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

Today we complete the construction of tensor relations describing the kinematics of a free rigid body. It so happened that over a sufficiently large number of articles we re-constructed part of the fundamental course of theoretical mechanics. These constructions, despite some abstractness, are useful both from a methodological point of view and from the point of view that, as applied to mechanics, the tensor approach, like a scalpel, reveals the true nature of familiar concepts, such as the laws of motion of material bodies, the speed of their points, angular velocity, angular acceleration. That's about the angular acceleration today and will be discussed.

We are getting bogged down deeper in the math matrix ...

1. Acceleration of the point of the body, making free movement. Angular acceleration enters the scene

In the article devoted to the tensor description of the kinematics of a solid body, we obtained that the components of the velocity of a point of a body making free motion in a connected coordinate system are determined by the relation
')


Where - components of the velocity vector of the pole in the associated coordinate system; - the angular velocity tensor. The superscript in parentheses means that the components of this tensor are represented in the associated coordinate system.

To get acceleration, firstly, we will move to the basic coordinate system - differentiation in it will be much easier to perform. But since the transformation of rotation is given for the contravariant components of vectors, we first of all raise the indices in (1)



and then, apply (2) the direct transformation of the rotation



and now we differentiate (3) with respect to time and obtain the expression of contravariant components of the acceleration of a body point



Where - contravariant components of pole acceleration in the base coordinate system

To interpret the result, we will come to the point where we started the way - to the connected coordinate system and covariant components.



The last expression in the chain of transformations contains the multiplier.



- the angular velocity tensor, therefore



- the invariant components of the acceleration of the point M of a solid with free movement. Now we will try to understand the meaning of the acceleration components (5). First, consider the last term, the angular velocity tensor in which you can paint through the angular velocity pseudovector



and, it is quite obvious that the derivative of the angular velocity tensor is represented through some pseudovector equal to the time derivative of the angular velocity pseudovector



From the course of theoretical mechanics it is known that the derivative of the angular velocity is called the angular acceleration of the body. So (7) is angular acceleration. Usually, the angular acceleration is usually denoted by another letter, which in LaTeX notation is written as \varepsilon . But this designation was “dragged away” by the Levi-Civita tensor, so we will use the symbol \epsilon which doesn’t look too impressive, but don’t we change the notation system because of such a trifle?

Based on the foregoing, we conclude that the time derivative of the angular velocity tensor is an antisymmetric angular acceleration tensor



for the designation of which we take the letter \xi , stylistically reminiscent of \varepsilon . Based on (8), the last term of (5) is equivalent to



or, in vector form



Where called the rotational acceleration point of the body .

Now we turn to the second term (5). In it, we write the angular velocity tensor through the pseudovector



Here we see a double vector product. Indeed, after all



the contravariant representation of the velocity vector of the point M is a relative pole, which participates in the subsequent vector multiplication by the angular velocity on the left. That is, the second term is the accelerating acceleration of a body point.



so we got the formula known from the course of theoretical mechanics

Acceleration of a point of the body during free movement is equal to the geometric sum of the pole acceleration, the rotational acceleration of the point around the pole and the accelerating acceleration of the point around the pole

And, finally, the first term in (5) can be written through the curvilinear coordinates of the pole, as was done in the article devoted to the kinematics and dynamics of the material point



and we get, in the most general form, the acceleration of a point of the body with free movement



Acceleration (10) is represented in its own (associated with the body) coordinate system. This expression is very general in nature, and the approach by which we came to it allows us to find out the true nature and the relationship between our usual kinematic parameters of motion. This is the theoretical value (10).

The practical significance of the resulting formula is such that it brings us one more step closer to obtaining the equations of motion of a rigid body in generalized coordinates.

2. Formal expression to calculate the angular acceleration through the turn tensor

To begin, calculate the angular acceleration tensor



Thus, the angular acceleration tensor is already determined by the second derivative of the rotation tensor. On the other hand, using the definition of the angular acceleration tensor (6), we can obtain an expression for the angular acceleration pseudovector



Well and, substituting (12) in (11) we finally get



Expression (13) looks spectacular and can be used, for example, to express the projections of angular acceleration on its own axes through the orientation angles of a solid (Euler, Krylov, plane angles, etc.). But for the most part it is theoretical - yes, here, look, as the angular acceleration is connected with the rotation matrix.

If we try to obtain the pseudovector of angular acceleration through the parameters of the final rotation, using (13), then this path can hardly be called optimal. Remember how much we carried with the angular velocity tensor ? That's the same thing! And here you can, in principle, do without SKA , it’s enough to turn to formula (7) and the article on the pseudo vector of angular velocity

3. Pseudovector of angular acceleration in the parameters of the final rotation

According to (7), we need only to differentiate the pseudovector of angular velocity, which is expressed in terms of the parameters of the final rotation as follows



and we get angular acceleration. This can be done manually.



Expression (15) can be slightly simplified. First, its second term is zero, since it contains the convolution of the Levi-Civita tensor with the same vector with two indices, which is equivalent to . Secondly, we can give similar terms, and we get the final expression



Now, using (8) from (16), we can pass to the angular acceleration tensor, but we will not do this. The actions to be performed are trivial, the resulting expression will be quite cumbersome. For practical purposes, formula (16) is sufficient for us.

If the axis of rotation does not change direction, then the derivatives of the orth of the axis of rotation vanish. This is possible when rotating around a fixed axis and in a plane-parallel motion. Then the angular acceleration vector looks trivial.



what gives the definition of the angular acceleration vector, which the teachers of theoremgech (including me), treat students to. In addition, it is clear from the last formula that the direction of this vector directly depends on the orientation of the basis of the coordinate system, and hence on the positive direction of rotation in it. This is well illustrated by the fact that the vector of angular acceleration is a pseudovector.

findings

Formulas (10), (14) and (16) are the last relations that close the construction of the kinematics of a solid body in arbitrary coordinates. We have come a long way - using the apparatus of tensor calculus, we have built anew all the kinematics of a solid body.

But we have not touched the main thing - how is it convenient to set the body position in space, which parameters to choose? How to connect these parameters with the kinematic characteristics of the motion of a solid body?

It would seem that the parameters of the final turn are bad? They are bad because they degenerate when the angle of rotation is zero. Recall how the rotation tensor is defined.



Resetting the angle of rotation in this expression we will come to the expression



We get that the rotation tensor is represented by the identity matrix. What is wrong with that, no turning, identical transformation? The bad thing is that it is impossible to obtain the components of the ort of the axis of rotation from such a rotation tensor. When integrating dynamic equations of motion, such a focus will lead to the collapse of the numerical procedure.

To build modeling systems, it is necessary to take parameters that do not undergo degeneration. These include the components of the rotation tensor itself, but there are nine of them. Plus three pole coordinates. Total - 12 parameters characterizing the position of the body in space. And the number of degrees of freedom of a solid is six. Thus, the six components of the rotation tensor are dependent quantities, which inflates the order of the system of equations of motion exactly twice.

Based on this consideration, the parameters of the final rotation are more profitable - there are four of them. And there is only one relationship equation.



and if not for the degeneration at they could be used.

However, non-degenerate parameters, with the help of which one can describe the orientation of a solid body in space, are, and they are directly related to the parameters of the final rotation. These are the parameters of Rodrigues Hamilton, which we will discuss in the next article.

Thanks

In the preparation of this article, for the input formulas, used the resource created by the user parpalak . In this regard, I want to thank him for creating and supporting such a useful service.

Well, traditionally, thank you for the attention of my readers!

To be continued...