ALGEBRAIC METHODS FOR CALCULATING BRANCHED MECHANICAL SYSTEMS FOR FORCED OSCILLATIONS

2020 ◽  
Vol 5 ◽  
pp. 12-20
Author(s):  
I.P. POPOV
Keyword(s):  
Differential Equations ◽  
Electrical Engineering ◽  
Mechanical Systems ◽  
Higher Order ◽  
Algebraic Methods ◽  
Forced Oscillations ◽  
Systems Of Equations ◽  
Second Order Differential Equations ◽  
Parallel Connections ◽  
Higher Order Differential Equations

Parallel-serial and serial-parallel connections of consumers of mechanical power are considered. According to the known parameters of systems and the disturbing harmonic effect, the velocities of the elements of mechanical systems and the forces applied to them are algebraically determined. A comprehensive representation of harmonic and related quantities is used. This approach is widely used in electrical engineering. For the considered branched mechanical schemes, the classical methods based on the solution of second-order differential equations become many times more complicated and require the solution of systems of equations, which are reduced to higher-order differential equations. The use of a symbolic (complex) description of mechanical processes and systems makes it possible to use instead simple and compact algebraic methods, the complexity of which is ten times less.

CALCULATION OF PARAMETERS OF BRANCHED MECHANICAL SYSTEMS WITH HARMONIC VIBRATIONS. Continuation

2021 ◽  
pp. 31-37
Author(s):  
I. P Popov
Keyword(s):  
Differential Equations ◽  
Mechanical Systems ◽  
Higher Order ◽  
Systems Of Equations ◽  
Calculation Of Parameters ◽  
Harmonic Vibrations ◽  
Second Order Differential Equations ◽  
Parallel Connections ◽  
Higher Order Differential Equations ◽  
Phase Relationships

Parallel-series and series-parallel connections of mechanical power consumers are considered. According to the known parameters of systems and the disturbing harmonic effect, the velocities of the elements of mechanical systems and the forces applied to them are algebraically determined. For the considered branched mechanical schemes, the classical methods based on solving second-order differential equations become many times more complicated and require solving systems of equations that reduce to higher-order differential equations. The use of a symbolic (complex) description of mechanical processes and systems makes it possible to use instead simple and compact algebraic methods, the complexity of which is ten times less. Vector diagrams, not being a necessary component of the study of mechanical systems, are of significant methodological importance, since they show quantitative and phase relationships between the parameters of systems.

SYMBOLIC REPRESENTATION OF FORCED OSCILLATIONS OF BRANCHED MECHANICAL SYSTEMS

2021 ◽  
pp. 118-130
Author(s):  
I.P. Popov ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Mechanical System ◽  
Degrees Of Freedom ◽  
Mechanical Systems ◽  
Time Instant ◽  
Forced Oscillations ◽  
Parallel Connections ◽  
Clear Idea ◽  
The Relationship

A calculation of dynamics of a mechanical system with n degrees of freedom, including inert bodies and elastic and damping elements, involves the derivation and integration of a system of n second-order differential equations, which are reduced to a differential equation of 2n order. An increase in the degree of freedom of the mechanical system by one increases the order of the resulting differential equation by two. The solution of higher-order differential equations is rather cumbersome and time-consuming. Integration of equations is proposed to be replaced with rather simpler algebraic methods. A number of relevant theorems that relate both active and reactive parameters of mechanical systems in the series and parallel connection of mechanical power consumers are proved. Using parallel-series and series-parallel connections as an example, the calculation methods for branched mechanical systems with any number of degrees of freedom, based on the use of symbolic or complex representation of forced harmonic oscillations, are shown. The phase relationships determining loading conditions and a possibility of its artificial change are considered. The vector diagrams of the amplitudes of forces, velocities and their components in a complex plane at a zero time instant are presented, which give a complete and clear idea of the relationship between these quantities.

scholarly journals Oscillation Results for Nonlinear Higher-Order Differential Equations with Delay Term

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 446
Author(s):  
Alanoud Almutairi ◽  
Omar Bazighifan ◽  
Youssef N. Raffoul
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Higher Order ◽  
Middle Term ◽  
Delay Term ◽  
Integral Averaging Technique ◽  
Equations With Delay ◽  
Oscillation Of Solutions ◽  
Comparison Technique ◽  
Higher Order Differential Equations

The aim of this work is to investigate the oscillation of solutions of higher-order nonlinear differential equations with a middle term. By using the integral averaging technique, Riccati transformation technique and comparison technique, several oscillatory properties are presented that unify the results obtained in the literature. Some examples are presented to demonstrate the main results.

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