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Vector method based on mass moment vectors and vector rotators coupled for pole and oriented axes is used for obtaining vector expressions for kinetic pressures on the shaft bearings of a rigid body dynamics with coupled rotations around axes without intersection. Mass inertia moment vectors and corresponding deviational vector components for pole and oriented axis are defined by K. Hedrih in 1991. These kinematical vectors rotators are defined for a system with two degrees of freedom as well as for rheonomic system with two degrees of mobility and one degree of freedom and coupled rotations around two coupled axes without intersection as well as their angular velocities and intensity. As an example of defined dynamics, we take into consideration a heavy gyrorotor disk with one degree of freedom and coupled rotations when one component of rotation is programmed by constant angular velocity. For this system with nonlinear dynamics, a series of tree parametric transformations of system nonlinear dynamics are presented. Some graphical visualization of vector rotators properties are presented too.

Well-known toy top or a tern is just a simple toy for many that has the unusual property that when it rotates sufficiently by high angular velocity about its axis of symmetry and it keeps in the state of stationary rotation around this axis. This feature has attracted scientists around the world and as a result of years of research created many devices and instruments, from simple to very complex structures, which operate on the principle of a spinning top that plays an important role in stabilizing the movement. Ability gyroscope that keeps the line was used in many fields of mechanical engineering, mining, aviation, navigation, military industry, and in celestial mechanics.

Gyroscopes’ name comes from the Greek words

Gyroscopes are very responsible parts of instruments for aircraft, rockets, missiles, transport vehicles, and many weapons. This gives them a very important role, and they need to be under the strict control of the design and inner workings because in case of damage they could lead to catastrophic consequences. Gyroscope (gyro, top) is a homogeneous, axis-symmetric rotating body that rotates by large angular velocity about its axis of symmetry and is now one of the most inertial sensors that measure angular velocity and small angular disturbances angular displacement around the reference axis.

Properties of gyroscopes possess heavenly bodies in motion, artillery projectiles in motion, rotors of turbines, different mobile installations on ships, aircraft propeller rotating, and so forth. The modern technique of gyroscope is an essential element of powerful gyroscopic devices and accessories, which are used to automatically control movement of aircraft, missiles, ships, torpedoes, and so on. They are used in navigation to stabilize the movement of ships in a seaway, to change direction, and direction of angular and translator velocity projectiles, and in many other special purposes.

There are many devices that are applied to the military, and their design is based on the principles of gyroscopes. Technical applications gyros today are so manifold and diverse that there is a need to get out of the general theory of gyroscopes allocates a separate discipline, called “applied theory of gyroscopes.”

An overview of optical gyroscopes theory with practical aspects, applications, and future trends is presented in [

The original research results of dynamics and stability of gyrostats were given in 1979 by Ančev and Rumjancev [

Three of papers [

Subjects of series of published papers (see [

By Cavalca et al. [

Each mechanical gyroscope is based on coupled rotations around more axes with one point intersection. Most of the old equipment was based on rotation of complex and coupled component rotations which resulting in rotation about fixed point gyroscopes.

The classical book [

No precisions and errors appear in the functions of gyroscopes caused by eccentricity and unbalanced gyrorotor body as well by distance between axis of rotations are reason to investigate determined task as in the title of our paper.

This vector approach proposed by us is very suitable to obtain new view to the properties of dynamics of pure classical task, investigated by numerous generations of the researchers and serious scientists around the world.

Using Hedrih’s (see [

Between them there are vector terms that present deviational couple effect containing vector rotators whose directions are the same as kinetic pressure components on corresponding gyrorotor shaft bearings.

Also, we can conclude that the impact of different possibilities to establish the phenomenological analogy of different physical vector models (see [

The primary-main vector is

Also, there are a number of the

For the case of a rigid body simple rotation about one axis there are two orthogonal vector rotators with same intensity depending on angular acceleration and angular velocity. Directions of these vector rotators are the same as components of kinetic pressures to shaft bearings. The vector rotators correspond to the rotation axis and one in the deviational plane through the axis and second orthogonal to the deviational plane and both with intensity

Organizations of this paper based on the vector method applications with use of the mass moment vectors and vector rotators for obtaining vector expressions for linear momentum and angular momentum and their derivatives of the rigid body coupled rotations around two axes without intersections. These obtained expressions are analyzed and series of conclusions are pointed out, all useful for analysis of the rigid body coupled rotations around two axes without intersections when system dynamics is with two degrees of mobility as well as with two degrees of freedom, or for constrained by programmed rheonomic constraint and with one degree of freedom.

By using two vector equations of dynamic equilibrium of rigid body dynamics with coupled rotations around two axes without intersection for two degrees of freedom it is possible to obtain two nonlinear differential equations in scalar form for rotations about each axes and also corresponding kinetic pressures in vector form bearing of both shafts.

The monograph [

Definitions of selected mass moment vectors for the axis and the pole, which are used in this paper are as follows.

Vector

Vector

Arbitrary position of rigid body coupled rotations around two axes without intersection. System is with two degrees of mobility (two freedom or one degree of freedom and one rheonomic constraint) where

For special cases, the details can be seen in [

This expression

In the case when the pole

The

Mass inertia moment vector

Collinear component

Let us to consider rigid body rotation around two axes first oriented by unit vector

When any of three main central axes of rigid body mass inertia moment is not in direction of self rotation axis, then we can see that rigid body is skew positioned. The angles

A plane in which lies the shortest distance, lenght

Vector rotators

In the rigid body, an elementary mass around point

By using basic definition of linear momentum and angular momentum as well as expresson for velocity of rotation elementary body mass

for linear momentum in the following vector form (see [

for angular momentum in the following vector form (see [

First term in expression (

Term

First term in expression (

By using expressions for linear momentum (

After analysis structure of linear momentum derivative terms, we can see that it is possible to introduce pure kinematic vectors depending on component angular velocitie and component angular accelerations of component coupled rotations that are useful to express derivatives of linear moment in following form

By using vector expressions for angular momentum (

After analysis structure of angular momentum terms, we can see, as in previous chapter for the derivatives of linear momentum, that it is possible to introduce pure kinematic vectors rotators depending on angular velocities and angular accelerations of component coupled rotations and that is used to express derivatives of angular momentum in the following shorter form:

We can see that in previous expression (

The first two vector rotators

Lets introduce notation

Angular velocity of relative kinematics vectors rotators

In Figure

Fourth vector rotator

This vector rotator

We can see that in previous vector expression (

The first

In Figure

Vector rotators

Rotators from first set are rotated around through pole

Let us introduce notation,

Angular velocity of relative kinematics vectors rotators

Also, it is possible to separate a few numbers of rotators and between the following:

Let us consider vector rotators for the special case when rigid body-disk rotate around two orthogonal axes without intersection.

Vector of relative mass center position

For this case unit vectors

Previous expressions for vectors rotators are derived with supposition that rigid body is disk and that unit vectors in different deviation planes are:

In Figure

Schematic presentation of deviational planes and component directions of the vector rotators of rigid body-disk dynamics with coupled rotation around two orthogonal axes without intersection. (a) Deviation plane containing body mass center

By use derived vector expressions of the vector rotators we can obtain some angles between corresponding vector rotator and basic vectors of corresponding movable coordinate systems coupled with corresponding compounding axis of component coupled rotations in the following form:

For the case that

We are going to take into consideration special case of the considered heavy rigid body with coupled rotations about two axes without intersection with one degree of freedom, and in the gravitation field. For this case generalized coordinate

Special case is when the support shaft axis is vertical and the gyrorotor shaft axis is horizontal, and all time in horizontal plane, and when axes are without intersection at normal distance

Model of heavy gyrorotor with two component coupled rotations around orthogonal axes without intersections.

Intensity of vector rotators

For that example, differential equation of the heavy gyrorotor-disk self rotation of reviewed model in Figure

Here it is considered an eccentric disc (eccentricity is

Relative nonlinear dynamics of the heavy gyrorotor-disk around self rotation shaft axis is possible to present by means of phase portrait method. Forms of phase trajectories and their transformations by changes of initial conditions, and for different cases of disk eccentricity and angle of its skew, as well as for different values of orthogonal distance between axes of component rotations may present character of nonlinear oscillations.

For that reason it is necessary to find first integral of the differential (

In the considered case for the heavy gyrorotor-disk nonlinear dynamics in the gravitational field with one degree of freedom and with constant angular velocity about fixed axis, we have three sets of vector rotators.

Three of these vector rotators

By use expressions (

One of the vectors rotators from the third set is

Intensity

(i) Intensity of the vectors rotators,

(ii) Vector rotators orthogonal to the self rotation axes are in the following vector forms:

Parametric equations of the trajectory of the vector rotators

Relative angular velocity

By using previous derived expression (

By using parametric equations, in the form (

Transformation of the trajectory of the vector rotator

In Figure

In Figure

In Figure

Transformation of the trajectory of the vector rotator

By using expression, in the form (

Relative angular velocity

Transformation of the trajectory of the vector rotator

In Figure

First main result presented is successful application the vector method by use mass moment vectors for investigation of the rigid body coupled rotation around two axes without cross-sections and vector decomposition of the dynamic structure into series of the vector parameters useful for analysis of the coupled rotation kinetic properties.

By introducing mass moment vectors and vector rotators we expressed linear momentum and angular momentum, as well as their derivatives with respect to time for the case of the rigid body coupled rotations around two axes without intersections. By applications of the new vector approach for the investigations of the kinetic properties of the nonlinear dynamics of the rigid body coupled rotations around two axes without intersections, we show that vector method, as well as applications of the mass moment vectors and vector rotators simples way show characteristic vector structures of coupled rotation kinetic properties.

Appearance, as it is visible, of the vector rotators, their intensity, and their directions as well as their relative angular velocity of rotation around component directions parallel to components of the coupled rotations, is very important for understanding mechanisms of coupled rotations as well as kinetic pressures on shaft bearings of both shafts.

Special attentions are focused to the vector rotators, as well as to the absolute and relative angular velocities of their rotations. These kinematical vector rotators of the heavy gyrorotor disk coupled rotations about two axes without intersection and their three parameter transformations are done as a second main result of this pepar.

A complete analysis of obtained vector expressions for derivatives of linear momentum and angular momentum give us a series of the kinematical vectors rotators around both directions determined by axes of the rigid body coupled rotations around axes without intersection. These kinematical vectors rotators are defined for a system with two degrees of freedom as well as for rheonomic system with two degrees of mobility and one degree of freedom and coupled rotations around two coupled axes without intersection as well as their angular velocities and intensity.

Parts of this research were supported by the Ministry of Sciences and Technology of Republic of Serbia through Mathematical Institute, SANU, Belgrade Grant ON174001 “Dynamics of hybrid systems with complex structures. Mechanics of materials”, supported by the Faculty of Mechanical Engineering University of Niš and Faculty of Mechanical Engineering University of Kragujevac.