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.

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.

scholarly journals Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Zhigang Huang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Small Functions ◽  
Linear Differential ◽  
The Relationship ◽  
Growth Of Solutions

This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomialsL(f)=d2f″+d1f′+d0fgenerated by solutions of the above equation, whered0(z),d1(z),andd2(z)are entire functions that are not all equal to zero.

scholarly journals On the integrable deformations of a system related to the motion of two vortices in an ideal incompressible fluid

2019 ◽  
Vol 29 ◽  
pp. 01015 ◽  
Author(s):  
Cristian Lăzureanu ◽  
Ciprian Hedrea ◽  
Camelia Petrişor
Keyword(s):  
Incompressible Fluid ◽  
Mechanical System ◽  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Mechanical Systems ◽  
Ideal Incompressible Fluid ◽  
First Integrals ◽  
Integrable Deformation ◽  
Integrable Deformations ◽  
The Given

Altering the first integrals of an integrable system integrable deformations of the given system are obtained. These integrable deformations are also integrable systems, and they generalize the initial system. In this paper we give a method to construct integrable deformations of maximally superintegrable Hamiltonian mechanical systems with two degrees of freedom. An integrable deformation of a maximally superintegrable Hamiltonian mechanical system preserves the number of first integrals, but is not a Hamiltonian mechanical system, generally. We construct integrable deformations of the maximally superintegrable Hamiltonian mechanical system that describes the motion of two vortices in an ideal incompressible fluid, and we show that some of these integrable deformations are Hamiltonian mechanical systems too.

scholarly journals Numerical simulation of oscillations of a nonlinear mechanical system with a spring-loaded viscous friction damper

2019 ◽  
Vol 265 ◽  
pp. 05030
Author(s):  
Viktor Nekhaev ◽  
Viktor Nikolaev ◽  
Marina Safronova
Keyword(s):  
Differential Equations ◽  
Mechanical System ◽  
Degrees Of Freedom ◽  
External Disturbance ◽  
Viscous Friction ◽  
Force Characteristic ◽  
Friction Damper ◽  
System Of Differential Equations ◽  
Nonlinear Force ◽  
Linear Spring

The dynamics of a mechanical system consisting of a parallel-connected main elastic element, an external disturbance compensator having a nonlinear force characteristic, and a viscous friction damper sprung by a linear spring are studied. The resulting system of differential equations describing the behavior of the system has one and a half degrees of freedom and has specific properties depending on the ratio of stiffness of the main spring and the spring suspension of a viscous friction damper. It is established that a single nonlinear system with one and a half degrees of freedom has either one or two harmonics. In the general solution of the system of differential equations, there are always two harmonics in the above-resonance zone, one of which is always equal to the disturbance frequency, and the second one is sufficiently close to the frequency k0. In the linear conservative case and the absence of suspension of the viscous friction damper, the natural frequency of the displacement of the system k0 =14.046 s-1.

Fundamental Exercises in Mechanical Systems Engineering Using Robot Manipulator

2011 ◽  
Author(s):  
Hideaki Takanobu
Keyword(s):  
Mechanical System ◽  
Linear Algebra ◽  
Systems Engineering ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Engineering Students ◽  
Robot Manipulator ◽  
Mechanical Systems ◽  
Forward Kinematics ◽  
Basic Learning

A five degrees-of-freedom (5-DOF) robot manipulator is used for the basic learning of mechanical system engineering. Students learned the forward kinematics as concrete applications of the mathematics, especially linear algebra. After making a manipulator, baton relay contest was done to understand the inverse kinematics by controlling the manipulator using a manual controller having five levers.

FORCED TORSIONAL VIBRATIONS IN THE PRESENCE OF RESISTANCE FORCES

2015 ◽  
Vol 8 (1) ◽  
pp. 9-11
Author(s):  
Беляев ◽  
Aleksandr Belyaev ◽  
Тришина ◽  
Tatyana Trishina
Keyword(s):  
Differential Equations ◽  
Mechanical System ◽  
Elastic Properties ◽  
Mechanical Systems ◽  
Torsional Vibrations ◽  
Rational Selection ◽  
Torsional Oscillations ◽  
Properties Of Materials ◽  
Resistance Forces ◽  
Selection Of

The work proposed differential equations describing the torsional oscillations of one- and two-mass mechanical systems taking into account the dissipative losses of various kinds and nature. The dependences for determining the equivalent rigidity of the elastic ties. Using the results of these studies can be realized rational selection of inertial and elastic properties of materials and components damper mechanical system

Hamiltonian formulation for mechanical systems with nonholonomic constraints

2014 ◽  
Vol 11 (03) ◽  
pp. 1450017
Author(s):  
G. F. Torres del Castillo ◽  
O. Sosa-Rodríguez
Keyword(s):  
Finite Number ◽  
Mechanical System ◽  
Infinite Number ◽  
Degrees Of Freedom ◽  
Hamiltonian Structure ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Hamilton Equations

It is shown that for a mechanical system with a finite number of degrees of freedom, subject to nonholonomic constraints, there exists an infinite number of Hamiltonians and symplectic structures such that the equations of motion can be written as the Hamilton equations, with the original constraints incorporated in the Hamiltonian structure.

The Optimized White Differential Equation of GM(1,1) Based on the Original Grey Differential Equation

2012 ◽  
Vol 166-169 ◽  
pp. 2971-2975 ◽  
Author(s):  
Rui Zhou ◽  
Jun Jie Li ◽  
Yao Chen
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Prediction Accuracy ◽  
High Growth ◽  
Growth Index ◽  
Simulation Accuracy ◽  
Modeling And Prediction ◽  
Standard Index ◽  
Grey Differential Equation ◽  
The Relationship

This paper starting from the original grey differential equations, through finding the relationship between the raw data and the derivative of its , constructed a new white differential equation which equal to the original grey differential equation, at the same time, getting the new GM(1,1)model which closer to the changes of data. Through the modeling and prediction of the standard index series, this model not only adapts to low growth index series, but also adapts to high-growth index series, and the simulation accuracy and prediction accuracy are high.

scholarly journals An algorithm for deriving equations of motion of constrained mechanical system

1999 ◽  
Vol 21 (1) ◽  
pp. 36-44 ◽  
Author(s):  
Dinh Van Phong
Keyword(s):  
Differential Equation ◽  
Mechanical System ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Computer Processing

The article deals with the form of equations of motion of mechanical system with constraints. For holozoic systems the number of differential equation is equal to the degrees of freedom, without regard to the number of chosen coordinates. The possibilities of computer processing (symbolical and numerical) are shown. Two simple examples demonstrate the described technique.

scholarly journals CONSTRUCTION OF DYNAMIC INSTABILITY ZONES FOR HIGH STUCTURES UNDER SEISMIC IMPACT

2020 ◽  
Vol 2 (2) ◽  
pp. 42-50
Author(s):  
V Fomin ◽  
◽  
І Fomina ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Coordinate System ◽  
Mechanical System ◽  
Dynamic Instability ◽  
Vertical Component ◽  
System Of Differential Equations ◽  
High Rise ◽  
Concentrated Masses ◽  
Seismic Impact

Seismic impacts create the possibility of parametric resonances, i.e. the possibility of the appearance of intense transverse vibrations of structure elements (in particular, of high-rise structures) from the action of periodic longitudinal forces. As a design model of a high-rise structure, a model is used which adopted in the calculation of high-rise structures for seismic effects, - a weightless vertical rod (column) rigidly restrained at the base with a system of concentrated masses (loads) located on it (Fig. 1). By solving the differential equation of the curved axis influence function for a rod is constructed by means of which influence coefficients are determined for the rod points, in which the concentrated masses are situated. These coefficients are elements of the compliance matrix . Next, the elements of the stiffness matrix are determined by inverting the matrix . Using a diagonal matrix of the load masses and matrix a system of differential equations of free vibrations of a mechanical system, consisting of concentrated masses, is constructed, and the frequencies and forms of these vibrations are determined. From the vertical component of the seismic impact, its most significant part is picked out in the form of harmonic vibrations with the predominant frequency of the impact. Column vibrations are considered in a moving coordinate system, the origin of which is at the base of the column. The forces acting on the points of the mechanical system (concentrated masses) are added by the forces of inertia of their masses associated with the translational motion of the coordinate system. The forces of the load weights and forces of inertia create longitudinal forces in the column, periodically depending on time. Further, the integro-differential equation of the dynamic stability of the rod, proposed by V. V. Bolotin in [8], is written. The solution to this equation is sought in the form of a linear combination of free vibration forms with time-dependent factors. Substitution of this solution into the integro-differential equation of dynamic stability allows it to be reduced to a system of differential equations with respect to the mentioned above factors with coefficients that periodically depend on time. For some values of the vertical component parameters of the seismic action, namely the frequency and amplitude, the solutions of these equations are infinitely increasing functions, i.e. at these values of the indicated parameters, a parametric resonance arises. These values form regions in the parameter plane called regions of dynamic instability. Next, these regions are being constructed. A concrete example is considered.

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