scholarly journals Theoretical Study on Synchronization of the Counter-Rotating Exciter in the Compound Vibrating Field

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Yisen Sun ◽  
Ming Su ◽  
Xu Huang ◽  
Rongchen Zhao ◽  
Rong Li ◽  
...  
Keyword(s):  
Differential Equation ◽  
Theoretical Study ◽  
Order Differential Equation ◽  
Periodic Coefficient ◽  
Vibrating System ◽  
Synchronization Problem ◽  
Small Parameters ◽  
Coupling Characteristics ◽  
The Impact ◽  
Theoretical Results

Aiming at the impact of the complex vibration environment generated by the integrated vibration equipment on the vibration testing equipment, this paper proposes a new method to solve the vibratory synchronization problem in the compound vibration environment. A new concept of the compound vibrating field is proposed, and a new simple vibrating system with a single counter-rotating exciter in a compound vibrating field is established. The motion differential equation of the system is established by the integral mean method with small parameters, and then the periodic coefficient differential equation is obtained through linearization. Based on the relevant theory of the second-order differential equation with periodic coefficient, the synchronization criterion and stability criterion of the vibrating system are derived. According to the theoretical criteria, the coupling characteristics of the exciter and the vibrating field are numerically simulated and analyzed, which supports the theoretical results. The proposed compound vibrating field provides a new way for studying vibratory synchronization.

scholarly journals Theory of Frequency Captured of Eccentric Rotor by Vibration Environment with Same Direction

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Su Ming ◽  
Li Rong ◽  
Xie Zhiping ◽  
Zheng Jiming
Keyword(s):  
Differential Equation ◽  
Stability Criterion ◽  
Periodic Coefficient ◽  
Second Order ◽  
Frequency Synchronization ◽  
Synchronization Problem ◽  
Eccentric Rotor ◽  
Small Parameters ◽  
The Stability ◽  
Rotating Bodies

Aiming at the frequency synchronization phenomena of oscillating or rotating bodies, this paper proposes a novel solution to address the self-synchronization problem of vibration systems. An integral mean method with small parameters and periodic coefficient (IMM-SPPC) is proposed, which converts the relative motion of the electrically driven eccentric rotor and the vibration environment into a second-order periodic coefficient differential equation. Through the calculation of the equilibrium point of the second-order periodic coefficient differential equation and the study of its stability, the synchronization criterion and the stability criterion of the eccentric rotor and the vibration environment are deduced. The simulation results show the validity of the deduced synchronization criterion and stability criterion. The proposed IMM-SPPC provides a new way for studying vibration synchronization.

scholarly journals Synchronization of Two Non-Identical Coupled Exciters in a Non-Resonant Vibrating System of Linear Motion. Part I: Theoretical Analysis

2009 ◽  
Vol 16 (5) ◽  
pp. 505-515 ◽  
Author(s):  
Chunyu Zhao ◽  
Hongtao Zhu ◽  
Ruizi Wang ◽  
Bangchun Wen
Keyword(s):  
Zero Solution ◽  
Moment Of Inertia ◽  
Moments Of Inertia ◽  
Vibrating System ◽  
Frequency Capture ◽  
Synchronization Problem ◽  
Small Parameters ◽  
Existence And Stability ◽  
Synchronous Operation ◽  
The Stability

In this paper an analytical approach is proposed to study the feature of frequency capture of two non-identical coupled exciters in a non-resonant vibrating system. The electromagnetic torque of an induction motor in the quasi-steady-state operation is derived. With the introduction of two perturbation small parameters to average angular velocity of two exciters and their phase difference, we deduce the Equation of Frequency Capture by averaging two motion equations of two exciters over their average period. It converts the synchronization problem of two exciters into that of existence and stability of zero solution for the Equation of Frequency Capture. The conditions of implementing frequency capture and that of stabilizing synchronous operation of two motors have been derived. The concept of torque of frequency capture is proposed to physically explain the peculiarity of self-synchronization of the two exciters. An interesting conclusion is reached that the moments of inertia of the two exciters in the Equation of Frequency Capture reduce and there is a coupling moment of inertia between the two exciters. The reduction of moments of inertia and the coupling moment of inertia have an effect on the stability of synchronous operation.

scholarly journals Cluster Projective Synchronization of Fractional-Order Complex Network via Pinning Control

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Li-xin Yang ◽  
Wan-sheng He ◽  
Fan-di Zhang ◽  
Jin-ping Jia
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Stability Theory ◽  
Order Differential Equation ◽  
Pinning Control ◽  
Projective Synchronization ◽  
Order Complex ◽  
Collective Phenomena ◽  
Fractional Order Differential Equation ◽  
Theoretical Results

Synchronization is the strongest form of collective phenomena in complex systems of interacting components. In this paper, the problem of cluster projective synchronization of complex networks with fractional-order nodes based on the fractional-order differential equation stability theory is investigated. Only the nodes in one community which have direct connections to the nodes in other communities are controlled. Some sufficient synchronization conditions are derived via pinning control. Numerical simulations are provided to show the effectiveness of the theoretical results.

Solving a Fractional-Order Differential Equation Using Rational Symmetric Contraction Mappings

2021 ◽  
Vol 5 (4) ◽  
pp. 159
Author(s):  
Hasanen A. Hammad ◽  
Praveen Agarwal ◽  
Shaher Momani ◽  
Fahad Alsharari
Keyword(s):  
Differential Equation ◽  
Fixed Points ◽  
Fractional Order ◽  
Metric Spaces ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
Contraction Mappings ◽  
Continuous Mappings ◽  
Theoretical Results

The intent of this manuscript is to present new rational symmetric ϖ−ξ-contractions and infer some fixed-points for such contractions in the setting of Θ-metric spaces. Furthermore, some related results such as Suzuki-type rational symmetric contractions, orbitally Υ-complete, and orbitally continuous mappings in Θ-metric spaces are introduced. Ultimately, the theoretical results are shared to study the existence of the solution to a fractional-order differential equation with one boundary stipulation.

scholarly journals Synchronization of a Dual-Mass Vibrating System with Two Exciters

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Yunshan Liu ◽  
Xueliang Zhang ◽  
Dawei Gu ◽  
Lei Jia ◽  
Bangchun Wen
Keyword(s):  
Numerical Simulation ◽  
Engineering Application ◽  
Rigid Bodies ◽  
Total System ◽  
Theoretical Derivation ◽  
Mass System ◽  
Vibrating System ◽  
Differential Motion ◽  
Coupling Characteristics ◽  
Theoretical Results

From the perspective of theoretical derivation, numerical simulation, and engineering application, the vibratory synchronization characteristics of a dual-mass vibrating system driven by two exciters, were studied. The differential motion equations of the total system were calculated using Lagrange’s equations, and the responses of the vibrating system in the steady state were derived by Laplace transform. The synchronization criterion between two exciters was deduced by using the averaging method. Based on the Hamilton principle, the stability criterion of the vibrating system in synchronous states is given. According to the theoretical results, the coupling characteristics between two exciters such as synchronization and stability were analyzed numerically. Some analyses of the numerical simulation of the system were carried out, which fully support the theoretical results. The rotational speed of two exciters, their phase difference, responses, and difference of responses of two rigid bodies were studied quantitatively in the subresonant state and super-resonant state of the system. This paper presents a practical example of vibratory synchronization of a dual-mass system driven by two exciters in engineering.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

THEORETICAL STUDY OF THE EFFECT OF GEOMETRICAL PARAMETERS AND LOCATION OF ROTOR IMPACT ELEMENTS OF AN IMPACT-CENTRIFUGAL GRINDER ON SPEED AND ANGLES OF CRUSHED PARTICLES FLIGHT

2020 ◽  
Keyword(s):  
Granulometric Composition ◽  
Theoretical Study ◽  
Process Efficiency ◽  
Geometrical Parameters ◽  
Grinding Process ◽  
Main Work ◽  
Impact Element ◽  
The Impact ◽  
Positive Effect ◽  
Efficiency Indicators

One of efficiency indicators of grain grinders is grain granulometric composition. The basis of mixed fodder is crushed grain, the particles of which must have a leveled granulometric composition for subsequent mixing and obtaining a high-quality feed mixture. In agricultural production, hammer crushers are widely used, in which the destruction of grain occurs due to the impact of a hinged hammer. The main disadvantage of these crushers is that not the entire surface of the hammers is involved in grinding, thus reduces grinding process efficiency. A slightly different principle of material destruction is laid down in the basis of the proposed design of the shock-centrifugal grinder. Main work is performed by flat impact elements located on the rotor, which serve to accelerate crushed particles with subsequent impact of them on the bump elements. An important step in the design of new constructions of shock-centrifugal grinders is to determine size and location of the impact elements on the rotor, without which the grinding process is not possible. In the calculation method presented, the dependencies for determining the velocities and angles of a single particle flight from the surface of a flat impact element for its specified dimensions are proposed. Two variants of an impact element location on the rotor are considered and analyzed: radial and at an angle in the direction of rotor rotation. As a result of research carried out, it is noted that in the case of inclined position of an impact element on the rotor an increase in flight speed and flight angles change in crushed particles, which gives the opportunity to have a positive effect on grinding process.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

scholarly journals Dynamics of the rumor-spreading model with hesitation mechanism in heterogenous networks and bilingual environment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuai Yang ◽  
Haijun Jiang ◽  
Cheng Hu ◽  
Juan Yu ◽  
Jiarong Li
Keyword(s):  
Lyapunov Method ◽  
Reproduction Number ◽  
Model Parameters ◽  
Rumor Spreading ◽  
Spreading Model ◽  
Reductio Ad Absurdum ◽  
The Stability ◽  
The Impact ◽  
Theoretical Results ◽  
The Basic Reproduction Number

Abstract In this paper, a novel rumor-spreading model is proposed under bilingual environment and heterogenous networks, which considers that exposures may be converted to spreaders or stiflers at a set rate. Firstly, the nonnegativity and boundedness of the solution for rumor-spreading model are proved by reductio ad absurdum. Secondly, both the basic reproduction number and the stability of the rumor-free equilibrium are systematically discussed. Whereafter, the global stability of rumor-prevailing equilibrium is explored by utilizing Lyapunov method and LaSalle’s invariance principle. Finally, the sensitivity analysis and the numerical simulation are respectively presented to analyze the impact of model parameters and illustrate the validity of theoretical results.

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