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

Starting to consider the dynamics of a solid body, we encountered an interesting tensor quantity, namely


called the inertia tensor of a solid . In addition, we found that the moment of inertia of a solid body, familiar from the course of theoretical mechanics, when it rotates around a fixed axis, is obtained from the inertia tensor using a simple formula
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Let us consider in more detail the properties of the tensor of inertia of a solid. And to begin with, we study the mechanical quantities, the calculation of which, as well as bringing the inertia forces to this center, leads to the concept of the inertia tensor.

1. The moment of momentum of a rigid body during rotation

Moment of momentum of the material point (MCD) relative to this center call a vector equal


For a solid body, when rotating around the pole MKD elemental volume


or in tensor form


Integrating (3) we obtain the MCD of a solid relative to the center


In accordance with (4), the inertia tensor is a linear operator connecting the MCD of a solid with its angular velocity.

2. The kinetic energy of a solid during rotation

The kinetic energy of the elementary volume of the body


which is equivalent to the tensor relation


Integrating the last expression over the entire volume of the body, we obtain the expression of kinetic energy


In expression (5), apparently, the inertia tensor appears again. . When the body rotates around a fixed axis, in accordance with the expression of the angular velocity through the final rotation of the body expression (5) is transformed into


Formula (6) is the kinetic energy of a solid during rotational motion, and , in accordance with (2), the moment of inertia of the body relative to the axis defined by the ort .

3. Covariant inertia tensor

It is easy to show that the tensor (1) is not symmetric. However, in Cartesian coordinates the tensor of inertia is a symmetric tensor, and based on this fact all its basic properties are derived. At the same time, we could not help noticing that in expressions (2) and (5) a value of the form


The obtained tensor (7) is a symmetric covariant tensor of the 2nd rank, since it is easy to verify that the equality . The tensor (7) will be called the covariant inertia tensor . Taking into account expression (7), it is possible to rewrite expression (2) for the axial moment of inertia through the contravariant components of the ort of the axis of rotation


The author did not meet the term just introduced in the literature, but since all the basic properties of the inertia tensor follow from symmetry (7), the introduction of this concept, as will be shown below, is fully justified.

4. Eigenvalues ​​and eigenvectors of the covariant inertia tensor

Let us show, to begin with, that the tensor (7), by virtue of its symmetry, has real eigenvalues. Let be - an arbitrary eigenvalue, which corresponds to an eigenvector . Then the ratio is fair


Assuming complex eigenvalues ​​and eigenvectors, multiply (9) from the left by the conjugate eigenvector


Perform complex conjugation (10)


Here we take into account that the components (7) are real numbers, and therefore the conjugation operation is equivalent to transposition. Since the tensor (7) is symmetric, , that is, taking into account (10)


or finally


Equality (12) holds if - real number.

Since the tensor (7) is represented by a matrix in three-dimensional space, it has three real eigenvalues corresponding to real eigenvectors accordingly, we can write the tensor relations


Multiply the scalar each of equations (13) by the corresponding eigenvector


Dividing both sides of equations (14) by the square of the modulus of the corresponding eigenvector, we get


It's obvious that


contravariant components of some orts. So in (15), according to (8), the eigenvalues ​​of the covariant inertia tensor are represented by the moments of inertia of the body with respect to the axes and


In addition, eigenvectors and form an orthogonal triple of vectors. Indeed, we will conduct a chain of transformations involving any pair of eigenvectors


Considering that get condition


which, due to the fact that in general , true when the scalar product of eigenvectors is zero


It means that . Repeating the proof for any pair of eigenvectors, we obtain that they are truly orthogonal to each other.

5. Main axes and main moments of inertia of a solid body

According to the results of the previous paragraph, we can say that an orthogonal coordinate system is connected with a solid, whose axes and are directed along the eigenvectors of the covariant inertia tensor. In these axes, in accordance with the definition of eigenvalues, the covariant inertia tensor is diagonalized.


There are moments of inertia in the diagonal calculated by the formulas (17). These moments of inertia are called the main moments of inertia of a solid , and the axes whose direction is given by vectors and - the main axes of inertia .

6. Huygens-Steiner theorem

Suppose that we know the central (calculated relative to the center of mass of the body) inertia tensor . Suppose we want to calculate the inertia tensor with respect to the point away from the center of mass in the direction of the known vector . In this case, the radius-vector of the elementary volume of the body relative to the point can be defined as the sum


Where - the radius of the vector of elementary volume of the body relative to the center of mass

Substitute (18) into (1)


Here we take into account that the integrals of the form


set the position of the center of mass of the body relative to the center of mass, that is, equal to zero. Finally, we obtain the expression for the inertia tensor


defining inertia tensor with respect to an arbitrary point through the inertia tensor with respect to the center of mass. Expression (19) is called the Huygens-Steiner theorem . The proof of this theorem is given in the most general form.

7. Inertia Tensor in Cartesian Coordinates

In Cartesian coordinates, the metric is given by the identity matrix, that is, formally


In this case, the expressions for the inertia tensor and the covariant inertia tensor coincide.


Therefore, in the Cartesian coordinates, the inertia tensor is also symmetric, and the above-mentioned properties are valid for it, which are associated with eigenvalues ​​and eigenvectors. In Cartesian coordinates, the inertia tensor is represented by a matrix


or


where the diagonal elements are called axial moments of inertia, and other elements - centrifugal moments of inertia.

Conclusion

The material of this article is the author's work. The literature is dominated by the approach to the study of the inertia tensor associated with the use of Cartesian coordinates. We also considered the most general approach and we were convinced that the properties of the inertia tensor and the Huygens-Steiner theorem can be obtained in arbitrary coordinates. All the formulas given in the article are transferred to the well-known from the course of theoretical mechanics when using the Cartesian metric.

Upd : Found a mention of the covariant tensor of inertia on some god forgotten site . Well, this confirms the idea that I used in this article.

To be continued...