Vector Rotators of Rigid Body Dynamics with Coupled Rotations around Axes without Intersection

Abstract

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.

1. Introduction

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 γυρο (turn) and σκοπεω (observed) and is related to the experiments that the 1852nd were painted by Jean Bernard Leon Foucault. The principle of gyroscope based on the principle of precession pseudoregular.

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 [1] written by Adi in 2006.

The original research results of dynamics and stability of gyrostats were given in 1979 by Ančev and Rumjancev [2].

Three of papers [3–5] written by Rumjancev related to stability of rotation of a heavy rigid body with one fixed point in S. V. Kovalevskaya's case, on the stability of motion of gyrostats and Stability of rotation of a heavy gyrostat on a horizontal plane pointed out important research results in this area.

Subjects of series of published papers (see [3–15]) are construction models, dynamics, and applications of gyroscopes as well as special phenomena of nonlinear vibration properties of the gyroscope, analysis of gyroscope dynamics for a satellites, analytical research results on a synchronous gyroscopic vibration absorber, inertial rotation sensing in three dimensions using coupled electromagnetic ring-gyroscopes, gyroscopes for orientation and inertial navigation, and others.

By Cavalca et al. [10] published in 2005 an investigation result on the influence of the supporting structure on the dynamics of the rotor system is presented.

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 [16] by Andonov et al. contains a classical and very important elementary dynamical model of heavy mass particle relative motion along rotate circle around vertical axis through its centre, whose nonlinear dynamics and singularities are primitive model of the simple case of the gyrorotor, and present an analogous and useful dynamical and mathematical model of nonlinear dynamics.

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 [17–22]) mass moment vectors and vector rotators, some characteristics members of the vector expressions of derivatives of linear momentum and angular momentum for the gyrorotor coupled rotations around two axes without intersection obtain physical and dynamical visible properties of the complex system dynamics.

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 [17, 20]) expressed by vectors connected to the pole and the axis and the influence of such possibilities to applications allows researchers and scientists to obtain larger views within their specialization fields. This is the reason for introducing mass moment vectors to the rotor dynamics, as well as vector rotators.

The primary-main vector is 𝔍⃗^{(𝑂)⃗𝑛} vector of the body mass inertia moment at the point 𝐴=𝑂 for the axis oriented by the unit vector 𝑛⃗ and there is a corresponding 𝔇⃗^{(𝑂)⃗𝑛} vector of the rigid body mass deviational moment for the axis through the point 𝐴 (see [17, 20]).

Also, there are a number of the vector rotators, pure kinematics vectors depending on angular velocity and angular acceleration of the body rotation as well as of the mass center position or deviational plane of the body in relation to the axis.

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 ℜ=√𝜔̇²+𝜔⁴. In the listed papers [17–22] as well as in others, written by first author of this paper, no listed heir, many applications of the discovered vector method by using mass moment vectors are presented for to express kinetic parameters of heavy rotors dynamics as well as of coupled multistep rotors dynamics and for gyrorotors dynamics.

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.

2. Mass Moment Vectors for the Axis to the Pole

The monograph [20], IUTAM extended abstract [17], and monograph paper [21] contain definitions of three mass moment vectors coupled to an axis passing through a certain point as a reference pole. Now, we start with necessary definitions of mass momentum vectors.

Definitions of selected mass moment vectors for the axis and the pole, which are used in this paper are as follows.(1)Vector 𝔖⃗^{(𝑂)⃗𝑛} of the body mass linear moment for the axis, oriented by the unit vector 𝑛⃗, through the point—pole 𝑂, in the following form (see Figure 1): 𝔖⃗=^{(𝑂)⃗𝑛def}_𝑉𝑛,⃗𝜌⃗𝑑𝑚=𝑛,⃗𝜌_𝐶⃗𝑀,𝑑𝑚=𝜎𝑑𝑉,(2.1) where 𝜌⃗ is the position vector of the elementary body mass particle 𝑑𝑚 in point 𝑁, between pole 𝑂 and mass particle position 𝑁.(2)Vector 𝔍⃗^{(𝑂)⃗𝑛} of the body mass inertia moment for the axis, oriented by the unit vector 𝑛⃗, through the point—pole 𝑂, in the following form 𝔍⃗=^{(𝑂)⃗𝑛def}_𝑉𝜌,⃗𝑛,⃗𝜌⃗𝑑𝑚.(2.2)

Figure 1 

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 𝜑1 and 𝜑2 are generalized coordinates fixed coordinate system and two moveable coordinate systems 𝑂1𝜉1𝜂1𝜁1=𝑂1𝜉1𝜂1𝑧 and 𝑂2𝜉2𝜂2𝜁2=𝑂2𝜉2𝜂2𝑧2 that are rotating with component angular velocities of rigid body coupled rotations: independent generalized (/or rheonomic) coordinates are 𝜑1 coordinate of precession rotation and 𝜑2 coordinate around self rotation axis. Vector rotators ℜ01, ℜ011, and ℜ022 are presented.

For special cases, the details can be seen in [17–22]. In the previously cited references, the spherical and deviational parts of the mass inertia moment vector and the inertia tensor are analysed. In monograph [20] knowledge about the change (rate) in time and the derivatives of the mass moment vectors of the body mass linear moment, the body mass inertia moment for the pole, and a corresponding axis for different properties of the body, is shown, on the basis of results from the first author’s reference [22].

This expression 𝔍⃗^{(𝑂)⃗𝑛}=𝔍⃗^{(𝑂1)⃗𝑛}+𝜌_𝑂⃗,𝔖⃗^{(𝑂1)⃗𝑛}+M^(𝑂1)𝐶,𝑛,⃗𝜌_𝑂⃗+𝜌_𝑂⃗,𝑛,⃗𝜌_𝑂⃗M(2.3) is the vector form of the theorem for the relation of material body mass inertia moment vectors, 𝔍⃗^{(𝑂)⃗𝑛} and 𝔍⃗^{(𝑂1)⃗𝑛}, for two parallel axes through two corresponding points, pole 𝑂  and pole 𝑂₁. We can see that all the terms in the last expression have the same structure. These structures are [𝜌_𝑂⃗,[𝑛,⃗𝑟_𝑂⃗]]M,[𝑟_𝐶⃗,[𝑛,⃗𝜌_𝑂⃗]]M, and [𝜌_𝑂⃗,[𝑛,⃗𝜌_𝑂⃗]]M.

In the case when the pole 𝑂₁ is the centre 𝐶 of the body mass, the vector 𝑟_𝐶⃗ (the position vector of the mass centre with respect to the pole 𝑂₁) is equal to zero whereas the vector 𝜌_𝑂⃗ turns into 𝜌_𝐶⃗ so that the last expression (2.3) can be written in the following form:𝔍⃗^{(𝑂)⃗𝑛}=𝔍⃗^{(𝐶)⃗𝑛}+𝜌_𝐶⃗,𝑛,⃗𝜌_𝐶⃗M.(2.4)

This expression (2.4) represents the vector form of the theorem of the rate change of the mass inertia moment vector for the axis and the pole, when the axis is translated from the pole at the mass centre C to the arbitrary point, pole O.

The Huygens-Steiner theorems (see [20, 21]) for the body mass axial inertia moments, as well as for the mass deviational moments, emerged from this theorem (2.4) on the change of the vector 𝔍⃗^{(𝑂)⃗𝑛} of the body mass inertia moment at point 𝑂 for the axis oriented by the unit vector 𝑛⃗ passing through the mass center 𝐶, and when the axis is moved by translate to the other point 𝑂.

Mass inertia moment vector 𝔍⃗^{(𝑂)⃗𝑛} for the axis to the pole is possible to decompose in two parts: first 𝑛(⃗𝑛,⃗𝔍⃗^{(𝑂)⃗𝑛}) collinear with axis and second 𝔇⃗^{(𝑂)⃗𝑛} normal to the axis. So we can write 𝔍⃗^{(𝑂)⃗𝑛}=𝑛⃗𝑛,⃗𝔍⃗^{(𝑂)⃗𝑛}+𝔇⃗^{(𝑂)⃗𝑛}=𝐽^{(𝑂)⃗𝑛}𝑛⃗+𝔇⃗^{(𝑂)⃗𝑛}.(2.5)

Collinear component 𝑛(⃗𝑛,⃗𝔍⃗^{(𝑂)⃗𝑛}) to the axis corresponds to the axial mass inertia moment 𝐽^{(𝑂)⃗𝑛} of the body. Second component, 𝔇⃗^{(𝑂)⃗𝑛}, orthogonal to the axis, we denote by the 𝔇⃗^{𝑂(⃗𝑛)}, and it is possible to obtain by both side double vector products by unit vector 𝑛⃗ with mass moment vector 𝔍⃗^{(𝑂)⃗𝑛} in the following form: 𝔇⃗^{(𝑂)⃗𝑛}=𝑛,⃗𝔍⃗^{(𝑂)⃗𝑛},𝑛=𝔍⃗^{(𝑂)⃗𝑛}𝑛,⃗𝑛⃗−𝑛⃗𝑛,⃗𝔍⃗^{(𝑂)⃗𝑛}=𝔍⃗^{(𝑂)⃗𝑛}−𝐽^𝑂⃗𝑛𝑛.⃗(2.6) In case when rigid body is balanced with respect to the axis the mass inertia moment vector 𝔍⃗^{(𝑂)⃗𝑛} is collinear to the axis and there is no deviational part. In this case axis of rotation is main axis of body inertia. When axis of rotation is not main axis then mass inertial moment vector for the axis contains deviation part 𝔇⃗^{(𝑂)⃗𝑛}. That is case of rotation unbalanced rotor according to axis and bodies skew positioned to the axis of rotation.

3. Linear Momentum and Angular Momentum Vector Expressions for Rigid Body Dynamic with Coupled Rotation around Axes without Intersection

3.1. Model of a Rigid Body Rotation around Two Axes without Intersection

Let us to consider rigid body rotation around two axes first oriented by unit vector 𝑛₁ with fixed position and second oriented by unit vector 𝑛₂ which is rotating around fixed axis with angular velocity 𝜔⃗₁=𝜔₁𝑛₁. Axes of rotation are without intersection. Rigid body is positioned on the moving rotating axis oriented by unit vector 𝑛₂ and rotate around self-rotating axis with angular velocity 𝜔⃗₂=𝜔₂𝑛₂ and around fixed axis oriented by unit vector 𝑛₁ with angular velocity 𝜔⃗₁=𝜔₁𝑛₁. Then, axes of rigid body coupled rotations are without intersection. The shortest orthogonal distance between axes is defined by length 𝑂₁𝑂₂ and it are perpendicular to both axes that is to the direction of angular velocities 𝜔⃗₁=𝜔₁𝑛₁ and 𝜔⃗₂=𝜔₂𝑛₂. This vector is 𝑟₀⃗=𝑂₁𝑂₂ (see Figure 1):𝑟₀⃗=𝑟₀𝑛₁,𝑛₂||𝑛₁,𝑛₂||=𝑟⁰𝑢₀₁,(3.1) and it can be seen on Figure 1.

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 𝛽_𝑖, 𝑖=1,2 are angles of skew position of rigid body to the self rotation axis. When center 𝐶 of the mass of rigid body is not on self rotation axis of rigid body rotation, we can say that rigid body is skew. Eccentricity of position is normal distance between mass center 𝐶 and axis of self rotation and it is defined by 𝑒⃗=[𝑛₂,[𝜌_𝐶⃗,𝑛₂]]. Here 𝜌_𝐶⃗ is vector position of mass center 𝐶 with origin in point 𝑂₂, and position vector of mass center with fixed origin in point 𝑂₁ is 𝑟_𝐶⃗=𝑟_𝑂⃗+𝜌_𝐶⃗.

A plane in which lies the shortest distance, lenght 𝑂₁𝑂₂, that is perpendicular to fixed axis of precession rotation by angular velocity 𝜔⃗₁=𝜔₁𝑛₁ is denoted as 𝑅_𝑛1. A plane that is formed by the shortest distance and fixed axis of component (transmission) rotation gyrorotor system is denoted as 𝑅₀ and in referent position with 𝑂₁𝑥 we denote axis of fixed coordinate system, with 𝑂₁𝑧 we denote axis in line with axis of component rotation by angular velocity 𝜔⃗₁=𝜔₁𝑛₁ while third axis 𝑂₁𝑦 is perpendicular to it. Lets choose a moveable axis 𝑂₁𝜉₁ in line to vector 𝑟₀⃗=𝑂₁𝑂₂, axis 𝑂₁𝜁₁=𝑂₁𝑧 that rotates by angular velocity 𝜔⃗₁=𝜔₁𝑛₁ around the moveable coordinate system is rotating 𝑂₁𝜉₁𝜂₁𝜁₁=𝑂₁𝜉₁𝜂₁𝑧 as it can be seen on Figure 2.

(a)

(b)

Figure 2 

Vector rotators ℜ1 (a) and ℜ2 (b) in relations to corresponding mass moment vectors 𝔍⃗(𝑂2)𝑛1 and 𝔍⃗(𝑂2)𝑛2, and their corresponding deviational components 𝔇⃗(𝑂2)𝑛1 and 𝔇⃗(𝑂2)𝑛2 as well as to corresponding deviational planes.

In the rigid body, an elementary mass around point 𝑁 we denote 𝑑𝑚 with position vector 𝜌⃗, and with origin in the point 𝑂₂ on the movable self rotation axis and with 𝑟⃗ vector positions of the same body elementary mass with origin in the point 𝑂₁ where point 𝑂₁ is fixed on the axis oriented by unit 𝑛₁ and 𝑂₂ is on self-axis rotation oriented by unit 𝑛₂ and both points are on the end of shortest orthogonal distance betwen axis of body coupled rotations. Position vector of elementary mass with origin in pole 𝑂₁ is 𝑟⃗=𝑟₀⃗+𝜌⃗, and velocity of mass particle 𝑑𝑚 is: 𝑣⃗=[𝜔⃗₁,𝑟₀⃗]+[𝜔⃗₁+𝜔⃗₂,𝜌]⃗.

3.2. Linear Momentum and Angular Momentum of a Rigid Body Coupled Rotations around Two Axes without Intersection

By using basic definition of linear momentum and angular momentum as well as expresson for velocity of rotation elementary body mass 𝑣⃗=[𝜔⃗₁,𝑟₀⃗]+[𝜔⃗₁+𝜔⃗₂,𝜌]⃗, we can write the following vector expressions: (a)for linear momentum in the following vector form (see [20, 23]): 𝔎⃗=𝜔⃗₁,𝑟₀⃗𝑀+𝜔₁𝔖⃗^{(𝑂2)𝑛}₁+𝜔₂𝔖⃗^{(𝑂2)𝑛}₂,(3.2) where 𝔖⃗^{(𝑂2)𝑛}₁=∭_𝑉[𝑛₁,𝜌]⃗𝑑𝑚 and 𝔖⃗^{(𝑂2)𝑛}₂=∭_𝑉[𝑛₂,𝜌]⃗𝑑𝑚 are correspond body mass linear moment of the rigid body for the axes oriented by direction of component angular velocities of coupled rotations through the movable pole 𝑂₂ on self-rotating axis;(b)for angular momentum in the following vector form (see [20, 23]): 𝔏⃗_𝑂1=𝜔₁𝑛₁𝑟²⁰𝑀+𝜔₁𝜌_𝐶⃗,𝑛₁,𝑟₀⃗𝑀+𝜔₁𝑟⁰⃗,𝔖⃗^{(𝑂2)𝑛}₁+𝜔²𝑟⁰⃗,𝔖⃗^{(𝑂2)𝑛}₂+𝜔¹𝔍⃗^{(𝑂2)𝑛}₁+𝜔₂𝔍⃗^{(𝑂2)𝑛}₂,(3.3) where 𝔍⃗=^{(𝑂2)𝑛}₁^def∭_𝑉[𝜌,⃗[𝑛₁,𝜌]⃗]𝑑𝑚 and 𝔍⃗=^{(𝑂2)𝑛}₂^def∭_𝑉[𝜌,⃗[𝑛₂,𝜌]⃗]𝑑𝑚 are corresponding rigid body mass inertia moment vectors for the axes oriented by directions of component rotations through the pole 𝑂₂ on self-rotating axes.

First term in expression (2.6) presents transmission part of linear momentum as if all rigid body mass is concentrate in pole 𝑂₂ on self-rotating axis and rotate around fixed axes with angular velocity 𝜔₁. This part is equal to zero in case when axes are with intersection. Second and third terms in expression for linear momentum present linear momentum of pure rotation, as relative motion around two axes with intersection in the pole 𝑂₂ on self rotation axes. This two parts are different from zero in all case.

Term 𝔖⃗^𝑂𝑛₁=[𝑛₁,𝑟_𝑂⃗]𝑀 is corresponding linear mass moment vector as if all rigid body mass 𝑀 is concentrate in pole 𝑂₂ on the self rotation axis for the axis oriented by direction of precision rotation, threw the pole 𝑂₁.

First term in expression (3.3) presents transmission part of angular momentum as if all rigid body is concentrate in pole 𝑂₂ on self-rotating axes and rotate arround fixed axis with angular velocity 𝜔₁. This part is equal to zero in case when axes are with intersection. First, second, third, and fourth members present transmission parts and fifth and sixth parts present relative angular momentum with respect to pole 𝑂₂ of pure rotation by two axes as they are with intersection in pole 𝑂₂ on self axis rotation. In case when axes are with intersection first four members in expression for angular moment are equal to zero.

3.3. Derivatives of Linear Momentum and Angular Momentum of Rigid Body Coupled Rotations around Two Axes without Intersection

By using expressions for linear momentum (3.2) after taking in account derivatives of parts, the derivative of linear momentum of rigid body coupled rotations around two axes without intersection, we can write the following vector expression:𝑑𝔎⃗𝑑𝑡=𝜔̇¹𝑛₁+𝜔̇₂,𝑟₀⃗𝑀+𝜔²¹𝑛₁,𝑛₁,𝑟₀⃗𝑀+𝜔̇₁𝔖⃗^{(𝑂2)𝑛}₁+𝜔²¹𝑛¹,𝔖⃗^{(𝑂2)𝑛}₁𝔖⃗^{(𝑂2)𝑛}₂+𝜔²²𝑛²,𝔖⃗^{(𝑂2)𝑛}₂+2𝜔¹𝜔₂𝑛¹,𝔖⃗^{(𝑂2)𝑛}₂.(3.4)

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 form𝑑𝔎⃗𝑑𝑡=ℜ⁰¹||𝑛₁,𝑟₀⃗||𝑀+ℜ₀₁₁|||𝔖⃗^{(𝑂2)𝑛}₁|||+ℜ₀₂₂|||𝔖⃗^{(𝑂2)𝑛}₂|||+2𝜔₁𝜔₂𝑛¹,𝔖⃗^{(𝑂2)𝑛}₂.(3.5)

By using vector expressions for angular momentum (4.1) after taking in account derivatives of parts, the derivative of angular momentum of rigid body coupled rotations around two axes without intersection, we can write the folowing expression:𝑑𝔏⃗_𝑂1𝑑𝑡=𝜔̇¹𝑟₀⃗+𝜔̇₁,𝑛₁,𝑟₀⃗𝑀+𝜔₁𝜔₂𝑟₀⃗,𝑛₁,𝑛₂,𝜌_𝐶⃗𝑀+𝜔₁𝜔₂𝑛₂,𝜌_𝐶⃗,𝑛₁,𝑟₀⃗𝑀+𝜔̇₁𝜌_𝐶⃗,𝑛₁,𝑟₀⃗𝑀−𝜔²¹𝜌_𝐶⃗,𝑟₀⃗𝑀+𝜔²¹𝑛₁,𝜌_𝐶⃗,𝑛₁,𝑟₀⃗𝑀+𝜔̇₂𝑟⁰⃗,𝔖⃗^{(𝑂2)𝑛}₁+𝜔²¹𝑛¹,𝑟₀⃗,𝔖⃗^{(𝑂2)𝑛}₁+𝜔²¹𝑟⁰⃗,𝑛₁,𝔖⃗^{(𝑂2)𝑛}₁+𝜔₁𝑟⁰⃗,𝔖⃗^{(𝑂2)𝑛}₂+𝜔²²𝑟⁰⃗,𝑛₂,𝔖⃗^{(𝑂2)𝑛}₂+𝜔¹𝜔₂𝑀𝑟₀⃗,𝑛₁𝜌_𝐶⃗,𝑛₂−𝑟₀⃗,𝜌_𝐶⃗𝑛₁,𝑛₂+𝜔̇₁𝜔₂𝑀𝑛₂𝜌_𝐶⃗,𝑛₁,𝑟₀⃗−𝜌_𝐶⃗𝑛₂,𝑛₁,𝑟₀⃗+𝑟₀⃗,𝑛₂𝑛₁,𝜌_𝐶⃗−𝑟₀⃗,𝜌_𝐶⃗𝑛₁,𝑛₂𝔍⃗^{(𝑂2)𝑛}₁+𝜔²¹𝑛¹,𝔍⃗^{(𝑂2)𝑛}₁+𝜔̇²𝔍⃗^{(𝑂2)𝑛}₂+𝜔²²𝑛²,𝔍⃗^{(𝑂2)𝑛}₂+2𝜔¹𝜔₂𝑛¹,𝔍⃗^{(𝑂2)𝑛}₂.(3.6)

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: 𝑑𝔏⃗_𝑂1𝑑𝑡=𝜒⃗¹²𝑟₀⃗+𝜔̇₁,𝜌_𝐶⃗,𝑀,𝜔̇₁,𝜔̇₂,𝜔₁,𝜔₂,𝑛₁,𝑛₂+𝜔̇₁𝑛₁𝑟²⁰𝑀+2𝜔₁𝜔₂𝑛¹,𝔍⃗^{(𝑂2)𝑛}₂𝑛¹,𝔍⃗^{(𝑂2)𝑛}₁𝑛¹+𝜔̇₂𝑛²,𝔍⃗^{(𝑂2)𝑛}₂𝑛²+ℜ₁|||𝔇⃗^{(𝑂2)𝑛}₁|||+ℜ₂|||𝔇⃗^{(𝑂2)𝑛}₂|||,(3.7) where the following denotation is used: 𝜒⃗₁₂=𝑟₀⃗𝜔,𝜌̇_𝐶1⃗,𝑀,𝜔̇₁,𝜔̇₂,𝜔₁,𝜔₂,𝑛₁,𝑛₂+𝜔̇₁𝜌_𝐶⃗,𝑛₁,𝑟₀⃗𝑀+𝜔²¹𝑛₁,𝜌_𝐶⃗,𝑛₁,𝑟₀⃗𝑀+𝜔₂₁𝑟⁰⃗,𝔖⃗^{(𝑂2)𝑛}₁+𝜔̇²𝑟⁰⃗,𝔖⃗^{(𝑂2)𝑛}₂+𝜔²¹𝑛¹,𝑟₀⃗,𝔖⃗^{(𝑂2)𝑛}₁+𝜔²²𝑛²,𝑟₀⃗,𝔖⃗^{(𝑂2)𝑛}₂+𝜔₂₁𝑛₁𝑟⁰⃗,𝔖⃗^{(𝑂2)𝑛}₁𝑀+𝜔²²𝑟⁰⃗,𝑛₂,𝔖⃗^{(𝑂2)𝑛}₂−𝜔¹𝜔₂𝑛¹,𝑟₀⃗,𝔖⃗^{(𝑂2)𝑛}₂𝑀+𝜔₂𝑀𝑛₁𝜌_𝐶⃗,𝑛₁,𝑟₀⃗+𝜔₁𝜔₂𝑛¹,𝑟₀⃗,𝔖⃗^{(𝑂2)𝑛}₂𝑀+𝜔₁𝜔₁𝑟⁰⃗,𝑛₁,𝔖⃗^{(𝑂2)𝑛}₂+𝜔¹𝜔₂𝑟⁰⃗,𝑛₂,𝔖⃗^{(𝑂2)𝑛}₁𝜔₂𝜌_𝐶⃗,𝑛₁𝑟₀⃗,𝑛₂𝑀−𝜔₁𝜔₂𝜌_𝐶⃗,𝑛₂𝑟₀⃗,𝑛₁𝑀.(3.8)

4. Vector Rotators of Rigid Body Coupled Rotations around Two Axes without Intersection

We can see that in previous expression (3.5) for derivative of linear momentum the following three vectors are introduced:ℜℜ₀₁₀₁=𝜔̇₁𝑢₀₁+𝜔²¹𝑣₀₁,=𝜔̇₁𝑛¹,𝑟⁰⃗𝑟₀+𝜔²¹𝑛¹,𝑛₁,𝑟⁰⃗ℜ𝑟₀₀₁₁,ℜ₀₁₁=𝜔̇₁𝑢₀₁₁+𝜔²¹𝑣₀₁₁,=𝜔̇₁𝔖⃗^{(𝑂2)𝑛}₁|||𝔖⃗^{(𝑂2)𝑛}₁|||+𝜔²¹⎡⎢⎢⎢⎣𝑛₁,𝔖⃗^{(𝑂2)𝑛1}|||𝔖⃗^{(𝑂2)𝑛}₁|||⎤⎥⎥⎥⎦=𝜔̇¹𝑛₁,𝜌_𝐶⃗||𝑛₁,𝜌_𝐶⃗||+𝜔²¹𝑛₁,𝑛₁,𝜌_𝐶⃗ℜ₀₂₂||𝑛₁,𝜌_𝐶⃗||,ℜ₀₂₂=𝜔̇₁𝑢₀₂₂+𝜔²¹𝑣₀₂₂,=𝜔̇₂𝔖⃗^{(𝑂2)𝑛}₂|||𝔖⃗^{(𝑂2)𝑛}₂|||+𝜔²²⎡⎢⎢⎢⎣𝑛₂,𝔖⃗^{(𝑂2)𝑛2}|||𝔖⃗^{(𝑂2)𝑛}₂|||⎤⎥⎥⎥⎦=𝜔̇²𝑛₂,𝜌_𝐶⃗||𝑛₂,𝜌_𝐶⃗||+𝜔²¹𝑛₂,𝑛₂,𝜌_𝐶⃗||𝑛₂,𝜌_𝐶⃗||.(4.1)

The first two vector rotators ℜ₀₁ and ℜ₀₁₁ are orthogonal to the direction of the first fixed axis and third vector rotator ℜ₀₂₂ is orthogonal to the self rotation axis. But, first vector rotator ℜ₀₁ is coupled for pole 𝑂₁ on the fixed axis and second and third vector rotators, ℜ₀₁₁ and ℜ₀₂₂, are coupled for the pole 𝑂₂ at self rotation axis and for corresponding direction oriented by directions of component angular velocities of coupled rotations. Intensity of two first rotators is equal and is expressed by angular velocity and angular acceleration of the first component rotation, and intensity of third vector rotators is expressed by angular velocity and angular acceleration of the second component rotation, and are in the following forms:ℜ₀₁=ℜ₀₁₁=𝜔̇²¹+𝜔⁴¹,ℜ₀₂₂=𝜔̇²²+𝜔⁴².(4.2)

Lets introduce notation 𝛾₀₁, 𝛾₀₁₁, and 𝛾₀₂₂ denote difference between corresponding component angles of rotation 𝜑₁ and 𝜑₂ of the rigid body component rotations and corresponding absolute angles of pure kinematics vector rotators about axes oriented by unit vectors 𝑛₁ and 𝑛₂. These angles are determined by the following relations:𝛾₀₁=𝛾₀₁₁=arctan𝜑̇²¹𝜑̈₁,𝛾₀₂=arctan𝜑̇²²𝜑̈₂.(4.3)

Angular velocity of relative kinematics vectors rotators ℜ₀₁,ℜ₀₁₁, and ℜ₀₂₂ which rotate about corresponding axes in relation to the component angular velocities of the rigid body component rotations are𝛾̇₀₁=𝛾̇₀₁₁=𝜑̇¹2𝜑̈₁−𝜑̇₁𝜑⃛₁𝜑̈²¹+𝜑̇⁴¹,𝛾̇₀₂=𝜑̇²2𝜑̈₂−𝜑̇₂𝜑⃛₂𝜑̈²²+𝜑̇⁴².(4.4)

In Figure 1. Vector rotators ℜ₀₁,ℜ₀₁₁, and ℜ₀₂₂ are presented.

Fourth vector rotator ℜ₀₁₂ is in the following vector form and with intensity ℜ₀₁₂:ℜ₀₁₂=2𝜔₁𝜔₂𝑛¹,𝔖⃗^{(𝑂2)𝑛}₂|||𝑛₁,𝔖⃗^{(𝑂2)𝑛}₂|||=2𝜔¹𝜔₂𝑛₁,𝑛₂,𝜌_𝐶⃗||𝑛₁,𝑛₂,𝜌_𝐶⃗||,|||ℜ⁰¹²|||=ℜ₀₁₂=2𝜔₁𝜔₂.(4.5)

This vector rotator ℜ₀₁₂ depends on both components of coupled rotations.

We can see that in previous vector expression (3.6) or (3.7) for derivative of angular momentum are introduced following two vectors rotators: ℜ₁=𝜔̇₁𝑢₁⃗+𝜔²¹𝑣₁ and ℜ₂=𝜔̇₁𝑢₂⃗+𝜔²¹𝑣₂ in the following vector form: ℜ₁=𝜔̇₁𝔇⃗^{(𝑂2)𝑛}₁|||𝔇⃗^{(𝑂2)𝑛}₁|||+𝜔²¹⎡⎢⎢⎢⎣𝑛₁,𝔇⃗^{(𝑂2)𝑛1}|||𝔇⃗^{(𝑂2)𝑛}₁|||⎤⎥⎥⎥⎦=𝜔̇¹ℜ₂𝑢₁⃗+𝜔²¹𝑣₁,=𝜔̇₂𝔇⃗^{(𝑂2)𝑛}₂|||𝔇⃗^{(𝑂2)𝑛}₂|||+𝜔²²⎡⎢⎢⎢⎣𝑛₂,𝔇⃗^{(𝑂2)𝑛2}|||𝔇⃗^{(𝑂2)𝑛}₂|||⎤⎥⎥⎥⎦=𝜔̇²𝑢₂⃗+𝜔²²𝑣₂.(4.6)

The first ℜ₁ is orthogonal to the fixed axis oriented by unit vector 𝑛₁ and second ℜ₂ is orthogonal to the self rotation axis oriented by unit vector 𝑛₂. Intensity of first rotator ℜ₁ is equal to intensity of previous defined rotator ℜ₀₁ and intensity of second rotator ℜ₂ is equal to intensity of previous defined rotator ℜ₀₂₂ defined by expressions (3.7). Their intensities areℜ₁=𝜔̇²¹+𝜔⁴¹,ℜ₂=𝜔̇²²+𝜔⁴².(4.7)

In Figure 2 vector rotators ℜ₁ (in Figure 2(a)) and ℜ₂ (in Figure 2(b)) in relations to corresponding mass moment vectors 𝔍⃗^{(𝑂2)𝑛}₁and 𝔍⃗^{(𝑂2)𝑛}₂, and their corresponding deviational components 𝔇⃗^{(𝑂2)𝑛}₁ and 𝔇⃗^{(𝑂2)𝑛}₂ as well as to corresponding deviational planes are presented.

Vector rotators ℜ₁ and ℜ₂ are pure kinematical vectors first presented in [20, 21] as a function on angular velocity and angular acceleration in a form ℜ=𝜑̈𝑢⃗+𝜑̇²𝑤⃗=ℜℜ₀. Also from Section 3.3 expressions (3.5) and (3.6) or (3.7) for derivatives for linear and angular momentum contain members with in tree types of different pure kinematical vectors rotators which rotate around first and second axis in corresponding directions of coupled rotation components, but with pole in 𝑂₁ or in 𝑂₂. These vector rotators are possible to separate by following criteria: (1) intensity of vector rotator is expressed by angular velocity 𝜔₁ and angular acceleration 𝜔̇₁ in the form ℜ₁=𝜔̇²¹+𝜔⁴¹ or angular velocity 𝜔₂ and angular acceleration 𝜔̇₂ in the form and ℜ₂=𝜔̇²²+𝜔⁴²; (2) intensity of the vector rotators is expressed by both angular velocity components 𝜔₁ and 𝜔₂, and no contain angular accelerations 𝜔̇₁ and 𝜔̇₂; (3) vector rotators are coupled by pole in 𝑂₁ or in 𝑂₂; (4) type of angular velocities components of vector rotators.

Rotators from first set are rotated around through pole 𝑂₂ axis in direction of first component rotation angular velocity and depend of angular velocity 𝜔₁ and angular acceleration 𝜔̇₁. There are two vectors of such type and all trees have equal intensity. Rotators from second set are rotated around axis in direction of second component rotation and depend of angular velocity 𝜔₂ and angular acceleration 𝜔̇₂. There are two vectors of such type and they have equal intensity.

Let us introduce notation, 𝛾₁ and 𝛾₂ denote difference between corresponding component angles of rotation 𝜑₁ and 𝜑₂ of the rigid body component rotations and corresponding absolute angles of pure kinematics vector rotators about axes oriented by unit vectors 𝑛₁ and 𝑛₂ through pole 𝑂₂. These angles are determined by following relations:𝛾₁=arctan𝜑̇²¹𝜑̈₁,𝛾₂=arctan𝜑̇²²𝜑̈₂.(4.8)

Angular velocity of relative kinematics vectors rotators ℜ₁ and ℜ₂ which rotate about axes in corresponding directions in relation to the component angular velocities of the rigid body component rotations through pole 𝑂₂ are𝛾̇₁=𝜑̇¹2𝜑̈²¹−𝜑̇₁𝜑⃛₁𝜑̈²¹+𝜑̇⁴¹,𝛾̇₂=𝜑̇²2𝜑̈²²−𝜑̇₂𝜑⃛₂𝜑̈²²+𝜑̇⁴².(4.9)

Also, it is possible to separate a few numbers of rotators and between the following:ℜ₁₂=2𝜔₁𝜔₂𝑛¹,𝔍⃗^{(𝑂2)𝑛}₂|||𝑛₁,𝔍⃗^{(𝑂2)𝑛}₂|||=2𝜔¹𝜔₂𝑢₁₂,(4.10) where 𝑢₁₂=[𝑛₁,𝔍⃗^{(𝑂2)𝑛}₂]/|[𝑛₁,𝔍⃗^{(𝑂2)𝑛}₂]| unit vector orthogonal to the axis oriented by unit vector 𝑛₁ and mass moment vector 𝔍⃗^{(𝑂2)𝑛}₂ for the axis oriented by unit vector 𝑛₂ through pole 𝑂₂, and intensity equal ℜ₁₂=2𝜔₁𝜔₂ twice multiplication of product of intensities of component angular velocities 𝜔₁ and 𝜔₂ of rigid body coupled rotations around exes without intersection.

5. Vector Rotators of Rigid Body-Disk Dynamics with Coupled Rotations around Two Orthogonal Axes without Intersection

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 𝜌_𝐶⃗ in relation to the pole 𝑂₂ and self rotation axis oriented by unit vector 𝑛₂, we can express in the movable coordinate systems with axes oriented by basic unit vectors: 𝑛₂,𝑢₀₂ and ^⇀𝑣₀₂ which rotate around self rotation axis with angular velocity 𝜔₂ in the form 𝜌_𝐶⃗=𝜌_𝐶(cos𝛽𝑛₂+sin𝛽𝑢₀₂), as well as by basic unit vectors 𝑢₀₁,𝑣₀₁ and 𝑛₁ which rotate around fixed axis oriented by unit vector 𝑛₁ with angular velocity 𝜔₁ in the following form: 𝜌_𝐶⃗=𝜌_𝐶⟨cos𝛽𝑢₀₁−sin𝛽cos𝜑₂𝑣₀₁+sin𝛽sin𝜑₂𝑛₁⟩. 𝛽 is angle between mass center vector position 𝜌_𝐶⃗ and self rotation axis oriented by unit vector 𝑛₂. Vector of the orthogonal distance between orthogonal axes without intersection is 𝑟₀⃗=−𝑟₀𝑣₀₁.

For this case unit vectors 𝑛₁ and 𝑛₂ are orthogonal, and after taking into account this orthogonality and corresponding formulas (4.1), (4.5), (4.6), and (4.10) for vector rotators we obtain the following vector expressions:ℜ₀₁₁=𝑣⁰¹𝜔̇₁cos𝛽−𝜔²¹sin𝛽sin𝜑₂−𝑢₀₁𝜔̇₁sin𝛽sin𝜑₂+𝜔²¹cos𝛽cos²𝛽+sin²𝛽sin²𝜑₂,|||ℜ₀₁₁|||=𝜔̇²¹+𝜔⁴¹ℜ₀₂₂=𝜔̇₂𝑣₀₂−𝜔²¹𝑢₀₂,|||ℜ₀₂₂|||=𝜔̇²²+𝜔⁴²ℜ₀₁₂=−2𝜔₁𝜔₂𝑢₀₁,|||ℜ₀₁₂|||=ℜ₀₁₂=2𝜔₁𝜔₂ℜ₁=−𝑢⁰¹𝜔̇₁cos𝛽+𝜔²¹sin𝛽cos𝜑₂+𝑣₀₁−𝜔̇₁sin𝛽cos𝜑₂+𝜔²¹cos𝛽√cos²𝛽+sin²𝛽cos²𝜑₂,|||ℜ₁|||=𝜔̇²¹+𝜔⁴¹ℜ₂=𝜔̇₂𝑣₀₂−𝜔²¹𝑢₀₂,|||ℜ₂|||=𝜔̇²²+𝜔⁴²ℜ₁₂=2𝜔₁𝜔₂𝑛¹,𝔍⃗^{(𝑂2)𝑛}₂|||𝑛₁,𝔍⃗^{(𝑂2)𝑛}₂|||=2𝜔¹𝜔₂𝑢₁₂,|||ℜ₁₂|||=ℜ₁₂=2𝜔₁𝜔₂.(5.1)

Previous expressions for vectors rotators are derived with supposition that rigid body is disk and that unit vectors in different deviation planes are:𝔇⃗^{(𝑂2)𝑛}₁|||𝔇⃗^{(𝑂2)𝑛}₁|||=−cos𝛽𝑢⁰¹−sin𝛽cos𝜑₂𝑣₀₁₁cos²𝛽+sin²𝛽cos²𝜑₂,⎡⎢⎢⎢⎣𝑛₁,𝔇⃗^{(𝑂2)𝑛1}|||𝔇⃗^{(𝑂2)𝑛}₁|||⎤⎥⎥⎥⎦=−cos𝛽𝑣⁰¹+sin𝛽cos𝜑₂𝑢₀₁cos²𝛽+sin²𝛽cos²𝜑₂𝑣₀₂=𝑢₂⃗=𝔇⃗^{(𝑂2)𝑛2}|||𝔇⃗^{(𝑂2)𝑛}₂|||,𝑢⁰²=𝑣₂=⎡⎢⎢⎢⎣𝑛₂,𝔇⃗^{(𝑂2)𝑛2}|||𝔇⃗^{(𝑂2)𝑛}₂|||⎤⎥⎥⎥⎦.(5.2)

In Figure 3. four schematic presentations of deviational planes and component directions of the vector rotators of rigid body-disk dynamics with coupled rotation around two orthogonal axes without intersection are presented. In Figure 3(a) deviation plane containing body mass center 𝐶, vector of relative mass center position 𝜌_𝐶⃗ in relation to the pole 𝑂₂ and self rotation axis oriented by unit vector 𝑛₂ is visible. In Figure 3(b) deviation plane containing self rotation axis oriented by unit vector 𝑛₂ and body mass inertia moment vector 𝔍⃗^{(𝑂2)𝑛}₂ and its deviational component vector of mass deviational moment 𝔇⃗^{(𝑂2)𝑛}₂ for self rotation axis and pole 𝑂₂ is visible. In Figure 3(c) two deviational planes through pole 𝑂₂: deviation plane containing self rotation axis oriented by unit vector 𝑛₂ and body mass inertia moment vector 𝔍⃗^{(𝑂2)𝑛}₂ and its deviational component vector of mass deviational moment 𝔇⃗^{(𝑂2)𝑛}₂ for self rotation axis and pole 𝑂₂ and deviation plane containing axis parallel to fixed axis oriented by unit vector 𝑛₁ and body mass inertia moment vector 𝔍⃗^{(𝑂2)𝑛}₁ and its deviational component vector of mass deviational moment 𝔇⃗^{(𝑂2)𝑛}₁ for axis oriented by unit vector 𝑛₁ and through pole 𝑂₂ are visible. In Figure 3(d) schematic presentation of the rigid body-disk skew and eccentrically positioned on the self rotation axis with corresponding mass moment vectors and deviation plane as a detail of the rigid body-disk coupled rotation around two orthogonal axes without intersection is visible. In all form of the parts in Figure 3. the component directions of the vector rotators components are visible.

(a)

(b)

(c)

(d)

Figure 3 

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 𝐶, vector of relative mass center position 𝜌𝐶⃗ in relation to the pole 𝑂2, and self rotation axis oriented by unit vector 𝑛2. (b) Deviation plane containing self rotation axis oriented by unit vector 𝑛2 and body mass inertia moment vector 𝔍⃗(𝑂2)𝑛2 and its deviational component vector of mass deviational moment 𝔇⃗(𝑂2)𝑛2 for self rotation axis and pole 𝑂2. (c) Two deviational planes through pole 𝑂2: deviation plane containing self rotation axis oriented by unit vector 𝑛2 and body mass inertia moment vector 𝔍⃗(𝑂2)𝑛2 and its deviational component vector of mass deviational moment 𝔇⃗(𝑂2)𝑛2 for self rotation axis and pole 𝑂2 and deviation plane containing axis parallel to fixed axis oriented by unit vector 𝑛1 and body mass inertia moment vector 𝔍⃗(𝑂2)𝑛1 and its deviational component vector of mass deviational moment 𝔇⃗(𝑂2)𝑛1 for axis oriented by unit vector 𝑛1 and through pole 𝑂2. (d) Schematic presentation of the rigid body-disk skew and eccentrically positioned on the self rotation axis with corresponding mass moment vectors and deviation plane as a detail of the rigid body-disk coupled rotation around two orthogonal axes without intersection.

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: 𝑡𝑔𝛾₁=𝜔̇¹𝜔²¹,𝑡𝑔𝛾₀₁₁=𝜔̇¹𝜔²¹=𝑡𝑔𝛾₁𝑡𝑔𝛾̃₁=1−𝜔̇¹/𝜔²¹𝑡𝑔𝛽cos𝜑₂𝜔̇₁/𝜔²¹+𝑡𝑔𝛽cos𝜑₂orintheform𝑡𝑔𝛾̃₁=1−𝑡𝑔𝛾¹𝑡𝑔𝛽cos𝜑₂𝑡𝑔𝛾₁+𝑡𝑔𝛽cos𝜑₂𝑡𝑔𝛾̃₀₁₁=𝜔̇¹/𝜔²¹−𝑡𝑔𝛽sin𝜑₂1+𝜔̇₁/𝜔²¹𝑡𝑔𝛽sin𝜑₂orintheform𝑡𝑔𝛾̃₀₁₁=𝑡𝑔𝛾⁰¹¹−𝑡𝑔𝛽sin𝜑₂1+𝑡𝑔𝛾₀₁₁𝑡𝑔𝛽sin𝜑₂.(5.3)

For the case that 𝜔̇₁=0, 𝜔₁=constant𝑡𝑔𝛾̃₀₁₁=𝜔̇¹/𝜔²¹−𝑡𝑔𝛽sin𝜑₂1+𝜔̇₁/𝜔²¹𝑡𝑔𝛽sin𝜑₂𝑡𝑔𝛾̃₁=𝑡𝑔𝛽sin𝜑₂,=1𝑡𝑔𝛽cos𝜑₂=𝑐𝑡𝑔𝛽1𝑜𝑠𝜑₂,(5.4) where 𝛾₁ is relative angle of rotation in comparison with angle of rotation 𝜑₁, when 𝛾̃₁ is absolute angle of rotor rotation about axis oriented by unit vector 𝑛₁, taking into account its rotation about axis oriented by unit vector 𝑛₂.

6. Dynamic of Rigid Body Coupled Rotation around Two Orthogonal Axes without Intersection and with One Degree of Freedom

6.1. Model Description of a Gyrorotor Coupled Rotations around Two Orthogonal Axes without Intersection and with One Degree of Freedom

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 𝜑₂ is independent, and coordinate 𝜑₁ is programmed. In that case, we say that coordinate 𝜑₁ is rheonomic coordinate and system is with kinematical excitation, programmed by forced support rotation by constant angular velocity. When the angular velocity of shaft support axis is constant, 𝜑̇₁=𝜔₁=constant, we have that rheonomic coordinate is linear function of time, 𝜑₁=𝜔₁𝑡+𝜑₁₀, and angular acceleration around fixed axis is equal to zero 𝜔̇₁=0.

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 𝑎. So we are going to consider that example presented in Figure 5. The normal distance between axes is 𝑎. The angle of self rotation around moveable self rotation axis oriented by the unit vector 𝑛₂ is 𝜑₂ and the angular velocity is 𝜔₂=𝜑̇₂. The angle of rotation around the shaft support axis oriented by the unit vector 𝑛₁ is 𝜑₁ and the angular velocity is 𝜔₁=constant. The angular velocity of rotor is 𝜔⃗=𝜔₁𝑛₁+𝜔₂𝑛₂=𝜑̇₁𝑛₁+𝜑̇₂𝑛₂. The angle 𝜑₂ is generalized coordinates in case when we investigate system with one degree of freedom, but system has two degrees of mobility. Also, without loss of generality we take that rigid body is a disk, eccentrically positioned on the self rotation shaft axis with eccentricity 𝑒, and that angle of skew inclined position between one of main axes of disk and self rotation axis is 𝛽, as it is visible in Figure 4.

Figure 4 

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

(a)

(b)

Figure 5 

Intensity of vector rotators ℜ022 and ℜ2, for different disk eccentricity (a) and for different initial conditions (b).

For that example, differential equation of the heavy gyrorotor-disk self rotation of reviewed model in Figure 4, for the case coupled rotations about two orthogonal axes, we can obtain in the following form:𝜑̈₂+Ω²𝜆−cos𝜑₂sin𝜑₂+Ω²𝜓cos𝜑₂=0,(6.1) whereΩ²=𝜔²¹𝐽^(𝐶)_2𝑢−𝐽^(𝐶)_𝑣2𝐽^{(𝐶)𝑛}₂,𝜆=𝑚𝑔𝑒sin𝛽𝜔²¹𝐽^(𝐶)_2𝑢−𝐽^(𝐶)_𝑣2,𝜓=𝐽2𝑚^{(𝐶)2𝑢}𝑒𝑎−s𝐽in𝛽^{(𝐶)𝑣2},𝜀=1+4𝑒𝑟².(6.2)

Here it is considered an eccentric disc (eccentricity is 𝑒), with mass 𝑚 and radius 𝑟, which is inclined to the axis of its own self rotation by the angle 𝛽 (see Figure 5.), so that previous constants (6.11) in differential equation (6.10) become the following forms:Ω²=𝜔²¹𝜀sin²𝛽−1𝜀sin²𝛽+1,𝜀=1+4𝑒𝑟²,𝜆=𝑔(𝜀−1)sin𝛽𝑒𝜔²¹𝜀sin²𝛽−1,𝜓=𝑒𝑟2𝜀𝑒si𝑎ns²i𝛽n𝛽−1.(6.3)

6.2. Phase Portrait of the Heavy Gyrorotor Disk Coupled Rotations About Two Axes without Intersection and Their Three Parameter Transformations

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 (6.10). After integration of the differential (6.3), the nonlinear equation of the phase trajectories of the heavy gyrorotor disk dynamics with the initial conditions 𝑡₀=0,𝜑₁(𝑡₀)=𝜑₁₀,𝜑̇₁(𝑡₀)=𝜑̇₁₀, we obtain in the following for𝜑̇²²=𝜑̇²⁰²+2Ω²𝜆cos𝜑²−12cos²𝜑₂+𝜓sin𝜑₂−2Ω²𝜆cos𝜑⁰²−12cos²𝜑₀₂+𝜓sin𝜑₀₂.(6.4) As the analyzed system is conservative it is the energy integral.

6.3. Kinematical Vector Rotators of the Heavy Gyrorotor Disk Coupled Rotations about Two Axes without Intersection and Their Three Parameter Transformations

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 ℜ₀₁, ℜ₀₁₁, and ℜ₁, from first set, are with same constant intensity |ℜ₀₁|=|ℜ₀₁₁|=|ℜ₁|=𝜔²¹=constant and rotate with constant angular velocity 𝜔₁ and equal to the angular velocity of rigid body precession rotation about fixed axis, but two of these three vector rotators, ℜ₀₁₁ and ℜ₁ are connected to the pole 𝑂₂ on the self rotation axis, and are orthogonal to the axis parallel direction as direction of the fixed axis. All these three vector rotators ℜ₀₁,ℜ₀₁₁, and ℜ₁ are in different directions (see Figures 3(a), 3(b), 3(c), and 4). Two of these vector rotators, ℜ₀₂₂ and ℜ₂, from second set, are with same intensity equal to ℜ₀₂₂=𝜔̇²²+𝜔⁴², and connecter to the pole 𝑂₂ and orthogonal to the self rotation axis oriented by unit vector 𝑛⃗₂ and rotate about this axis with relative angular velocity 𝛾̇₂ defined by second expression (4.9), 𝛾̇₂=(𝜑̇₂(2𝜑̈²²−𝜑̇₂𝜑⃛₂))/(𝜑̈²²+𝜑̇⁴²), in respect to the self rotation angular velocity 𝜔₂. These two of these vector rotators, ℜ₀₂₂ and ℜ₂ are oriented in the following directions:ℜ₀₂₂=𝜔̇₂𝑛₂,𝜌_𝐶⃗||𝑛₂,𝜌_𝐶⃗||+𝜔²²𝑛₂,𝑛₂,𝜌_𝐶⃗||𝑛₂,𝜌_𝐶⃗||,ℜ²=𝜔̇₂𝔇⃗^{(𝑂2)𝑛}₂|||𝔇⃗^{(𝑂2)𝑛}₂|||+𝜔²²⎡⎢⎢⎢⎣𝑛₂,𝔇⃗^{(𝑂2)𝑛2}|||𝔇⃗^{(𝑂2)𝑛}₂|||⎤⎥⎥⎥⎦.(6.5)

By use expressions (5.1) we can list following series of vector rotators of the gyrorotor-disk with coupled rotation around orthogonal axes without intersection and with 𝜔₁=constant: ℜ₀₁₁ℜ₀₁=𝜔²¹𝑣₀₁,|||ℜ₀₁|||=𝜔₂₁,=−𝜔²¹⟨sin𝛽sin𝜑₂𝑣₀₁+cos𝛽𝑢₀₁⟩√cos²𝛽+sin²𝛽sin²𝜑₂,|||ℜ₀₁₁|||=𝜔₂₁ℜ₀₂₂,=𝜔̇₂𝑣₀₂−𝜔²¹𝑢₀₂,|||ℜ₀₂₂|||=ℜ₀₁₂𝜔̇²²+𝜔⁴²,=−2𝜔₁𝜔₂𝑢₀₁,|||ℜ₀₁₂|||=ℜ₀₁₂ℜ₁=2𝜔₁𝜔₂,=−𝜔²¹𝑢₀₁⟨sin𝛽cos𝜑₂⟩+𝑣₀₁⟨cos𝛽⟩√cos²𝛽+sin²𝛽cos²𝜑₂,|||ℜ₁|||=𝜔₂₁ℜ₂,=𝜔̇₂𝑣₀₂−𝜔²²𝑢₀₂,|||ℜ₂|||=ℜ₁₂𝜔̇²²+𝜔⁴²,=2𝜔₁𝜔₂𝑛¹,J⃗^{(𝑂2)𝑛}₂|||𝑛₁,J⃗^{(𝑂2)𝑛}₂|||=2𝜔¹𝜔₂𝑢₁₂,|||ℜ₁₂|||=ℜ₁₂=2𝜔₁𝜔₂.(6.6)

One of the vectors rotators from the third set is ℜ₀₁₂ with intensity |ℜ₀₁₂|=2𝜔₁𝜔₂ and direction: ℜ₀₁₂=2𝜔₁𝜔₂[𝑛₁,[𝑛₂,𝜌_𝐶⃗]]/|[𝑛₁,[𝑛₂,𝜌_𝐶⃗]]|=−2𝜔₁𝜔₂𝑢₀₁. This vector rotator is connecter to the pole 𝑂₂ and orthogonal to the axis oriented by unit vector 𝑛₁ and relative rotate about this axis. Intensity of this vector rotator expressed by generalized coordinate 𝜑₂, angle of self rotation of heavy disk, taking into account first integral (6.4) of the differential equation (6.1) obtain the following form:|||ℜ₀₁₂|||=2𝜔₁𝜑̇²⁰²+2Ω²𝜆cos𝜑²−12cos²𝜑₂+𝜓sin𝜑₂−𝔘,(6.7) where 𝔘 denotes 2Ω²(𝜆cos𝜑₀₂−1/2cos²𝜑₀₂+𝜓sin𝜑₀₂).

Intensity ℜ₀₂₂=𝜔̇²²+𝜔⁴² of two of these vector rotators, ℜ₀₂₂ and ℜ₂, from second set, depends on angular velocity 𝜔₂ and angular acceleration 𝜔̇₂. For the considered system of the heavy gyrorotor-disk dynamics, for obtaining expression of intensity of vector rotators, ℜ₀₂₂ and ℜ₂, from second set, in the function of the generalized coordinate 𝜑₂, angle of self rotation of heavy disk self rotation, we take into account a first integral (6.4) of nonlinear differential equation (6.1), and by using these result and previous expressions (6.6) of vector rotator we can write the following.

(i) Intensity of the vectors rotators, ℜ₀₂₂ and ℜ₂, connected for the pole 𝑂₂ and rotate around self rotation axis, in the following form: |||=ℜ|||ℜ₀₂₂|||₀₂₂=Ω₂𝜑₂|||−𝜆−cos𝜑₂sin𝜑₂+𝜓cos𝜑₂²+𝜑̇²⁰²+2Ω²𝜆cos𝜑²−12cos²𝜑₂+𝜓sin𝜑₂−𝔘².(6.8)

(ii) Vector rotators orthogonal to the self rotation axes are in the following vector forms: ℜ₀₂₂𝜑₂=Ω²−𝜆−cos𝜑₂sin𝜑₂+𝜓cos𝜑₂𝑛²,𝜌_𝐶⃗+Ω₂||𝑛₂,𝜌_𝐶⃗||−𝜑2̇Ω²⁰²+₂2Ω²𝜆cos𝜑²−12cos²𝜑₂+𝜓sin𝜑₂𝜆cos𝜑⁰²−12cos²𝜑₀₂+𝜓sin𝜑₀₂𝑛²,𝑛₂,𝜌_𝐶⃗ℜ||𝑛₂,𝜌_𝐶⃗||,(6.9)₂𝜑₂=+ΩΩ²₂−𝜆−cos𝜑₂sin𝜑₂+𝜓cos𝜑₂|||𝔇𝔇⃗⃗^{(𝑂2)𝑛2}_{(𝑂2)𝑛2}|||−𝜑2̇Ω²⁰²+₂2Ω²𝜆cos𝜑²−12cos²𝜑₂+𝜓sin𝜑₂𝜆cos𝜑⁰²−12cos²𝜑₀₂+𝜓sin𝜑₀₂⎡⎢⎢⎢⎣𝑛²,|||𝔇𝔇⃗⃗^{(𝑂2)𝑛2}_{(𝑂2)𝑛2}|||⎤⎥⎥⎥⎦.(6.10)

Parametric equations of the trajectory of the vector rotators ℜ₀₂₂ and ℜ₂ are in the following same forms:𝑣_ℜ𝑢_ℜ𝜑₂=Ω²−𝜆−cos𝜑₂sin𝜑₂+𝜓cos𝜑₂,𝜑₂=Ω²−𝜑2̇Ω²⁰²+₂2Ω²𝜆cos𝜑²−12cos²𝜑₂+𝜓sin𝜑₂𝜆cos𝜑⁰²−12cos²𝜑₀₂+𝜓sin𝜑₀₂,(6.11) but it is necessary to take into consideration that is not in same directions, but is in the same plane orthogonal to the axis oriented by unit vector 𝑛₂ and through pole 𝑂₂.

Relative angular velocity 𝛾̇₂ of both vector rotators ℜ₀₂₂ and ℜ₂ in plane orthogonal to the axis oriented by unit vector 𝑛₂ and through pole 𝑂₂. in relation on angular velocity of self rotation, 𝜔₂=𝜑̇₂ is possible to express by using second expression (4.9), 𝛾̇₂=(𝜑̇₂(2𝜑̈²²−𝜑̇₂𝜑⃛₂))/(𝜑̈²²+𝜑̇⁴²), and we can write the following:𝛾̇₂=±ℎ+Ω²2𝜆cos𝜑₂−cos²𝜑₂2Ω⁴𝜆−cos𝜑₂²sin²𝜑₂−𝜑⃛₂−Ω²𝜆−cos𝜑₂sin𝜑₂²+ℎ+Ω²2𝜆cos𝜑₂−cos²𝜑₂².(6.12)

By using previous derived expression (6.8) for intensity of the vectors rotators, ℜ₀₂₂ and ℜ₂, connected for the pole 𝑂₂ and rotate around self rotation axis, oriented by unit vector 𝑛₂ in the orthogonal plane through pole 𝑂₂ and by changing some parameters of heavy gyrorotor structure, as it is eccentricity 𝑒, angle of disk inclination 𝛽, orthogonal distance between axes 𝑎, as well as parameter 𝜓 contained in the coefficients of the nonlinear differential equation (6.1) and presented by expressions (6.5), we obtain series of the graphical presentation, and some of these are presented in Figure 5.

By using parametric equations, in the form (6.11), of the trajectory of the vector rotators ℜ₀₂₂ and ℜ₂ connected for the pole 𝑂₂ and rotate around self rotation axis, oriented by unit vector 𝑛₂ in the orthogonal plane through pole 𝑂₂ and by changing some parameters of heavy gyrorotor-disk, as it is eccentricity 𝑒, angle of disk inclination 𝛽, orthogonal distance between axes 𝑎, as well as parameter 𝜓 contained in the coefficients of the nonlinear differential equation (6.1) and presented by expressions (6.5), we obtain series of the graphical presentation, and some of these are presented in Figure 6.

(a)

(b)

Figure 6 

Transformation of the trajectory of the vector rotator ℜ022 (and ℜ2) in the plane through pole 𝑂2 and orthogonal to the self rotation axis for different values of parameter 𝜓.

In Figure 6. transformation of the trajectory (hodograph) of the vector rotator ℜ₀₂₂ (and ℜ₂) in the plane through pole 𝑂₂ and orthogonal to the self rotation axis for different values of parameter 𝜓 is presented.

In Figure 9. transformation of the trajectory (hodograph) of the vector rotator ℜ₀₂₂ (and ℜ₂) in the plane through pole 𝑂₂ and orthogonal to the self rotation axis for different values of parameter 𝜆 is presented.

In Figure 7. transformation of the trajectory (hodograph) of the vector rotator ℜ₀₂₂ (and ℜ₂) in the plane through pole 𝑂₂ and orthogonal to the self rotation axis, for different values of parameter 𝑎, orthogonal distance between axes of gyrorotor-disk coupled component rotations is presented.

(a)

(b)

Figure 8 

Transformation of the trajectory of the vector rotator ℜ022 (and ℜ2) in the plane through pole 𝑂2 and orthogonal to the self rotation axis, for different values of parameter 𝑎, orthogonal distance between axes of gyrorotor-disk coupled component rotations.

By using expression, in the form (6.12), of relative angular velocity 𝛾̇₂ of the vector rotator ℜ₀₂₂ (and ℜ₂) rotation in the plane through pole 𝑂₂ and orthogonal to the self rotation axis, oriented by unit vector 𝑛₂ and by changing some parameters of heavy gyrorotor structure, as it is eccentricity 𝑒, angle of disk inclination 𝛽, orthogonal distance between axes 𝑎, as well as parameter 𝜓 contained in the coefficients of the nonlinear differential (6.1) and presented by expressions (6.5), we obtain series of the graphical presentation, and some of these are presented in Figure 8.

(a)

(b)

Figure 9 

Relative angular velocity 𝛾̇2 of the vector rotator ℜ022 (and ℜ2) in the plane through pole 𝑂2 and orthogonal to the self rotation axis, for different values of parameter 𝑎, orthogonal distance between axes of gyrorotor-disk coupled component rotations.

(a)

(b)

Figure 7 

Transformation of the trajectory of the vector rotator ℜ022 (and ℜ2) in the plane through pole 𝑂2 and orthogonal to the self rotation axis for different values of parameter 𝜆.

In Figure 8. relative angular velocity 𝛾̇₂ of the vector rotator ℜ₀₂₂ (and ℜ₂) in the plane through pole 𝑂₂ and orthogonal to the self rotation axis, for different values of parameter 𝑎, orthogonal distance between axes of gyrorotor-disk coupled component rotations is presented.

7. Concluding Remarks

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.

Acknowledgments

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.