Flexible Mechanisms ◽  
Spatial Degrees Of Freedom

Abstract A finite element method in time is presented for the periodic solution of vibrating elastic mechanisms with clearances. The solution of motion is made possible by utilizing time finite elements which discretize the forcing time period into a number of time intervals. During each interval, the solution form is derived from a Hamilton’s law of varying action. The periodic response is described in terms of a set of temporal nodes of all spatial degrees of freedom of the system, yielding a block-diagonal nonlinear algebraic system to be solved iteratively. The suggested method is applied to an example problem of cam-driven valve train, demonstrating the effectiveness of the method in dealing with multiple clearance nonlinearities.

Dynamic Analysis of Flexible Mechanisms With Clearances

1996 ◽  
Vol 118 (4) ◽  
pp. 592-594 ◽  
Author(s):  
Yu Wang ◽  
Zaiping Wang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Periodic Solution ◽  
Algebraic System ◽  
Degrees Of Freedom ◽  
Valve Train ◽  
Time Intervals ◽  
Nonlinear Algebraic System ◽  
Flexible Mechanisms ◽  
Spatial Degrees Of Freedom

A finite element method in time is presented for the periodic solution of vibrating elastic mechanisms with clearances. The solution of motion is made possible by utilizing time finite elements which discretize the solution period into a number of time intervals. The periodic response is described in terms of a set of temporal nodes of all spatial degrees of freedom of the system, yielding a block-diagonal nonlinear algebraic system to be solved iteratively. The suggested method is applied to an example problem of cam-driven valve train, demonstrating the effectiveness of the method in dealing with multiple clearance nonlinearities.

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