Content

1. What is a tensor and what is it for?
2. Vector and tensor operations. Ranks of tensors
3. Curved coordinates
4. Dynamics of a point in the tensor representation
5. Actions on tensors and some other theoretical questions
6. Kinematics of free solid. Nature of angular velocity
7. The final turn of a solid. Rotation tensor properties and method for calculating it
8. On convolutions of the Levi-Civita tensor
9. Conclusion of the angular velocity tensor through the parameters of the final rotation. Apply head and maxima
10. Get the angular velocity vector. We work on the shortcomings
11. Acceleration of the point of the body with free movement. Solid Corner Acceleration
12. Rodrig – Hamilton parameters in solid kinematics
13. SKA Maxima in problems of transformation of tensor expressions. Angular velocity and acceleration in the parameters of Rodrig-Hamilton
14. Non-standard introduction to solid body dynamics
15. Non-free rigid motion
16. Properties of the inertia tensor of a solid
17. Sketch of nut Janibekov
18. Mathematical modeling of the Janibekov effect

Introduction

The dynamics of a solid body is a section of mechanics, which at one time set a clear vector for the development of this science. This is one of the most difficult sections of the dynamics, and the problem of integrating the equation of spherical motion for an arbitrary case of body mass distribution has not been solved yet.

In this article, we begin to consider the dynamics of a rigid body using the apparatus of tensor algebra. This pilot article on dynamics will answer a number of fundamental questions relating, for example, to such an important concept as the center of mass of the body. What is the center of mass, what distinguishes it from other points of the body, why are the equations of motion of the body mainly relative to this point? The answer to these and some other questions is under the cut.
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Integrating the equations of motion of this children's toy is one of the still unsolved problems of mechanics ...

1. Old, like the world, d'Alembert principle

First, consider the movement of a material point. Directly from the axioms follows the basic equation of the dynamics of a point



the acceleration multiplied by the mass is the vector sum of the forces applied to the point. And the forces that are applied to the point need to talk more. In the section of mechanics, called analytical mechanics, the forces applied to the points of the mechanical system are subject to strict classification.

The forces on the right side of (1) are divided into two groups

1. Active forces . This group of forces can be given the following definition
   

   Active is called the force, the magnitude of which can be determined from the conditions of the problem
   

   
   In formal terms, the active force is determined by the vector function
   
   
   
   
   
   Where - the generalized coordinate of a point; - generalized point velocity. From this expression it is clear that, starting from the solution of the motion problem and having initial conditions (time, position and speed), it is possible to immediately calculate the active force.
   
   Gravity, elasticity, Coulomb force of interaction of a charge with an electric field, Ampere force and Lorentz force, viscous friction force and aerodynamic resistance are all examples of active forces. The expressions for their calculation are known and these forces can be calculated, knowing the position and velocity of a point.
   
2. Reactions of communications . The most unpleasant forces that you can think of. I recall one of the axioms of statics, called the axiom of relations
   

   Connections applied to the body can be dropped, replacing their action with a force, or a system of forces
   

   
   The point depicted in the figure is not a free point. Its movement is limited by the connection, conventionally represented in the form of a certain surface, within which the trajectory of movement is located. The axiom given above makes it possible to remove the surface by applying a force to a point whose action is equivalent to having a surface. At the same time, this force is not known in advance - its value satisfies the constraints on the position, velocity and acceleration imposed by the connection, and, of course, the reaction vector depends on the applied active forces. Relationships to be determined in the process of solving the problem. The reactions of bonds also include dry friction, the presence of which, even in a simple task, significantly complicates the process of its solution.
   

Based on this classification, the equation of motion of a point (1) is rewritten in the form



Where - resultant of active forces applied to a point; - resultant reactions imposed on a point of bonds.

And now let's do the simplest focus - we will move the acceleration with mass to another part of equation (2)



and we introduce the notation



Then, equation (2) turns into



The force represented by the vector (3) is called the d'Alembert inertia force . And equation (4) expresses the d'Alembert principle for a material point

The material point is in equilibrium under the action of active forces, coupling reactions and inertia forces applied to it.

Let, what kind of equilibrium can we talk about if a point moves with acceleration? But after all, equation (4) is an equilibrium equation, and by applying force (3) to a point, we can replace the motion of a point by its equilibrium.

It is quite common to dispute whether the inertial forces (3) are physical forces. In engineering practice, the concept of centrifugal force is used, which is the force of inertia associated with centripetal (or selective) acceleration, which bends the point trajectory. My personal opinion is that the forces of inertia are the mathematical focus shown above, which allows one to proceed to the consideration of equilibrium instead of moving with acceleration. The inertia force (3) is determined by the acceleration of a point, but it, in turn, is determined by the action on the point of the forces applied to it, and in accordance with Newton's axiomatics, the force is primary. Therefore, there is no reason to speak of any "physicality" of inertia forces. Nature knows no active forces depending on acceleration.

2. Principle d'Alembert for a solid body. Main vector and main moment of inertia

Now we extend equation (4) to the case of motion of a rigid body. In mechanics, it is considered as an immutable mechanical system consisting of a set of points, the distance between which at each moment of time remains unchanged. All points of the body move along different trajectories, but the equation of motion of each point corresponds to (2)



The forces acting on a specific point can be divided into external active external reaction and internal forces , representing the forces of interaction of the point in question with the other points of the body (in essence, internal reactions). All the mentioned forces are resultant of the corresponding group of forces applied to the point. Apply the D'Alembert Principle to this equation.



Where - the force of inertia applied to a given point of the body.

Now that all points of the body are in equilibrium, we can use the equilibrium condition of a rigid body, which gives us the static

A solid body is in equilibrium under the action of a system of forces applied to it, if the main vector and the main moment of this system of forces, relative to the chosen center O, are wound to zero

The main vector of the system of forces is the vector sum of all forces applied to the body. The sum of the forces applied to each point of the body is determined by the last equation, therefore adding the equations for all points, in its left part we get the main vector



At the same time, the sum of the internal forces is zero, as a consequence of Newton's third law. Similarly, we calculate the sum of the moments of all forces relative to the chosen arbitrary center O, which gives us a zero moment of the main moment of the system of forces



moreover, as shown in the classical course of dynamics, the sum of the moments of internal forces applied to the system of material points is zero, that is, . Equations (5) and (6) already express the d'Alembert principle for a solid body, but with only one necessary amendment.

The number of active forces and bond reactions in equations (5) and (6) is, of course. Most of the terms in the corresponding amounts are equal to zero, since active external forces and reactions of external relations, generally speaking, are applied only at certain points of the body. What can not be said about the forces of inertia - the forces of inertia are applied to each point of the body. That is, the sum of the forces of inertia, and the sum of their moments relative to the chosen center, are sums integral. The system of inertia forces can be reduced to the main vector and the main point and we can write that



the main vector and the main moment of inertia forces applied to a solid body. Integrals (7) and (8) are taken over the entire volume of the body, and - the radius vector of the body point relative to the selected center O.

Based on this consideration, we can rewrite (5) and (6) in the final form



Equations (10) and (11) express the dalamber principle for a solid body

A tevrd body is in equilibrium under the action of external forces applied to it, coupling reactions, the main vector and the main moment of inertia forces.

In fact, (10) and (11) is a form of writing the differential equations of motion of a rigid body. They are often used in engineering practice, but from the point of view of mechanics, this form of writing the equations of motion is not the most convenient. Indeed, the integrals (7) and (8) can be calculated in general form and come to more convenient equations of motion. In this regard, (10) and (11) should be considered as the theoretical basis for the construction of analytical mechanics.

3. The center of mass and the inertia tensor appear on the scene.

Let us return to our tensors and with their help we calculate integrals (7) and (8) for the general case of motion of a rigid body. As the center of the cast, choose the point O ₁ . This point is chosen as a pole and it defines a local basis for the coordinate system associated with the body. In one of the previous articles, we defined a tensor ratio to accelerate the point of the body in such a movement.



Multiplying (12) by the mass of a point with a minus sign, we get the inertia force applied to the volume element of a solid body



Expression (13) is a covariant representation of the inertia force vector. We rewrite the double vector product in (12) in a more convenient form using the Levi-Civita tensor and pseudovectors of angular velocity and angular acceleration



Substituting (14) into (13) and taking the triple integral over the entire volume of the body, given that the angular velocity and angular acceleration are the same at each point of this volume, that is, they can be taken out of the integral sign



The integral in the first term is the body mass. The integral in the second term is a more interesting thing. Recall one of the formulas of the course of theoretical mechanics:



Where - contravariant components of the radius-vector of the center of mass of the body in question. Not in the sense of the concept of the center of mass, we simply replace the integrals in (15) in accordance with formula (16), taking into account that the covariant components are used in the second term (15).



Yeah, expression (17) is also familiar to us, let's imagine it in a more familiar vector form.



The first term in (18) is the inertial force associated with the translational motion of the body along with the pole. The second term is the centrifugal inertia force associated with the natural acceleration of the center of mass of the body as it moves around the pole. The third term is the rotational component of the main vector of inertia forces associated with the rotational acceleration of the center of mass around the pole. In general, everything is in accordance with the classical relations of the theorem.

An inquisitive reader will say: “Why use tensors to get this expression, if in a vector form it would be obtained in a less obvious way?”. In response, I will say that obtaining formulas (17) and (18) was a warm-up. Now we get the expression of the main moment of inertia with respect to the selected pole, and here the tensor approach manifests itself in all its glory.

Take equation (13) and multiply its vectorially from the left by the radius of the body point vector relative to the pole. Thus, we obtain the moment of inertia applied to the elementary volume of the body



Perform the substitution (14) in (19) again, but we are not in a hurry to take the integral



I do not know about you, but my eyes are dazzled, even with my familiarity with such formulas. The components are arranged in a more natural order - the rotational and centrifugal components are swapped. In addition, from the first to the second term, the complexity of the transformative calculations increases. We will simplify them in turn, first simplify the first, immediately taking the integral



Here again appeared the radius of the vector of the center of mass. There is nothing complicated here - we have one acceleration of the pole and we carried it beyond the integral sign. We will deal with the interpretation a bit later, but for now we will transform the second term (20). In it we can perform the convolution of the product of Levi-Civita tensors on the silent index k



Here we used the property of the Kronecker delta to replace the free vector / covector index when performing convolution. Now take the integral, taking into account that the angular acceleration is constant for the entire volume of the body



In both! The incomprehensible “crocodile” collapsed into a compact formula by means of formal tensor transformations. I dissemble, we introduced a new designation:



But this is not just an abstract formula. The structure of expression (24) shows that it reflects the distribution of body mass around the pole and is called it - the inertia tensor of a solid . This value is truly fundamental for mechanics, and we will talk about it in more detail, so far I will only say that (24) is a second-rank tensor whose components are the axial and centrifugal moments of inertia of the body in the chosen coordinate system. It characterizes the inertness of a solid during rotation. I draw the reader's attention to how quickly we obtained the expression for the inertia tensor, in essence, acting in a formal way. With vector ratios without breaking the brain can not do, I was convinced of this on personal experience.

Finally, we turn to the last term (20). When taking the integral in it, the inertia tensor should also turn out, and we will transform it in such a way as to achieve this goal. In this part of expression (20), the relation between the inertia tensor and the angular velocity of the body should appear. Let's start, for the beginning turning the product of Levi-Civita tensors



There is a significant simplification of the expression - due to the properties of the Kronecker delta and the fact that the vector product . But the inertia tensor in (25) is not visible. In order to get it, we will conduct a series of equivalent transformations.



Here we again took into account that , used the properties of the Kronecker delta and the operation of raising / lowering indices when multiplied by the metric tensor. And, now we integrate (26)



Here we again see the inertia tensor:



taking into account which we obtain a compact expression for the component of the main moment of inertia associated with centrifugal forces



Expression (27) is equivalent to the vector-matrix relation:



And though I am overwhelmed with pretentious phrases, I will postpone them for later, but now I will carefully write out the final result in vector form.

In the general case of the motion of a solid body, the main vector and the main moment of inertia applied to a solid body are equal to



And now let's still admire - in spite of the fact that the above transformations are similar to Egyptian hieroglyphs, they are formal , we just performed the actions on tensor indices and used the properties of tensor operations. We did not have to practice with vectors, paint vector operations in components and reduce the resulting projections of vectors to the results of matrix operations. All matrix and vector operations of finite expressions are automatically issued by us. In addition, such fundamental characteristics as the coordinates of the center of mass of the body and the inertia tensor are naturally obtained.

Lecturing to students, I set myself the goal of deriving (29) and (30) using vectors. After I translated a stack of paper and pretty brains, I came to the result. Take a word - the above transformations are simply seeds, compared to what you have to go through without using tensors.

In addition, expressions (29) and (30) were obtained by us for an arbitrary center of reduction of forces, as which we took the O1 pole. These expressions will help us understand what is the center of mass of the body and its importance for mechanics.

4. The special role of the center of mass

Using formulas (29) and (30), we return to equations (10) and (11) and, completing the substitution, we come to the differential equations of motion of a rigid body



What are these bad equations? And the fact that they depend on each other - the acceleration of the pole will depend on the angular acceleration and the angular velocity of the body, the angular acceleration - on the acceleration of the pole. Vector determines the position of the center of mass of the body relative to the pole. And what if we choose the pole right in the center of mass? Then because and equations (31), (32) will take a simpler form



Do you recognize these equations? Equation (33) is a theorem on the motion of the center of mass of a mechanical system, and (34) is the dynamic Euler equation of spherical motion. And these equations are independent of each other. Thus, the center of mass of a solid is a point relative to which the inertia forces are reduced to the simplest form. The translational motion together with the pole and the spherical around the pole are dynamically unleashed. Body inertia tensor, calculated relative to the center of mass and is called the central inertia tensor .

Equations (33), (34) in the foreign literature are called Newton-Euler equations, and, at present, they are very actively used to build software designed for modeling mechanical systems. In the framework of the cycle of tensors, we still remember about them.

Conclusion

The article you read has two goals - in it we introduced the basic concepts of the dynamics of a rigid body and illustrated the power of the tensor approach while simplifying cumbersome vector relations.

In the following, we will dwell on the inertia tensor and study its properties. Having plunged into the wilds of analytic mechanics, we reduce equations (31) - (34) to the equations of motion in generalized coordinates. In general, there is still something to tell about. In the meantime, thank you for your attention!

To be continued...